Journal of Mathematics Teacher Education_2

EDITORIAL

STORIES AND THE CHALLENGE FOR JMTE

Stories have always been a part of the human tradition. Children want tohear stories told by their parents. Young people want to hear stories abouttheir heroes. Stories define people, religions, and even countries. Sit in ateacher’s lounge and you won’t have to wait long to hear stories aboutstudents. Conversations with teacher educators quickly yield stories aboutteachers and their classroom experiences. Stories can provide compellingevidence about the educational process. Consider, for example, what welearned when we encountered Benny (Erlwanger, 1975); Lynn, Jeannie,and Kay (Thompson, 1984); and, more recently, characters presented instories told inJMTE(see, e.g., Jaworski, 1998; Schifter, 1998).

The challenge forJMTE is to tell stories about mathematics teachereducation that not only inform us about educating teachers but that alsoextend our knowledge in some theoretical way regardless of whether themethodologies used are qualitative or quantitative. We would hope that ourstories lead us toward a clearer theoretical rationale that guides our workwith teachers.

I am reminded of the following words by Mitroff and Kilmann (1978):

The best stories are those which stir people’s minds, hearts, and souls and by doing sogive them new insights into themselves, their problems, and their human condition. Thechallenge is to develop a human science that more fully serves this aim. The question thenis not, “Is storytelling science?” but “Can science learn to tell good stories?” (p. 93)

Their point is that good stories are not a product of a particular method-ology but rather that good stories provide the insights we need to movethe field forward. Although the paradigm wars (Gage, 1989) are no longerraging, I see a new sort of tension emerging, a more fundamental one thatgoes to the heart of whether science is telling good stories. This tensionfocuses on the notion of critical analysis, and whether our stories are theproduct of analyses that honor the notion of doing science.

Science, storytelling or otherwise, should have a theoretical orienta-tion that supports both explanation and prediction. The issue becomeswhether our explanations are viable (von Glaserfeld, 1987) and whether

Journal of Mathematics Teacher Education2: 1–2, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

2 STORIES AND THE CHALLENGE FOR JMTE

our predictions, either statistical or naturalistic, provide insight into theteaching condition. Mason & Waywood (1996) emphasize the inseparabil-ity of theory and research and note that the role of theory is to describe,explain, predict, and inform. Combined, these outcomes can both confirmand extend our understanding of teacher development. Consider how manySaras (Elbaz, 1983) we encounter in our work with teachers and howSara’s practical knowledge enhances our own understanding of teacherdevelopment. Stories are meaningful to us so long as we can identify withthe characters and develop a certain interpretation and expectation of theiractions.

It is the intent ofJMTE to meet the challenge of publishing storiesthat can provide a context for conceptualizing ways of working with andeducating teachers. It is my hope that theJMTE stories of today willprovide the substance for tomorrow’s wisdom. That would makeJMTEthe significant journal our profession desires and deserves.

REFERENCES

Elbaz, F. (1983).Teacher thinking: A study of practical knowledge. New York: Nichols.Erlwanger, S. (1975). Benny’s conceptions of rules and answers in IPI mathematics.

Journal of mathematical behavior, 1(2), 7–25.Gage, N. (1989). The paradigm wars and their aftermath, a “historical” sketch of research

on teaching since 1989.Educational Researcher, 18(7), 4–10.Jaworski, B. (1998). Mathematics teacher research: Process, practice and the development

of teaching.Journal of mathematics teacher education, 1, 3–31.Mason, J. & Waywood, A. (1996). The role of theory in mathematics education and

research. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.),International handbook of mathematics education(1055–1089). Dordrecht: Kluwer.

Mitroff, I. & Kilmann, R. (1978). Methodological approaches to social science. SanFrancisco: Jossey-Bass.

Schifter, D. (1998) Learning mathematics for teaching: From a teachers’ seminar to theclassroom.Journal of mathematics teacher education, 1, 55–87.

Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics andmathematics teaching to instructional practice.Educational Studies in Mathematics, 15,105–127.

Von Glasersfeld, E. (1987).The construction of knowledge. Seaside, CA: IntersystemPublications.

THOMAS J. COONEYThe University of Georgia105 Aderhold HallAtheus, USA

RUHAMA EVEN

THE DEVELOPMENT OF TEACHER LEADERS AND INSERVICETEACHER EDUCATORS∗

ABSTRACT. This article discusses the development of teacher leaders and inserviceteacher educators whose role it is to promote teacher learning about mathematics teachingin the process of changing school mathematics. The Manor Program for the developmentof teacher leaders and teacher educators is used as a vehicle for addressing this issue. Thearticle focuses on aspects of curriculum design, discusses the theoretical rationale for thelearning opportunities provided by the program, and considers several problematic aspectsencountered.

INTRODUCTION

Calls for reform in school mathematics, as presented in several docu-ments in the last decade (e.g., Australian Education Council, 1990;National Council of Teachers of Mathematics, 1989; Superior Commit-tee on Science, Mathematics and Technology Education in Israel, 1992),recommend major changes in the way mathematics is taught in school.Emphasis is placed on presenting connected and rich mathematics, onproviding students with problem situations meaningful both to the studentand mathematically, on connecting instruction with students’ conceptionsand ways of thinking, and on developing classroom cultures that supportand promote reasoning, understanding, and cooperative examination ofmathematical ideas.

The history of educational reform in general and of mathematicslearning and teaching in particular suggests that meaningful changes ineducational practice are hard to achieve. One reason for the slow progressof educational changes might be that a common response to calls for

∗ The major part of this article was written while on sabbatical leave at the Universityof Wisconsin-Madison.

The author is grateful to Tom Romberg, Chris Hartmann, Gwen Fisher, Abraham Arcavi,and Maxim Bruckheimer for their helpful criticisms and suggestions.

The Manor Program team: Ruhama Even (director), Hasida Bar-Zohar, Mercedes BenAv (left 1996), Orly Gottlieb, Nili Hirshfeld, Neomi Robinson, and Josephine Shamash-Smith.


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educational reform consists of an exclusive focus on curriculum develop-ment in accordance with what is regarded as desirable school mathematics.A good curriculum is certainly a necessary step, but it is by no meanssufficient because teachers rarely use curriculum materials as intended bytheir developers (Romberg & Pitman, 1990). Teachers play a critical rolein the ways reform ideas are realized in the classroom, but the need forteachers tolearn to teach in new ways is often neglected (Cohen & Barnes,1993).

Thus, inservice teachers need to be provided with real opportunitiesto learn about mathematics learning and teaching. At first glance, themagnitude of the task may seem to be an obstacle to the realization oflarge-scale teacher learning, but actually this magnitude implies a changein the way one conceives the roles of many people in contact with teachers,such as teacher mentors, mathematics coordinators in school, or staff ofprojects for improving mathematics teaching. In many cases these peopledo not regard themselves as teachers of teachers, nor do they usuallysupport genuine teacher learning. Yet they have the appropriate condi-tions to become authentic inservice teacher educators. Consequently, theyshould be prepared toteach teachers, i.e., to initiate, guide, and supportteacher learning.

In this article I discuss the development of a professional group ofteacher leaders and inservice teacher educators whose role it is to promoteteacher learning about mathematics teaching. The report and discussionare situated in the context of an Israeli experience, theManor Project.The Manor Project, funded by the Israeli Ministry of Education, is partof a national endeavor to improve mathematics teaching and learning inIsrael, initiated byTomorrow 98– the report of the Superior Committee onScience, Mathematics and Technology Education in Israel (1992).

Until recently, there was no organized, formal preparation of teacherleaders and teacher educators in Israel. Teachers in leadership roles andinservice teacher educators usually were (and many still are) experi-enced teachers who had acquired a reputation for being excellent teachers.However, being a good teacher does not necessarily imply the ability tohelp others develop their teaching, just as being a good mathematiciandoes not necessarily imply the ability to help others learn mathematics.Therefore, there was a need to create a special preparation program for theeducation of teacher leaders and teacher educators.

Teacher leaders and educators require not only adequate preparationbut also adequate resources. Similar to the need for instructional materi-als for both children and teachers in every classroom, teacher leaders andeducators require materials developed for the purpose of planning learning

THE DEVELOPMENT OF TEACHER LEADERS 5

experiences in teacher education programs or in professional develop-ment activities. The Manor Project responds to both these needs. In thisarticle I focus on the first component of the Project, the teacher lead-ers’ and educators’ preparation program with its emphasis on curriculumdesign. I discuss the theoretical rationale for the kinds of learning oppor-tunities included in the program and analyze several problematic aspectsencountered.

DESCRIPTION OF THE PROGRAM

Program Objectives and Focus

The program is based on several assumptions: that the participants arethoughtful learners; that they are prepared to be professional teacher lead-ers or inservice teacher educators who view teachers as thoughtful learners;that teacher leaders and educators should enhance the professionalism ofteachers; that after completing the program the participants will need todevelop their own ways of supporting teacher learning and professionalismaccording to their specific contexts, the teachers involved, and personalpreferences. Consequently, we decided to emphasize the following:

• the development of understanding about current views of mathematicsteaching and learning;• the development of leadership and mentoring knowledge and skills and

of work methods with teachers;• the creation of a professional reference group.

More specifically, the program focuses on cognitive, curricular, techno-logical, and social aspects of teaching different mathematical topics suchas algebra, analysis, geometry, the real numbers, probability, and statistics;it examines critical educational issues such as alternative assessment andteaching in heterogeneous classes; it enhances mathematical knowledge; itemphasizes the development of leadership skills and methods for workingwith teachers; and it encourages discussion of practical difficulties anddilemmas. In the final year, the program also focuses on initiating changein school mathematics teaching and learning.

Participants

In this article, I focus on the first group of teacher leaders and inser-vice teacher educators who started the program in the 1993–94 academicyear. Thirty mathematics educators were selected from approximately 100applicants, with selection based on the following criteria: (a) a first degree

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either in mathematics or in a mathematics-related field, such as a B.Ed.with a mathematics major, or a B.S. in Chemistry; (b) experience in math-ematics teaching and inservice work with mathematics teachers, at leastone of them in grade nine or above; (c) agreement to conduct weeklyinservice work with a group of secondary mathematics teachers duringthe program; (d) reputation as a successful teacher with the potential tobecome a good teacher leader or teacher educator; and (e) a reasonablespread of participants across the country.

Participants’ teaching experience varied from 5 to 29 years with a meanof 18 years. About one fourth of the participating teachers taught at thejunior-high level only, another fourth at the senior-high level only, andthe rest had teaching experience at both levels. About two thirds of theparticipants held only a bachelors’ degree, one third had a masters’ degree,and one teacher held a Ph.D.

About one half of the participants had less than five years of experiencein conducting inservice work with secondary teachers or as mathematicscoordinators in their schools, almost one-half had between five to ten yearsof experience, and a few (three) had more than 15 years of experience.Almost all had participated in many inservice courses on mathematicsteaching; about half had also completed courses for school coordinatorsor courses dealing with leadership skills, although the leadership courseswere not specific to mathematics or mathematics education.

Operation of the Program

The program extended over three years in an effort to allow sufficient timefor the participants to learn, experience, and experiment with the topics andideas encountered. Further, there was a need for development and growthin the participants’ conceptions, beliefs, and dispositions about the natureof mathematics learning and teaching and about teaching teachers (Even,1994). Such changes require time to be effective (Guskey, 1986; Lappan etal., 1989; Loucks-Horsley, Hewson, Love & Stiles, 1998).

During each school year, the participants met weekly for four hourswith project staff and guest lecturers. In addition, they conducted weeklytwo-hour professional development activities, some explicitly focused oninitiating change in mathematics teaching and learning. As an overallassignment for each year, the participants prepared portfolios that docu-mented their learning experiences. They received feedback on partial draftsseveral times throughout the year, both from project staff and from theirpeers. Upon completion, the portfolios provided resources for materialsto be used by the project’s and other mathematics teacher leaders andeducators in the preparation of teacher development activities.


OPPORTUNITIES FOR LEARNING

In this section, I analyze three issues that exemplify the complexity asso-ciated with educating teacher leaders and teacher educators. The first issuefocuses on mathematical tasks. The second issue is related to cognitiveresearch in mathematics education. The third issue, in contrast to the firsttwo which are closely related to the role of the mathematics teacher, centerson aspects directly related to the teacher leader’s role.

The analysis is based on several questionnaires administered to allthe participants, interviews with a sample of the participants, video-documentation of selected course meetings, documentation of the bi-weekly project staff meetings, participants’ weekly reports on their teacherdevelopment meetings, staff observations of several teacher developmentmeetings conducted by the participants, and the participants’ portfolios.

Issue 1: Considering What Constitutes a Good Problem

Algebra is one of the most problematic topics in school mathematics.Many teachers and textbooks tend to confine their presentation to mechan-ical symbol manipulation and memorization of procedures. Thus, typi-cally, students simplify, solve equations, and substitute, using algorithmsacquired without meaningful understanding of variables, expressions, andequations. Many other topics are taught in a similar manner. We wanted thecourse participants to put forward their ideas about teaching algebra andmathematics in general, to re-examine their implicit assumptions aboutwhat constitutes good learning experiences in algebra (mathematics), andto develop richer, more empowering vision about good mathematicalactivities.

Acquaintance with theoretical background. Thus, a great deal of the firstyear of the program focused on various aspects related to the teachingand learning of algebra. The main issues were: (a) an historical viewon the development of algebra, (b) various and sometimes confusingmeanings of letters, and (c) students’ understanding of algebraic conceptswith an emphasis on operational and structural approaches. A large partof the whole-group meetings included presentations and discussions ofresearch findings that focused on students’ and teachers’ conceptionsand ways of thinking (e.g., Arcavi, 1994; Arcavi & Schoenfeld, 1988;Harper, 1987; Küchemann, 1981; Lee & Wheeler, 1986; Tirosh, Even& Robinson, 1998; Sfard, 1995; Sfard & Linchevski, 1994). Alternativeways of conceptualizing school algebra were presented and discussed,as well as several innovative curriculum development projects that were

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Figure 1. A figure composed of six squares.

technology-based, investigation-based, function-based, and process- ratherthan product-based.

Conceptualizing characteristics of good problems. At the beginning ofthe second year, emphasis was placed on connecting these rather broadtheoretical ideas to practice. Thus the participants were asked: “What is agood problem in school algebra and school mathematics?” The followingcharacteristics were offered: A good problem should (a) be clearly phrased,(b) stimulate thought, (c) be interesting, (d) be connected to everyday life,(e) not require previously-learned knowledge, (f) cause conflict, (g) not befrustrating.

Are the above really the most critical attributes of a goodalgebraprob-lem or, more generally, a goodmathematicsproblem? Is it really importantthat a problem be clearly phrased? What does it mean for a problem to beinteresting? Don’t we want students to use previously-learned knowledge?In an attempt to answer these questions, we decided to examine a specialcase of a problem that felt like a good problem, thePerimeter-18 Prob-lem (see also Even & Lappan, 1994). Six squares were arranged on theoverhead projector as in Figure 1, and the participants were asked to addsquares so that the figure would have a perimeter of 18 (It was assumedthat the length of the side of a small square was 1 unit and that eachsquare added had to be placed exactly along at least one side of anothersquare).

In the discussion that followed the exploration, participants pointed outthat, unlike traditional textbook problems which usually have one correctsolution, this problem had many correct solutions. The question then arose,“Are there some solutions that are more interesting or more important thanothers?” The participants chose to concentrate on the extreme cases: Whatis the least number of squares that must be added to make the perimeter18? What is the greatest number of squares that can be added?

While working on these questions, the participants were astonishedby the unexpected relations they found between area and perimeter. Forexample, they found that, if the area was increased by adding one square,the perimeter might remain unchanged or even decrease. They also found


that by adding one area unit, the perimeter could change by either +2, 0,–2, or –4 units and that these changes were the only ones possible.

The realization that the shape of the figure using the most squares mustbe a rectangle made it easy to check all four rectangles with whole-numberdimensions and a perimeter of 18 and to decide that the 4× 5 rectanglehad the greatest area. This finding, again, raised additional questions, suchas, “What will happen if we allow the dimensions of the small squares tobe any real number?”

Most of the participants knew that the solution should be the 4.5×4.5 square, and they suggested to prove the conjecture with calculus. Theinstructor then raised the question, “What would be a convincing argumentfor a junior-high school student?” Tabular and graphical representationsof the area and the corresponding lengths of the sides were accepted asuseful tools for pattern recognition and for arriving at the conjecture thatthe square was the solution. The participants also agreed that pattern recog-nition might be used as convincing evidence at some levels. However,they felt that a real proof was also needed. This brought in algebra, andthe participants suggested several proofs which they felt would convincestudents.

The participants felt that the 18-Perimeter Problem was a good prob-lem. They then modified their notion of a good problem to include thefollowing characteristics:

• The problem situation provides a meaningful context.• The problem supports extended discussion and allows an investigation

of a big problem.• Multiple correct solutions are possible.• Mathematical connections can be made between representations

(concrete, pictorial, verbal, numeric, graphic, algebraic) and domains(algebra, geometry, measurement).• Various levels of presentation and solution are possible.• There is a genuine need for use of algebra.

The participants compared these characteristics with the ones they hadsuggested earlier, and they reflected on their learning of what constitutes agood problem.

Using good problems to teach mathematics. For the concluding stage of theactivity, the aim was to further connect theory with practice. The partici-pants’ discourse during the previous stage had revealed that very few hadever thought about what makes a problem good, and not many had expe-rienced mathematics teaching with such characteristics. Even those whohad conducted such activities had used them only on special occasions,

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such as before holidays. Therefore, the participants were asked to conducta class activity based on a good problem during a regular mathematicsclass.

This task turned out to be much more difficult than we had anticipated.Some participants claimed that they did not have time to incorporate goodproblems into their regular mathematics teaching. This claim may reflecta traditional approach to mathematics teaching, but it may also reflect afeeling of insecurity. The covert message could be, “I don’t know how toincorporate good problems into my teaching.” In any case, lack of incorpo-ration may indicate that the experiences during the first half of the programwere insufficient to help some of the participants feel comfortable with theuse of good problems in their teaching.

Many of the participants claimed that they were unable to find a goodproblem for their classes. For some this claim meant that they could notfind a problem that would satisfy all the outlined criteria, as if those criteriacomposed a checklist. For others the claim meant that they were unable toinvent such a problem, as if the use of a ready-made problem was notlegitimate. These kinds of difficulties could be overcome quite simply.But there were also more substantial obstacles. Many participants expe-rienced real difficulties in identifying a good problem. These difficultiesmay reflect an actual limitation of the materials available to teachers, butthey may also reflect the participants’ inability to see the potential in evenordinary problems; i.e., to see through the problem into the activity thatcould emerge from it. These difficulties were not easy to overcome, as welearned from the analysis of the second-year portfolios.

As part of their second-year portfolios the participants reported on andanalyzed the good problem activity they had conducted. They explainedwhy they chose the problem and why they defined it as a good prob-lem. Some stuck to the criteria developed in the course whereas otherseliminated some aspects and/or added others. For example, several of theparticipants added an emphasis on the potential of the problem to helpstudents to develop as problem solvers, as indicated by the followingcomment: “Work on this problem encouraged students to ask additionalquestions and to expand their investigation. They also became motivatedto deal with problem situations” (AI, portfolio, Year 2).

Almost all participants considered the explorative aspect of the problemas an important characteristic, as illustrated in the following excerpt:

Students did not have a prescribed algorithm for solving the problem. Therefore, they didnot know how to solve the problem at first glance. They really needed to search and inves-tigate, to look for patterns and generalize, to hypothesize and then justify their hypothesesor refute them, and to check their answers. (GL, portfolio, Year 2)


This comment may indicate professional growth, and indeed many ofthe participants began using mathematically rich problems with a varietyof possible solutions. Still, there were some who focused only on theexploratory aspect of the activity and did not pay much attention to themathematics involved.

Many of the participants emphasized that their students were able todevelop different solutions according to their knowledge, abilities, andpreferences. The following comment was typical:

There are many possible solutions to the problem which vary by approaches and levels.This makes the problem suitable for a large range of students. Some of these solutionsrequire only limited knowledge of mathematics (e.g., several students found the sum byactually adding the numbers on a calculator); other solutions involve deep and broadknowledge (e.g., making connections to algebra and geometry or translating from onerepresentation to another). (GL, portfolio, Year 2)

Most of the participants chose a problem that was not closely relatedto a specific topic, as is exemplified in the following comment: “Thecontent of the problem is broad and is not narrowly tied to the mathemat-ical topic currently dealt with in class. The problem can be connected tovarious topics and approached in various representations” (RM, portfolio,Year 2).

Such an approach was, in many cases, combined with the choice ofa problem that did not require previously-learned knowledge. The choiceof a problem that was not closely connected to the specific mathematicstopic being taught may reflect a mature view of mathematics, in contrastto the all too common view of mathematics as a collection of unrelatedtopics, concepts, and procedures. But when combined with the beliefthat no previously-learned knowledge should be required, it may indi-cate that a good problem was still not regarded as an integral part ofthe mathematics curriculum. On the other hand, an illuminating exam-ple of the possibilities of a more integrated approach was presented byone of the participants who had chosen a standard trigonometry problemstraight from the textbook. She let her class work on it for almost an hour,encouraging them to use any tools they wanted, and was amazed by herstudents’ innovative solution processes. All solutions were based on ratheradvanced previously-learned mathematics knowledge, but not necessarilylast week’s classroom instruction.

Overall, the good problem activity seemed to have been a worth-while learning experience. Many participants were surprised by theirstudents’ responses and the unexpected learning outcomes. These unex-pected outcomes made them more aware of the potential of such activities;more importantly, the participants became more acutely aware of theneed for careful consideration when choosing or designing activities for

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students. They began to perceive mathematics problems in light of theactivities that can emerge from them. Most participants recalled the Whatis a good problem? activity as one of their most important learning expe-riences. One teacher summarized her experience this way: “What is agood problem? developed my awareness of what I am teaching. It alsodeveloped my ability and my knowledge to transform a plain probleminto a good one” (SH, questionnaire, Year 3). In addition to their ownlearning, several participants conducted teacher development meetings onthis topic, even though they were not required to do so as part of thecourse. This reflects the importance they assigned to the issue, as wellas further connections between theory and practice that support teacherlearning about mathematics teaching.

Issue 2: Focusing on Students’ Conceptions and Ways of Thinking

To make appropriate decisions for helping and guiding students in theirknowledge construction certainly requires awareness of students’ concep-tions and ways of thinking about the subject matter. A teacher who paysattention to students’ conceptual understanding can develop activities thatchallenge student thinking and support student learning. Such teacherknowledge about students may be generated through a teacher’s practicalexperiences as well as through acquaintance with relevant mathemat-ics education literature on students’ conceptions and learning processes.These two sources of knowledge are very different in nature and, as Lein-hardt, Young & Merriman (1995) argued, “The integration of knowledgelearned in the academy with knowledge learned in practice is neither trivialnor is it obvious how this integration should be accomplished” (p. 402).The authors claim that true integration of these two kinds of knowledgeinvolves examination of one kind while using ways of thinking related tothe other. This can be done by “asking learners to particularize abstracttheories and to abstract principles from particulars” (p. 403).

In our program we tried to follow the model proposed by Leinhardt etal. (1995), assuming that the synthesis of theoretical and practical sourcesof knowledge is more complete and informative than either one alone. Wewanted the participants to build upon and interpret their experience-basedknowledge using research-based knowledge. Vice versa, we wanted theparticipants to examine theoretical knowledge acquired from reading andto discuss research in the light of their practical knowledge.

Acquaintance with theoretical background. To help the participantsbecome familiar with relevant research literature, a large part of the whole-group meetings included presentations and discussions of research findings


on learners’ conceptions and ways of thinking in mathematics. The mainissues raised and discussed were

• discrepancies between concept image and concept definition offunction (e.g., Even, 1993; Vinner & Dreyfus, 1989),• difficulties in translating and making connections between different

representations of function (e.g., Bell & Janvier, 1981; Even, 1998),• the objective and subjective multi-facets of variable (e.g., Arcavi &

Schoenfeld, 1988; Küchemann, 1981),• cognitive development in algebra (e.g., Sfard, 1995),• students’ conceptions of the derivative (e.g., Amit & Vinner, 1990),• “proofs that explain and proofs that prove” (e.g., Hanna, 1990),• hypotheses and proofs in technological environments (e.g.,

Hershkowitz & Schwarz, 1995),• various levels and aspects of geometrical thinking (e.g.,

Hershkowitz, 1990),• conceptions of irrational numbers andπ in particular (e.g., Tall &

Schwarzenberger, 1978),• statistical thinking in a technological environment (Ben-Zvi &

Friedlander, 1997).

Analyses of the videotaped course sessions, of the yearly summaryquestionnaires, and of the interviews with a sample of the participants atthe end of each year indicate that most of the participants felt that becom-ing acquainted with research findings helped them learn about students’conceptions and ways of thinking. A quote from a first year interviewillustrates this: “The fact that I was forced to read research on the teachingof mathematics brought me closer to the way children think.”

Mini-studies. After completing one year of the program, the project stafffelt it was time for the participants to deepen their knowledge aboutstudents’ conceptions and to make connections between their experience-based knowledge and what they had learned from the more theoreticalliterature. Therefore, the participants were asked to choose one of the stud-ies presented in the course, replicate it or a variation of it with students andteachers, and then write a report that would describe the subjects’ waysof thinking and difficulties, and compare the results with the results of theoriginal study.

Most teacher leaders chose to start from a study that was in some wayrelevant to their actual work in the field. They used the original study as abase for developing a study that would help them answer questions thatwere important to them. For example, the topic of generalizations andjustifications in algebra was presented during the course from different

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perspectives and made several participants curious about how their ownstudents would behave.

Many of the participants were surprised by the results of their study,usually for two different reasons. Some found that the students coulddo much more and reach much higher levels of thinking than they hadexpected. On the other hand, there were others who expected their studentsto perform better than the ones in the original study, assuming that therewas something wrong with theteachingof the material in the originalstudy, whereas they themselves taught it well. When planning their studies,these participants tried to prove that their students would do better, but,in many cases, they were surprised to find that their students had similarthinking patterns and mistakes as the ones in the original study.

Initially, almost all the participants expressed a lack of interest forconducting the mini-study. They claimed that either for technical reasonsthey would not be able to conduct the mini-study or that there was nothingto be learned from replicating a study. However, after completing the task,most of the participants felt that they learned a lot. They referred to twokinds of benefits. One was academic: They felt that replicating a studyexpanded their theoretical knowledge and helped them develop a betterunderstanding of the issues presented and discussed in the articles theyhad read. The other kind of benefit involved learning aboutreal studentsin a situation relevant to their practice.

The mini-studies were very demanding. Therefore, the project staffoffered a great deal of support, such as reading and giving written andoral feedback on the research proposal and on several drafts of parts of thework throughout the year. Also, a few of the course meetings were devotedto discussing the proposals and ways of presenting research results, and todeveloping the discussion section. In addition, individual meetings werescheduled at the participants’ request.

Support did not come from the project staff only. The participants wereallowed and even encouraged to conduct the mini-study in pairs. Also,several weeks before the final work and the whole portfolio were to begraded, the participants exchanged portfolios with each other to obtainfeedback from a colleague. Both types of support – from colleagues andfrom project staff – were reported as important by the participants.

One kind of difficulty deserves special attention. Articulating whatthey had learned from the mini-study and presenting it in written formwas hard for most of the teacher leaders and teacher educators. Writinga paper is hard for many people, but there seemed to be more to thedifficulties of writing the mini-study report than initially met the eye. Itwould seem that their difficulties were, in a way, what Leinhardt, Young


& Merriman (1995) describe as a transformation of knowledge learned inone location into forms associated with another. The participants neededto transform knowledge learned in practice into forms usually associatedwith the academy, and vice-versa, which is not trivial.

Connecting theory to practice: Working with other teachers. Beforeconducting the mini-studies, several of the interviewees stated that theywanted to help their colleagues become aware of the theoretical materialto which they had been introduced in the course, but they were not surehow to go about it. After they had learned first-hand about real students’conceptions in a specific topic, had developed their knowledge and under-standing in various domains, and had become more skillful in work withother teachers, several participants decided to work in the same directionin their weekly teacher development meetings. At first, the teachers werenot very interested in learning about research, and they objected to theapparently non-practical nature of the activity, as the following quotationillustrates. “At first the teachers objected: ‘It’s a waste of time. Instead,one can prepare another worksheet.’ But today they look forward to theexposure to research and articles” (SH, interview, Year 3).

Indeed, many of the participants started to be aware of and appreciatethe possibilities theoretical knowledge offered in that it provided a contextfor teachers to study their practices. This appreciation is illustrated by thedescription of how one teacher leader directed the attention of the teachersin her school to topics that seemed important to her, how they analyzed thecurriculum differently as a result of newly acquired theoretical knowledge,and how her school’s practice was changing.

Actually, our objective is to change some elements in the actual teaching. To introduceelements that may be familiar to some of the teachers but the solutions of how to implementand bring them to work in the field are not well known. For example, one of the studies thatI worked on was on the concept of the algebraic expression as a variable. I was exposed to itlast year and I brought it [to the staff]. And it was simply amazing, the students’ responses.And then we gave these things to the class and we raised additional questions which areactually already in the textbook. But [this time] we concentrated on them and therefore thestudents gained some more . . . . So we need to apply what we learn in the field. Because ifthere is no impact in the field then there is no point in studying this research. (SH, interview,second year)

Issue 3: Becoming a Teacher Leader and Teacher Educator

In this section I focus on a more global view of the development of teacherleaders and educators. I focus on three types of development that seemessential to their role: personal, professional, and social. Adapting Bell& Gilbert’s (1994) use of these terms from the context of teacher devel-

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opment to the context of teacher leader/educator development, I take thetermpersonal developmentto mean an affective development that involvesattending to feelings about the change process, about being a teacherleader, and about mathematics education and teacher education.Profes-sional developmentinvolves changing concepts and beliefs about mathe-matics education and teacher education, and changing teacher educationactivities.Social developmentinvolves working with and relating in newways to other teacher leaders and educators, to teachers, principals, andsuperintendents. These three types of development were also integral to theactivities described earlier, but I focus on them in this section as they seemcentral to the more comprehensive roles of the teacher leader and teachereducator. Personal, professional, and social developments are interactiveand interdependent, and “development in one aspect cannot proceed unlessthe other aspects develop also” (Bell & Gilbert, 1994, p. 494). Nonetheless,for reasons of clarity, I discuss each type separately.

Personal development. An important goal of the personal developmentaspect of the program was to help participants develop professional senseand confidence. All course participants had already conducted inservicework with secondary teachers in various projects or had served as math-ematics coordinators in their own schools. However, when entering theprogram, many did not consider themselves teacher leaders or teachereducators. In many cases, they were not sure what their role really entailed,and they felt that they did not have the knowledge nor the skills to leadteachers towards learning about mathematics teaching and changing thetraditional practice of school mathematics.

In an effort to promote the participants’ professional self-esteem, wehelped them expand their knowledge in several fields related to theirwork, as exemplified in the previous sections. Several additional strategieswere used. For example, when the program started, mathematics teacherleader and teacher educator preparation was an innovative activity in Israel.Therefore, this was an opportunity to set high standards for the program.Hence, an advanced academic component for which the participants wouldreceive graduate credit was included. In Israel, inservice courses rarelycarry academic credit of any kind. This component encouraged participantcommitment and work investment into the whole program, as well as thedevelopment of professional self-esteem.

In addition, we emphasized the message, in words and in actions, thatthey were expected to be part of the leadership to improve mathematicsteaching in the country and that society counted on them to perform therole for which they were being prepared. We approached the participants


as professionals, expecting them to take their work in the course and in thefield very seriously and to respect each other’s contributions.

Analysis of the final questionnaires and interviews suggests that theparticipants felt that they had made progress in their personal develop-ment as expressed by self-confidence, desire for continuous learning anddevelopment, and willingness to accept challenging leadership roles, asillustrated in the following two comments:

As a teacher leader today I have more confidence in what I say and in the kinds of activity Iconduct with the teachers. (SR, interview, Year 3) I constructed a professional philosophy,both solid and flexible. I can defend it and also be flexible as changes occur. (BN, wholeclass discussion, Year 3)

Professional development. At the beginning of the third year, the partic-ipants were told that, because they were expected to assist in improvingthe teaching and learning of mathematics, most of the final year wouldbe devoted to their learning how to plan, conduct, and evaluate changeinitiatives. Emphasis was put on connecting what was learned in the othercourse components with the issue of actual change in school mathematics.Each participant chose an aspect of school mathematics in which she or hewanted to work with the teachers in the school. Those who chose the sametopic formed a team coordinated by a staff member or one of the partic-ipants. Based on the interests and expertise of the team coordinators, theparticipants were offered five topics that seemed central to current effortsto improve mathematics teaching and learning in Israel:

• Building a mathematics room• Developing a program for students to work on projects in mathe-

matics• Teaching in heterogeneous classes• Using new technologies in the teaching of mathematics• Helping to prevent at-risk high school students from dropping out

and not taking the matriculation exam

Each team member had to work within the framework of the team topicbut had autonomy to plan, conduct, and evaluate the project, accordingto the work conditions, the teachers involved, the student population, andpersonal preferences. It should be emphasized that the participants wereto work with the teachers in the school and not directly with the students.Team members met to discuss their work, plan activities, consult with oneanother, share and discuss ideas, support each other, and explore ways ofimplementing their plans and evaluating the implementation.

In addition, several whole-group meetings were devoted to theoreti-cal aspects of initiating change in school mathematics, such as planning

18 RUHAMA EVEN

change initiatives, the critical role of the teacher in the success of long-termeducational change, fundamental barriers to change related to the nature ofteaching, and various ways of evaluating change initiatives. To encouragethe participants to reflect on their experiences, they were required to submita detailed report as the main part of the third year’s portfolio.

Despite the differences between teams related to the nature of theirprojects, two issues were common to all. One issue was workingwith theteachers, which is closely related to social development and is discussedin the following section. The other issue dealt with participants’ emergingunderstanding that change in school mathematics is a slow and compli-cated process. When starting their projects, many set very ambitiousobjectives and expected to reach them quickly and smoothly. When thisdid not happen, many felt frustrated and unsuccessful. During the yearthey learned to set, in addition to the overall objectives, short-term andmore manageable operative objectives, to appreciate small progress, andto better understand what such an endeavor entails.

At the end of the year almost all felt satisfied with their work, eventhough they achieved only a fraction of what they had planned first.

I learned that one can dare and initiate. One must not be afraid. (GL, questionnaire,Year 3) Creating something from nothing is an extraordinary experience for me. (TL,questionnaire, Year 3)

Social development. Social development involves learning to work withother people in the educational system in new ways. The initiationof change provided numerous opportunities for such development, andindeed, many participants reported on such learning in their questionnairesand interviews. For example:

I learned that the success of the initiative depends on recruiting different factors in thesystem, such as, principals, superintendents, home-room teachers, etc. (GL, questionnaire,Year 3) Dealing with technical and organizational problems, financial resources, and schoolpolitics is new to me. I learned a lot from it. (MR, questionnaire, Year 3)

If initiating change is not a one-person project, then more attentionmust be paid to teamwork. At the beginning of the year, quite a few ofthe participants tried totell the teachers what to do and how to do it.Being pressed to show results, many of the participants embraced thecommon didactic approach. After overcoming the stage of complainingabout the teachers’ lack of cooperation, they learned that it was importantfor the teachers to have a sense of ownership, and the participants startedto encourage the teachers to participate in the planning, decision making,assignment of roles, setting of time-tables, and sharing of responsibilities– key points for successful professional development work with teachers


(e.g., Clarke, 1993; Loucks-Horsley et al., 1998). Although many used aless didactic approach to the teachers, it seems that in cases where notice-able results were expected, it was hard for the participants to approach theteachers as thoughtful learners. Instead, they tended to focus on achievingthe goals they set, which were in many cases too ambitious for the timeand resources available.

Teamwork also means working together with other teacher leadersand educators. Several studies (e.g., Fullan, 1990; Loucks-Horsley et al.,1998; McLaughlin, 1991) suggest that teacher collegiality and collabora-tive work environments are critical to change. We expanded this idea toteacher leaders/educators’ collegiality and collaborative work. This featureof the program appeared to be important, as, for example, reported in thefollowing excerpts:

The discussions and the search for solutions together with other team members gave mestrength in situations where I wanted to give up, thinking that it was impossible to changethe teachers in this area, and that nothing would help. It gave me a lot of strength. (SR,questionnaire, Year 3) My colleagues [the other team members] were an important andprofessional source of support, feedback and consultation. (RB, questionnaire, Year 3)

These initiatives were not the only time that participants worked coop-eratively. Throughout the three-year program we emphasized the develop-ment of a professional community. For example, they conducted severaltasks in pairs (e.g., the mini-research), and made team presentations. Weencouraged the participants to open their work to colleagues both forcritique and use. They read each other’s yearly portfolios for feedback andgave oral presentations to other course participants, to other mathematicseducators in national conferences and meetings, and to administrators suchas school principals and superintendents.

The reason for encouraging the course participants to present their workto others was three-fold. First, the work they carried out was important andinnovative, and influenced the field of mathematics teaching and learn-ing in positive ways. Second, because of the difficulties associated withattempts to achieve change, we wanted administrators and other influentialfigures in the educational system to know and appreciate the quality of thework the participants were doing and would be able to continue if theyreceived appropriate support. In other words, we wanted to help our grad-uates in their future work by opening doors for them. Finally, we wanted tohelp the participants develop a sense of professionalism in which makingone’s work public, so that colleagues can learn from, criticize, and improveit, is part of the game.

20 RUHAMA EVEN

DISCUSSION

At the end of the program, participants had grown professionally andpersonally. Such growth could be detected in the participants’ reports onand our observations of the weekly teacher development activities theyconducted throughout the three-year program. They gradually paid moreattention to the teachers’ needs and desires, and they were able to identifyalternative possible strategies and make sound choices. The content andthe topics of the teacher development activities also changed: The activitiesbecame richer. They teacher leaders paid more attention to teacher learningabout learning processes and students’ conceptions and ways of thinking,they examined student assessment seriously, and they included coopera-tive analysis of events that the teachers in the group had experienced. Inconducting the teacher development meetings, the participants graduallyencouraged active participation of the teachers, started to use technologi-cal tools, and emphasized the development of teamwork. Their reflectiveabilities also developed considerably until they were able to criticize theteacher development activities they conducted and to suggest modifica-tions. However, the balance between the need to support teacher learningand the need to initiate change seems to be a problematic issue that remainsa challenge to both the participants and the project staff.

Lessons Learned

The program participants were not the only learners during this three-yearprogram. The program staff learned how to conduct a program for thedevelopment of teacher leaders and educators, what its focus, content, andnature might be, or how to operate it. In the following I attempt to sharesome of our insights.

Regular and frequent staff meetings. It sounds almost trivial but it ishard to over-emphasize the importance of regular frequent staff meetings.At the start of the program, the six staff members (all have doctoral ormasters degree) possessed extensive prior experience, mostly in curricu-lum development and implementation, and in school-based inservice workwith teachers. However, the mission of the Manor Program was differ-ent. The bi-weekly staff meetings, although not always appreciated byall staff members in the beginning, became crucial to the developmentof a common vision and a feeling of shared ownership. In these meet-ings we discussed aims and ways of operation; we planned together, andlater reflected on, specific activities and the general program; we madeexplicit previously implicit beliefs, guidelines, and desires; we negotiated


alternative directions; we consulted each other when trying new ideas; wehelped solve problems encountered during the operation of the program;and we divided responsibilities. In addition to contributing enormously tothe development of a common vision and a feeling of shared ownership,these meetings served also as the means for the actual operation of theeveryday life of this complicated program.

Feedback and support. Based on results of studies on teacher developmentand learning (e.g., Bell & Gilbert, 1994; Loucks-Horsley et al., 1998),attention was given to support and feedback throughout the program, bothfrom the project staff and the participants, as was partly illustrated in thisarticle. The following example highlights a facet not described earlier;it illustrates our developing understanding of the importance and possi-ble nature of effective feedback and support. Throughout the programthe participants were required to report frequently and regularly in writ-ing on the weekly teacher development activities they conducted. Duringthe first year of the program, the staff read these reports and conductedgroup discussions with the participants on several problematic issues thatemerged from these reports, such as rationale and aims, ways of workingwith teachers in general and on specific mathematical topics in particu-lar, and so forth. At the beginning of the second year we added anotherkind of feedback, an individual written feedback on each report. Althoughthis required considerable staff time, its impact exceeded our expectations.During the first year there was moderate improvement in the quality of theactivities, and only a slight change in the participants’ ability to describecoherently what they did and why, and in their ability to analyze andreflect on the activity. During the second year, however, these dimensionsimproved greatly until the participants were able to criticize the activi-ties they conducted and, during the third year, to suggest modifications.Consequently, with the second group we have included individual writ-ten feedback on each report from the beginning of the first year, andindeed, the above changes have been noticed much sooner. In general,we found that both individual and group feedback and support are impor-tant, as are cognitive and affective feedback and support. Indeed, thesewere mentioned often by the participants as important characteristics ofthe program which enabled them to learn, explore, experiment, share, andgrow.

Connecting theory and practice. Another factor that seemed to contributeto the participants’ learning has to do with the connection between theoryand practice that characterized many of the learning experiences offered

22 RUHAMA EVEN

in the program. Theoretical issues were always illustrated by examplesfrom practice, and, vice versa, analyses of specific cases of practice weretied to theory. I have described two representative examples, the “Whatis a good problem?” activity and the mini-studies. The importance of thisconnection seems to be rooted in its potential to offer new ways of exam-ining and understanding practice. It helped the participants, as Leinhardt,Young & Merriman (1995) describe, make their situational, intuitive, andtacit practical knowledge more formal and explicit, and to make theoret-ical knowledge more available for use in practice. The importance of theconnection between theory and practice seems to be also related to thedialectic of beliefs and practice (e.g., Cobb, Wood & Yackel, 1990), in thatchange in one domain is connected to change in the other.

CONCLUDING COMMENTS

The need to work with inservice secondary teachers as part of the processof changing school mathematics is mentioned in several documents (e.g.,Romberg, 1984; Superior Committee on Science, Mathematics and Tech-nology Education in Israel, 1992). However, the fact that teacher leadersand teacher educators also need adequate preparation is often neglected.Although School mathematics: Options for the 1990’s(Romberg, 1984)did recognize the need to develop such programs, the literature has verylittle to offer about possible ways to construct such programs. This arti-cle describes central aspects of one such a program, in an attempt toexpand and enrich understanding about professional preparation of teacherleaders and teacher educators. We based the program on the followingprinciple guidelines: that the participants are thoughtful learners and thatthey should be prepared to be professional teacher leaders and inserviceteacher educators who view teachers as thoughtful learners, whose learn-ing should have an impact on school mathematics. These guidelines arereflected in the choice of the main foci of the program as well as inthe opportunities for learning we designed. Learning is never a simpleprocess, and learning to become a teacher leader or teacher educator is noexception.

REFERENCES

Australian Education Council (1990).A national statement on mathematics for Australianschools. Melbourne: Curriculum Corporation.


Amit, M. & Vinner, S. (1990). Some misconceptions in calculus: Anecdotes or the tip ofan iceberg? In G. Booker, P. Cobb & T.N. de Mendicuti (Eds.),Proceedings of the 14thinternational conference for the psychology of mathematics education, Vol. 1(37–44).Mexico: Program Committee.

Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics.For thelearning of mathematics, 14(3), 24–35.

Arcavi, A. & Schoenfeld, A.H. (1988). On the meaning of variable.Mathematics Teacher,81, 420–427.

Bell, B. & Gilbert, J. (1994). Teacher development as professional, personal, and socialdevelopment.Teaching and teacher education, 5, 483–497.

Bell, A. & Janvier C. (1981). The interpretation of graphs representing situations.For thelearning of mathematics, 2(1), 34–42.

Ben-Zvi, D. & Friedlander, A. (1997). Statistical thinking in a technological environ-ment. In J.B. Garfield & G. Burrill (Eds.),Proceedings of the international associationfor statistical education round table conference(45–55). Voorburg, The Netherlands:International Statistical Institute.

Clarke, D.M. (1993).Influences on the changing role of the mathematics teacher.Unpublished doctoral dissertation, University of Wisconsin-Madison.

Cobb, P., Wood, T. & Yackel, E. (1990). Classrooms as learning environments for teachersand researchers. In R.B. Davis, C.A. Maher & N. Noddings (Eds.),Constructivist viewson the teaching and learning of mathematics(125–146). Reston, VA: National Councilof Teachers of Mathematics.

Cohen, D.K. & Barnes, C.A. (1993). Pedagogy and policy. In D.K. Cohen, M.W.McLaughlin & J.E. Talbert (Eds.),Teaching for understanding: Challenges for policyand practice(207–239). San Francisco: Jossey-Bass Publishers.

Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospec-tive secondary teachers and the function concept.Journal for research in mathematicseducation, 24, 94–116.

Even, R. (1994).Tel-Aviv project for improving mathematics teaching in junior-highschools: A teacher development approach (1990–1993). Unpublished manuscript,Rehovot, Israel: Weizmann Institute of Science (in Hebrew).

Even, R. (1998). Factors involved in linking representations of functions.Journal ofmathematical behavior, 17(1), 105–121.

Even, R. & Lappan, G. (1994). Constructing meaningful understanding of mathematicscontent. In D.B. Aichele & A.F. Coxford (Eds.),Professional development for teachersof mathematics, 1994 Yearbook(128–143). Reston, VA: National Council of Teachers ofMathematics.

Fullan, M.G. (1990). Staff development, innovation, and institutional development. InB. Joyce (Ed.),Changing school culture through staff development(3–25). Alexandria,VA: Association for Supervision and Curriculum and Development.

Guskey, T.R. (1986). Staff development and the process of teacher change.EducationalResearcher, 15(5), 5–12.

Hanna, G. (1990). Some pedagogical aspects of proof.Interchange, 21(1), 6–13.Harper, E. (1987). Ghosts of Diophantus.Educational studies in mathematics, 18, 75–90.Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher &

J. Kilpatrick (Eds.),Mathematics and cognition(70–95). Cambridge, UK: CambridgeUniversity Press.

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Hershkowitz, R. & Schwarz, B.B. (1995).Reflective processes in a technology-based math-ematics classroom. Paper presented at the annual meeting of the American EducationalResearch Association, San-Francisco.

Küchemann, D.E. (1981). Algebra. In K.M. Hart (Ed.),Children’s understanding ofmathematics(102–119). London: John Murray.

Lappan, G., Fitzgerald, W., Phillips, E., Winter, M.J., Lanier, P., Madsen-Nason, A., Even,R., Lee, B, Smith, J. & Weinberg, D. (1988).The middle grades mathematics projectfinal report to the National Science Foundation for grant #MDR8318218. East Lansing,MI: Michigan State University.

Lee, L. & Wheeler, D. (1986). High school students’ conception of justification in alge-bra. In G. Lappan & R. Even (Eds.),Proceedings of the seventh annual meeting of thePME-NA(94–101). East Lansing, MI: Michigan State University.

Leinhardt, G., Young, K.M. & Merriman, J. (1995). Integrating professional knowledge:The theory of practice and the practice of theory.Learning and Instruction5, 401–408.

Loucks-Horsley, S., Hewson, P.W., Love, N. & Stiles, K.E. (1998).Designing professionaldevelopment for teachers of science and mathematics. Thousand Oaks, CA: CorwinPress.

McLaughlin, M.W. (1991). Enabling professional development: What have we learned? InA. Lieberman & L. Miller (Eds.),Staff development for education in the 90s(61–82).New York: Teachers College Press.

National Council of Teachers of Mathematics (1989).Curriculum and evaluation stan-dards for school mathematics. Reston, VA: Author.

Romberg, T.A. (1984).School mathematics: Options for the 1990’s. Chairman’s report ofa conference. Washington, DC: U.S. Government Printing Office.

Romberg, T.A. & Pitnam, A.J. (1990). Curricular materials and pedagogical reform:Teachers’ perspective and use of time in the teaching of mathematics. In R. Bromme& M. Ben-Peretz (Eds.),Time for teachers: Time in schools from the practitioner’sperspective(189–226). New York: Teachers College Press.

Sfard, A. (1995). The development of algebra: Confronting historical and psychologicalperspectives.Journal of mathematical behavior, 14, 15-39.

Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification – the case ofalgebra.Educational Studies in Mathematics26, 191–228.

Superior Committee on Science Mathematics and Technology Education in Israel (1992).Tomorrow 98: Report. Jerusalem, ISRAEL: Ministry of Education, Culture and Sport(English edition: 1994).

Tall, D.O. & Schwarzenberger, R.L.E. (1978). Conflicts in the learning of real numbersand limits.Mathematics Teaching, 82, 44–49.

Tirosh, D., Even, R. & Robinson, N. (1998). Simplifying algebraic expressions: Teacherawareness and teaching approaches.Educational studies in mathematics, 35, 51–64.

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Department of Science Teaching,Weizmann Institute of Science,Rehovot 76100,ISRAEL

BRENT DAVIS

BASIC IRONY: EXAMINING THE FOUNDATIONS OF SCHOOLMATHEMATICS WITH PRESERVICE TEACHERS

ABSTRACT. This article reports on an inquiry into what tends to be taken for grantedwith regard to the teaching and learning of mathematics. The inquiry, undertaken in thecontext of a course on methods for mathematics teaching, was developed around an exam-ination of the mathematical notions that infuse conventional theories of cognition and thatpermeate the structures and practices of school mathematics. In particular, concepts drawnfrom or aligned with Euclidean geometry were examined. Specifically, alternatives drawnfrom fractal geometry were explored. The importance of interrogating the often-transparentfigurative underpinnings of our thinking about thinking is highlighted.

THE PROBLEM OF THE BASICS

As a mathematics educator and a teacher educator, among the things I havefound most troubling in my professional life are teachers’ and prospectiveteachers’ thoughts on the nature of mathematics and on the role of math-ematics in their lives – or, more precisely, their lack of thoughts on thesetopics.

The issue was brought home in a recent introductory level course onmethods for teaching mathematics. Early in the term, I requested a groupof 19 future secondary teachers to compose for presentation in class a briefresponse to the question,What is mathematics?The assignment includedthe qualification that I was not after a formal definition. Rather, the inten-tion was to spark their thinking about what it was they would later beteaching. Because all of the teacher candidates had completed a baccalau-reate degree in mathematics or a mathematics-related field, and most ofthem had taken a course in the history and philosophy of mathematics, Iexpected that their responses would prompt some interesting discussionand debate.

The following week’s class began with the reading of the descriptions,during which I collected key points and common themes on the chalkboardfor later discussion. It quickly became apparent, however, that there wasnot going to be much to mull over, as the contributions were easily cat-egorized according to three repeated ideas: mathematics as “the study of


26 BRENT DAVIS

relationships”, a notion that had been suggested by another instructor theprevious week; mathematics as “the study of pattern”, which had comeup in an assigned reading; and formal definitions drawn verbatim fromdictionaries of mathematics. Not one person had undertaken to composeher or his own description. My impression was that only a few suspectedtheir contributions might be lacking in any way. Whether due to misin-terpretation of my intentions or to some other factor, their descriptionsfailed to demonstrate an awareness of the complexity of thought and thediversity of opinion that are present in discussions of what constitutesmathematics.

Personally convinced that there is little hope for substantive changein the culture of school mathematics without deepened appreciations ofenacted beliefs on the nature of the discipline – and, of course, of therationales for teaching the subject matter that fall out of those beliefs –I set about to prompt more critical understandings of both the nature ofmathematical knowledge and the relevance for teachers of engaging withthat issue. This article represents, in some ways, a report of my ongoingefforts toward these ends.

The article is developed around a teaching strategy I call abasic irony,borrowed from Rorty’s (1989) notion of irony. In brief, Rorty describedan irony as the “opposite of common sense” (p. 74) which involves thedeliberate interrogation and reformulation of mindsets and worldviews. Asan intellectual stance, to be ironic is to be willing to question the familiar,to trouble the taken-for-granted, to seek out the transparent prejudices thatinform perception.

Irony, in this sense, is a deliberate effort to turn something onto itself.In his analysis, Rorty (1989) focused on the development of new figura-tive language as a means of revealing and interrupting the metaphors andimages that are used to make sense of various concepts and phenomena.Such efforts, he argued, render commonsensical notions more opaque andmore available for critical examination. By way of example, a popularmetaphor in discussions of education and psychology isbrain as computer.Although originally intended as an analogy that might offer insight intoparticular cognitive processes, the figurative dimension of this notion isoften forgotten. In fact, brain as computer is now commonly presented asa literal truth. In recent years, however, there has been considerable effortdevoted to revealing the limited and limiting nature of this metaphor, adiscussion which, ironically, relies on the invention and deployment of newmetaphors.

For Rorty (1989), such ironies simultaneously depend on and revealthe play of language. In the process, they highlight the uncertainty and

BASIC IRONY: EXAMINING FOUNDATIONS 27

volatility of our language-dependent interpretations of the world. Ironies,however, need not be strictly focused on vocabulary. Gödel’s (1931)Incompleteness Theorems, for example, are instances of ironies in thecontext of mathematics. In essence, Gödel used a privileged form of argu-mentation to reveal its own limitations. In a similar, but less sophisticatedmanner, this paper is intended to be ironic. In it, I use a linear argument tochallenge the appropriateness of thinking about mathematics teaching inlinear terms.

In my teaching methods course, the focus of the exercise in irony wasthe manner in which mathematics has permeated modern and Westernsensibilities. The context for the exercise was an introductory study offractal geometry. In brief, I sought to adapt the strategy of turning thingsonto themselves through an examination of the history and some of thesurprising results of this recent branch of mathematical inquiry. My hopewas to engage class members in a process of uncovering and recognizinglong-established mathematized notions that have come to infuse uncriticalunderstandings of the nature of the subject matter and that have seepedbeyond the conventional bounds of the discipline to serve as common sensenotions for describing and explaining the world.

My use of the termbasicto qualify irony is also intended to be ironic.As will be developed, one of the preoccupations of mathematics educa-tion has been with the basics, albeit the notion is subject to a diversityof interpretations. I employ it here both to point to the ongoing worriesabout basics and as a reminder that even our basics seem to have a basis.I maintain that it is the more fundamental category, the one that tends tobe accepted uncritically, that should be the subject of our basic concerns.I personally find it ironic that a field that is so preoccupied with basicsis, for the most part, so out of touch with the foundations of its ownactivity.

I use a teaching episode to frame this discussion, but my intention indoing so is not to prescribe a course structure for others. As I hope will beclear, the classroom happenings were dependent on a host of contingen-cies, including who was present, what some had seen on television, andwhat other instructors were suggesting. Moreover, references to my ownteaching should not be read as attempts to present evidence in support ofan argument. They are, rather, for the purpose of developing the notion of abasic irony as a pedagogical device – one which, in its capacity to uncoverunconscious knowings and doings, may help to transform how we thinkabout mathematics, education, learning, and teaching.

Although not a principal purpose of this article, I feel it important toattempt some articulation of the attitude toward teaching and formal edu-

28 BRENT DAVIS

cation that infuses this writing. In brief, I do not regard schooling as aninnocent activity. It is always and already implicated in the transformationof culture whether or not we are consciously aware of it. Moreover, aswith all realms of human activity, it is complicit in the manner in whichhumanity understands and acts out its relationship to what is perceivedas the non-human part of the universe. Mathematics has played a keyrole in the dichotomization of humanity and nature, just as more recentdevelopments in mathematics have worked to problematize the distinctionwe draw between our species and the rest of the biosphere. As such, Iwrite from the perspective that educators must be more deliberate partic-ipants in making culture, and this entails conscious efforts to erase theperceived and enacted distance between ourselves and the complex livingworld.

USING FRACTALS TO EXPLORE THE ISSUE OF BASICS

There were two reasons for choosing the topic of fractal geometry as thecontext for this inquiry. First, fractal geometry is a relatively new domainof mathematical inquiry. As such, it provides a location for examining thedynamic character and unpredictable directions of mathematics research.Given the unprecedented number and the currency of the histories of thisbranch of mathematics, fractal geometry provides a rich site for wonderingabout what it is that mathematicians do, how mathematical research pro-ceeds, how mathematics itself evolves, and how our formal mathematicsshapes and reshapes our more unformulated perceptions and involvementsin the world. For the mathematics teacher, for example, fractal geometryhas the capacity to interrupt common sense, as it provides a set of imagesand metaphors that might be used to examine some of the transparentbackdrop of conventional educational practice. The Euclidean notions thatunderpin linear lesson plans, hierarchical curricula, grid-like classroomarrangements, and rigid boundaries of subjects become more apparent inthe context of forms that are nonlinear and dynamic, and whose bound-aries slip between dimensions. In the same way that cultural idiosyncrasiesand worldviews are revealed when a society’s customs are cast againstthose of other cultures, so the usually invisible aspects of one mathemat-ical system become more apparent when that system is juxtaposed withanother.

The second reason for developing the basic irony around fractal geom-etry was more local: As a topic that was relatively new to the methodsstudents, fractal geometry provided an opportunity to monitor our ownprocesses of mathematics learning. In terms of the issue guiding the


Figure 1. A Fractal Card. (The photograph is courtesy of Barbara Budd of the VancouverSchool District.)

inquiry, I felt it critical to be immersed in mathematical activity whilewondering about the nature of mathematics. A tenet of interpretive inquiryis that one cannot stand outside the object of one’s investigation, but must,rather, consider how one is shaped while participating in shaping thatobject.

The basic irony was introduced through the construction of FractalCards, an activity that involves establishing and recursively applying asequence of folds and cuts on paper in order to create a three-dimensionalrepresentation of a fractal image (see Simmt & Davis, 1998). It beganwith little formal introduction. Several completed cards were presentedand some terms that pertained to their construction were defined before theclass was launched into the guided construction of the example pictured inFigure 1. Within a few minutes, all present had generated a card, and agood portion of the class had begun experimenting with other cuts, folds,and combinations.

Most of the introductory three-hour time block was devoted to playingwith the activity: exploring variations, attempting to recreate completedcards, noticing relationships, generalizing patterns, and identifying topicsin mandated curriculum documents that might be addressed through thisactivity. As it turns out, virtually every topic in our provincial curriculumwas at least touched on.

30 BRENT DAVIS

UNCOVERING ASSUMPTIONS

Prior to the second class into the topic, I had assembled some questionsto assist with orienting the discussion of the fractal cards activity withthe intention of getting at the issue of the nature of mathematics: How isthis activity mathematical? How does it problematize the earlier defini-tions? What does it say about the process of mathematical research? Is thisdomain of inquiry about discovery? Creation? Both? Neither?

The discussion went in quite a different direction, however. The firstand, as it turned out, the most persistent issue to arise had to do withfitting the activity into the established school curriculum. Other topics ofconcern included questions around structuring the activity to reduce ambi-guity, ensuring that all necessary background topics had been covered, andevaluating student performances.

In other words, there was nothing ironic about the discussion: Theintended ground of the task became the stubborn figure of student con-cern. The fractal cards activity, rather than serving as the backdrop of aninterrogation of what tends to be taken for granted in mathematics teach-ing, was treated like any other classroom activity. It became an item tobe dissected, located, and catalogued within the matrix of the mandatedcurriculum.

Rather than despairing of my plans and giving in to the desire to controlthe direction of the learning – which would have amounted to a fallingin to the same mindset that I was seeking to avoid – I chose to let thediscussion continue around these issues for the entire first half of the class.As it turned out, several important considerations did come up, includingmaking teaching more attentive and responsive to learner actions and mov-ing the teaching focus away from isolated concepts to contexts wherein anarray of ideas arise. These served as the starting place of the post-breakportion of the session.

In that second half, I more deliberately attempted to prompt an inver-sion of the figure and ground of the earlier discussion, seeking to usefractal geometry as a vehicle for experiencing some of the preoccupationsabout teaching that had just been identified. The first point of examinationwas the unquestioned assumption that an official curriculum documentprovided the final word on which topics were to be addressed and inwhat order they should be covered. I began by recounting some of thedetails of the emergence of fractal geometry as a domain of mathemat-ical inquiry. The fits-and-starts nature of its evolution, its tumultuousstatus within a more formalist culture, and its reliance on electronic tech-nologies and trial-and-error are qualities that suggest the development offractal geometry is better thought of in terms of its own images than in


terms of the linear logic that is more typically used to describe traditionalmathematics.

There was no disagreement with this idea, and a few students suggestedthat the analogy could be taken a step further: The activities of a singlemathematician might be seen as a similar phenomenon to the more generalrealm of mathematics research. Notions of non-linearity and recursivityseemed to be appropriate in describing both. Additionally, qualities of thecomplex relationship between mathematician and mathematics were nicelyhighlighted by the fractal image. The whole (of mathematics) is not a mereassemblage of the parts (mathematicians); rather the whole seems to unfoldfrom the part and to be enfolded in the part.

The discussion then turned to a comparison of the fractal cards activityto the dominant structures of mathematics lessons in the students’ owneducational histories. The lack of rigidly specified learning goals, the rangeof mathematical notions addressed, the participatory role of the teacher,the unpredictable but sophisticated directions of inquiry, and the mannerin which the classroom investigation seemed to reflect many aspects ofresearch in mathematics were all cited as features that separated the activ-ity from prior experiences. After these and other points were discussed, Iventured to suggest that, like the structure of mathematical inquiry, mathe-matics learning and school curricula might be better thought of in terms ofthe nonlinear imagery and recursive structures offered by fractal geometrythan in terms of the lines, grids, and neatly distinguished areas aligned withmore Euclidean notions. That is, fractal imagery was useful for describ-ing phenomena that ranged from mathematics research to the activities ofindividuals in the mathematics class.

This suggestion was readily accepted with regard to individual cogni-tion and mathematics research. Such phenomena, it was agreed, are far toocomplex to be understood in terms of simple trajectories or linear causes-and-effects. The analogy, however, met with resistance on the issue offormal mathematics curricula. Despite the emergent structure of the fractalcards activity, it was argued that a formal curriculum represents an attemptto order existence. Unlike the complex and unpredictable nature of theresearch process, schooling is intended to tame rather than to embrace thechaotic backdrop of life. And this assertion is supported by virtually allof the textbooks, the teacher’s manuals, and the programs of study thatline classroom bookshelves. The purpose of mathematics teaching, it wasargued, is to equip learners with the basics for living, not to alert learnersto the obvious complexities of existence.

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RECONCEPTUALIZING THE NOTION OF BASICS

The termbasics, in fact, arose with a surprising frequency. Suspecting thatthere was consensus on neither the meaning of the term nor on the empha-sis that should be afforded the basics in the mathematics classroom, I endedthe second class by assigning the task of preparing a brief description ofwhat each person meant by the wordbasics.

By the following week, most of the students noticed that they had beenusing the term somewhat ambiguously. For the most part, their use wasin reference both to those concepts that are considered fundamental toother concepts, e.g., in the way that counting is basic to addition, and tothose competencies that have been deemed necessary for an adult citizenin our society. It was further noted that, in both cases, what is basic isclearly neither pre-given nor stable, but dependent on particular interestsand social circumstances.

Despite the apparent variations in meaning, however, the similarity ofboth senses was quickly noted by students. Each seems to derive froma conception of knowledge as rooted in fundamentals that can be spec-ified and upon which more sophisticated understandings can be erected.Hardly confined to discussions of school mathematics, this conception ofknowledge is deeply inscribed in Western academic traditions. The mod-ern desires “to get to the bottom of things”, “to identify root causes”, “toreduce to first principles”, and – ironically, as one student noted with regardto this exercise in irony – “to interrogate the ground” of particular activitiesare affiliated with this conception to some extent.

Modern mathematics serves as the principal, although not exclusive,model for the conception of knowledge that suggests that all valid claimscan be traced to a few self-evident premises. The teaching of mathemat-ics has borrowed this assumed structure – which is to say, mathematicspowerfully informs its own teaching as it offers not just a collection oftopics to be studied but a relational structure that has been interpretedas a somewhat prescriptive learning sequence. Although the myth of thatpristine and unambiguous structure has been punctured by mathematicsitself, the rational argument is sobasic to Western mindsets that it con-tinues as the dominant image after which the teaching of mathematics ispatterned.

In this context, I suggested an alternative interpretation of basics, viz.,as the knowledge that has slipped into unformulated activity, as opposed toa list of concepts or competencies. Basics, in this sense, are the tacit groundof activity and perception. This idea was discussed by Grumet (1995) whocontrasted it to more popular interpretations:


The basics [as popularly conceived] are drawn from a fantasy of communion, and weproject this wish on to history, mistaking the rounded edges of the past for a perfect circleof consensus. When the fantasy shifts location from historical sentimentality to a currentcurriculum, it becomes an agenda of control imposed on a community whose diversitysplinters the steady rhythms of shared lives. Rather than describing an existing consensus,the basics are deployed to create an arbitrary compact. (p. 15)

Whether the basics are regarded as a narrow band of testable conceptsor a more hazily defined collection of social competencies, they remainarbitrary compacts, intended to deny the complexities of existence.

In other words, both sides of the ever-present back-to-basics debatemight be making the same error, which amounts to an ironic failure toattend to the basics – that is, to the predominant worldview that is tacitly,but pervasively, enacted in our schools. This point was highlighted in arecent exchange between Lynne Cheney and Thomas Romberg in the edi-torial pages ofThe New York Times(1997, August 11, p. A13). Cheney’sfamiliar argument is summed up in her final sentence: “If we want ourchildren to be mathematically competent and creative, we must give thema base of knowledge [read: a mastery of particular procedures] upon whichthey can build.” Unlike Cheney, Romberg attempted to interrupt some ofthe uninterrogated ground of these sorts of debates, delving into the natureof mathematics and rationales for teaching it. However, he ended up inalmost the same place: “Unless we reform math education so that ourchildren can be prepared for the immense technological changes alreadyoccurring, our nation will lose – and so will our children.”

Granted, these two discussants argue for slightly different sets of com-petencies – Cheney for what is more classically considered as basic,Romberg for the abilities to “communicate, reason, compute, generalizeand formalize 20th century experience, and to serve 21st century goals.”But both amount to a desire to articulate a set of universal basics and, fromthere, to ensure that those basics are learned. It is not surprising, then, thattheir closing sentences sound suspiciously similar. Neither writer seemsto have gotten at what is being taken for granted, at those assumptionsthat support “the steady rhythms of shared lives” (Grumet, 1995, p. 16),at those basics that we don’t need to go to school to learn as they “arethreaded through our body knowledge” (ibid.).

Both arguments suggest a conception of life processes as analogous tothe structure of the linear, logical argument whereby, in order to progresstoward predicted ends, one must begin with solid foundations. The implicitcommitment to direct and linear conceptions of human existence aredemonstrated most obviously in the desire to manage the future through abasiceducation today. The underlying view of history and progress is thatof logical, linear, and predictable unfoldings. As demonstrated in the brief

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history of fractal geometry undertaken in class, this Newtonian conceptionof the universe has been losing ground in much of contemporary academicinquiry in which there has been a broad shift in metaphor away from thelanguage of causal connections and formalist structures and toward moreorganic notions (see Brockman, 1995). The same progression of movingaway from employing mathematics or physics as the model of knowledgetoward taking up biological notions to describe human knowing has beensweeping across most domains of psychological and social inquiry, includ-ing mathematics education. This shift inbasics is powerfully revealedin the constructivist-oriented investigations that now dominate mathemat-ics education research – but which, it must be admitted, have not muchaffected standard school curricula.

Such points made for some lively class discussions. In the end, however,my students were reluctant to set aside the linear curriculum guides and theeven more linear lesson plans that had been so fundamental to their livesin formal educational settings. As they cited needs for effectiveness, effi-ciency, uniformity, testability, and control, our third class period devotedto this exercise in irony ended with a general agreement that the notionof basics as embodied or enacted suggests a fractal-like character thatcompels consideration of both agent and collective as similar, dynamic,and nested. However, the formulated basics of curriculum guides, howeverartificial, were argued to provide a necessary structure for formal educa-tion, a perspective grounded in the preservice teachers’ perceived needsfor predictable outcomes. It bears mentioning that there was also consider-able agreement on the suggestion that mathematics teaching would likelyfollow the evolutions in mindset that are more evident in other culturaldomains. Put differently, participants agreed that mathematics will likelycontinue to inform its own teaching through the assumed structure of itsown knowledge claims. As postmodern sensibilities are taken up insideand outside of mathematics, new possibilities for curriculum and teachingwill likely arise.

There was no great consensus, however, on the issue of the mathematicsteacher’s role in effecting such changes. But the variations in educationalphilosophy were not split between those who argued that teachers mustseek to preserve the status quo through working to transmit culture andthose who felt teachers must be critical or radical instigators of socialtransformation. On the contrary, it seemed that everyone agreed that teach-ers were inevitably agents of social change. The difference in opinion layin whether that agency is enacted as an inevitable part of a larger cul-tural evolution, or whether it should be taken up more deliberately andconscientiously.


EXPLORING INDIVIDUAL COGNITION

Our basic irony, over the course of just a few classes, had thus taken usinto discussions of the nature and evolution of knowledge, the processesof social change, varied philosophies of education, and cognition. As ateacher, I was pleased with the way students were beginning to interrogatewhat was being taken for granted and to propose alternatives for thinkingand acting. However, feeling the need to bring a little focus to the discus-sions, and wanting to address issues of learning more directly, I thoughtthat it might be a good time to look at theories of cognition.

Given the complexity of current discussions, cognition is a topic that Ihave had trouble presenting in an accessible manner to preservice teachers.Framing the topic in terms of fractal geometry, however, proved to be aneffective introduction. In brief, I began by noting some of the most com-mon foci of conventional educational research, citing issues of learning,the joint production of knowledge, and the role of society in determiningthe character of the individual. As had already been taken up in class,these phenomena are hardly distinct, but seem to be intertwined, in muchthe same way that mathematician and mathematics are caught up in oneanother. In particular, the way that each seems to be as complex a phenom-enon as the others is analogous to the way that the intricateness of fractalimages does not vary with scale.

The lesson was decidedly more structured and teacher-centered, as Ielected a principally lecture format to present some background infor-mation on theories of cognition. The presentation was to be interspersedwith moments of discussion, intended to provide students with an oppor-tunity to articulate, reformulate, and re-present understandings, as wellas to point to possibilities for further investigation. The following repre-sents a brief summary of the key points of the lecture and the ensuingdiscussion.

The differences and similarities of the varied constructivisms, the issuesthey address, and their objects of inquiry have been discussed in sev-eral contexts (e.g., Bereiter, 1994; Davis, 1996; Davis & Sumara, 1997;Spivey, 1997; Steffe & Gale, 1995) as researchers have looked for sharedassumptions, similar processes, common logics, and parallel products.These constructivisms share, for example, the goal of questioning ourcommon sense about thinking. All of them tend to highlight uncriticalmachine-based (e.g., mind as computer), ocular-centric (e.g., understand-ing as seeing), and corporatist (e.g., knowledge as capital) metaphors.Each questions the ways that cognition has been cut up and located.Mind/body, self/other, individual/collective, nature/nurture are some of thedichotomies that have been rendered problematic by interpretations of

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cognition that embrace the inseparability of activity, understanding, andidentity.

Given their prominence in current mathematics education research, Ibegan with an introduction of those constructivist discourses that focuson issues of individual cognition. A central thesis of these discourses,viz., that the cognizing agent’s basis of meaning is found in her or hisdirect experience with a dynamic and responsive world, was illustratedthrough reference to an unanticipated outcome of the fractal cards activ-ity. Earlier in the term, several students had reported that they had “seenfractals everywhere” – in trees, in salad bars, in landscapes, and so on –after they had been introduced to the new category of geometric forms.Conception and perception had become conflated for these students inthis event. In this way, the event served as support for the constructivistassertion that cognition is a process of maintaining an adequate fit withone’s ever-changing circumstances, as opposed to progressing toward anoptimal internal representation of an external world.

A more sensual, embodied attitude toward cognition was also readilyappreciated in the context of our fractal cards activity. Construction ofthe cards provided an opportunity to attend to the kinesthetic dimensionsof understanding. The importance of attending to the physical ground ofone’s understandings proved to be a point of tremendous interest to thepre-service teachers. The balance of this class was spent in small group dis-cussion of the sorts of physical experiences that might underlie or influencemathematical competencies.

EXPLORING COLLECTIVE COGNITION

I began the next class with the reminder that subject-centered construc-tivisms not only posited a more sensual cognition, but a cognition that wasnot strictly internal. That is, not only do other parts of the body partic-ipate in thought, knowing is also distributed across the objects of one’sworld. Although a departure from the commonsensical notion that thoughtand memory reside in the brain, the stretching of the cognition beyondneural processes and physical activity to include various artifacts seemedto be appreciated. One student’s example of the necessity of a pencil andpaper to think through mathematical notions was particularly helpful indeveloping the idea.

For the purpose of stimulating discussion, I suggested that one’s cogni-tion should not only include one’s physical experiences and the artifactsof one’s world, but other agents in one’s world. There was immedi-ate resistance to the notion, expressed in the acceptance that others


might influence your thinking, but thoughts are ultimately personal andbounded.

Not wanting to launch into an extended debate, I proceeded with apresentation of a second category of constructivist discourses, social con-structivisms, which focus more on small groups, e.g., pairs of students,a teacher and a pupil, or a classroom, as learners build shared under-standings. This perspective on cognition is thus much less focused onthe individual than more subject-centered constructivisms and tends to beconcerned with conversation patterns, relational dynamics, and collectivetraits.

The fractal cards activity and our subsequent discussions served as anillustration of the manner in which individual understandings are caughtup in the movement of the collective. In each activity, for example, par-ticular strands of inquiry and interest arose and spread, while others werepassed over. The directions of investigation, more often than not, were notdeliberate. Rather, a simple question, a chance remark, a surprise happen-ing would occasion the collective interest to follow one path instead ofanother of an infinite range of possibilities. That is, the character of thecollective activity was similar (in the mathematical sense) to the characterof individual cognition.

Despite these noted similarities, the suggestion that cognitive processesmight not be strictly individual was at first rejected by most class mem-bers. The fractal geometric notion of self-similarity, however, proved quitehelpful in this regard. In the same way that a part of a fractal imageresembles the whole – but is not identical to the whole – so the cognitiveprocesses of the individual might be thought to resemble the dynamicsof a group. Within these nested dynamics, as with the fractal image, thepart, that is, the individual, can be seen as a whole unto itself, with itsown particular integrity. Of course, the phenomenon of knowledge, con-sidered on the level of the individual sense-maker, is something differentthan knowledge considered on the level of social grouping. But, in thecontext of the fractal cards activity, it was obvious that these were tightlyrelated and interdependent. Personal understandings were clearly inextri-cable from the emergent foci of the collective over the preceding classes.Within our discussion, it became evident that the relationship betweenindividual cognition and collective cognition was not a simple matter ofa back-and-forth or dialectical relationship. Rather, individual and collec-tive cognition appeared to be knitted together fractally. That is, as with afractal card, there seemed to be a self-similarity between smaller and largerelements.

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In keeping with the emphasis on irony, a large portion of this discussionfocused on the contrast between these fractal-informed notions and morepopular conceptions of cognition. In particular, the dependence of morepopular theories on rigid divisions, well-bounded regions, hierarchies, andlinear processes helped to uncover some of the mathematized sensibilitiesthat underpin and infuse much of current thinking about thinking.

In terms of theoretical discussions, as was highlighted in the ensuingclass discussion, this self-similarity is demonstrated in the shared use ofevolutionary and ecological metaphors – and, in particular, in the centralityof the notion of adequatefit with prevailing circumstances as the measureof individual or collective knowledge. The principal point of departure ofsocial constructivist theories from subject-centered constructivist theories,then, is not on matters of process or product, but on the phenomenological-and-biological order of the object of inquiry, i.e., collective rather thanindividual activity.

EXPLORING CULTURAL COGNITION

My impression was that the fractal imagery was essential to drawing atten-tion to the similarities of these two categories of constructivism and toenabling an understanding of cognition as a much broader and more com-plex phenomenon than had been previously assumed. Further, the nexttopic of discussion, critical and sociocultural theories and their contribu-tions to thinking about thinking, was supported by the fractal image. Thetopic was broached through the question, “What happens if you pull thecamera back even further on this nested interpretation of individual andcollective cognition?”

Unlike subject-centered constructivist discourses, which tend to bemost interested in how the individual shapes an understanding of the world,cultural constructivists are generally more interested in how the worldshapes the understanding of the individual. Rooted in critical and inter-pretive philosophic traditions, these discourses began their inquiries intothe complex characters of culture and identity long before ComplexityTheory came together as a field of study. Moreover, although predatingfractal geometry and non-linear dynamics, critical theorists have long beenarguing for very fractal-like notions to trouble the rigidly logical and linearsensibilities that permeate Western thought.

With reference to mathematics education, some of the topics dis-cussed by cultural constructivists have included the hidden agendas ofthe classroom, such as the establishment and maintenance of gender andracial norms, the enactment of tacit social contracts among educators


and learners, and the privileging of mathematics and mathematized sys-tems of knowledge. The actual in-class discussion of these issues tookup more than two sessions, as we examined postmodernist, feminist, andneo-Marxist critiques of education, generally, and mathematics education,specifically.

Notably, these discussions were characterized by what I interpreted tobe a deep appreciation of the complex intertwinings of individual and cul-ture. This understanding was highlighted in a part of the discussion whereseveral students, making figurative use of the notion of self-similarity,pointed to the centrality of the notion of body in subject-centered, social,and cultural constructivisms alike. The figure of the body is different ineach case – with subject-centered constructivisms focusing on the bodybiologic, social constructivisms focusing on the body epistemic, and cul-tural constructivisms focusing on the body politic – and these intertwiningbodies might be considered in terms of the different orders of phenomenathat serve as the foci of the varied constructivisms.

This metaphoric commitment to the body across the interpretive frame-works proved vital to understanding their shared logics. If one imaginesthe notion of body as analogous to a region of a fractal figure – that is,as an element which can be regarded as an object unto itself, as a fractalcomponent of a larger object, or as an assemblage of smaller objects –the relationships among the varied constructivisms become more apparent.Further, the body metaphor recalls the fact that one is always, at best,studying only a part of the whole. But that part, since it is a fractal andnot a mere fragment, has the capacity to point beyond itself.

I ended the multi-session study of constructivisms and cognition byasking, “Where does cognition happen?” The question served to highlightthree common themes of constructivist discourses: First, as just mentioned,each focuses on a body (biologic, epistemic, and/or politic) as the site ofcontestation. Second, they all draw on evolutionary theory to describe thedynamics of their subjects/objects of inquiry, whether the individual, thesocial group, or a culture. Third, cognition is not seen as locatedin a body,but as a means of describing the relationships that make that body cohere orthat enable that body to maintain its viability and integrity within a larger,similarly dynamic and responsive context.

The question also served as a reminder of the ironic nature of the dis-cussion. The assertion that cognition could not be unambiguously locatedprompted a brief discussion of our discomfort with phenomena that haveno tidy edges, that refuse linear characterizations, that disallow reductivedescriptions and explanations – in brief, that do not fit with concept, forms,or analytic tools of classical mathematics.

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RECONCEPTUALIZING COGNITION

In terms of the theories of cognition and their implications for mathematicsteaching, my original intention was to limit examinations to subject-centered, social, and cultural constructivisms. The discussions, however,spilled beyond these bounds – in part, ironically, because of the frac-tal analogy used to draw attention to the common ground of the variedinterpretive frameworks.

The fractal logic developed above suggests that the tendency to limitdiscussions of cognition to the domain of human sociality is troublesome.This point was, in fact, made by several of the students in the context ofour discussion of cultural constructivisms. If cognition is fractal-like, itwas argued, we should be able to extend the analysis in two directions:both to sub-human processes and beyond humanity. From my position asinstructor, this was an important moment in the basic irony, as it seemedthat the fractal notion had moved beyond its intended role of an illustra-tive analogy and allowed, and perhaps even compelled, us to examine themulti-tiered dynamics of cognition. That is, the possibility of a furtheriteration of the analysis was made evident and was encouraged throughthe emergent understandings of fractal geometry.

In response to students’ questioning the possibility and relevance ofextending these discussions of cognition, I elected to use a small portionof class time to present some details on other domains of inquiry that havetaken up similar strands of thought. On the matter of sub-human processesthat that might be aligned with this fractal logic, for example, I pointed totwo areas of inquiry, viz., studies of brain structure and research into theimmune system.

On the former, recent research has demonstrated that the brainseems to be fractally structured and that the activities at each level oforganization resemble those at every other level (see Calvin, 1996). Suchanalysis renders the Euclidean model of a pyramid-shaped hierarchy,with individual neurons at the base and the functioning brain at the apex,inadequate, because each level of functioning seems to have its ownparticular autonomy and integrity. That is, cognition is not merely a globalprocess that emerges in the amalgamated activities of neurons, but aprocess that is embodied in each element, be it neuron, minicolumn, orhemisphere. In a related manner, recent HIV/AIDS-prompted researchhas demonstrated that the immune system learns, forgets, hypothesizes,errs, and recovers in a complex dance with other bodily systems(Varela, 1989). Neither fully autonomous nor a mere mechanical compo-nent of a larger whole, then, it seems that a person’s immune system is


related to the person in the same way that the individual is related to thecollective.

In the other direction, current work in global ecology has demonstratedthe devastating consequences of a conceptual separation of humanity fromnature. Perhaps finding its clearest articulation in the Gaia Hypothesis (seeLovelock, 1979), and gaining in popularity as a way of understandinghumanity’s role in the biosphere, this conception of life on Earth positsthat our species is a mere sub-system of a larger organismic relationality– a grander body of which we are part. The implications for formal edu-cation are immediate in this regard: Knowledge is not merely knowledgeof the world, but knowledgein the world, wholly complicit in shaping theconditions and realities of the planet.

Notably, in our discussions, these extensions were not merely matters ofstretching an analogy. Rather, the fractal cards activity itself, like so manyfractal images, prompted us toward this way of thinking as many of thecards that had been generated bore striking and unexpected resemblancesto objects of the natural world: trees, snails, faces, mountains, an arteryseparating into capillaries and then rejoining into a vein, to name a few.As each of these appeared, the ideal realm of mathematics was pulled tothe ground of our physical engagement with the not-human part of theworld. This, in fact, might have been the most significant aspect of theexercise in irony. The habit of thinking of mathematics as existing on anideal Platonic plane (or, more recently, on a social plane) was interruptedwith the realization that our mathematical knowledge does not separate usfrom but knits us together with the rest of the biosphere. As Kline (1980)has suggested,

Unexpected . . . uses of mathematical theories arise because the theories are physicallygrounded to start with and are by no means due to the prophetic insight of all-wise math-ematicians who wrestle solely with their souls. The continuing successful use of thesecreations is by no means fortuitous. (p. 295)

Kline might be interpreted as suggesting a sort of fusing of subject-centered, social, and cultural constructivisms. Beyond that, he is clearly sit-uating humanity, through mathematics, in conversation with a responsiveand dynamic universe.

With the same sort of insight, our basic irony into the nature of mathe-matics moved beyond questions of personal, social, and cultural activityinto wondering about existence itself. That is, there was a recognitionof the inextricability of issues of epistemology and ontology. This samesensibility is part of recent ecological theories of cognition, such as enac-tivism (Varela, Thompson & Rosch, 1991). Briefly, such theories mightbe described as further iterations of constructivisms. They simultaneously

42 BRENT DAVIS

apply the same sorts of logics, metaphors, and images at phenomenal andbiological levels that range from the sub-cellular to the planetary, neces-sarily collapsing questions of cognition with questions of life. The idea isconcisely captured in aphorisms offered by Maturana and Varela (1987):Knowing is doing; knowing is being. The attentiveness to the biologicalrepresents an important interruption of the conventional emphases on inter-preted experience and formulated knowledge. The role of bodily needs andphysical drives – phenomena that, within those discourses that are boundedby human sociality, tend to be disregarded, reduced to social constructions,or seen as base instincts to be overcome – thus play a renewed and vitalrole in ecological thinking about thinking.

The import of these matters with regard to the original focus of the basicirony (i.e., the nature of mathematics) was not lost in our discussions. Withthe recognition of the self-similar intertwinings of subjective knowing andcollective knowledge, mathematics, whether considered in terms of a man-ner of inquiry or a body of knowledge, came to be seen as inseparable fromthe activities of the agents who are enacting mathematized sensibilities.A return to the question, “What is mathematics?” thus prompted us toconsider how we position ourselves in relation to the perceived-to-be not-human part of the world and, within that stance, to ask how we make senseof our preoccupations and motivations.

Ensuing discussions focused on the relevance of understanding cur-riculum as a complex form, rather than an artificial structure imposed tomanage complexity. In other words, we examined the potential for think-ing of curriculum as another iterative layer, lodged in the fractal form thatnow included individual, collective, culture, and biosphere. This imaginingprompted the conversation toward, for example, the need for more fluidunderstandings of learning goals, lesson plans, and teaching approaches– a dramatic shift in thinking from the earlier noted resistance to morecomplexified understandings of curriculum.

The balance of the course was more focused on what it might mean toenact such a complex curriculum, with a significant portion of our timedevoted to examining the role of the teacher. As I have developed theseissues elsewhere (e.g., Davis, 1994, 1996, 1997), I will limit my remarksto describing the consequent attitude toward pedagogy as more participa-tory, more mindful, more attentive, and more tentative than the controllingmanner of mathematics teaching that is now widely critiqued.


STUDENT RESPONSE

The final assignment in this course involved the crafting of a position paperon mathematics teaching. Students were asked to take up, among othermatters, issues of the nature of mathematics and processes of cognitionin a discussion of their emerging conceptions of what it means to teachmathematics. In this section, I draw from some of this written work in aneffort to represent how, at the course’s end, the issues addressed in theexercise in irony were being taken up by these preservice teachers.

On the issue of the nature of mathematics, students now demonstratedconsiderable appreciation for the relevance of this topic, in their responsesto the question, “What is mathematics?” The following comment wastypical:

When we started, I knew what math was, and I couldn’t figure out why you would ask usthat. Now [I] realize that I don’t know the answer. The weird [ironic] thing is that I alsorealize mynot knowing is going to make me a better math teacher.

Such comments were generally followed by attempts to take up a fractalanalogy to discuss the nature of mathematics. Interestingly, for most stu-dents, the analogy compelled them to also talk about individual cognizingagents. A second person had the following to say:

The more I think about it, the more I realize that the question, “What is mathematics?”,isn’t really the right one to be asking. Phrased that way, it makes me think that mathematicsmust be something, out there, separate from us. A better question, at least for those of uswho will be teaching the subject, seems to be “What does mathematics do?”

This student went on to examine how mathematical sensibilities participatein shaping perceptions and world views. Her paper ended with a examina-tion of how a discussion of the nature of mathematics is really a discussionof ourselves. This conflation of the body of mathematical knowledge withthe bodies of knowers, she pointed out, is a conclusion that must be drawnif one thinks in terms of fractal geometry rather than use the more tidilybounded regions of Euclidean figures to “cut up the world.”

In fact, and to my surprise, the idea of understanding the relationshipbetween collective knowledge and individual knowing in terms of onebeing nested in the other was perhaps the most common theme in thestudents’ papers. It was certainly more prominently represented in theirwritings than it was as a topic of class discussion. In almost every case,similarities were noted between the complex and unpredictable ways thatmathematics has evolved and the ways that individual mathematical under-standings emerge. Although few students made explicit use of ecologicaland evolutionary notions in their examinations of mathematics and cogni-

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tion, for the most part they demonstrated well developed appreciations ofthe core principles of the varied constructivisms.

They also demonstrated that they were able to bring these under-standings to bear on their thinking about teaching. As one student putit,

Instead of thinking of the classroom as a collection of discrete units, it’s interesting to thinkof it in terms of a single amoeba-like body that’s made up of many smaller, distinguishable,but not really separable amoeba-like bodies.

The struggle for an alternative metaphoric frame is evident in this state-ment. My own interpretation is that this student was attempting to artic-ulate a conception of personal knowing and collective knowledge thatabandons the language of classical physics and adopts the language ofbiology. The shift might also be represented in terms of leaving behindEuclidean notions and taking up a more fractal geometric frame. Thisinterpretation was borne out in his paper, in which he critiqued a cause-and-effect mentality that he saw as underpinning the desires to predictoutcomes and control behaviors. In its place, metaphors drawn from lifeprocesses and other complex phenomena were used. His closing paragraphhighlighted this transition:

When I think about my own mathematics learning, most of that happened when therewasn’t a teacher anywhere near me. But that didn’t mean that teachers didn’t matter. Theydid. Profoundly. It’s just that they nevercausedme to learn what I learned. So what itcomes down to for me is this: remembering that my students’ learning will depend on whatI do, but will never be determined by it. As a teacher, then, I’m a participant in students’learning in pretty much the same way that, as a group, my mathematics classroom becomespart of the body of mathematics.

Although not explicitly stated, once again the concepts of nestedness, eco-logical intertwinings, and complex evolutions are intuited. I read thesestatements as demonstrations of the students’ abilities to be ironic – that is,as noted earlier, to deliberately interrogate the ground of one’s assumptionsby turning the language and the logic of one’s thinking onto itself.

BACK TO BASICS

In retrospect, the course that I taught was not so much guided by an exam-ination of the nature of mathematics, as I had originally intended, as bythe sort of thinking about thinking announced by McCulloch (1963) whenhe asked, “What is number that man may know it, and a man that he mayknow number?” (p. 1).


In posing this reflexive question, McCulloch was identifying mathe-matics as a humanity, as the potential site of anthropological inquiry. Andit is precisely that attitude that I have sought to bring to my own teaching.For me, the study of mathematics is a study of ourselves. Understand-ing mathematics, particularly for teachers, involves an appreciation of themanner in which mathematized notions are woven through our bodies, thatis, the ways that mathematics is continuously enacted beneath the surfaceof conscious awareness.

For the prospective mathematics teacher, the sort of teaching emphasisrepresented by a basic irony has a manifold purpose: compelling exam-ination of the subject matter, its nature, its contribution to perception,thought, and activity; prompting interrogation of the place of schoolingas part of a mathematized culture and mathematics as part of a culture ofeducation; occasioning an awareness of our complicity, through our math-ematics, in the conditions of the planet; fostering a mindfulness towardthe ways knowledge is enacted and the manners in which mathematicsteaching, like the fractal image, always points beyond what is immediatelypresent.

There are many readily available sites for such investigations. Besideswidely debated issues around notions of the basics, one might examinethe pervasive presence of number, the primacy of rationality over othermodes of knowing, the ubiquitous presence of the line (in time lines,story lines, lines of reason), the mathematized notions that infuse ourlanguage and underpin cultural ideals (e.g., equality, independence, auton-omy, order), and the new infusion of mathematized sensibilities that derivefrom computer use and computer-based metaphors. Or one might engagein a mathematical anthropology, borrowing from Sumara’s (1996) notionof “literary anthropology” (p. 231). This involves an analysis of a commonartifact or activity for what it reveals about sensibilities that have fadedinto transparency, that prejudice perceptions, that permeate identities. Vir-tually any article can serve as the basis for such an inquiry, as each of thethingsof our world “is modeled to the human action which it serves. Eachone spreads round it an atmosphere of humanity” (Merleau-Ponty, 1962,p. 347). Further, any topic in a grade school curriculum will do as the focalpoint of a basic irony or a mathematical anthropology – as will, for thatmatter, the actual forms in which curricula, textbooks, concepts, problems,and evaluation schemes are presented. Or one might simply examine one’ssurroundings and their shaping influences on perception. As Abram (1996)suggests:

The superstraight lines and right angles of our . . . architecture . . . make our animal senseswither even as they support the abstract intellect; the wild, earth-born nature of the materi-

46 BRENT DAVIS

als – the woods, clays, metals, and stones that went into the building – are readily forgottenbehind the abstract and calculable form. (p. 64)

Indeed, in our study of fractal images, one of the most engaging tasksinvolved stepping out-of-doors to identify and examine the self-similarityof trees, rivulets of water, flowers, and other natural forms. Most studentswondered why they had been unable to see this self-similarity prior to ourinquiry, agreeing that the Euclidean world that has been erected aroundus had likely played an important role in dulling perceptions of theseforms.

In terms of using a basic irony as a pedagogical device, a critical ele-ment is an engagement in some aspect of mathematical inquiry because,by definition, an exercise in irony relies on turning a mode of thinkingonto itself. Possible topics for such inquiry include fuzzy logic, non-lineardynamics, and knot theory. For me, fractal geometry has proven very use-ful, even in such contexts as afternoon workshops where there is littleopportunity for sustained engagement. Its utility, I believe, arises from twofactors: the ease with which its core principles can be illustrated and theready body of natural forms that are fractal-like but that are not popularlyperceived as geometric. Combined, these qualities remind us of the roleof knowledge in perception and thus open the door to examinations of theways that mathematics infuses world views and mind sets – that is, to theways that our knowing also involves forgetting, an allowing of metaphorsand analogies to slip into literalness (Rorty, 1989).

It is this forgetting that most prompts my interest in basic ironies. Con-ceived as a deliberate re-cognizing of the fractional dimensions of knowingand knowledge, a basic irony is a rethinking of what it means to knowand to teach mathematics. Re-iterating what others have already suggested(e.g., Skovsmose’s (1985) “mathematical archaeology” and Frankenstein’s(1983) “critical mathematics”), and pushing the central worry beyondhumanity’s technologies and power structures, this pedagogical empha-sis reminds us of the very ground of our activity, that is, of the basics.Following Grumet (1995),

We don’t need to go to school to learn these basics. They are threaded through body-knowledge, and no amount of resolve can make them disappear. What is basic to educationis neither the system that surrounds us nor the situation of each individual’s livedexperience. What is basic to education is the relation between the two. (p. 16)

With this assertion, I find myself, ironically, where I and my methods stu-dents began in the quest to better understand the nature of mathematics:with a concern for relations.

But the re-emergence of the idea is not a simplerecurrence. Rather,having undertaken the basic irony, the notion of relation is now a very


different one. And so its return is arecursiveevent; it underscores thatan irony is not a matter to be resolved or concluded, but an occasion forturning thinking and acting back onto themselves.

REFERENCES

Abram, D. (1996).The spell of the sensuous: Perception and language in a more-than-human world. New York: Pantheon Books.

Bereiter, C. (1994). Constructivism, socioculturalism, and Popper’s world.EducationalResearcher, 23(7), 21–23.

Brockman, J. (Ed.) (1995).The third culture: Beyond the scientific revolution. New York:Touchstone.

Calvin, W.H. (1996).How brains think: Evolving intelligence, then and now. New York:Basic Books.

Cheney, L. (1997 August 11). Once again, basic skills fall prey to a fad.The New YorkTimes, p. A13.

Davis, B. (1994). Mathematics teaching: Moving from telling to listening.Journal ofCurriculum and Supervision, 9, 267–283.

Davis, B. (1996).Teaching mathematics: Toward a sound alternative. New York: Garland.Davis, B. (1997). Listening for differences: An evolving conception of mathematics

teaching.Journal for Research in Mathematics Education, 28, 355–376.Davis, B. & Sumara, D.J. (1997). Cognition, complexity, and teacher education.Harvard

Educational Review, 67, 105–125.Frankenstein, M. (1983). Critical mathematics education: An application of Paulo Freire’s

epistemology.Journal of Education, 163, 315–339.Gödel, K. (1962[1931]).On formally undecidable propositions of Principia Mathematica

and related systems. New York: BasicBooks.Grumet, M.R. (1995). The curriculum: What are the basics and are we teaching them?

In J.L. Kincheloe & S.R. Steinberg (Eds.),Thirteen questions: Reframing education’sconversation(2nd ed., 15–21). New York: Peter Lang.

Kline, M. (1980). Mathematics: The loss of certainty. New York: Oxford UniversityPress.

Lovelock, J. (1979).Gaia, a new look at life on earth. New York: Oxford UniversityPress.

Maturana, H. & Varela, F. (1987).The tree of knowledge: The biological roots of humanunderstanding. Boston, MA: Shambhala.

McCulloch, W. (1963).Embodiments of mind. Cambridge, MA: The MIT Press.Merleau-Ponty, M. (1962).Phenomenology of perception. London: Routledge.Romberg, T. (1997 August 11). Mediocre is not enough.The New York Times, p. A13.Rorty, R. (1989).Contingency, irony, solidarity. New York: Cambridge University Press.Simmt, E. & Davis, B. (1998). Fractal cards: A space for exploration in geometry and

discrete mathematics.Mathematics Teacher, 91 (February), 102–108.Skovsmose, O. (1985). Mathematical education versus critical education.Educational

Studies in Mathematics, 16, 337–354.Spivey, N.N. (1997).The constructivist metaphor: Reading, writing, and the making of

meaning. San Diego, CA: Academic Press.Steffe, L. & Gale, J. (Eds.) (1995).Constructivism in education. Hillsdale, NJ: Erlbaum.

48 BRENT DAVIS

Sumara, D.J. (1996).Private readings in public: Schooling the literary imagination. NewYork: Peter Lang.

Varela, F. (1989).Principles of biological autonomy. New York: Elsevier North Holland.Varela, F., Thompson, E. & Rosch, E. (1991).The embodied mind: Cognitive science and

human experience. Cambridge, MA: The MIT Press.

Faculty of Education,York University,4700 Keele Street,Toronto, Ontario M3J 1P3,[email protected]

RAFFAELLA BORASI, JUDITH FONZI, CONSTANCE F. SMITH and BARBARA J.ROSE

BEGINNING THE PROCESS OF RETHINKING MATHEMATICSINSTRUCTION: A PROFESSIONAL DEVELOPMENT PROGRAM?

ABSTRACT. This article describes a professional development program that introducesmiddle school teachers to an inquiry approach to mathematics instruction as a vehicle toreform their teaching of mathematics. The program is characterized by the use of a fewillustrative units that provide an integrated context for experiences as learners and expe-riences as teachers. Project participants consisted of mathematics and special educationteachers in school-based support teams led by school facilitators. Results suggest that theillustrative units were of considerable value in promoting reform. The benefits of includingan heterogeneous group of teachers in the same professional development program and ofinvolving teachers from the same school are also discussed.

INTRODUCTION

Mathematics educators throughout the world have called for more student-centered instruction, with increased emphasis on problem solving andconceptual understanding and a greater appreciation for mathematicsas a cultural phenomenon (e.g., HMSO, 1982; Bishop, 1988; NCTM,1989, 1991; Grouws, 1992). Recently, these recommendations have foundsupport in the results of the Third International Mathematics and ScienceStudy (U.S. Department of Education, 1996). The results of TIMMSemphasize the importance of curriculum choices that engage students indepth with a few important mathematical ideas, and of teaching practicesthat center around the class struggling as a community with complex andopen-ended tasks.

The proposed changes ofwhatmathematics should be taught andhowit should be taught require teachers to rethink both their teaching practicesand the very goals of teaching mathematics. As increasingly recognizedin the mathematics teacher education literature (e.g., Clarke, 1994; Friel& Bright, 1997; Fennema & Nelson, 1997; Loucks-Horsley et al., 1998),

? This study was funded by a grant from the National Science Foundation (award #TPE-9153812). The opinions and conclusions reported here, however, are solely the authors’.


50 RAFFAELLA BORASI ET AL.

such instructional changes are not easy to accomplish, may take severalyears, and require appropriate professional development.

A critical element of any professional development program designedto promote school mathematics reform, however, is its initial stage. Thenature and extent of teachers’ participation throughout the program willdepend to a great extent on their understanding of what constitutes reform.Teachers need to reconsider and in many cases challenge their currentbeliefs about mathematics and its teaching, something that has been widelyrecognized as one of the biggest challenges for teacher education (e.g.,Thompson, 1992).

A number of programs have proved successful in initiating schoolmathematics reform (e.g., Schifter & Fosnot, 1993; Fennema & Nelson,1997; Friel & Bright, 1997; Schifter, 1998). These programs have shownthat there is no unique way to successfully begin the process of chang-ing mathematics teachers’ beliefs and practices. For example, programsinformed by Cognitive Guided Instruction have taken students’ mathemat-ics thinking as their main focus, and evaluation studies of these programshave documented significant changes in participants’ beliefs and practices(e.g., Carpenter & Fennema, 1991). In the various programs developedby Schifter and her colleagues (SummerMath for Teachers, Schifter &Fosnot, 1993; Teaching to the Big Ideas, Schifter, 1998), experiencesthat engage teachers in learning activities and challenge their conceptionsof mathematics and its teaching play a key role; this approach has alsoled to measurable changes in teachers’ beliefs (Simon & Schifter, 1991).This article examines the nature and results of a professional developmentprogram that took yet another approach to reform by introducing middleschool teachers to an inquiry approach to mathematics instruction suitablefor all students.

The program employed a number of practices that have been recognizedin the mathematics teacher education literature to be effective in encour-aging teachers to rethink their pedagogical beliefs and practices. Thesepractices can be identified as follows:

• engaging teachers as learners in mathematics learning experiences thatmodel the content and pedagogy promoted by the program, so that theycan personally experience the power and drawbacks of such activitiesand thus be in a better position to evaluate the potential value of theactivities for their own students;• including supported field experiences as an integral part of the

program, in order to provide the participating teachers with the struc-ture and support necessary to begin to put into practice their new visionfor mathematics instruction;

RETHINKING MATHEMATICS INSTRUCTION 51

• providing teachers with multiple opportunities to reflect on their expe-riences as learners and as teachers, by means of both discussions andwriting tasks, so that they can better appreciate the significance ofthese experiences and their implications for their beliefs and practices;• encouraging the participation of teams of teachers from the same

school, in order to begin to develop a support community that couldsustain teachers’ long term efforts towards instructional innovation.

At the same time, the program we examine is also characterized by thefollowing novel features:

• A few illustrative inquiry units are used as a common context todevelop several of the practices listed above.• The target audience includes both mathematics and special education

teachers.• School-based support teams are established and led by a mathematics

teacher educator (hereafter referred to asschool facilitator).

We describe how two variations of this program played out, evaluate theoutcomes of these implementations, and finally articulate what we learnedabout effective professional development practices. Methodologically, wewould characterize this study as a reflection on our practice informed bythe systematic analysis of our data. As such, we believe that our contri-bution should be received more in the spirit of the reflective practitionerliterature (e.g., Schön, 1983) than as a typical program evaluation or evena research report.

THE PROFESSIONAL DEVELOPMENT PROGRAM AND ITSIMPLEMENTATIONS

Program Development

The professional development program was developed within the largerproject “Supporting Middle School Learning Disabled Students in theMainstream Mathematics Classroom”, funded by a grant from the NationalScience Foundation (Woodward, Borasi & Packman, 1991). This largerproject aimed at supporting middle school mathematics teachers inrethinking their teaching goals and practices in light of theStandards(NCTM, 1989, 1991), while also helping them address the learning needsof all students in their classes, including mainstreamed learning disabledstudents. An inquiry approach to mathematics instruction (as articulated inBorasi, 1992, 1996) was adopted as the main vehicle to achieve this goal.


A preliminary research component of the project sought to developa better understanding, in the context of middle schoolinclusive class-rooms, of how an inquiry approach to mathematics instruction couldrespond to the recent call for school mathematics reform and to the needsof diverse students. (Note: the terminclusive will be used to indicatemainstream classrooms that include some students traditionally servedin self-contained special education classes). Three mathematics inquiryunits that focused on tessellations, area, and remodeling, respectively,were designed by a collaborative team of mathematics teacher educators,mathematics and special education teachers, and an expert in learningdisabilities. In addition to introducing middle school students to impor-tant geometry and measurement concepts, each unit was also informed bymore general goals such as (a) experiencing being part of a communityengaged in meaningful mathematical activities, (b) seeing the relevanceof mathematics to real-life, (c) developing problem-solving and problem-posing skills, (d) improving mathematical reasoning and communication,and (e) fostering confidence in one’s ability to do mathematics. Carefullymonitored implementations of each unit in a few different instructionalcontexts confirmed the potential of these units to promote key instructionalgoals supported by the NCTM Standards with both regular students andstudents with learning disabilities (Borasi, 1995; Callard, 1996; Thornton& Langrall, 1997).

Informed by this preliminary classroom research, the team who haddesigned the illustrative units created a year-long professional develop-ment program for both mathematics and special education teachers. Thegoals of this program were to (a) promote the rethinking of mathematicsteaching in middle school inclusive classrooms in the spirit of the NCTMStandards, (b) develop an awareness about students’ learning differencesand disabilities, and (c) support the participants as they began to put intopractice their new vision of teaching mathematics in inclusive classrooms.

The successful implementation of the three illustrative inquiry unitsled to the decision to organize the field experience component of theprogram around the teaching of one of these units, as a way to scaffoldteachers’ first experiences teaching mathematics through inquiry. Sets ofinstructional materials, including detailed reports of selected implementa-tions, were created to support the planning and implementation of theseunits.Experiences as learnerswere designed to engage teachers in similarlearning activities. These experiences as learners, in turn, provided thecatalyst for activities designed to introduce the participants to an inquiryapproach to mathematics instruction and prepare them for the supportedfield experiences, i.e.the experiences as teachers.


TABLE I

Key Elements of the Professional Development Program

Summer institute experiences (6 days plus independent work):

• Experiences as learners of illustrative units, each followed by explicit reflectionsfrom complementary perspectives

• Readings, reflections, and discussions on aspects of mathematics, learning, andteaching

• Informative sessions on learning disabilities and their implications for mathe-matics instruction

• Activities to prepare for the implementation of an illustrative unit as theparticipants’ first experience as teachers

Supported field experiences expectations (during the following school year):

• Adapt and implement one illustrative unit in at least one class at the verybeginning of the school year

• Adapt and implement at least one more inquiry unit (either another illustrativeunit or preferably a unit designed by the participant)

• Meet regularly with a support team during the school year

• Attend 3–4 project-wide day-long follow-up meetings, scheduled at strate-gic points during the school year, to share experiences and receive furtherprofessional development

• Final reflection on the field experiences

The constraint of working with teachers volunteering their time led tothe decision to organize the program as a 6-day intensive summer institute,followed by a semester or year-long field component. A summary of theprogram’s key elements is given in Table I.

The program was field-tested with some variations two consecutivetimes, referred to as Implementation A and Implementation B. In bothcases, the summer institute was facilitated by the team of four mathematicsteacher educators and an expert in learning disabilities who had designedthe program. The four mathematics teacher educators and an additionalfifth educator served as school facilitators during the field experiencecomponent. Key variations in the two implementations of the program aresummarized in Table II.

The following description is based on Implementation B and is intendedto provide an image of how the key elements of the program played outand of the nature of one of the illustrative inquiry units that constituted thebackbone of the program.


TABLE II

Key Variations in the Two Implementations of the Program

Implementation A Implementation B

Number andcomposition ofthe classroomteachers whocompleted theprogram

14 classroom teachers in teamscomprising at least a mathteacher and a special educationteacher from the same school

25 classroom teachers (teamsfrom the same school wereencouraged but not required)

Incentivesprovided for theparticipants

Stipend for attending summerinstitute and follow-up meet-ings, and for preparing writ-ten reflections and documentingtheir field experiences

Attend the professional deve-lopment program free of charge

Trainingexperiences

Summer institute (includingonly 1 experience as learner onTessellation and 3 days devotedto unit planning) +4 follow-upmeetings over the school year(1 of which was devoted to anexperience as learner on Area)

Summer institute (includingsome experience as learnersfor all three units and 1 dayfor unit planning) +3 follow-upmeetings over the first semesterof the next school year

Fieldexperiencesexpectations

Over the school year, imple-ment and document at least twoof our illustrative units plus aninquiry unit of their own design

Over the first semester of theschool year, implement at leastone of our illustrative units, plusanother inquiry unit (possiblyof their own design)

Supportduring fieldexperiences

(Over the entire school year)Occasional in-class support bya facilitator assign to the school;weekly meetings with the facil-itator and all the participantsfrom the same school (in somecases including teachers whoparticipated in the program theprevious year)

(Over the first semester ofthe school year) Occasionalin-class support by a facilitatorassigned to a group of closelylocated schools; weekly meet-ings with the facilitator andall the participants from thegroup of schools (in some casesincluding teachers who partici-pated in the program in previ-ous year(s))


The Summer Institute

The summer institute began with a few experiences intended to set thestage for the program’s two major strands, i.e., teaching mathematicsthrough inquiry and addressing the needs of diverse learners. After somepreliminary reading assignments and activities designed to introduce keyissues in school mathematics reform and learning disabilities, the partic-ipating teachers engaged as learners in a 5-hour-long exploration oftessellations, designed along the lines of one of the illustrative units. Wechose tessellations as the theme for this first inquiry experience becausewe expected this topic to be new to most participants, regardless of back-ground or training. Thus, mathematics, special education, and elementaryteachers alike could feel like genuine learners.

The teachers’ inquiry began with some activities around the definitionof tessellation. Prior to the summer institute, participants had received thedefinition, “A tessellation is the repeated use of any one closed figure thatcovers a flat surface without gaps or overlaps”, together with the task offinding two examples of tessellations that fit the definition. On the first dayof the summer institute, the participants were asked to determine whetherthe examples were tessellations or not, first in small groups and then asa large group. Because of the somewhat ambiguous nature of the givendefinition, this initial activity was intended to stimulate controversy andto raise the question of what should count as a tessellation. The partici-pants eventually came to agree, at least "for the time being”, on a modifiedversion of the original definition: “A tessellation is the repeated use of anyone closed figure thatcould cover the planewithout gaps or overlaps.”Some readings and follow-up discussions helped the participants identifywhat they had learned about the nature of mathematical definitions andrecognize that mathematics is a product of human activity.

The initial activity was also intended to raise genuine questions abouttessellations that the teachers may be interested in pursuing further. Inorder to personally experience what it means to engage in mathemati-cal inquiry, the teachers were invited to first articulate and then exploretheir own questions and conjectures about tessellations. This task wasconducted in small groups with the support of a variety of manipula-tives and the participants’ own examples of tessellations. The conjecturesraised frequently required participants to examine the properties of vari-ous geometric figures and/or geometric transformations. Public sharingof conjectures and results provided an opportunity not only to learn fromother peoples’ results, but also to make explicit what is involved in gener-ating and testing conjectures – a process that is at the heart of much ofmathematicians’ activities.


Throughout the summer institute, facilitators purposefully modeledteaching practices that could support an inquiry approach to teaching math-ematics. These strategies included, for example, starting with meaningfuland complex problems, accommodating different learning modalities, andvalidating student contributions during discussions by recording them onlarge sheets of paper and referring back to them.

The session immediately following the inquiry on tessellations wasdevoted to eliciting and sharing participants’ own perspectives as learnersin this experience, so that they could better appreciate the pedagogicalapproach illustrated by the activity as well as the effect it could haveon different learners. As a vehicle for this first reflection, the facilitatorengaged the group in a “walk down memory lane”, that is, participantswere first asked to identify key events in the tessellation inquiry, examinewhat helped or hindered their learning, and then share these reflectionswith the rest of the group.

In the next session, teachers who had implemented an original inquiryunit on tessellation in their classrooms were asked to provide testimo-nials from their experiences, in order to address participants’ concernsabout how such an experience would play out with students. Partici-pants were introduced to the instructional materials for the tessellationunit and assigned some readings, selected to provide further images ofwhat happened when the tessellation unit was implemented in differentinstructional contexts. The next day, the readings were discussed to addressadditional questions about the implementation with middle school studentsand to share ideas about how to read and use these instructional materials.

Later in the week, the cycle of activities was repeated around the illus-trative unit on area. That is, participants engaged in an inquiry on area,developed once again along the lines of the illustrative unit for students,reflected on this experience, and read and discussed selections from theinstructional materials accompanying the area unit. These activities weredesigned to complement the ones developed around the tessellation unitin a number of ways. Because area is a topic covered in the traditionalmathematics curriculum, this illustrative unit was used to show that aninquiry approach should not be limited to non-traditional topics. The focusof the reflection on the second experience as learners was on its math-ematical content and goals, in order to draw participants’ attention toother important aspects of teaching mathematics through inquiry. Simi-larly, the readings from the instructional materials were selected to exposeparticipants to content and formats other than those previously assigned.

In addition to the activities discussed so far, the summer institute alsoincluded a brief introduction to a unit on remodeling, a series of sessions


addressing learning disabilities, the discussion of the characteristics andimplications of the teaching mathematics through inquiry approach, andshared journals. The use of journals was a daily practice in which eachparticipant was required to contribute a reflective entry which was to becopied and distributed to the rest of the group.

All these activities helped the participants become familiar with theunit they would teach as their first field experience requirement. Towardsthe end of the summer institute, additional activities were offered to scaf-fold the teaching experience even further. Specifically, after the intensiveweek of the summer institute, participants chose which illustrative unitthey intended to implement, did further reading from the supporting mate-rials for that unit, and then drafted a tentative plan for implementing theunit. These preliminary plans were discussed and further refined during aday-long meeting scheduled a week later.

The Supported Field Experiences

Participants in the summer institute had committed up front to implement-ing one of the illustrative inquiry units at the beginning of the school year,and at least one other unit later in the year. The program attempted toaddress the teachers’ need for support during the implementation in threecomplementary ways.

First, support teams of 5–8 participants from the same school (or nearbyschools when necessary) were formed, and a mathematics teacher educatorwas assigned to each team as facilitator. Each team was expected to meetweekly over the school year to provide a forum for discussion of the fieldexperiences.

Second, the team’s school facilitator provided individual classroomsupport during the implementation process. This usually included individ-ual meetings to discuss preliminary plans for implementing the unit, occa-sional classroom visits followed by a debriefing meeting, and even phonecalls in which immediate concerns and questions could be addressed.Third, three additional project-wide meetings were offered. These day-long meetings were carefully scheduled at critical points of the fieldexperiences in order to provide opportunities for sharing experiences andtools as support for the next stage. More specifically, this componentinvolved:

• a first meeting scheduled very soon after most participants hadcompleted their first experience as teachers, in which participantscould share and discuss the results of these experiences, the concernsthey raised, and the insights gained;


• a second meeting scheduled before most of the participants engagedin their second experience as teachers, in which issues about planningnew inquiry units were explicitly addressed;• a final meeting in which the participants presented the inquiry unit

they designed, and described and reflected on its implementation, sothat the whole group could gain new instructional ideas.

Participants were encouraged to document the implementation of theinquiry units, not only for their own sake, but also to support the plan-ning of similar experiences in other classrooms. A written reflection wasexpected from each participant at the end of the field experiences.

HIGHLIGHTS FROM THE PROGRAM EVALUATION

The evaluation of a complex professional development program such asthe one described is no easy task. To accomplish this task, we chose toaddress the following questions:

1. To what extent did the teachers participate in various aspects of theprogram? Because both implementations involved volunteer teachers,we believe that this information can shed light on what the participantstruly valued about the program.

2. What were the effects of the program in terms of changes in partic-ipants’ beliefs and practices? The answer to this question may beconsidered the ultimate measure of success for teacher developmentprograms.

3. Did the program promote further efforts towards school mathematicsreform? Looking at what happened to the program participants in lateryears is especially important in this case, given the program’s goal ofinitiating a long-term process of instructional innovation.

In what follows, we first summarize our response to each of thesequestions based on the analysis of the data collected for each of thetwo implementations (as listed in Table III). We then briefly discuss thestrengths and limitations of the program as suggested by the analysis.

Teachers’ Participation in Various Aspects of the Program

Teachers’ participation in different components of the program presentedsome interesting trends. Participation in the summer institute was veryhigh. Only two of the 56 participants who began a summer institute did notcomplete it. With only one exception (a teacher who was getting marriedthe next day), none of the other participants missed even one full day.


TABLE III

Data Collected in the Two Implementations of the Program

Data collected: Implementation A: Implementation B:

Initial survey (teaching position, experi-ence, etc.)

x x

Audio/video-taping of all S.I. sessionsand follow-up meetings

x x

Journals and other artifacts producedduring S.I.

x x

Anonymous feedback questionnaire onS.I. activities

x (10/15) x (21/39)

Audiotapes and fieldnotes of weekly teammeetings

x x

Fieldnotes of classroom observations x x

Weekly journals written during the fieldexperiences

(some) N/A

Written reflections at the end of theprogram

x (some)

Written documentation on the inquiryunits taught

x (some)

Final survey (asking for self-reported dataon inquiry units implemented and theirduration; instructional strategies used;adaptations made for learning disabledstudents; aspects of thinking and practicemost effected by project experiences; anddissemination efforts)

x (only classroomteachers)

“Summary form” on individual partici-pants’ activities and growth completed byhis/her school facilitator

x x

Audiotapes of presentations (some) N/A

Audio/videotapes of testimonials at S.I. (some) (some)

Math teaching autobiography written afew years after completion of the program

(some) (some)


These data are indicative of the value participants attached to the summerinstitute, especially when one considers that Implementation B participantsreceived no stipend. Also, the feedback in the anonymous evaluation ques-tionnaires at the end of the summer institutes was positive. Although someparticipants offered constructive criticisms about a few specific activities,there was no negative comment about the experience as a whole. Thefollowing quotes are representative of the participants’ overall feelingsabout the summer institute:

A tremendous experience. You obviously put a lot of time and energy into organization andimplementation. Well done! (anonymous evaluation questionnaire)

Overall, I thought the workshop was very successful; much more so than I hadanticipated. (anonymous evaluation questionnaire)

Of the 54 participants who completed a summer institute, 42 partici-pants, 14 from Implementation A and 28 from Implementation B, wereclassroom teachers in a position to be able to participate in all of thecomponents of the field experiences during the following school year. Ofthe 42 teachers, 3 teachers in Implementation B and none in Implementa-tion A dropped from the program. The extent to which the remaining 39participants fulfilled their field experience requirements, however, variedconsiderably.

In contrast to the almost perfect attendance of the summer institutes,attendance at meetings during the school year left more to be desired.Although all 39 participants attended their weekly team meetings at leastoccasionally, only Implementation A participants and about half of Imple-mentation B participants were present on a regular basis, that is, theymissed no more than 3 meetings over the school year. Absences at theproject-wide follow-up meetings, held on Saturdays during the school year,were common in both implementations. It is difficult, however, to deter-mine the extent to which these absences were due to participants’ lack ofinterest or to the difficulty presented by the logistics of attending meetingson Saturday or in a different school at the end of a school day.

Data about the implementation of innovative units, as part of the fieldexperiences, are reported in Table IV. These data show that all 14 Imple-mentation A participants implemented two illustrative units as well as aunit of their own design and 5 went beyond this minimum requirement.All of the Implementation B participants implemented at least one illus-trative unit, and more than half of them fulfilled their commitment ofimplementing at least two inquiry units. A few teachers in this group alsoelected to engage in additional innovative experiences in their classes. Weconsider these results a good indication that participants were encour-aged by the summer institute to attempt the implementation of one of


TABLE IV

Data about Innovative Units Implemented by Classroom Teachers who Completed theProgram

Team1 Participant2 Total number Number of Number of Total Percentof different illustrative new units duration of instructionalinnovative units units (in changeunits weeks) (given a 40implemented week year)

IMPLEMENTATION A

I A1-m 6 2 4 21 52.5

I A2-m 6 2 4 21 52.5

II C1-m 3 2 1 8 20

II C2-sp 3 2 1 8 20

(w/C1)

II C3-sp 4 2 2 11.5 28.75

II C4-m 4 2 2 11.5 28.75

III D1-m 3 2 1 8.5 21.25

III D2-sp 4 2 2 11 27.5

(w/D1,3)

III D3-m 3 2 1 10 25

IV E1-m 3 2 1 8 20

IV E2-sp 3 2 1 8 20

(w/E1)

V F1-m 3 2 1 11 27.5

V F2-sp 3 2 1 11 27.5

(w/F1)

V F3-m 3 2 1 11 27.5

Total # 14 Ave 11.4 Ave 28.1

IMPLEMENTATION B

I A4-m 3 1 2 9.5 23.75

I A5-el 1 1 3.5 8.75

I B2-el 4 2 1 17 42.5

III G1-m 2 2 3.5 8.75

III G2-sp 1 1 .5 1.25

(w/G1)

III G3-sp 1 1 3 7.5

(w/G1)

III G4-m 2 2 3 7.5

III G5-sp 2 2 3 7.5

(w/G4)


TABLE IV

Continued

Team1 Participant2 Total number Number of Number of Total Percentof different illustrative new units duration of instructionalinnovative units units (in changeunits weeks) (given a 40implemented week year)

IV E3-m 2 1 1 5 12.5

IV E4-sp 2 1 1 11 27.5

V F4-sp 2 2 5 12.5

(w/F3)

V F5-sp 1 1 2 5

V H1-m 5 2 3 12 30

V H2-sp 5 2 3 12 30

(w/H1)

VI I1-m 3 1 1 16 40

VI I2-sp (alone 5 2 1 24 60

& w/I1)

VI I3-m 3 3 13 32.5

VI J1-m 3 1 1 8.5 21.25

VI K1-m 1 1 4 10

VII L1-m 1 1 8 20

VII M1-sp 2 2 6 15

VII N1-m 1 1 4 10

VII N2-sp 1 1 4 10

(w/N1)

VII O1-el 2 1 1 5.5 13.75

VII P1-el 2 2 3.5 8.75

Total # 25 Ave 7.5 Ave 18.7

1 Team Code indicates the support team the teacher participated in during the Field Expe-rience.2 Participant Code indicates the teacher’s school and area of specialization (m – math, sp– special education, el – elementary).

the illustrative units in their classes; and most of them found this experi-ence valuable enough to pursue further opportunities to teach mathematicsthrough inquiry, despite the high demands in terms of time and effort theseexperiences required. Overall, feedback from participants in both groups,as provided in journals, written reflections, and spontaneous comments atmeetings, suggests that many teachers found these experiences as teachersvaluable.


It is also worth noting that only 11 out of the 14 Implementation Aparticipants and 9 of the 25 Implementation B participants complied withthe requirement to document and reflect on their field experiences in writ-ing. Although all those who completed this task acknowledged its value,many teachers shared that they felt so overwhelmed by the demands ofplanning inquiry experiences that they could not find the time to keep agood record of what was happening and to write reflections.

Effects of the Program in Terms of Changes in Participants’ Beliefs andPractices

We believe that the program was quite successful in accomplishing its maingoal of initiating the process of rethinking beliefs and practices, althoughthe extent of the changes in beliefs and practices that resulted from theparticipation in the program varied considerably among individuals.

The data reported in Table IV show that each of the 39 teachers whocompleted the program engaged in substantial instructional change in atleast one of their classes. More specifically, Implementation A participantsdesigned and taught some new and innovative units from a minimum of 8up to 21 weeks of instruction (corresponding to 20–52.5% of the schoolyear), and Implementation B participants, with two exceptions, did so fora minimum of 3 to a maximum of 24 weeks (corresponding to 7.5–60%of the school year). The occasional classroom visits conducted by theschool facilitators suggest that these experiences represented a substan-tial step forward toward implementing the vision for school mathematicsarticulated in the NCTM (1989) Standards, although they could not all beconsidered legitimate examples of inquiry teaching.

Survey results also suggest that many of the participants continuedto use some of the strategies advocated by reform proponentsoutsideoftheir implementation of our illustrative units. As summarized in Table V,more than a third of the respondents reported their continued use ofmanipulatives and reflections on the content and process of specific learn-ing experiences, and more than half reported continued use of journalsand other writing assignments, cooperative groups, class discussions/largegroup processing, and public recording of ideas on newsprint. The docu-mentation put together by Implementation A teachers for the innovativeunits they had implemented further supports these self-reported data.

Changes in participants’ beliefs are more difficult to measure. However,participants’ journals and written reflections, as well as their contribu-tions in meetings, provide plenty of anecdotal evidence that they hadbegun to question their views of mathematics, teaching, and learning. Forexample, several participants expressed an increased appreciation for the


TABLE V

Teachers’ Reported Use of Instructional Strategies Beyond Our Illustrative Units

Strategy: Number of teachers Percentagereporting its use (outof 36 who completedthe survey)

writing, journals 32 88%

cooperative groups 31 86%

use of newsprint to record strategies and discussions 21 56%

large group processing 21 58%

think/pair/share 19 53%

projects/presentations 19 53%

reflections on what learned 15 42%

use of manipulatives 13 36%

others∗ 21 58%

At least one of the above 34 94%

7 or more of the above 13 37%

In this category we grouped participants who reported the use of at least one of the follow-ing strategies: alternative assessments such as portfolios, the use of open-ended problems,creating a “need to know,” sharing results and strategies, student centered discussions, lessemphasis on teacher’s answers, use of peer tutors.

more humanistic aspects of mathematics, as illustrated by the followingquotes:

The tessellation unit opened my eyes and dispelled my beliefs and perceptions aboutmathematics. Never before had I been asked to think about what is mathematics and whatmathematicians do. The tessellation unit stretched my global perspective and more impor-tantly thepossibilitiesin mathematics in relationship to innovative and cross-curricularideas. (anonymous evaluation questionnaire)

I’d always thought that math was one of the last absolutes on earth and now (laughing)my faith is shaken. (transcript from a summer institute video)

Other comments revealed the assumption of new goals for mathematicsinstruction, along with a new awareness for students’ abilities and learningneeds:

I suppose what surprised me most about teaching the units was how little even the brighteststudents conceptually understood WHAT they were doing mathematically before they wereexposed to this new approach. . . . I feel that all of our students are now gaining a newconceptualization of WHY, WHAT and HOW they are doing mathematics. (H2-sp, finalreflections)

By using the inquiry method of teaching, I was amazed at what the students came upwith for ways to arrive at the solution. In most cases they were very original and even


enlightened me on many different approaches to solve the problem at hand. (D1-m, finalreflections)

A more systematic study of the participants’ growth with respect to aninquiry approach to teaching mathematics was conducted for Implemen-tation A teachers only. This study involved the identification of elementsthat could be considered characteristic of the inquiry approach promotedby the program. For each of ten key elements, as listed in the first columnof Table VI, the data available for each of the teachers, including jour-nals, final reflections, unit documentations, and testimonials provided inthe context of summer institutes or other presentations, were examined tofind evidence of whether that teacher had come tovalueand/orpracticethat aspect of teaching mathematics through inquiry. The results of thisanalysis are summarized in Table VI.

This analysis shows that at least half (and generally many more) ofthe teachers showed evidence of valuing and practicing the characteris-tic elements of inquiry we had identified, with the only exception beingItem 5. These results are even more significant when we consider whatthey tell us about individual teachers.

For all but five teachers (C1-m, C2-sp, D2-sp, D3-m, F3-m, who hadsubmitted very little written material) we could find evidence ofvaluingatleast 7 of the 10 key elements of inquiry. For all but two teachers (C1-m,C2-sp) there was evidence ofpracticingat least 7 of these elements. Theseresults confirm the school facilitators’ judgment, based on the long-terminteraction with the participants in the weekly meetings and classroomobservations, that, with the exception of the pair C1-m and C2-sp, the otherteachers had come to value an inquiry approach to mathematics instructionand had made big steps towards putting it into practice in some of theirteaching. The teachers’ progress is also shown by the number of weeksof instructional innovation implemented (see Table IV) and the quality ofthese innovative units, as revealed by the analysis of the lesson plans andartifacts they collected to document a selection of these experiences.

It is worth noting that the two teachers constituting the exceptionwere a mathematics teacher and a special education teacher who hadbeen assigned to teach a blended mathematics class for the first time andwere having serious personality and philosophy conflicts. The mathematicsteacher in the team (C1-m) was the only Implementation A participant whowas not convinced of the value the inquiry approach. Her special educationpartner (C2-sp) showed more evidence in her writing that she valued theinquiry approach. Yet, she felt that her role in the blended classroom didnot empower her to influence instruction; therefore, it was not even possi-ble for us to collect evidence about her practice of any inquiry approach.


TABLE VI

Evidence of Practicing and Valuing Specific Elements of an Inquiry Approach byImplementation A Teachers

Characteristics Teachers with evidence % Teachers with evidence %

supportingvaluing (of 14) supportingpractice1 (of 13)

Students actively A1-m, A2-m, C1-m, 100 A1-m, A2-m, C1-m, 100

engaged in the C2-sp, C3-sp, C4-m, (14) C3-sp, C4-m, D1-m, (13)

construction of D1-m, D2-sp, D3-m, D2-sp, D3-m, E1-m,

mathematical E1-m, E2-sp, F1-m, E2-sp, F1-m, F2-sp,

knowledge F2-sp, F3-m F3-m

Students develop A1-m, A2-m, C3-sp, 79 A1-m, A2-m, C3-sp, 92

ownership of the C4-m, D1-m, D3-m, (11) C4-m, D1-m, D2-sp, (12)

inquiry E1-m, E2-sp, F1-m, D3-m, E1-m, E2-sp,

F2-sp, F3-m F1-m, F2-sp, F3-m

The class acts as a A1-m, A2-m, C2-sp, 57 A1-m, A2-m, C3-sp, 61

community of C3-sp, C4-m, D3-m, (8) D1-m, D2-sp, D3-m, (8)

inquirers E1-m, E2-sp E1-m, E2-sp

Math is portrayed A1-m, A2-m, C2-sp, 71 A1-m, A2-m, C3-sp, 69

as the product of C4-m, D1-m, D3-m, (10) D1-m, D2-sp, E1-m, (9)

human activity E1-m, E2-sp, F1-m, E2-sp, F1-m, F2-sp

F2-sp

Ambiguity, errors, A1-m, A2-m, E1-m, 28 A1-m, A2-m, C3-sp, 92

uncertainty are E2-sp (4) C4-m, D1-m, D2-sp, (12)

valued as stimulus D3-m, E1-m, E2-sp,

for inquiry F1-m, F2-sp, F3-m

Priority is given to A1-m, A2-m, C3-sp, 86 A1-m, A2-m, C1-m, 100

developing C4-m, D1-m, D2-sp, (12) C3-sp, C4-m, D1-m, (13)

problem solving, D3-m, E1-m, E2-sp, D2-sp, D3-m, E1-m,

understanding, F1-m, F2-sp, F3-m E2-sp, F1-m, F2-sp,

confidence F3-m

The above priorities A1-m, A2-m, C3-sp, 57 A1-m, A2-m, C3-sp, 85

are reflected in C4-m, D1-m, E1-m, (8) C4-m, D1-m, D2-sp, (11)

appropriate E2-sp, F1-m D3-m, E1-m, E2-sp,

assessment of F1-m, F2-sp, F3-m

student learning


TABLE VI

Continued

Characteristics Teachers with evidence % Teachers with evidence %

supportingvaluing (of 14) supportingpractice1 (of 13)

Rich mathematical A1-m, A2-m, C3-sp, 71 A1-m, A2-m, C1-m, 100

situations provide C4-m, D1-m, E1-m, (10) C3-sp, C4-m, D1-m, (13)

opportunities for E2-sp, F1-m, F2-sp, D2-sp, D3-m, E1-m,

learning F3-m E2-sp, F1-m, F2-sp,

F3-m

Teacher facilitates A1-m, A2-m, C1-m, 86 A1-m, A2-m, C1-m, 100

student learning C3-sp, C4-m, D1-m, (12) C3-sp, C4-m, D1-m, (13)

by the use of D2-sp, E1-m, E2-sp, D2-sp, D3-m, E1-m,

appropriate F1-m, F2-sp, F3-m E2-sp, F1-m, F2-sp,

teaching F3-m

strategies

Teacher listens to A1-m, A2-m, C3-sp, 71 A1-m, A2-m, C3-sp, 77

students as they C4-m, D1-m, E1-m, (10) C4-m, D1-m, E1-m, (10)

construct meaning E2-sp, F1-m, F2-sp, E2-sp, F1-m, F2-sp,

F3-m, F3-m

1The total number is 13 as one teacher felt she did not have any responsibility overteaching her assigned blended class.

In contrast, simply by complying with the program’s requirements, themathematics teacher did put into practice several key characteristics ofinquiry during the teaching of her three innovative units. But, because shewas not valuing the principles informing such an approach, these attemptsremained isolated and did not influence the rest of her teaching, as capturedin her partner’ s reflections at the end of the program:

Looking back at the units from the project. . . what surprised me the most was how differentthey were from the traditional math units which are typically seen in a middle school math-ematics classroom . . . . The units were miles away from the usual, run of the mill math textand the ditto/worksheet math task format. Unfortunately, what I saw happening was thatupon completion of [these] units there was little or no change in the mainstream classroom. . . the classroom quickly reverted back to a very traditional text/worksheet format – almostas if in relief – leaving no trace of evidence that the units had been used or that any changein style could have possibly occurred. (C2-sp, final reflections)

In contrast, the engagement in the field experiences of the other 12Implementation A participants led to some rethinking of their wholeapproach to teaching, although to an extent and with implications that


differed considerably among individual teachers. Consider, as illustra-tive of this effect, the following quote from one of the teachers’ finalreflections:

It was not long into the actual teaching of the Tessellations unit before I realized thatmy teaching would never really be the same again, nor for that matter would my view ofmathematics or my role in the classroom . . . . This was not simply about developing andteaching 2 or 3 innovative units. Much more is involved. . . . Somehow all of my units havebeen colored by the experience. I now plan for much more student involvement, look for theteachable moments, allow kids to discover and question more, raise questions and problemsto solve, and try to instill in them a ‘mathematician’s spirit’. (E2-sp, final reflections)

Promotion of Further Efforts Toward School Mathematics Reform

We can respond to the third question only with regard to Implementation Aparticipants, for whom we collected systematic data beyond the year theyparticipated in the professional development program. Of the 12 teacherswho experienced a change in beliefs and practices all but one, who leftteaching to become a principal, sustained and even increased the instruc-tional changes observed during the field experiences in successive years.They all continued to teach most of the innovative units they had devel-oped, in all the classes in which such units were appropriate. Five teachersdesigned some new units as well. Ten of these teachers also continued theirinvolvement in school reform by volunteering their support to Implementa-tion B participants, by providing testimonials at the summer institutes, bycontinuing to participate in the weekly team meetings, and/or by providingclassroom support to colleagues in their same school.

Additional evidence of the program’ s success in terms of initiatinga long-term process of instructional innovation can also be found at theschool level. In at least six of the participating schools, participation inthe program also coincided with establishing blended mathematics classesfor the first time, an experiment that was deemed successful by teachers,administrators, parents, and, in particular, special education students. Simi-larly, it is important to mention that four schools among those with thehighest number of participants actively pursued the possibility of continu-ing in the process of school mathematics reform through their participationin an NSF-funded Local Systemic Change project.

Strengths and Limitations of the Program

Our analysis suggests that overall this professional development programwas quite successful in initiating a long-term process of rethinking one’spedagogical beliefs and practices, as well as in promoting some immedi-ate instructional change, at least for the great majority of the participants.


Participants’ feedback and degree of attendance, combined with our ownobservations, suggest that the activities organized in the summer instituteand the scaffolded experience of implementing an illustrative unit werethe most successful components of the program. The mixed success of thesupport teams and follow-up meetings calls for some rethinking of thesecomponents in order to make the support they were intended to offer moreaccessible and effective. We also believe that the greater extent of instruc-tional change achieved by Implementation A participants (on average, over50% more than what was achieved by Implementation B participants) isrelated to the different support provided to the group during their fieldexperiences and thus points to the value of such support, however costly.

INSIGHTS GAINED ABOUT MATHEMATICS TEACHEREDUCATION

Although we hope that the successful results reported so far may inviteother teacher educators to offer similar programs, we believe that ourexperiences can offer yet another contribution to the mathematics teachereducation literature. As we reflected on our program implementations andtheir outcomes, we came to identify a number of ways in which the moreunique features of the program, i.e., the role of the illustrative inquiryunits, the participation of a diverse group of teachers, and the role of theschool facilitator, contributed to the effectiveness of acknowledged profes-sional development practices that had informed the design of the program.In what follows, we offer these insights as working hypotheses that webelieve are worth further exploration. In the meantime, we hope that thesereflections on our practice will help other practitioners as they need tomake decisions in designing their professional development programs.

Insights on Orchestrating Successful Experiences as Learner

The value of experiences as learners as a means to challenge teachers’pedagogical beliefs and to illustrate the benefits of alternative instructionalapproaches has long been recognized in the literature (e.g., Simon, 1994;Schifter, 1998). This value was confirmed by the participants’ responsesto the tessellation and area inquiries developed within our summer insti-tute. Several participants even identified these experiences as the mostinfluential component of the summer institute for them.

At the same time, we believe that our experiences as learners aroundtessellation and area were especially successful because they weredesigned around an illustrative inquiry unit. More specifically, partici-


pants’ motivation in engaging in these experiences was greater becausethey knew they would teach one of these units to their own students, andthus they wanted to become intimately familiar with the unit’s design. Inorder to be illustrative of an entire unit, these experiences as learners werenot reduced to a series of short and well-defined mathematical activities,but rather involved the participants in open-ended learning experienceswhich developed over a considerable period of time. By doing so, theyillustrated the role of important elements such as homework, transitionsamong different components of the unit, synthesis, and reflections.

Our experiences as learners also benefited considerably from the pres-ence of a very diverse group of participants, as our programs includedspecial education and elementary teachers along with teachers special-izing in secondary mathematics. Although we had been worried that thedifferences in mathematics background and confidence represented by thisgroup might present a problem, this diversity turned out to be a veryeffective element. All participants were surprised at how successfully theyhad managed to learn together, benefiting from both the greater mathe-matical knowledge of some individuals and the creativity of those whohad a more limited mathematical background. The situation served as amodel of an heterogeneous mathematics class, and thus was very effectivein convincing the teachers of the feasibility and value of implementingsimilar units in inclusive classrooms. In addition, these learning experi-ences were eye-openers for several of the special education and elementaryteachers who had previously disliked and even feared mathematics. As thenon-mathematics specialists shared their enthusiasm at discovering thatmathematics could be much more meaningful, interesting, and accessiblethan they had previously thought, the mathematics teachers in the groupwere in turn energized by the realization that similar experiences couldhave the same effect on their students.

Insights on Scaffolding Experiences as Teachers

Once again, our experiences confirmed the two-way interaction betweenbeliefs and practice highlighted in the teacher education literature (e.g.,Thompson, 1992). In several cases, the positive experience of teaching anillustrative unit spurred participants to rethink more radically their ownteaching practices. Seeing what their students could do when mathemat-ics instruction was approached differently was what truly motivated theseteachers to continue in the process of instructional innovation, as shown bythese quotes taken from participants’ reflections on their field experiences:

The bottom line is that these approaches work. They work for the LD students and theywork for all the students in the classroom who are actively involved in their own learning


. . . . Many students finally developed a realunderstanding of mathematics, as opposed tojust being able to do a few simple computations. One student, that I did not feel I wassuccessful with over most of the year, has frequently told other teachers that he finallyunderstands mathematics as a result of the class. (F1-m, final reflections)

I think I was most surprised by the fact that the students thought of some of the activitiesas being fun. I haven’t heard eighth grade students refer to math as fun in the past! Mostof the time they were interested and extremely involved in the activities. (F2-sp, finalreflections)

Yet, field experiences can also turn out to be counterproductive if theteachers do not immediately see positive outcomes in their students’ learn-ing as a result of the new instructional approach. This is not an uncommonoutcome because the teacher is still learning and experimenting with unfa-miliar pedagogy. This realization, in turn, makes it crucial to find effectiveways to support teachers in their first field experiences, so that theseinstructional experiences will be successful and can then further the teach-ers’ confidence and understanding of the reform being promoted, ratherthan discourage them from ever trying again.

The results of our field testing have confirmed our belief that it is impor-tant that teachers experience an illustrative unitas learnersbefore theyimplement itas teachersas a first field experience. We think it is quitetelling that in Implementation A, where the participants had engaged inonly one inquiry unit, the one on tessellations, all teachers chose to imple-ment the tessellation unit as their first field experience. In contrast, whenImplementation B participants were introduced to all three units during thesummer institute, the choice of what unit to implement in the field experi-ence fell quite evenly among the three illustrative units. School facilitatorsalso repeatedly noticed how teachers relied heavily on their memories ofwhat they had experienced in the summer institute when they planned theirown implementations of our illustrative units, a practice that sometimespresented drawbacks, as the participants’ memories were not always accu-rate or complete. In addition, several participants explicitly mentioned howtheir experiences as learners in the summer institute greatly increased theirconfidence as they moved to plan and teach their first inquiry unit.

Participants also identified other experiences and resources that hadcontributed to the success of their first field experience. First, severalparticipants mentioned the value of hearing the testimonials of teacherswho had actually implemented these units in their middle school classes.This helped them have what someone called a reality check, and gavethem opportunities to ask some burning questions about putting an inquiryapproach into practice with middle school students. Second, the instruc-tional materials we had created to support the planning and facilitationof each illustrative unit also played an important role. These materials


included detailed reports of a few different implementations of the unit,with examples of the instructional materials created by the teachers whohad implemented the units in middle school classrooms. In most of theparticipants’ units, we could recognize a great number of tasks and work-sheets which had been borrowed, with appropriate modifications whenneeded, from these reports. One teacher’s comment reflects the apprecia-tion of having access to the materials because: “You did not have to createthis whole thing from scratch, which was helpful too, because what youwere wrestling with is a new approach” (transcript of one of Team VImeetings).

However important, these preliminary experiences still did not elimi-nate the need for some classroom support as individual teachers movedto plan and implement an illustrative unit. School facilitators especiallynoticed the value of preliminary planning sessions and of observing aseries ofconsecutivelessons.

Insights on Teachers’ Engagement in Productive Reflections

Reflection has long been recognized as a key element in teachers’ learning.Simon (1994), in particular, has made reflection an integral component ofeach of the learning cycles he has identified in his model of mathematicsteachers’ professional development. Depending on the focus of each cycle,such reflection can address the mathematics of an experience as learner, itspedagogy, student thinking, or complementary aspects of the teacher’ sown practice.

Indeed, just engaging in experiences as learners and experiences asteachers per se may have limited value unless teachers are able to drawfrom these experiences larger implications for their beliefs and practice.The Implementation A teacher who could successfully implement someillustrative inquiry units in her class, and yet afterwards went back toher traditional teaching approach as if nothing had happened, is a goodillustration of what can happen as a result of practice alone. The needfor reflecting on experiences as learners is well captured by the followinginsightful comment by a participant:

This [reflection] is a necessity because the process is critical but when you are part ofthe process you focus on the learning rather than what the process that brought about thelearning was. We need to have it pointed out that you were modeling various teachingmethods, especially since many of these ideas were new to us. (anonymous evaluationquestionnaire)

How can such reflections be best promoted and facilitated within aprofessional development program? We believe that our experiences haveallowed us to identify some effective practices to this regard.


First of all, we found that just providing time and opportunity forreflection after an experience as learner or after a field experience is notsufficient. Rather, it is important that reflective sessions and writing assign-ments be carefully focused by the facilitator by means of specific promptsor tasks. The “walk down memory lane” activity described earlier is agood illustration of this point. After they had engaged in an inquiry ontessellations, the participants were asked to (a) reconstruct in detail thekey components of the activities they had experienced, (b) write downwhat helped/hindered their learning in each of these components (whichrequired them to examine their experience critically and with a specificfocus, thus moving them beyond mere recollection), and (c) share anddiscuss what they wrote with the rest of the group. All these componentswere instrumental to making explicit the different reactions that variouslearners had to the same activity. This, in turn, enabled all participants tobetter appreciate some of the pedagogical decisions made by the teacherin designing and/or teaching the unit.

This example also illustrates another important element of most ofour reflective sessions: offering opportunities to make one’s reflectionspublic. We believe this is important both to encourage participants tobetter articulate their thoughts and to enable them to benefit from otherpeople’s perspectives. The latter was especially true in our case, becauseof the different backgrounds and expertise represented by the mathematics,elementary, and special education teachers. For example, when examiningan experience as learner in which they all had participated, the math-ematics teachers often contributed a more sophisticated articulation ofthe mathematical ideas that had been embedded in these experiences andthus helped the other participants to appreciate both the legitimacy andsignificance of what they had done. Special education teachers, in turn,were more sensitive to the learning differences demonstrated by variousparticipants, and their comments helped their colleagues become aware ofthese differences and their implications. They also were often quicker tonotice and describe the pedagogical strategies and decisions the facilita-tors had made to support the participants’ inquiry and learning and to seehow these strategies could support some of their students in a mainstreamclassroom. Elementary teachers’ experience in teaching all subject matters,in turn, provided unique contributions to the whole group, as they couldmake valuable connections between the new approach to mathematics justexperienced and similar paradigm shifts with which they were familiar inlanguage arts or other subject areas.

Although sharing may happen naturally in the context of large groupdiscussions, this is usually not the case when participants are asked to


reflect in writing. Yet, we have found the practice of shared journals,where participants’ individual entries are read by the entire group, quiteeffective. In fact, several participants reported that, although they did notlike writing the journals, they continued to do so because they got so muchfrom reading other peoples’ journals. Perhaps, finding opportunities forsharing other kinds of written reflections might help participants placegreater value on these experiences. As discussed earlier, although teacherswho complied with the requirement of writing a reflection recognized thatit had greatly contributed to their growth, most participants chose not tocomply with this field experience requirement, and thus were never able toexperience its value.

Because of the difficulties we often experienced when assigning writ-ten reflections, we think that it was especially important that teachersbe offered opportunities to reflect on their field experiences that do notalways require writing. In our program, for example, such reflections wereimplicitly promoted as participants were asked to share their experiences atweekly meetings with their support group and in the project-wide follow-up meetings. Another valuable stimulus for reflection came from a moreunexpected source, as some of the Implementation A participants wereoffered opportunities to present on their experiences at professional confer-ences and/or to various audiences within their district. Although at firstthese engagements were accepted with considerable reservation, teacherseventually came to identify these experiences as valuable opportunities tothink about and make sense of their experiences of attempting instructionalinnovation, as expressed in the following quote:

[My colleague] and I were asked by people coordinating the grant to be presenters at [aconference]. I remember us both being very reluctant because we did not feel remotelyqualified. This was a tremendous growth experience for me in several ways. First, I wasforced to truly think about what I had learned that I felt was important to pass on. Second,I had to come up with a vehicle to pass the information on. Finally, I had to deal with myfears of presenting to not only my peers but teachers with many more years of experiencethan me. (E1-m, autobiography)

Insights on the Participation of Teachers From the Same School

As we recruited teachers for our program implementations, we were wellaware of the recommendation in the literature that teachers should comewith at least one colleague from the same school (e.g., Clarke, 1994).Indeed, we made this a requirement for Implementation A participants.Many of the teachers in Implementation B who did not have a partnerexperienced more difficulty in their field experiences.


The mere presence of a colleague from the same school or workingin the same class, however, does not guarantee support, as illustratedby the experience of the two Implementation A teachers for whom ourprogram was not successful (C1-m and C2-sp). Rather, any form of collab-oration will require a certain amount of mutual respect, shared views,and compatible working styles. In addition, we observed that the mostsuccessful collaborations occurred when pairs of participants were respon-sible for teaching some common class or course in the same school,so that they could truly plan instruction together and have a genuinemotivation for sharing and reflecting together on what happened in theirclassrooms. A particularly interesting form of collaboration occurredwhen a mathematics and a special education teacher, who co-taught ablended class, attended the program together, as noted by several schoolfacilitators:

[I2-sp] and [I1-m] have worked as a truly collaborative team from the very beginning –they respect each other and trust each other completely. [I2-sp] is teaching [I1-m] aboutjournaling, and writing in general, and about rethinking/ modifying the way [I1-m] wouldtypically do things to better meet the needs of LD (really ALL) students – [I1-m] is teaching[I2-sp] about thebig ideas in mathematics . . . . The actual work in their classroom continuesto grow and improve through these efforts. (Team VI school facilitator’s field notes)

Whether they came to the summer institute with a partner or not,participants in our program also benefited from having colleagues in theirschool who had previously participated in the program. Even if in mostcases these more experienced teachers were not in a position to plantogether with the new participant, it was quite comforting to have themaccessible as resources on a daily basis and in the same building. Theextent to which these teachers played a mentoring role, however, variedconsiderably among individuals.

Additional benefits were experienced when not just pairs but a criti-cal mass of teachers from the same school participated in the program,that is, at least 3 to 4 teachers, which usually represented one fourth toone third of the teachers responsible for mathematics instruction in thebuilding. When this happened, we observed that the participants could notonly support each other’s attempts at improving their teaching practicesmore effectively, but more importantly they could occasionally join forceswhen some school-wide policy threatened to undermine these attempts.For example, the participants from School A (who represented about halfof the mathematics faculty in the school) were able to take a leadershiprole in rethinking the distribution of topics across grade levels so that fewertopics would be taught each year but in more depth; this decision providedthem with the time that they felt they needed to implement inquiry units. A


higher concentration of program participants in the same school also effec-tively meant that their support team would focus mostly on their school.As a result, the school facilitator could not only provide in-class supportmore effectively, but also actively work at building a support communitywithin the school.

The role played by the school facilitator in charge of a support team,especially when the team centered around a specific school, merits somefurther comments. As each participant’s energy and attention was taken bythe challenges presented by rethinking their own practices, there seemedto be a definite need for someone to coordinate group initiatives such asthe support team weekly meetings. There were also definite advantagesto having an outsider (rather than a more experienced teacher from thesame school) playing these roles. This not only avoided the creation ofa hierarchy among teachers from the same school, which could disruptoften fragile relationships among peers, but also provided a different set ofperspectives and expertise. School facilitators, however, met with differentlevels of success in different teams. Our experience suggests that schoolfacilitators should play some instructional role in the summer institute; inthe two cases when this did not happen, the school facilitators reported agreater difficulty in establishing a rapport and mutual respect with theirteam members.

SUMMARY

The professional development program featured in this article has shownhow the process of school mathematics reform can be successfully initiatedby using illustrative inquiry units as the common context for both experi-ences as learners and scaffolded practice intended to provide the catalystfor changes in beliefs and practices. We suggest that the use of illustra-tive units can enhance the value of established professional developmentpractices such as engaging teachers as learners in instructional experi-ences modeling a novel pedagogy, offering a supported field experiencecomponent, and providing multiple opportunities for reflection.

As the program involved both mathematics and special education teach-ers, our experiences also revealed the benefits of working with suchan heterogeneous group. The diversity in backgrounds and perspectivesamong the participants not only enriched their experiences as learn-ers and the reflective sessions that followed these experiences, but alsomodeled the viability and benefits of inclusive classrooms. Furthermore,this arrangement encouraged the collaboration of special education andmathematics teachers within the same school.


Finally, the attempts made within this program to create a supportcommunity for the participants as they engaged in instructional innovationhave pointed out the values and limitations of the popular recommendationof involving groups of teachers from the same school in a professionaldevelopment program. Most importantly, our experience suggests thata mathematics educator external to the school can play a critical rolein promoting and supporting individual and school-wide efforts towardsschool mathematics reform.

REFERENCES

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nary report to the National Science Foundation for the project #TPE-9153812.Borasi, R. (1996).Reconceiving mathematics instruction: A focus on errors. Norwood, NJ:

Ablex.Callard, C. (1996).Investigating tessellations to learn geometry: An example of planning

and implementing an inquiry unit. Preliminary report to the National Science Foundationfor the project #TPE-9153812.

Carpenter, T. & Fennema, E. (1991). Research and Cognitively Guided Instruction. InE. Fennema, T. Carpenter & S. Lamon (Eds.),Integrating research on teaching andlearning mathematics(1–16). Albany, NY: State University of New York Press.

Clarke, D. (1994). Ten key principles from research for the professional development ofmathematics teachers. In D.B. Aichele & A.F. Coxford (Eds.),Professional develop-ment for teachers of mathematics(37–48). Reston, VA: National Council of Teachers ofMathematics.

Fennema, E. & Nelson, B.S. (1997).Mathematics teachers in transition. Mahwah, NJ:Lawrence Erlbaum Associates.

Friel, S. & Bright, G. (1997).Reflecting on our work: NSF teacher enhancement in K-6mathematics. Lanham, MA: University Press of America.

Grouws, D. (1992).Handbook of research on mathematics teaching and learning. NewYork: MacMillan Publishing.

HMSO (1982).Mathematics counts. London, UK: Author.Loucks-Horsley, S., Hewson, P.W., Love, N. & Stiles, K.E. (1998).Designing professional

development for teachers of science and mathematics. Thousand Oaks, CA: CorwinPress.

National Council of Teachers of Mathematics (NCTM). (1989).Curriculum and evaluationstandards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (1991).Professional standards forteaching mathematics. Reston, VA: Author.

Schifter, D. (1998). Learning mathematics from teaching: From a teachers’ seminar to theclassroom.Journal of Mathematics Teacher Education, 1, 55–87.

Schifter, D. & Fosnot, C.T. (1993).Reconstructing mathematics education: Stories ofteachers meeting the challenge of reform. New York: Teachers College Press.


Schön, D.A. (1983).The reflective practitioner: How professionals think in action. NewYork: Basic Books.

Simon, M. (1994). Learning mathematics and learning to teach: Learning cycles inmathematics teacher education.Educational Studies in Mathematics, 22, 309–331.

Simon, M. & Schifter, D. (1991). Towards a constructivist perspective: An interventionstudy of mathematics teacher development.Educational Studies in Mathematics, 22,309–331.

Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research . In D.Grouws (Ed.),Handbook of research on mathematics teaching and learning(127–146).New York: MacMillan Publishing.

Thornton, C. & Langrall, C. (1997). Mathematics instruction for elementary students withlearning disabilities.The Journal of Learning Disabilities, 30(2), 142–150.

U.S. Department of Education, National Center for Education Statistics. (1996).Pursu-ing excellence: A study of U.S. eighth-grade mathematics and science achievement ininternational context. Washington, DC: Author

Woodward, A., Borasi, R. & Packman, D. (1991).Supporting middle school learningdisabled students in the mainstream mathematics classroom. A proposal to the NationalScience Foundation.

Raffaella BorasiUniversity of Rochester,Warner School of Education,Dewey Hall,Rochester, NY 14627,USA

Judith FonziUniversity of Rochester,Warner School of Education,Dewey Hall,Rochester, NY 14627,USA

Constance F. SmithState University of New York at Brockport,Education Department,Brockport, NY 14420,USA

Barbara J. RoseRoberts Wesleyan College,Mathematics Department,Rochester, NY 14624,USA

JEFFREY A. FRYKHOLM

THE IMPACT OF REFORM: CHALLENGES FOR MATHEMATICSTEACHER PREPARATION

ABSTRACT. The mathematics education community has been widely recognized as aleader in the standards-based reform movement. Despite the widespread interest and atten-tion that reform documents such as the NCTMStandardshave generated, what has yet tobe fully understood is the impact of these types of reform documents on the preserviceteacher preparation process. This paper examines the state of the standards-based reformeffort in mathematics teacher education by presenting the results of a three-year study ofsix cohorts of secondary mathematics student teachers (n = 63). Examined in particularare the ways in which these student teachers’ professed beliefs about and knowledge ofthe NCTM Standards contrast with their teaching practices. Possible explanations for themismatch between belief and knowledge statements and teaching practices are examined,along with other implications for mathematics teacher preparation.

THE IMPACT OF REFORM: CHALLENGES FORMATHEMATICS TEACHER PREPARATION

Motivated and shaped by a variety of influences, the mathematics educa-tion community has been a leader in the recent educational reformmovement (Massel, 1994; Apple, 1992). Well-recognized catalysts inthe ongoing reform of mathematics education in America have beenthe National Council of Teachers of Mathematics’ [NCTM]Standardsdocuments (NCTM, 1989, 1991, 1995). These documents, which outlinegoals for reforming the content, pedagogy, and assessment of schoolmathematics, are commonly thought of as representative of the reformmovement in mathematics education (Brown & Borko, 1992). Other coun-tries, for example England, have also revised goals, curriculum, andteaching approaches in mathematics education through updates in nationalcurriculum guidelines (Department for Education and Employment, 1995).

In the midst of widespread acclaim and attention about mathematicseducation reform, however, questions exist about the impact on class-rooms and teachers. Despite the ways in which reform agendas have beenwidely publicized, classroom practices in general have been slow to change(Weiss, 1995). Similarly, it is still unclear how the reform movement inmathematics education has impacted the preparation of preservice teachers


80 JEFFREY A. FRYKHOLM

(Frykholm, 1996), a topic that will be further explored in the followingpages.

The purpose of this article is to shed light on the learning-to-teachprocess in mathematics education. Explored are the experiences of sixcohorts of preservice teachers including: their perceptions of the reformmovement; their difficulties in implementing reform-based teaching; theirinsights as to why they did not teach in ways more consistent with thepreparation program; and their reactions to the disparity between the idealspresented in the university preparation experience and the realities of theschool place.

RATIONALE FOR INCREASED ATTENTION TO PRESERVICEPREPARATION

Nearly a decade after the mathematics education community in NorthAmerica embraced a vision of reform-oriented teaching and learning in ourschools, there is cause for concern that the typical mathematics classroomof today remains similar to those of fifteen or twenty years ago. Supportfor this statement is found in Weiss’ (1995) examination of the most recentNational Survey of Science and Mathematics Education. Weiss reportedthat only a little more than half (56%) of the inservice teachers surveyed(grades 9–12) were “well aware” (p. 4) of the primary tenets of reform inmathematics education. Of that group, nearly half reported that they werenot prepared to explain the philosophy and specific recommendations ofreform to a colleague. Moreover, when asked to describe “various strate-gies that definitely should be a part of mathematics instruction” (p. 7), theteachers surveyed responded in the following way:

• 50% supported the use of calculators;• 18% reported that they took students’ preconceptions of a topic into

consideration when planning curriculum/instruction;• 20% emphasized writing about mathematics;• 20% agreed that mathematics subjects (algebra, probability, geometry,

etc.) should be integrated and taught together;• 27% advocated the use of cooperative learning groups;• 18% advocated performance-based assessment;• 26% advocated hands-on/manipulative activities.

The topics represented in the previous list are important elements in thereform movement in mathematics education (NCTM, 1989). That so fewteachers appear to endorse these key foundations of reform is problem-atic. Moreover, the percentages above were reflective of what strategies

CHALLENGES FOR MATHEMATICS TEACHER PREPARATION 81

the teachers feltshouldbe a part of mathematics instruction. The data saynothing about the percentage of teachers who actuallyare implementingthese strategies. In all likelihood, these percentages would be even lower.

The fact that many inservice teachers have not adopted the goals andassumptions of the reform movement in mathematics education is notsurprising when viewed within the context of the research literature onteacher change. Duffy & Roehler (1986), for example, have suggestedthat teachers contemplate new ideas and instructional strategies by filter-ing them through their existing conceptions about teaching and learning.Although innovative, reform-based ideas and activities often seem appeal-ing to teachers at first glance, the ideas often do not survive the filteringprocess. If they do survive the process, Duffy and Roehler contend, thesenew ideas are often distilled or significantly altered by teachers to theextent that they lose what it was that made them innovative. As Brown,Cooney & Jones (1990) have suggested, “Inservice teachers’ resistance tochange and preservice and beginning teachers’ reversion to teaching stylessimilar to those their own teachers used are legendary” (p. 649).

Hence, despite the notable efforts of the mathematics educationcommunity to promote change in school classrooms, the reform of instruc-tion continues to be problematic. Simply put, many mathematics teachersare not prepared to implement reform-based pedagogy. A number of schol-ars have agreed in principle with this statement, and they have thereforesuggested that researchers and educators undergo a shift in focus to placegreater attention on the preservice preparation process (Brown, Cooney &Jones, 1990; Brown & Borko 1992). As Weiss (1995) concluded in herreport on the status of reform in mathematics education, “It is clear thatsubstantial changes are needed inpre-service[emphasis added] educationif future generations of teachers are to be prepared to teach mathematics asenvisioned in theStandards” (p. 16). Researchers and mathematics teachereducators should examine the preservice preparation process in order tocontribute to a literature base that, as Cooney (1994) suggested, is “still inits infancy” (p. 613).

TEACHERS’ KNOWLEDGE, BELIEFS AND PRACTICES

The research literature on mathematics teacher preparation has been struc-tured around several primary themes (Brown & Borko, 1992). Fundamen-tal to the process of learning to teach are previous experiences, knowledgestructures, and belief systems that preservice teachers bring to the prepara-tion process. The relationships between beliefs, knowledge, and teachingpractices have been widely explored (e.g., Fennema & Franke, 1992;


Putnam, Lampert & Peterson, 1990; Thompson, 1992, 1984). Althoughmost of the research cited in these works involved inservice elementaryteachers, it nevertheless can provide a lens for considering research onsecondary preservice teachers.

Content Knowledge and Teaching Practices

There is a body of research suggesting that preservice teachers lack richunderstandings of mathematics (Brown, Cooney & Jones, 1990). Forexample, Ball (1990a,b) reported that preservice teachers are often unableto give explanations for correct answers to mathematics problems. More-over, they do not regularly see the connections between mathematicaltopics. These deficits prohibit preservice teachers from adopting practicesthat depart from those they experienced as learners (Ball, 1990a; Lampert,1985, 1986). Specifically, novice teachers often do not have the knowl-edge to interrupt the common practice of compartmentalizing mathematics(Ball, 1990a).

Borko and colleagues (Borko, Livingston, McCaleb & Mauro, 1988)found other relationships between content knowledge and teaching prac-tices. Student teachers with strong content knowledge spent less timepreparing for classes, focused more on instructional strategies than oncontent, and were more flexible and confident in their teaching. Similarly,Shulman and Grossman (1988) found that “prior subject matter knowledgeand background. . . affect the ways in which teachers select and structurecontent for teaching, choose activities and assignments for students, anduse textbooks and other curriculum materials” (p. 12). That is, teachersmake decisions about curriculum and instruction based largely on previousknowledge – knowledge often established in environments quite differentthan those in which they find themselves teaching.

Beliefs and Pedagogical Knowledge

A second theme explored in research about preservice mathematics teach-ers involves the relationship between beliefs and pedagogical knowledge.Although it seems that beginning teachers’ beliefs about mathematicsteaching and learning are typically altered throughout the preparationprocess (Ball, 1988; Schram, Wilcox, Lappan & Lanier, 1989), it is equallyclear that “many of their beliefs about mathematics and its teaching are inplace before they begin their teacher education programs” (Brown, Cooney& Jones, 1990). These often deeply ingrained beliefs can confound thedevelopment of a reform-based perspective for the classroom (Frykholm& Brendefur, 1997).


Schram and colleagues (Schram et al., 1989) studied changes in beliefsof a cohort of preservice teachers that entered the preparation programwith a traditional view of mathematics as an abstract and mechanical setof rules and symbols. By the end of the program, the students were begin-ning to question their previous views as they developed more conceptualunderstandings of mathematics. Despite this growth in their own think-ing, however, they did not change their beliefs about how children shouldlearn mathematics. Rather than teaching for conceptual understanding, theparticipants used instructional methods that fostered the idea of mathe-matics as a hierarchically ordered set of facts and procedures. Similarly,Feiman-Nemser & Buchmann (1986, 1987) examined how the prepara-tion program, the student teaching setting, and preservice teachers’ beliefsshape teacher learning. They found that student teachers often missedopportunities to help their students engage significantly in the mathematicsat hand and that they had difficulties making a transition to the type ofpedagogical thinking that characterizes reform in mathematics education.

The three constructs of content knowledge, pedagogical knowledge,and existing belief systems are central to this report and might be concep-tualized in the following way. First, reform-based instruction requiresteachers to possess a rich conceptual understanding of mathematics. Iflearners are to reason, communicate, make connections, and problemsolve, then teachers must be comfortable enough with the mathematicsat hand to provide meaningful learning opportunities for such activities.Further, beginning teachers bring to their preparation experiences deeplyingrained notions about mathematics and mathematics instruction. It isimportant to consider if and under what circumstances preservice teachersbegin to explore overlaps in their beliefs, content knowledge, and peda-gogical knowledge during the preparation experience – a process that isconfounded by the often competing philosophies and practices at play inschool and university settings. An examination of these complexities andof the ways beliefs and practices emerge during the transition from studentto teacher is at the heart of this article.

CONTEXT OF THE STUDY

Participants and Program Description

The research study took place over three years during which the studentteaching experiences of six cohorts of preservice teachers (n = 63) wereexamined. The students were enrolled in a secondary mathematics teachereducation program designed extensively around the themes and documents


that have been the hallmarks of the recent reform movement in mathemat-ics education (e.g., Steen, 1990; MSEB, 1989; NCTM, 1989, 1991, 1995).Data were gathered during the semester-long student teaching internshipthat came at the conclusion of a two-course sequence on mathematicseducation.

The first of the two methods courses addressed the secondary mathe-matics curriculum. As such, one of the guiding resources was the NCTM(1989) Curriculum and Evaluation Standardswhich the students readand studied at great length. The second course focused on pedagogicalissues, guided largely by the NCTM (1991)Professional Standards forTeaching Mathematics. During both courses, students engaged in read-ings, dialogue, writings, and classroom activities that examined the role ofcurriculum and instruction in fostering students’ conceptual understandingand mathematical power (NCTM, 1989). In addition to regular readingsand classroom discussions about current issues in mathematics educa-tion, assignments included: textbook reviews, mathematical concept maps,student interviews, teacher interviews, classroom observations, technologyexplorations, reflection papers, and micro-teaching opportunities.

At the conclusion of the two preparation courses, students partici-pated in a semester-long teaching internship. They were placed in localsecondary schools to work in the classrooms of experienced teachers whohad volunteered to serve as cooperating teachers. After initial days ofacclimation, orientation, and observation, the student teachers graduallyassumed full responsibility for the teaching of three classes. It was duringthis internship that the data reported in this article were collected.

Data Collection

Data were collected in a variety of formats including lesson observa-tions, post-lesson conferences, interviews, survey questionnaires, seminarsessions, and informal conversations and interactions. Each of these datasources is described below.

Lesson observations. Each student was observed a minimum of four timesduring the student teaching internship. During the observations, copiousfield notes were recorded by the college supervisor. Included in the fieldnotes were descriptions of the sequence of events of the class period, thetime spent on each activity, teacher behaviors, student behaviors, curricu-lar emphases, and other observations regarding particular elements of thelesson that were often recalled in the post-lesson conference. Data werecollected in this manner during 205 lesson observations and kept for lateranalysis.


Post-lesson conferences. Each observation was followed by a 30–45minute post-lesson conference. Discussed during the conference wereissues related to the class period including (but not limited to): thestrengths and possible weaknesses of the lesson, student engagement,lesson planning, learning activities, questioning strategies, mathematicseducation reform issues, and areas for future focus. Throughout theseconferences, efforts were made to help the student teachers reflect on theirclassroom practices and their beliefs about mathematics instruction. As anexample, it was not uncommon for the supervisor to recount an event in theclass period and then ask the student teacher to analyze his or her actions,the students’ responses, and other possible courses of action that mighthave been pursued. When possible, these conferences were audio-tapedfor later transcription and analysis. In every case, however, field notes weretaken during the conference in order to summarize the topics that had beendiscussed by the supervisor and student teacher.

Interviews. For each cohort, a subset of the participants participated in twointerviews conducted during the student teaching experience. Althougheach student teacher in the study was invited to engage in the interviews,participation was voluntary. In all, 24 interviews were conducted withtwelve students. The first interview was conducted near the midpoint ofthe semester, and the second interview took place at the conclusion ofthe internship. All interviews were audio-taped for later transcription andanalysis. The intent of the interviews was to provide a more detailed exam-ination of several individual cases in a way that was not possible with all63 subjects in the study.

The interviews were structured around Spradley’s (1979a) descrip-tion of the ethnographic interview – conversations guided by “explicitpurpose[s], ethnographic explanations, and ethnographic questions”(p. 59). Spradley described ethnographic interviews as “involv[ing]purpose and direction”, which implies a delicate balance of questions,responses, and explanations by both interviewer and participant. Althoughthe particular questions themselves were dependent on the students’settings and experiences, the types of questions asked were similar for eachinterview.

For example, to initiate an interview, Spradley suggested to plan onasking a blend of descriptive questions (e.g., “Could you describe theapproach you take when planning for your lessons?”), structural questions(e.g., “What are all the factors that influence your decisions about planningfor a lesson?”), and contrast questions (e.g., “Can you think of other waysyou might have taught that concept?”). Throughout these mini-tour ques-


tions, the interviewer makes use of hypothetical situations, uses follow-upquestions to clarify meaning, expresses interest, expresses ignorance, veri-fies questions and answers, and incorporates the participant’s terms andphrases (Spradley, 1979a, pp. 67–68).

Survey questionnaire. An anonymous survey questionnaire was adminis-tered at the conclusion of the student teaching experience. Each participantin the study completed and returned a questionnaire for analysis (n = 63).The purpose of the questionnaire was to examine not only the conceptionsheld by the student teachers about a number of issues related to reform inmathematics education, but also to document the amount of agreementamong students on various issues. Half of the 26 questions were free-response items, and the remaining questions were based upon a five-pointLikert scale. The following questions represent a subset of the thirteenopen-response items:

• What did you find most challenging about teaching mathematics?• How (if any) did your philosophy of mathematics teaching and

learning change throughout the semester?• Describe the ways in which your teaching practices did or did not

match your philosophy of mathematics education.• Describe factors that influenced the teaching strategies you adopted in

the classroom.• What impact did the two methods courses have upon the way you

taught and interacted with students in your classes?

Back-to-campus seminars. At least five seminar sessions were held duringthe student teaching experience for each cohort of student teachers. Thesetimes of reflection were loosely structured and provided students with aforum to discuss their frustrations, share their successes, and work onideas for lessons and activities with peers. The seminars became valuablesources of data as students interacted and conversed freely with each other.Field notes were recorded during each of these seminar sessions detailingthe topics of conversation, perspectives, and insights shared during themeetings.

Data Analysis

Erickson (1986, p. 146) suggested that the analysis of data from qual-itative studies involves a process of “generat[ing] empirical assertions,largely through induction” and “establish[ing] an evidentiary warrant” forthese assertions through a systematic search for confirming data. To aid


in completing this task, the entire data set was examined repeatedly andthoroughly. Based largely on the recommendations of Wolcott (1993),Strauss (1987), and Spradley (1979a,b), analytical tools, including theexpansion of field notes, writing memos, and coding the data, were appliedthroughout the process.

Expansion of field notes. Spradley (1979a) emphasized the importanceof building on original field notes. He suggested, “As soon as possibleafter each field session [the researcher] should fill in details and recordthings that were not recorded on the spot” (p. 75). Based on his recom-mendations, field notes taken during observations, conferences, interviews,and meetings were immediately expanded, i.e., reworked, filled-in, andcompleted, so as to retain a more accurate and complete description ofthe field experience for later analysis.

Writing memos. Throughout the data collection and analysis, particularattention was given to Strauss’ (1987) suggestion to include a memoingprocess for the purpose of capturing insights, questions, and understand-ings as they occurred throughout the research. Following his recommen-dations, a separate file for random thoughts, insights, ideas, or possibleinterpretations of the data was kept as data were both collected andanalyzed. Several of these memos were later pursued and proved to beparticularly helpful in understanding relationships among the data.

Coding the data. Strauss (1987) recommended a coding process be appliedto the data in the early stages of the analysis. By carefully reading thetranscripts, field notes, and observational records, regularities and patternsin the data were explored. These themes were then organized and labeledas either external codes, that is, larger, theoretical concepts in the data, orinternal codes, i.e., particular themes within an external code. For exam-ple, one of the primary external codes that emerged was Beliefs aboutthe Standards. Within this external code were a number of sub-codes, forexample, the Standards as content, value statements about the Standards,and knowing the Standards.

Analysis. The data sets were first analyzed by cohort group. The entiredata set for each cohort of students was analyzed at the conclusion of thestudent teaching experience, independently of data from any of the othercohorts. As the independent data sets for each cohort were examined, aniterative process was applied that entailed a systematic fracturing of thedata that led to generative questions, and ultimately, to the discovery ofcore categories or themes that were represented across the multiple data


sources. Erickson (1986) referred to this primary analysis as a process ofsearching for links in the data that consist of “patterns of generalizationwithin the case at hand, rather than for generalizations from one case orsetting to another” (p. 148).

Upon completion of an initial analysis of the data sets for each indi-vidual cohort, the entire data record for the three years was examined. Asa starting point, common themes that had emerged across cohort groupswere identified. For example, for each cohort, codes were established tohelp understand students’ perceptions of their cooperating teachers. Thesecodes from the independent data sets were then aggregated into one file, inwhich further analysis across the entire study was completed. Part of thissubsequent analysis entailed the organization of excerpts from the data intodomainsor classes(Schatzman & Strauss, 1973) in which relationshipswere drawn, where possible, between themes and across data sets. Thisprocess of establishing domain analyses (Spradley, 1979a) helped orga-nize the emerging themes across the three years of the study which ledto the primary assertions for this article. Excerpts from the data recordare presented below to illustrate the interpretations that were made and toprovide descriptions of the context within which these preservice studentswere engaging in the learning-to-teach process.

FINDINGS

The findings of this study are organized around two primary themes. Thefirst theme, students’ reported views of reform, explores the knowledgeclaims and belief statements the student teachers made with respect to thereform movement, as well as the value statements they made about theircommitment to innovative instruction. The second theme, the realities ofpractice, emerged from the teaching experiences of the students as theyendeavored to reconcile their notions of reform with the challenges of theschool classroom. Notable throughout the investigation was the contrastbetween these two themes: how the vision of reform that many studentteachers articulated was seldom evidenced in their teaching practices.

Student Teachers’ Views of Reform: Teaching and the Standards

Although the preservice teachers had varying degrees of appreciation forreform, there was little doubt that the students had become familiar withthe vision expressed in the NCTM Standards (1989, 1991) documents. Infact, “the Standards” became a sort of catch-all phrase that was used torefer to or describe most aspects of the reform movement.


Knowledge of the Standards. Many student teachers spoke of the Standardswith glowing affirmation. One student in the initial cohort went so far asto say that “the Standards are the Bible of mathematics education” (David,Cohort 1). Others, however, responded with some disdain at the ways inwhich the preparation program had indoctrinated them to the reform move-ment. “Readings, readings, readings of standards, standards, standards,”said one student, “we had that shoved down our throats for two semesters”(Michael, Cohort 2).

When confronted with a hypothetical scenario in which the studentswere asked to describe the primary tenets of mathematics education reformto an uninformed colleague, many of the student teachers framed theirresponses in terms of the NCTM Standards. For example, one studentresponded by suggesting that reform was “a collection of standards inteaching and assessment that we should try to apply in the classroomand that include methods and ideas for getting kids really involved inmath” (Amy, Cohort 2). Another student referred to the NCTM documentsexplicitly by stating,

Reform is like the NCTM Standards – a set of rules or guidelines that all mathematicsteachers should follow in order to keep their students up to date with the rest of the worldas far as in mathematics capability. These standards emphasize what should be importantwithin the classroom, and they de-emphasize what shouldn’t be. They are the ideals thatshould be taught in any mathematics curriculum. (Sarah, Cohort 3)

A fairly rigid perception of the Standards – as a set of rules – wasseen throughout the study. The students repeatedly referred to the Stan-dards as rules or steps that were to be followed closely as a means ofimproving mathematics instruction. As one student noted, “The Standardsare a set of ‘new rules’ that a council of teachers thinks would help uskeep the students engaged. They emphasize four things: problem solving,reasoning, communication and making connections” (Ben, Cohort 2).

Another perception that surfaced repeatedly was the notion that knowl-edge of the reform movement, or the ability to talk about reform coher-ently, was almost as important as actually being able to apply the tenets ofreform in the classroom. That is, the student teachers felt the need to knowabout basic reform goals and visions as a measure of their competenceand ability to meet the expectation of future peers and administrators. Forexample, one student reported that the mathematics methods courses “gaveme the most formal indoctrination to the Standards – I got to pick up all myprofessional jargon” (Aaron, Cohort 4). Another student talked about hisnewly developed ability “to talk a good game about the Standards. This isimportant in order to get a job by surviving the interview process” (Steven,Cohort 3). In a similar vein, a third student teacher spoke about the impact


TABLE I

Student teachers’ perceptions of the Standards

Question 1: How much introduction to the theory, philosophy, and recommendationsof theStandardsdo you feel you received in your preparation program?

Answer options: very little, if at all a little some a good amount a great deal

Responses: 0 3 7 22 31

Question 2: As a whole, of how much value are theStandardsdocuments to you inyour practice of teaching mathematics?


Responses: 0 1 21 25 16

Question 3: How closely did your teaching embody the recommendations and chal-lenges of theStandards? (i.e., How much did you actually teach like theStandardsrecommend?)



Question 4: How conscious of theStandardswere you as you planned for yourlessons? (i.e., Did what you know about theStandardsaffect how andwhat you planned for your classes?)



Question 5: As a whole, how much practicality do you see in theStandards? (i.e.How feasible and practical is it to teach in a real classroom according tothose recommendations?)


Responses: 3 16 22 15 7

Question 6: How much practical training and experience (aimed at implementing ateaching style consistent with theStandards) do you feel you receivedin your preparation program?)


Responses: 6 20 22 11 4

of his supervisor on his development. “Working with [my supervisor] hasbeen great. He talks about the Standards a lot which really helps me,because I need to know that stuff for interviews” (Brad, Cohort 6).

Survey data confirmed the exposure to the reform movement that thestudent teachers received, as well as the their reported knowledge of andcommitment to the Standards. As illustrated in Table I, students reported


that they not only had received an extensive introduction to issues ofreform via the NCTM Standards, but that they had come to value theStandards highly as an influence on their teaching practices.

Feasibility and necessity of implementing reform. The student teachersexhibited various levels of optimism with respect to implementing innova-tive teaching strategies. Almost to a person, the students admitted that theylacked the experience and expertise to implement the Standards fully in theclassroom. Nevertheless, many of them remained confident that they couldin fact model their classes and teaching practices after reform-based ideas.“I think implementing the Standards in the classroom is very feasible – andneeded”, said one student. “After seeing class after class being torturedwith worksheets and lectures the past couple of weeks it is easy to seethat the kids are sick of what they are doing and need something fresh andexciting” (Julie, Cohort 2). Another student responded similarly by saying,

I still like the Standards and I definitely don’t think that it is ‘pie-in-the-sky’ theory. I thinkthat it would be possible to implement the Standards on a regular basis and plan to workmy tail off to do it!! I am seeing now how absolutely terrible traditional methods are andevery little bit can help! (Angie, Cohort 3)

In contrast, other students shared their concerns about whether or notit was realistic to teach in a manner consistent with reform recommenda-tions. These students were quick to point to a number of limiting factorsthat prohibited extensive and regular implementation of reform strategies.For example, the students cited issues of limited class time, curricularrestraints, unruly or unmotivated students, a lack of innovative resourcesand ideas, lack of knowledge about technology, and excessive planningtime as factors which made reform-based teaching unrealistic. Externalcircumstances were often cited as impediments to reform in the classroomas well, as suggested in the following excerpt.

Sometimes, circumstances just aren’t good enough to be able to use the Standards. TodayI was talking to a first year teacher at my school. She told me that she was all about theStandards mentally, but she said it is almost impossible to teach like that because she isjust trying to get comfortable with everything and make it through the year. She knows herstudents get bored and she hates that but yet they can’t handle the more exciting lessons. . . . She told me that the way almost all of us [student teachers] think now, and how criticalwe are, will change when we are the one’s in our own classroom. (Bryan, Cohort 6)

This particular excerpt points toward the perspective that, regardlessof how much one might believe in the vision of reform, it is simplyimpossible, given the constraints of the school system, to follow reformrecommendations on a daily basis in the classroom.

Another student shared a similar perspective:


Okay, we’re being honest here, right?. . . So what do I think? I think the Standards are anice idea and offer good suggestions. However, I do not think they are the end all, be allwhen it comes to teaching. I think circumstances and situations will arise where teachingwith the Standards in mind is next to impossible. So, all in all, I think they need to be takenwith a grain of salt. (James, Cohort 3)

Notable about comments that student teachers made about implement-ing reform was the degree to which their impressions were tied to theteaching strategies of their cooperating teachers. Some of the preserviceteachers were inspired by the ways in which their cooperating teacherswere implementing reform. “My teacher is a great example”, reported onestudent, “she is an encouragement to me . . . . Much of what she does itright in synch with the Standards” (Mark, Cohort 6). Those students whoworked with cooperating teachers committed to reform strategies attestedto the feasibility of teaching in a manner consistent with the Standards.Other students, however, seemed to be negatively influenced by their coop-erating teachers. Many who did not see reform as viable in the classroomreported that their cooperating teachers rarely implemented innovativestrategies.

In some cases, students reacted strongly to the traditional mind sets oftheir cooperating teachers. Due to the frustration they felt as a result ofthe rigidity of their cooperating teachers, these students were inspired todo something completely different in the classroom than what they wereobserving. As one student suggested,

My teacher teaches off the overhead. Uses no group work. I have yet to see her use thecalculators. Her attitude with the students is very strict, not very warm, and for the mostpart, confrontational . . . . I think Ms. Smith’s classroom is causing me to actually view,first-hand, what not to do. It is an anguishing thing to have to sit through. . . . At first I wasreally upset . . . and part of me still is upset. But now I’m looking past it and am seeing itmore as an experience to learn from. (Becky, Cohort 6)

Whether the student teachers saw reform-based strategies implementedin the classroom or not, the evidence suggests that the beliefs and teachingstrategies of the cooperating teachers impacted the thinking of the studentteachers. As illustrated in Table III, the student teachers largely reportedthat their teaching practices modeled those of their cooperating teachers.Moreover, they overwhelmingly suggested that their cooperating teacherswere the most significant influence on the development of their teachingphilosophy and instructional practices. Certainly, these responses do notdiffer from results reported in the research literature (see Zeichner & Gore,1990). Student teachers have long attested to the impact of cooperatingteachers. What is problematic in this era of reform is that the cooperatingteachers often do not ascribe to a reform-based perspective. Evidence forthis assertion may be found in Table III, where students reported that their


TABLE II

Types of lessons observed

direct enhanced direct reform unclassifiable total

instruction instruction based

Fall 1993 18 10 5 3 36

Spring 1994 29 6 3 — 38

Fall 1994 15 10 3 1 29

Spring 1995 26 18 4 2 50

Fall 1996 18 11 5 0 34

Spring 1996 7 7 3 1 18

Totals 113 62 23 7 205

cooperating teachers generally did not discuss, emphasize, or model theNCTM Standards in their work with the student teachers.

The Standards as content. The preservice teachers in the study understoodthe primary tenets of reform, talked about them knowledgeably, and recog-nized what reform-based instruction should look like. They certainly knewthe Standards well enough to be able to maintain an opinion as to whetheror not their cooperating teachers followed them. What was noteworthy,however, was the way that the Standards became synonymous, if not areplacement for, a broader conception of reform.

“The Standards” was a term that was repeated often, many times inreference to any instructional theory or practice that deviated from whatthe students understood to be traditional instruction. Some began to treatthe Standards as if the document represented a body of content knowl-edge. Many students said, “I teach the Standards”, just as they mighthave said, “I teach Algebra I.” Similarly, they approached the Standardsmuch like they thought about their understanding in a content area like,for example, geometry. One student even thought of the Standards as acurriculum when he stated, “I would say that the Standards are meant tobe similar to a national curriculum” (James, Cohort 5). In some cases,students suggested that their knowledge of how to apply the Standards wasinsufficient. Statements such as, “I don’t know enough of the Standardsyet to teach them” (Andrea, Cohort 4) were quite common and similar toinstances in which students expressed uncertainty in their mathematicalknowledge in particular content areas.


The preservice teachers came to view the Standards as a body of contentthat was to be learned as part of the curriculum in the preparation program.There appeared to be little recognition of the Standards as simply oneof many representations of a philosophy of education. Their view of theStandards as a content area led to some frustration in that they could neverseem to “get the answer” for good mathematics instruction. They viewedreform-based instruction as they might have viewed their performance ina higher level mathematics course – feeling the need to know the contentof the Standards and to be able to demonstrate their competence in theclassroom as if it were on a test.

The Realities of Practice

A majority of the students reported not only knowledge of the reformmovement, but also that they valued the NCTM Standards as an impor-tant aspect of their developing philosophies and practices as mathematicsteachers. Other findings in this study, however, provided a stark contrast tothese belief and knowledge statements made by the student teachers.

First, whereas the students reported receiving large doses of theoryrelated to the reform movement in their preparation classes, they suggestedthat they had not received enough practical advice and experience on howto implement them. One student remarked, “We need more direct interac-tion and practical experience with the Standards. Right now, we are heavilyfocused on a theoretical approach . . . . [There was] not enough practi-cal, useful ideas presented in methods, and not enough time to practicepractice-teaching” (Linda, Cohort 1). Others made similar statements, forexample, “We need more practical applications to go along with the theory.We always talk about providing real-life experiences and connections inmath classes. Shouldn’t the same thing be done for us in math-ed classes?”(Kirsten, Cohort 2)

Survey data confirmed these concerns (see Table I). Almost two thirdshad questions as to thepracticality of the Standards, i.e., they were uncer-tain about whether or not it was “feasible or practical to teach in a realclassroom according to the Standards recommendations” (Survey, Ques-tion 5). Moreover, over three fourths of the students felt that they hadnot received adequate preparation to implement a teaching style consistentwith the Standards. In short, the student teachers reported that, althoughthe Standards were valuable inasmuch as they articulated a compellingvision for what mathematics instruction could be, there had been littleoffered in the way of practical advice and examples of innovative peda-gogy that could be used as a model for implementing such instructionalstrategies.


TABLE III

Cooperating teachers, the preparation program, and the standards∗

Question 1: How much pressure did you feel from the university program (supervisors,professors, classes, etc.) to teach like theStandardsrecommend?


Responses: 0 3 13 19 20

Question 2: “How much pressure did you feel from your cooperating teacher to teachlike theStandardsrecommend?”



Question 3: How often do you discuss theStandardswith your cooperating teacher?



Question 4: Did you pay more attention to theStandardsthan your cooperating teacherdid?

Answer options: yes no

Responses: 47 8

Question 5: How closely did your own teaching style and/or philosophy mirror theteaching style/philosophy of your cooperating teacher?



Question 6: Which of the following influences affected the development of yourteaching philosophy during your student teaching the most?

Answer options: educ. program coop. teachers instructors supervisor other

Responses: 10 28 6 5 14

∗The first cohort of students (n = 8) did not receive questions 1–4.

Lesson observations confirmed the difficulty student teachers hadteaching in a manner that reflected the reform movement (see Table II).When viewed holistically, there was a great deal of conformity in the waysthat the student teachers were teaching. Of the 205 lesson observations inwhich data were collected, 113 (roughly 55%) mirrored the descriptionof teacher-centered mathematics instruction articulated by Welch (1978)nearly two decades ago.

In all math classes that I visited, the sequence of activities was the same. First, answerswere given for the previous day’s assignment. The more difficult problems were worked


on by the teacher or the students at the chalkboard. A brief explanation, sometimes noneat all, was given of the new material, and the problems assigned for the next day. Theremainder of the class was devoted to working on the homework while the teacher movedaround the room answering questions. The most noticeable thing about math classes wasthe repetition of this routine. (p. 6)

These 113 lessons followed an identical pattern: Solutions to selectedhomework problems were presented, the student teacher conducted a 15–20 minute lecture of new material, and students spent the remaining timeworking the next homework assignment.

Another 62 lessons (roughly 30%) matched this description with onlya few differences. Small adjustments were made to the classroom routinesuch as problem-oriented activities at the beginning of class, or group-work on homework assignments. The primary focus of these lessons,however, continued to be teacher-directed instruction, i.e., presentationsof prior homework problems and lectures on new material.

In contrast to the 185 teacher-centered class periods, only 23 (roughly11%) of the remaining lessons (seven lessons did not fit any of these clas-sifications) deviated significantly from the direct instructional approachdescribed above. These 23 lessons contained significant departures fromteacher-dominated instruction and reflected general goals for reform-basedinstruction. That is, the student teachers created learning opportunitiesand activities that reflected specific recommendations contained in reform-based documents such as the Standards. Some lessons included opportuni-ties for students to do some process writing about their solution strategies,some lessons incorporated technology activities (primarily with graph-ing calculators) that encouraged students to explore a mathematical topicfrom a technological perspective, some lessons allowed for students tocommunicate about mathematics or their understandings of mathematicalconcepts in cooperative groups, and some lessons implemented an authen-tic assessment task. These examples are representative of other instances inwhich students deviated significantly from the traditional, teacher-directed,expository lesson.

Many factors relating to the nature of the student teaching experi-ence, such as expectations of cooperating teacher, lack of experience, ormandatory departmental guidelines, may be viable explanations for therelatively few innovative lessons that were observed. Nevertheless, it isworth noting the large percentage of the lessons that mirrored the tradi-tional direct instructional model, particularly given the student teachers’professed commitment to the reform movement.

Several students expressed concern over the mismatch in the way theytaught their classes with the way they believed they should have taughtthem. One student reported, “Every day I feel that I am not doing what


my education has taught me to do in the classroom” (Becky, Cohort 6).Another offered the following comments:I’m very disappointed in the difficulties I had putting my Standards-based ideas of a class-room into action. It’s not happening most of the time. Maybe we need coaching to be a littlemore aggressive and have more confidence that the Standards are good. Perhaps we wouldif we entered student teaching with more ammunition under our belts. (Dean, Cohort 3)

The preceding two quotes are representative of others in which thestudent teachers expressed frustration as they tried to reconcile the visionof reform-based classrooms as depicted in their preparation courses withthe realities of teaching in the real classroom. A common reaction of manystudent teachers was surprise as to how demanding it was to teach in a waythat was consistent with the reform movement.

DISCUSSION AND RECOMMENDATIONS

This research was designed to examine the role and impact of a reformmovement – in the context of this American study, the NCTM Standardsdocuments – on the teacher preparation process. Whereas researchers havedetailed the effects of content knowledge and existing belief structures onpreservice teachers, this study focused on the impact of a larger reformmovementitselfon the development of a generation of beginning teachers.In the following sections, I focus on various aspects that emerged fromthis research and on how the issues are situated within the context of theresearch on mathematics teacher preparation.

Providing Reform Oriented Experiences

Most of the preservice teachers in the study reported that they had neverengaged in the type of mathematics experiences advocated by the reformmovement. Many openly admitted that their own secondary school andcollege mathematics classes had been teacher dominated and in lectureformat, and therefore they had little or no experience to add perspectiveto the reform theories they encountered in the preparation program. Onestudent echoed this concern in the following way.You give us all this theory at the university, but we have no experience from which to drawon to help us make sense of it all . . . . In thinking back on my training at the university it isclear that I had no modeling of how to actually teach a math class. The only teacher I got tosee in action was my cooperating teacher, and that was for the five days prior to being putin front of a class. A student can reach his/her student teaching experience with basicallyno idea of what to do. (Michael, Cohort 2)

Although it is probably not the case that this student hadno modelingof how to teach a math class prior to his student teaching, the concern


is nevertheless worthy of attention. Many preservice teachers have notexperienced the learning of mathematics from a reform-based perspective.Therefore, it is difficult for them to envision how they might teach in such afashion in the midst of the many other adjustments they are making as theyundergo the transition from student to classroom teacher. Moreover, themajority of these students reported that they excelled in their high schoolmathematics classes. Some of these students admitted that, because theydid well under traditional instruction, they struggled with the notion thatthere was a need to teach in a different way. As one student reported,

I like the Standards and want to use them in my teaching. It is just that I have been taught theold traditional way of learning mathematics and have done well and now am being askednot just to learn something new, but also teach it in a brand new style as well. (Michelle,Cohort 1).

Such comments point to the need to provide experiences during thepreparation process in which prospective teachers engage in mathematicsas learnersin ways that we hope they will one day implement as teachers.

Examining the Placement Dilemma

The findings of this study confirm that the field placement process inmathematics education continues to be problematic. There is a wealthof research literature on the role of cooperating teachers in the studentteaching process and, in particular, the impact cooperating teachers havein socializing beginning teachers (see Zeichner & Gore, 1990). A topicthat surfaced repeatedly in this study was the disparity between whatthe students experienced in methods courses and what they saw andexperienced in their placements.

The apprenticeship model continues to characterize most field experi-ences, as was the case in this program, and contributes to “experience[s]for student teachers . . . [that] are often determined by the luck of the draw,and not as a planned part of a curriculum” (Zeichner, 1996, p. 219). Meade(1991) has suggested that the selection of cooperating teachers most oftencomes down to a decision based simply on those inservice teachers whohave volunteered to accept a student teacher. The selection of cooperat-ing teachers in this manner leads to cases in which student teachers are“frequently placed in classrooms where the teaching they are exposed tooften contradicts what they are taught in the colleges” (Zeichner, 1996,p. 219). One can easily understand the frustrations of beginning teacherswho feel caught between what one student described as “a rock and ahard place.” While feeling the pressure to perform for university super-visors in ways that reflected the goals and guidelines of the preparationprogram, these student teachers also reported feeling the weight of the


daily assessments made by cooperating teachers who not only maintaineda great degree of power in the relationship, but also quite often operatedunder a different set of assumptions than those advocated by the university.

These differing philosophies and expectations between the universityand the school place are significant given what the research literaturereports about the ways in which preservice teachers are often socialized bytheir cooperating teachers (Zeichner & Gore, 1990). Despite recognizingtheir cooperating teachers’ general lack of commitment to reform-basedpedagogy, a majority of the student teachers nevertheless reported thattheir cooperating teachers had been the most significant influence onthe development of their thinking and teaching strategies. The place-ment process certainly deserves continued attention and examination asit remains perhaps the most significant factor in the preparation process. Inlight of the findings of this research, two recent suggestions in the literaturemay be worthy of continued focus and attention.

First, there is general support for reexamining the role of the universitysupervisor in the student teaching experience. As Zeichner (1996) noted,

the literature has shown that very little deep thinking about teaching and learning goes on insupervisory conferences with teacher education students; that there is an unwillingness onthe part of many collaborating teachers and university supervisors to discuss controversialissues or to offer critical feedback for fear of upsetting the delicate interpersonal balanceof the triad. (p. 223)

Certainly, the mathematics education reform movement, when takenseriously, demands deep thinking and conversation on controversial issues.Yet, as Zeichner suggested, it is rarely the case that supervisors in thefield engage in these types of conversations with students and cooperatingteachers.

Borko & Mayfield (1995) have suggested a reconceptualization of therole of the university supervisor that may, in part, address Zeichner’sconcerns. They propose that university supervisors

use their limited time in schools to help cooperating teachers become teacher educators. Forexample, they can model ways of observing student teachers and strategies for conductingconferences that focus on teaching and learning and help student teachers to become reflec-tive about their practice. They can also provide support and guidance for student teachersto integrate theoretical and research-based ideas from their university courses into theirteaching. (p. 517)

This suggestion by Borko and Mayfield is one that has yet to be fullyexplored, and certainly deserves the continued attention of researchers andteacher educators.

A second idea receiving increased attention in the literature that mightcorroborate with the findings of this study is the potential impact ofcommunity-based teacher education experiences for prospective teachers


(Zeichner & Melnick, 1996). An example of community-based teachereducation may be found in my work with beginning mathematics teach-ers (Frykholm, 1998). In an attempt to better support student teachersthroughout their student teaching experience, to counterbalance the power-ful socialization forces at play in the school place, and to more heavilyinvolve supervisors in the teacher education process, student teachers andsupervisors engaged in an intense and intentional community experiencethroughout the student teaching internship. Through weekly observationsand community gatherings, both student teachers and supervisors alikegrappled with controversial issues, engaged in authentic problem solving,and supported one another throughout the semester. As I suggested,

Only when beginning teachers continually find themselves in discussions about learners,pedagogy, mathematics and reform – when they, out of habit, develop a critical conscious-ness about teaching – only then will they be able to interrupt the traditional expositionalmodel that has been perpetuated for decades in mathematics classrooms (p. 320).

The student teachers in this study were quite vocal in suggesting thatthey needed more support and guidance from the preparation programwhile in the field. Perhaps a more intensive community experience wouldhelp offset the difficulties inherent in the present placement process.

Avoiding the Standards as Content

We know that mathematical content knowledge plays a role in the qualityof instruction implemented by beginning teachers (Ball, 1990a,b; Lampert,1985, 1986). The more richly connected mathematical content knowledgea teacher possesses, the more likely it is that the teacher will providemeaningful mathematical explanations and activities in the classroom. Ina parallel argument, it would seem as though a deep understanding of thereform movement would enable beginning teachers to teach in ways thatare more consistent with reform recommendations. That did not appearto be the case in this study, however. As students engaged in readings,discussions, and reflections about reform ideals throughout the preparationexperience, some notable outcomes emerged.

First, as Ball (1990a) has suggested, beginning teachers with limited,unconnected, or strictly algorithmic understandings of mathematics tendto think of mathematics as a fragmented set of facts. In a similar sense,the preservice teachers in this study often compartmentalized elements ofthe reform movement. As they described their teaching beliefs and prac-tices, they often isolated particular aspects of the Standards. For example,students regularly spoke about feeling the need to implement problemsolving, a testament to their knowledge of one facet of the Standards.They seemed to lack, however, a richer understanding of the relationship


of problem solving to other primary tenets of reform, as well as the math-ematical contexts at hand. That is, the findings suggested that perhaps theknowledge of the reform movement that these student teachers possessedwas not as rich or comprehensive as we might be led to believe fromtheir comments and reported beliefs. Moreover, as evidenced repeatedly inclassroom activities, the student teachers rarely facilitated authentic prob-lem solving in the classroom. So, although they had surface knowledge ofreform ideals, this knowledge was not connected enough to allow them tofacilitate problem-oriented instruction in a coherent and integrated way.

Many of the student teachers came to think of the NCTM Standardsas synonymous with the reform movement. Perhaps due in part to theformat of the Standards documents, they tended to view the Standards ascontent, separated into chapters much like a mathematics textbook mightbe organized. This tendency to think of reform as a series of distinctrecommendations led them to perceive their instruction in certain ways.Students often viewed their teaching as successful only to the extent that itmirrored particular aspects of the Standards. Missing from this perspectiveis a recognition that the Standards, and reform in general, are really moreabout student learning than they are about teacher performance. Of course,the two go hand in hand. The point, however, is that the student teacherswere much more likely to reflect on how much their teaching might have“lived up to the Standards” as they were to reflect on students’ learning –about how well their teaching led students to engage in the mathematics athand.

As noted earlier, this study took place in an American setting in whichthe NCTM Standards were the primary reform documents under examina-tion. Certainly, the mathematics education community in North Americaneeds to carefully examine how the Standards are received and inter-preted by both prospective teachers and teacher educators. Generalizablefrom this study, however, is the wider concern that any reform documentis susceptible to being treated as content if not carefully presented andthoughtfully examined by participants in the preparation process. Teachereducators must be cautious not to present any particular document as theonly representation of reform that prospective teachers experience. In thecontext of this American study, the NCTM documents offer a signifi-cant and compelling vision of what constitutes mathematics education.Yet, prospective teachers need to see the ways in which the Standardsmovement is nested within the context of broader educational reforms andconstructivist theories of learning. Until such a time, they may continueto perceive the Standards as the exclusive path to good mathematicseducation. As long as they view the Standards in this way, the danger of


preservice teachers treating the Standards as the content material of themethods courses will persist.

Recognizing and Dealing with Duality

I close this section with one final implication that emerged from this study.Throughout the methods classes, the students in the study began to thinkin new ways about mathematics teaching and learning. Their emergingbeliefs, however, did little to impact instruction. Although the relation-ship between newly developed belief structures and instruction has beenexplored previously, this study adds to the research literature by providingevidence that studentsrecognizethis duality between their thinking andpractice. That is, the student teachers in this study were well aware of thefact that their teaching often was not consistent with the teaching strategiespromoted in the methods courses and reinforced in mathematics educa-tion reform literature. They repeatedly expressed their frustrations overthis mismatch between what they knew to be possible in a mathematicsclassroom and what they were actually doing.

One implication that follows is that teacher educators must, in bothpreparation course work and during field experiences, push students toexamine the contradictions between their beliefs and instructional deci-sions. The research literature suggests that beginning teachers often fallback on previous beliefs and teaching models that they experienced aslearners (see Thompson, 1992). The findings of this study point towarda window of opportunity in which teacher educators might first challenge,and then support, these beginning teachers as they make the transition to anew way of thinking about mathematics learning and teaching. Perhapsthis might best be done through supportive environments such as thesupervision model described (Frykholm, 1998).

Summary

This study exposes some of the difficulties facing teacher educators inmathematics education. Beginning teachers continue to adopt the instruc-tional practices of their cooperating teachers, many of whom still modeland encourage traditional, direct instruction. Yet, it does appear thatthese same beginning teachers are uncomfortable perpetuating the typeof instruction the mathematics education community has endeavored tochange. The findings presented in this article suggest that they are eagerto gain knowledge of reform, they search for new models of instruction toemulate and, perhaps most importantly, they recognize the ways in whichtheir emerging beliefs often run counter to their teaching practices.


The findings of the study point to several areas which could benefitfrom continued examination. Namely, we need focused research on theimpact of methods courses and field experiences on student teachers and onthe connections between the two experiences. We need to closely examinepreservice teachers’ perceptions of reform and the duality that can impacttheir thinking. Above all, this research confirms the need for a closer exam-ination of the ways in which beginning teachers challenge their own beliefstructures as they engage in the process of becoming a mathematics teacherin this exciting time of change and reform.

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TEACHER EDUCATION AROUND THE WORLD

ASPECTS OF MATHEMATICS TEACHER EDUCATION IN JAPAN:FOCUSING ON TEACHERS’ ROLES1

Yoshinori Shimizu

INTRODUCTION

One of the characteristics of mathematics lessons in Japanese elementaryand lower secondary schools relates to the frequent exposure of students toalternative solution methods for a problem (e.g. Becker, Silver, Kantowski,Travers & Wilson, 1990; Stevenson & Stigler, 1992; Lee, Graham &Stevenson, 1996). Japanese mathematics teachers, particularly in elemen-tary schools, often plan to organize an entire lesson around the multiplesolutions to a single problem in a whole class instructional mode (Nagasaki& Becker, 1993; Shimizu, 1996; Stigler, Fernandez & Yoshida, 1996).Alternative solution methods for the problem are usually presented byseveral of the 30 to 40 students in a class.

In this setting, a teacher has to pose the problem and anticipate students’responses to it. In other words, the teacher has to consider, both in planningand during the lesson, the diversity of experience and knowledge studentsbring into the classroom. How do Japanese mathematics teachers selectthe problems for their lessons? Do they have certain techniques for dealingwith the diversity of their students? If there are such techniques, how doprospective and beginning teachers learn and develop them?

In this article aspects of mathematics teacher education in Japan arediscussed. Rather than presenting an outline of the entire education ofJapanese teachers, I will focus on the teachers’ key roles during the lessons,roles that prospective and beginning teachers are supposed to learn eitherin a teacher preparation program or through the interaction with theircolleagues. In an effort to provide a context for this aspect of mathe-matics teacher education, a sample lesson and the typical organizationof a mathematics lesson shared by Japanese teachers are described first.Next, I introduce several Japanese pedagogical terms that refer to teachers’key roles during the lessons. The importance of these roles for educating


108 YOSHINORI SHIMIZU

teachers is discussed. Also, the importance of workshops as a means ofpromoting these roles is emphasized. The lesson on division by two-digitnumbers is used for reference throughout the article.

LOOKING INTO A JAPANESE MATHEMATICS LESSON

Consider a fourth grade classroom in Tokyo in which the teacher, Mr.Matsumaru, introduces division by two-digit numbers. This is the first timethat his students are faced with a problem that requires division by a two-digit number, although they have previously learned to solve problems thatrequire division by single-digit numbers.

At the beginning of the lesson, Mr. Matsumaru refers to an earlieractivity in which students had planted bulbs on the school grounds. Theteacher uses this activity as context and poses the following problem to thestudents:

We are going to plant 128 bulbs of tulips into 16 planters. The same number of bulbs areto be planted in each planter. How many bulbs will be planted in each planter?

After he has presented the problem with a picture and a model thatrepresent the setting, Mr. Matsumaru asks his students to think about howthey could express the situation in mathematical terms. Based on the mean-ing of division asdividing into equal parts, students share the expression128÷ 16 as a mathematical expression for the setting. Consequently, thetask at hand is to find the answer to 128÷ 16. Mr. Matsumaru encourageshis students to explore many different ways of finding the answer. Afterthe students have worked on the problem individually, several solutionprocesses and solutions are shared and discussed.

Typical Organization of Mathematics Lessons

A mathematics lesson in Japan lasts 45 minutes in elementary schoolsand 50 minutes in secondary schools and is typically divided into severalsegments (Becker et al., 1990; Stigler et al., 1996). A common lessonorganization consists of segments that often serve as the steps or stagesboth in teachers’ planning and in the teaching-learning processes (Shimizu,1996):

• Presentation of a problem;• Individual problem solving by students;• Whole-class discussion about the methods for solving the problem;

and• Summing up by the teacher (Exercises/Extensions).

ASPECTS OF MATHEMATICS TEACHER EDUCATION IN JAPAN 109

As in the Mr. Matsumaru’s introductory lesson on division by two-digitdivisors, mathematics lessons usually begin with a practical problem or aword problem written on the chalkboard or taken from the textbook. Afterthe problem is presented and read by students, the teacher confirms thatthe problem is understood by the students. If not, the teacher may ask thestudents to read again or, in some cases, she or he may ask a few studentsto show their initial ideas about the methods to solve the problem. Then,about 10–15 minutes are assigned for the students to solve the problem ontheir own.

While students are working on the problem, the teacher moves about toobserve students’ work. During this time period, the teacher gives sugges-tions or helps individually those who are having difficulties. The teacheralso watches for students who have good ideas, with the intention of callingon those students – in a certain order – in the subsequent discussion.

During the discussion, students spend the majority of their time listen-ing to the solutions being proposed by their classmates, as well as present-ing their own ideas. When discussing solutions to the problem, the teacherasks students to present alternative methods to solve the same problem.Presenting an idea, even a wrong one, is strongly encouraged and praised.In some cases, the teacher may select an incorrect solution for presentationin order to make a point. Finally, the teacher reviews and summarizesthe lesson, and, if necessary, presents an exercise which will apply whatstudents have learned.

Teachers’ Roles During the Lessons

The following pedagogical terms are commonly used to describe theteachers’ key roles within a lesson:Hatsumon, Kikan-shido, Neriage, andMatome.

Hatsumon.Hatsumon meansasking a key questionthat provokes students’thinking at a particular point in the lesson. At the beginning of the lesson,the teacher may ask a question to probe or promote students’ understandingof the problem. During the whole-class discussion, on the other hand, heor she may ask, for example, about the connections among the proposedapproaches to solving the problem or the efficiency and applicability ofeach approach.

Mr. Matsumaru, after students had shared the expression 128÷ 16 asa mathematical expression for the problem setting, asked the followingquestion: “With which number as a divisor, instead of 16, could you findthe answer?” With this question, he was trying to emphasize the difference


between what the students had learned in the previous grade and what theywere faced with now.

Kikan-shido.Kikan-shido meansinstruction at students’ deskand includesa purposeful scanning by the teacher of the students’ individual problem-solving processes. While the teacher moves about the classroom, silentlymonitoring students’ activities, he performs two important activities thatare closely tied to the whole-class discussion that will follow the individ-ual work. First, the teacher assesses students’ problem-solving progress.In some cases, the teacher suggests a direction for students to followor gives hints for approaching the problem. Second, the teacher makesmental notes as to which students used the expected approaches and whichstudents used different approaches to the problem. These students will beasked to present their solutions later. Thus, in the period of the purposefulscanning, the teacher considers questions like, “Which solution methodsshould I have students present first?” or “How can I direct the discussiontowards an integration of students’ ideas?” Some of the answers to suchquestions might have been prepared in the planning phase, but some arenot.

Neriage.The term Neriage describes the dynamic and collaborative natureof the whole-class discussion during the lesson. In Japanese, the termNeriage meanskneading upor polishing up. In the context of teaching,the term works as a metaphor for the process of polishing students’ ideasand of developing an integrated mathematical idea through the whole-classdiscussion. Japanese teachers regard Neriage as critical for the success orfailure of the lesson.

Based on the teacher’s observations during Kikan-shido, he or she care-fully calls on students to present their solution methods on the chalkboard,selecting the students in a particular order. The order is quite importantboth for encouraging those students who found naive methods and forshowing students’ ideas in relation to the mathematical connections amongthem. In some cases, even an incorrect method or error may be presentedif the teacher thinks this would be beneficial to the class. Once students’ideas are presented on the chalkboard, they are compared and contrastedorally. The teacher’s role is not to point out the best solution but to guidethe discussion toward an integrated idea.

In the case of Mr. Matsumaru, he selected several students’ solutions forthe presentation and focused on the relationships among those solutions.He used the figure of the planters in various arrays as representations.He spent, in particular, a fair amount of the time discussing the idea that


“We can divide both the dividend and divisor by 2 without changing theresult.” This idea had been proposed by a student and was then explainedby another student as he used the figure of planters. The idea was expandedby other students who used the numbers 4 and 8 as common divisors of 128(the dividend) and 16 (the divisor). The idea that “one can divide both thedividend and divisor by the same number without changing the result” wasa main target of the whole-class discussion.

Matome.The Japanese term Matome meanssumming up. Japanese teach-ers think that this stage is indispensable for a successful lesson. TheMatome stage is identified as a critical difference between the U.S.and Japanese classroom activities (Fujii, Kumagi, Shimizu & Sugiyama,1998). According to the U.S.-Japan comparative analysis, the Matomestage Japanese teachers tend to make a final and careful comment onstudents’ work in terms of mathematical sophistication.

In general, in the Matome stage the teacher reviews what studentshave discussed in the whole-class discussion and summarizes what theyhave learned during the lesson. Mr. Matsumaru summarized the regularityof division students’ had found and discussed as follows: “The answerremains the same when we divide both the divisor and dividend by thesame number.” Also, Mr. Matsumaru emphasized the usefulness of theidea for reducing the problem with division by a two digit number intoones involving only single-digit divisors.

ASPECTS OF MATHEMATICS TEACHER EDUCATION INJAPAN

Mentoring Beginning Teachers

To become a teacher in the Japanese educational system, a student mustobtain a teacher’s certificate by completing the subjects in a universitycourse, in accordance with the provisions of the Educational PersonnelCertification Law. Although the teaching certificate is valid throughout thecountry, each local government is responsible for hiring teachers. With theteacher’s certificate, prospective teachers take an examination offered byeach local board of education. Successful prospective teachers are hiredby the local board.

Beginning teachers who have been hired are considered to be noviceswho need the support of their experienced colleagues. All beginning teach-ers are required to participate in the induction training program for oneyear after their appointment. For each beginning teacher, a master teacher,


who might be a head teacher or another experienced teacher, is assigned tohelp the novice make a successful start of their educational service and tolearn and practice the different roles the teacher assumes in the course of alesson.

During the induction period, workshops are offered both at the nationaland at local levels. The induction program includes approximately 300hours of closely supervised and monitored teaching, with some of theclassroom lessons being observed. In addition, the novices attend at least20–30 full or partial days of further training at educational centers run bythe regional prefecture or the local boards.

There are other opportunities for beginning teachers to learn from theirexperienced colleagues. Workshops of a particular style,Jugyo Kenkyu-kai(lesson study meeting), are regularly held for both beginning and expe-rienced teachers. The workshops include an actual lesson observed bythe attending teachers as well as an extended discussion after the lesson.Teachers exchange ideas about the lesson with a focus on the contenttaught and on the teacher’s roles assumed during the lesson. Experiencedteachers or mathematics educators are sometimes invited to comment onthe development of the observed lesson, on the interpretations of thetopic taught, and on how the lesson could be improved. In addition, thereare many informal circles of 10 to 20 practicing teachers. The teachersgather after school once a month to discuss, for example, how they weresuccessful or not in their teaching of mathematics, to introduce interestingproblems or topics, and to examine the proposed lesson plans from variousviewpoints.

The physical arrangement of the school promotes the interaction amongcolleagues. All teachers share a large room, the teachers’ room, where eachteacher has a desk. In addition to classroom teaching, teachers spend aconsiderable time in the teachers’ room. This situation allows them to shareinformation about students, ideas about mathematical topics, and instruc-tional materials. Also, teachers in public schools are required to move fromone school to another several times in their careers, possibly every three toten years, within their regional prefecture. In some prefectures, teacherseven move from elementary schools to lower secondary schools, and viceversa. Teachers also move within a school from one grade to another eachyear. These moves may be beneficial for beginning teachers as they helpthem become familiar with the content taught in different grades.

Educating Beginning Teachers About Lesson Plans

Throughout the educational process, lesson plans are used as vehicleswith which teachers can learn and communicate about the topic to be


taught, possible students’ approaches to the problem presented, and theimportant teacher roles. Preservice teachers are intensively taught how towrite lesson plans well. Inservice teachers also do have opportunities towrite or examine lesson plans at workshops and the informal circles afterschool.

In general, lesson plans are written in detail. Writing lesson plans isa critical exercise for preservice teachers, although they can easily accesssample lesson plans in a teachers’ edition or in other books. For a particulartopic, these lesson plans also include expected students’ responses to theproblems. During this period of education, prospective teachers are learn-ing through intensive coaching to write and polish their lesson plans byusing a particular format (Figure 1).

Steps Main Anticipated Remarks

learning activities students’ responseson teaching

• Posing a problem

• Students’ individual

problem solving

• Whole-class

discussion

• Summing up

(Exercise/Extension)

Figure 1. A Common Framework for Lesson Plans.

An important part of planning for a lesson isKyozai-kenkyu. Kyozai-kenkyu refers to the careful analysis of the topic in accordance with theobjective(s) of the lesson. It includes analyses of the mathematical connec-tions both among the current and previous topics (and forthcoming ones,in some cases) and within the topic. Also included are the anticipationof students’ approaches to the problem and the planning of instructionalactivities based on the anticipated responses. For example, the currentNational Course of Study (Ministry of Education, 1989) emphasizes theimportance of understanding and using regularities of four operations. Inaccordance with the Ministry’s intentions, Mr. Matsumaru had planned toimplement a lesson that focused on the importance of regularities of divi-sions, such as a÷ b = (a ∗ c)÷ (b ∗ c) or a÷ b = (a ÷c)÷ (b÷ c), basedon his interpretation of the topic: These regularities of division would beuseful for the students later when they would study division by decimal


fractions (in fifth grade) and by a common fraction (sixth grade): 0.8÷ 0.2= (0.8∗ 10)÷ (0.2∗ 10) = 8÷ 2; 2/5÷ 3/4 = ((2/5)∗ 20)÷ ((3/4) ∗ 20)= 8÷ 15.

It should also be noted that Mr. Matsumaru intentionally had selectedboth the particular numbers 128 and 16 and the problem which requireda partitive division for introducing the topic. Consciously, he had chosennot to use numbers like 120÷ 20 or a problem setting that required aquotitive division. He thought that if he would have used 120÷ 20 forintroducing division by a two-digit number, students’ attention would beconfined to dividing the two numbers each by ten. Also, if he had used aprime number like 17 for the divisor, no regularity of division would havebeen noticed by his students.

For any given problem, prospective teachers in a teacher preparationprogram will be expected to anticipate several student responses to theproblem. For the division problem, possible strategies included:

1. Guessing.2. Repeated subtraction: “How many “16s” are there in 128?” (128 –

16 – 16 – 16 – . . . – 16 = 0).3. Repeatedly substituting numbers into the expression1 ∗ 16 = 128

(1 ∗ 16 = 16, 2∗ 16 = 32, 3∗ 16 = 48, . . . 8∗ 16 = 128).4. “Dividing by 16” means “dividing by 8 first and then by 2” (128÷

16 = 128÷ (8 ∗ 2) = (128÷ 8)÷ 2 = 16÷ 2 = 8).5. Dividing both the dividend and divisor by the same numbers, e.g.,

128÷ 16 = (128÷ 2)÷ (16÷ 2) = 64÷ 8 = 8, or 128÷ 16 = (128÷ 4)÷ (16÷ 4) = 32÷ 4 = 8.

6. When multiplying the divisor by 2, the quotient is halved (128÷ 4= 32, 128÷ 2 = 64) so the answer of 128÷ 16 is half of 128÷ 8 =16.

The success of a lesson depends heavily on the interpretation of thetopic. Thus, Kyozai-kenkyu is a crucial part of the lesson planning forJapanese teachers. This kind of analysis is heavily emphasized in pre-service teacher training courses at the university. Kyozai-kenkyu is alsoemphasized as student teachers are supervised by experienced teachersduring their practice teaching. In summary, the educational value of acareful content analysis is considered very important.

FINAL REMARKS

In this article, several pedagogical ideas commonly shared by Japaneseteachers were described. The discussion of multiple solutions to a problem


in a whole-class mode is a common style for teaching mathematics inJapanese schools. Teachers play several key roles for making this teachingstyle work well. These key roles are learned by prospective and beginningteachers through both formal and informal educational settings.

A mathematics teacher in the Japanese educational system must notonly finish a teacher preparation course and pass a examination, butalso learn from their colleagues the teachers’ roles and the educationalvalue of a careful subject-matter analysis. Beginning teachers in Japan areexpected to continue learning teachers’ roles through the informal interac-tion with their experienced colleagues as well as in a formal educationalsetting.

NOTE

1 An earlier version of this paper was presented at the 74th Annual Meeting of the NationalCouncil of Teachers of Mathematics which was held in San Diego, CA, 25–28 April1996. The author would like to thank Jerry P. Becker, Southern Illinois University, CathyBrown, Indiana University, and Toshiakira Fujii, Yamanashi University, for their thoughtfulcomments on earlier drafts of the paper.

REFERENCES

Becker, J.P., Silver, E.A., Kantowski, M.G., Travers, K.J. & Wilson, J.W. (1990). Someobservations of mathematics teaching in Japanese elementary and junior high schools.Arithmetic Teacher, 38(2), 12–21.

Fujii, T., Kumagai, K., Shimizu, Y. & Sugiyama, Y. (1998). A cross-cultural study ofclassroom practices based on a common topic.Tsukuba Journal of Educational Studyin Mathematics, 17, 185–194.

Lee, S.Y., Graham, T. & Stevenson, H.W. (1996). Teachers and teaching: Elementaryschools in Japan and the United States. In T.P. Rohlen & G.K. Letendre (Eds.),Teachingand learning in Japan(157–189). Cambridge, UK: Cambridge University Press.

Ministry of Education, Science, Sports and Culture, (1989).The national course of study.Tokyo: Author.

Nagasaki, E. & Becker J.P. (1993). Classroom assessment in Japanese mathematics educa-tion. In N. Webb (Ed.),Assessment in the mathematics classroom(40–53). Reston, VA:National Council of Teachers of Mathematics.

Shimizu, Y. (1996). Some pluses and minuses of “typical pattern” in mathematics lessons:A Japanese perspective.Bulletin of the Center for Research and Guidance for TeachingPractice, 20, 35–42. Tokyo Gakugei University.

Stevenson, H.W. & Stigler, J.W. (1992).The learning gap: Why our schools are failingand what we can learn from Japanese and Chinese education.New York: SummitBooks.


Stigler, J.W., Fernandez, C. & Yoshida, M. (1996). Cultures of mathematics instruction inJapanese and American elementary classrooms. In T.P. Rohlen & G.K. Letendre (Eds.)Teaching and learning in Japan(213–247). Cambridge, UK: Cambridge UniversityPress.

Department of Mathematics and InformaticsTokyo Gakugei UniversityKoganei, Tokyo, 184-8501 Japane-mail: [email protected]/Fax: +81-423-29-7471

Guest Editorial

BARBARA JAWORSKI

MATHEMATICS TEACHER EDUCATION, RESEARCH ANDDEVELOPMENT: THE INVOLVEMENT OF TEACHERS

Most teacher education programmes are designed to enhance mathematicslearning experiences through the reform or development of teaching. Thus,an important question for any programme is: What is the role of the teacherin this programme?

There are three possible roles: pupil, participant, and partner.

1. Pupils play no part in determining the philosophy, objectives, design,or delivery of the programme. The programme is totally determinedby those who lead it, who are seen to be in the best position to know,theoretically and practically, what is needed and the most appropriateways to interpret it in the programme structure. All the power restswith the leaders and none with the pupils.

2. Participants are included partially in planning and delivery. Theirviews are sought on needs and expectations, and their feedback is usedto modify and improve the programme. However, the leaders make thekey decisions and largely control philosophy and design. The powerstill rests mainly with the leaders.

3. Partners are engaged in equal status with leaders in determining theprogramme. They may take different foci depending on the level ofexpertise and on the agreed objectives. The programme is democratic-ally designed for the benefit of those taking part and capitalizes on theagreed strengths of all its participants. The balance of power may stillbe with the leaders, but the partner teachers play a significant role indetermining their programme.

Of course, such a distinction of roles is a polarization, and programmesrarely fit exactly one of these patterns. Nevertheless, this polarizationserves to address a key issue of how development or reform is relatedto ownership. Pupils have little ownership, participants some, partners agreat deal. I see parallels with the three-fold framework for researcher-


118 BARBARA JAWORSKI

practitioner cooperation in educational research presented by Wagner(1997). In this framework practitioners range from subjects to co-learningpartners of the research.

Leaders of teacher education programmes tend to be educators andresearchers in university or college departments. Their involvement inresearch and other academic affairs means that they operate at highlytheoretical levels. They develop expertise in relationships between thetheories of mathematics education and the practices which are supposed toresult in enhanced mathematical experiences for pupils. Sadly, few of theseeducators teach mathematics to students in classrooms; consequently, thereis little opportunity to interpret the practices they expect to be effective.Instead they work with teachers in a sincere effort to achieve enhancedlearning environments.

The translation of theories of teaching for effective mathematicallearning into practice is one stage removed from the research and theoriz-ing from which the theories were initiated. In general, teachers arenot interpreting their own theories into classroom practice, with all thecomplexity this entails; rather, educators are finding ways to enableteachers to interpret external research and theory into classroom practice.We should not be surprised that the results are not what we might hope, thatreform principles and standards are not achieved according to our visions,that teachers, in the realities of schools and pressures of community andsocial practice, do not act in ways we try to promote. I situate myself firmlywithin this “we,” as I know, from many years of experience, the issuesand tensions produced by these theory-practice interfaces with teacherssomewhere in the middle.

Recently, David Hargreaves from the University of Cambridge (UK)offered a trenchant challenge to the educational research community(Hargreaves, 1996). He spoke of a gap between researchers and prac-titioners: “It is this gap. . . which betrays the fatal flaw in educationalresearch. For it is the researchers, not the practitioners, who determinethe agenda of educational research” (p. 3). He suggested that educa-tional research as it is currently conceived neither reflects the realityof classrooms nor provides an “evidence-based corpus of knowledge”regarding effective classroom practice. Hargreaves’ comment caused afurore in the educational research community and a long-running debateabout links between educational research and teaching development (see,for example, Hammersley, 1997).

Undoubtedly there is a gap, maybe a number of gaps, between teachersand educators, between teachers and researchers, between teachers andthose who determine educational agendas. Teachers are treated as pupils,

GUEST EDITORIAL 119

or participants, but rarely as partners in the educational process. Yet, it isultimately the teachers who work with students in classrooms, and whohave the power to transform mathematics education. Thus, whether devel-opment programmes are research based or not, it seems that for successthey need to treat teachers more fully as partners.

In some parts of the world teachers take responsibility for developingmathematics teaching and researching its development (e.g., Krainer,1998). In some cases, teachers and educators work alongside each otherin complementary ways, each contributing to and learning from theprocesses and practices involved (e.g., Dawson, 1999). Thus, developmentoccurs through a reflexivity of interaction and mutual respect in teachingand researching in mathematics classrooms. We are only just starting toperceive the issues and tensions in these practices, or ways in which theymight permeate mathematics education more widely. Perhaps this is aresearch agenda for the new millennium.

REFERENCES

Dawson, S. (1999). The enactive perspective on teacher development: A path laid whilewalking. In B. Jaworski, T. Wood & S. Dawson (Eds.),Mathematics teacher education:Critical international perspectives(pp. 148–162). London: Falmer Press.

Hammersley, M. (1997). Educational research and teaching: A response to DavidHargreaves’ TTA lecture.British Educational Research Journal, 23(2), 141–161.

Hargreaves David H. (1996, April).Teaching as a research based profession: Possibil-ities and prospects. Teacher Training Agency Annual Lecture. London: Teacher TrainingAgency.

Krainer, K. (1998). Dimensions of teachers’ professional practice: Action, reflection,autonomy and networking. In T. Breiteig & G. Brekke (Eds.),Theory into practicein mathematics education. Proceedings from NORMA 98(pp. 36-43). Kristiansand,Norway: Agder College Research Series.

Wagner, J. (1997). The unavoidable intervention of educational research: A framework forreconsidering research-practitioner cooperation.Educational Researcher, 26(7), 13–22.

BARBARA JAWORSKI

Associate Editor,JMTE

OLIVE CHAPMAN

INSERVICE TEACHER DEVELOPMENT IN MATHEMATICALPROBLEM SOLVING?

ABSTRACT. A humanistic perspective provided the basis for a problem-solving orientedteacher inservice program. The program was designed to provide opportunities thatallowed elementary teachers to focus on personal experience as a way of achieving self-understanding and a way of reconstructing their personal meanings about problem solvingand problem-solving instruction. Impact of the program was studied through interviewswith the six participants and observations of their teaching. The results indicated that theprogram had a positive effect on the participants’ beliefs about and teaching of problemsolving.

Studies on mathematics teachers’ knowledge, beliefs, attitudes, practices,and professional development are forming a growing body of literatureabout the mathematics teacher (e.g., Chapman, 1997; Cooney, 1985; daPonte, 1994; Even & Tirosh, 1995; Knapp & Peterson, 1995; Simon &Schifter, 1991; Thompson, 1984; Wood, Cobb & Yackel, 1991). One ofthe underlying themes of these studies is to gain an understanding of howchange in the teaching of mathematics comes about. It is in the context ofthis theme that this article is framed. The article discusses inservice math-ematics teachers’ change in the context of teaching mathematical problemsolving. The focus is on a humanistic approach to teacher development andthe effects of this approach on the thinking and practice of experiencedelementary teachers.

Lester (1994) drew attention to the ongoing under-representation ofresearch on the teaching of problem solving. In particular, he pointed outthat very little of the literature on mathematical problem-solving instruc-tion discusses the specifics of the teacher’s role or the teacher’s perspectiveof that role. But equally under-represented in the literature are studies onteacher development in problem-solving instruction. This gap in the liter-ature is inconsistent with the prominent role problem solving plays in thecurrent reform movement in mathematics education (NCTM, 1989, 1991).

? This article is based on a project supported by a grant from The Alberta AdvisoryCommittee for Educational Studies.


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Current reform recommendations emphasize a problem-solving per-spective to mathematics instruction. This perspective necessitates thatteachers understand genuine problem solving. Many inservice teachers,however, might not have experienced mathematical problem solving as adistinct topic or a way of thinking. They are more likely to have experi-enced algorithmic word problems under the heading of problem solving.Thus, for them, teaching from a problem-solving perspective could meansomething quite different from what is intended by reform recommen-dations. Consequently, these teachers are unlikely to facilitate problem-solving thinking or help students to think for themselves. According toLester, “The primary purpose of mathematical problem-solving instructionis not to equip students with a collection of skills and processes, but ratherto enable them to think for themselves” (1985, p. 66). Teachers who under-stand problem-solving instruction in terms of this primary purpose aremore likely to teach mathematics from this problem-solving perspective.Problem solving as a distinct topic could play an important role in helpingteachers develop this perspective. It is in this context that problem solvingis being considered as important for teacher development and change.

Another factor to consider in teacher development is the relationshipbetween a teacher’s beliefs and practice. There is growing evidence thatteachers’ beliefs about mathematics and its teaching play a significant rolein shaping their classroom behaviors (Chapman, 1997; da Ponte, 1994;Thompson, 1984). Thus, as Ernest (1989) argued, significant and mean-ingful teaching reforms are unlikely to take place unless teachers’ deeplyheld beliefs about mathematics and its teaching and learning change.This relationship between teachers’ beliefs and practices suggests that aform of professional development that focuses on self-understanding, inregard to teachers’ thinking and behaviors, may be of relevance to mathe-matics education. The study reported in this article was concerned withinvestigating this form of professional development in the context of aproblem-solving inservice program.

THE PROBLEM SOLVING INSERVICE [PSI] PROGRAM

Theoretical Perspective

The goal of the PSI program was to provide opportunities that allow tradi-tional mathematics teachers to focus on personal experience as a way ofachieving self-understanding and a way of reconstructing their personalmeanings about problem solving and problem-solving instruction. Theprogram had a humanistic emphasis based on concepts of lived experience

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(Carr, 1986; Dewey, 1938), personal meaning (Polanyi, 1958, 1975), andnarrative reflection (Bruner, 1986; Polkinghorne, 1988).

As lived experience, a teacher’s classroom behavior has a past-futurestructure. In terms of Dewey’s (1938) theory of experience, this structureexists in part because of what is brought to it and in part because of its influ-ence on the future brought about by the alterations that occur in internaland environmental conditions of an experience. In relation to this past-future structure, Carr (1986) described three critical dimensions of humanexperience: significance, value, and intentions. In general terms, the pastconveys significance, the present conveys value, and the future conveysintentions. If teachers focus on experience, then, they could learn aboutthemselves and their teaching in terms of the meanings associated withthe past, the present, and a possible future that may shape their classroombehaviors.

A teacher’s classroom behavior can also be conceptualized in termsof personal meaning. According to Polanyi (1958), personal meaning isconstructed from experience and reflects a dialectic blend of the indi-vidual and the social. Personal meaning is also a basis for organizing one’sknowledge of the world and one’s behavior in it. Personal meaning, then,can be used as a basis for considering and modifying the ways in whichteachers perceive and execute their professional tasks. A teacher’s personalmeaning embodies what the teacher feels, thinks, believes, and wants(Elbaz, 1990). A mode for accessing this meaning is narrative knowing(Bruner, 1986).

Narrative, or narrative knowing, is considered essential for under-standing human experience (Bruner, 1986; Polkinghorne, 1988). Bruner(1986) described narrative knowing as being concerned with the expli-cation of human intentions in the context of actions. In general terms,narrative embodies both personal meaning and the past-future structure ofexperience. Narrative has been promoted as an especially relevant modefor expressing teachers’ practical understanding (Carter, 1993; Elbaz,1990).Narrative reflectionrefers to reflection framed in narrative knowing.It is a way of living out one’s story and a way of storying one’s experience.Like narrative inquiry (Connelly & Clandinin, 1990), narrative reflection isa process of collaboration that involves mutual storytelling and re-storyingamong participants. It requires an “I-Thou” (Buber, 1969) relationshipbetween the audience and the story that is told. For example, the audiencecannot treat the story as an object to be analyzed. Narrative reflection, then,focuses more on the expression than on the analysis of meaning. In thePSI program, narrative reflection based on self-stories of past, present, and

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possible future experiences formed the basis for meaning recovery andre-construction of teachers’ actions.

Program Activities

The activities of the PSI program were organized into four stages.

Stage 1: Introduction.The introduction focused on the nature of narrativereflection and on how to facilitate depth in personal reflections, sharing,and collaboration. The teachers were coached, for example, on how toshare personal experiences in terms of “I” and on how to focus on selfby resonating with someone else’s story. Reflections were based on theteachers’ experiences with real-life problems. The need for non-criticalacceptance of each other was also emphasized.

Stage 2: Reflection on personal meaning.This stage focused on theteachers’ histories, actions, and intentions in the context of teachingand learning problem solving prior to entering the PSI program. Themain activity was narrative reflection with a focus on self, content,teaching, and learning. Self, for example, involved self-stories thatreflected personal feelings about problem solving and self as problemsolver. Content involved self-stories that reflected the nature of problemsand problem solving. Teaching and learning involved self-stories thatreflected instructional strategies and the learner’s role in problem solving,respectively.

Stage 3: Problem-solving experiences.This stage focused on non-algorithmic problem solving based on the teachers’ problem solvingexperiences with a variety of problems that were non-routine for them.These problems were framed in both concrete and abstract contexts as inthe following two examples:

(a) Using all of the construction paper (30 cm square) and no more than 3cm of adhesive tape, build the tallest free-standing structure possible.

(b) Two bicycles, 90 km apart, are ridden toward each other at rates of 20km/hr. A fly going at 30 km/hr starts on the nose of one cyclist, fliesto the other’s nose, back and forth, losing no time on the turns untilthe cyclists meet. How many kilometres does the fly fly?

To solve the problems, the teachers were instructed to use any approachthat made sense to them. They were also told that the process of solvingthe problem was more important than the solution itself and that it was


acceptable to not arrive at an answer or a correct one providing an effortwas made to solve the problem.

Three themes were used to guide the problem-solving experiences. Thefirst theme focused on the problem-solving process of an individual andinvolved the following activities:

1. Solving two problems and preparing written summaries of the solutionprocesses.

2. Working in groups of two as problem solver and observer. Whilethinking aloud the problem solver solved the observer’s problem. Theobserver listened and summarized the problem solver’s process.

3. Thinking aloud while solving a problem, including all emotionalaspects of the experience, and audio-taping the process.

4. Solving a problem and writing a detailed, narrative journal of theprocess, including all emotional aspects of the experience.

These activities were accompanied by narrative reflection with a focus onthe nature of the problem-solving process.

The second theme focused on the problem-solving process of a smallgroup. The main activities required that the teachers solve problems whileworking in groups of two, three, and six. Each group discussed andprepared summaries of the group process. These summaries formed thebasis for narrative reflection on small-group problem solving.

The third theme focused on teacher-student interactions while studentssolved problems. The main activities required that the participants work ingroups of two and three in the roles of teacher and student(s) based on thefollowing four scenarios:

1. Teacher works with one student, student thinks aloud while solving aproblem, teacher is silent, i.e., cannot provide help.

2. Teacher works with a group of two students, students collaborate andthink aloud while solving a problem, teacher is silent, i.e., cannotprovide help.

3. Same as (1), but teacher can now intervene, e.g., provide help.4. Same as (2), but teacher can now intervene, e.g., provide help.

The role of student included reflection from the student’s perspective onwhen help was or was not needed, on the form in which help was needed,and on feelings when the teacher intervened. The role of the teacherincluded reflection from the teacher’s perspective on when, how, and whythe teacher wanted to intervene, or actually intervened. Post-scenarioreflections were based on the participants’ experiences regarding theintervention.

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Stage 4: Reflection on problem-solving experiences.This stage focusedon what the teachers learned as a result of the experiences in Stage 3.The main activity was narrative reflection with a focus on self, content,teaching, and learning, as described in Stage 2.

THE STUDY

The study investigated the effects of the PSI program on teachers’ personalmeanings about problem solving and its teaching. In particular, two keyquestions guided the research: (a) How do the PSI activities influenceinservice elementary teachers’ personal meanings of themselves as mathe-matical problem solvers, of the nature of mathematical problem solving,and of its teaching and learning; and (b) how do the PSI activities influenceclassroom behaviors in the teaching of problem solving; if meaningfuland fundamental changes occur to the teachers’ personal meanings andteaching, then how can these changes be attributed to the PSI activities?

Given the humanistic focus of the PSI program, the study was framed ina humanistic perspective of research (Bogdon & Taylor, 1975). The focusof this perspective is to identify themes in the participants’ experiences inorder to convey the essence of the experience for them.

Project Participants

The six participants were selected from volunteers who wanted to learnmore about the teaching of problem solving. They came from five schools.Mary and Pam taught Grade 3, Susan Grade 4, Wendy Grade 5, and Roseand Ann Grade 6 (all names are pseudonyms).The criteria for selectionwere: (a) they were not mathematics majors, (b) they were not mathematicsspecialists, (c) they did not have strong mathematics backgrounds, and (d)they had negative attitudes towards mathematics. These criteria were usedbecause they likely reflected characteristics of elementary teachers moregenerally. The volunteers were interviewed by telephone to determinewho satisfied all of the required criteria. This was a structured inter-view of approximately 15 minutes that focused on the level and type ofmathematics the volunteers completed as students, their performance inmathematics, their feelings toward mathematics, and their participation inmathematics, and their teaching of problem solving.

The participants had little or no experience solving non-routine prob-lems as learners. However, they all were exposed to problem solving basedon the approaches prescribed by their textbooks. These approaches gener-ally involved a demonstration of a strategy, followed by the presentation of


several problems that could be solved with the strategy. In the year beforethe PSI project, the participants had started to include lessons on problemsolving, based on their interpretations of the textbook’s approach, into theirteaching. This shift in their teaching was in anticipation of changes to theirschool curriculum that would reflect the curriculumStandards(NCTM,1989).

Participants’ Involvement

The participants worked on the PSI activities under the supervision ofthe researcher. They spent 2 hours on the program introduction (Stage1), 4 hours on reflection on personal meaning (Stage 2), an average of22 hours per participant on the problem-solving experiences (Stage 3),and 2 hours on reflection on problem-solving experiences (Stage 4). Thisdistribution was not predetermined. It evolved as the participants indicatedthat they were ready to move on. All of the teachers participated fully inthe PSI activities. Throughout the study, they participated willingly andtalked freely about their thinking and teaching. To encourage this level ofinvolvement in the study, the teachers were assured that the goal of theinvestigation was not to judge them or their teaching. In addition, the PSIexperience seemed to have allowed them to establish a sense of trust inworking with the researcher that continued throughout the study.

Data Collection

Data for the study came from three sources: open-ended interviews,classroom observations, and the PSI activities. Each participant was inter-viewed five times during the study. The length of the interviews rangedfrom 60 to 90 minutes each. The interviews were framed in a paradigmaticcontext (Bruner, 1986) and a narrative context (Bruner, 1986; Mishler,1986). The paradigmatic context focused on analytical explanations basedon questions framed in a theoretical mode (e.g., “What is the teacher’srole when students are solving problems?”). The narrative context focusedon descriptions of lived experiences based on a variety of scenarios (e.g.,“Tell a story of a particular situation involving you solving a genuine math-ematics problem that had special meaning for you.”). All of the interviewswere audio-taped and transcribed.

The participants were interviewed once before (Pre-PSI Interview) andonce immediately after (Post-PSI Interview 1) the PSI experience whichoccurred during the summer of 1994. The focus in these interviews wason thinking about self, content, teaching, and learning with respect toproblem solving. The participants were later interviewed three times (Post-PSI Interviews 2, 3, and 4) during the 1994–1995 school year. The focus

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in these interviews was on actual post PSI teaching, with emphasis onthe relationship between practice and the personal meanings constructedduring the PSI experience. The number and timing of these interviews weremutually agreed on with the teachers. Post-PSI Interview 2 was conductedclose to the middle of the first term when the teachers felt that their routineswould be in place and they could talk about the events that had happened ingetting to that point. Post-PSI Interviews 3 and 4 were conducted towardsthe end of the first and second terms, respectively, when, for the teachers,there was a sense of completion of a teaching cycle. Thus they were betterpositioned to look back in a more meaningful way. These three inter-views, along with classroom observations, were also intended to track theteachers’ behavior over two terms in order to identify some stable pattern.

Classroom observations of each participant’s teaching, when studentswere explicitly being engaged in problem solving, were conducted onceprior to and three times after the PSI experience. Post-PSI observationsoccurred about three-quarters into the first and second terms of Year 1 andthe middle of the first term of Year 2 after the program. The number andtiming of these observations were influenced by the agreement with theteachers to observe them approximately the same time in each school termand for the same number of times. Only one observation per term was prac-tical in order to accommodate all of the teachers and the researcher withinthe same two-week period and not to disrupt their normal teaching sched-ules. The observations documented what the teachers and students actuallydid during the lessons. This process was guided by categories (e.g.,problem presentation, teacher intervention) identified from Post-PSI Inter-view 1 that reflected significant shifts in the teachers’ personal meaningsabout teaching problem solving. The teacher-student verbal interactionsduring these lessons were audio-taped and transcribed. Following eachobservation, the teacher was asked to share anything about the lesson shewanted to highlight or clarify and anything on her thinking about anydeviations observed between her actions and post-PSI personal meaning.

The PSI activities were also a direct source of data that provided infor-mation about the nature of the knowledge the teachers were constructingabout self, content, teaching, and learning in the context of problemsolving. All of the teachers’ discussions while working on the PSI activitieswere audio-taped and transcribed. Copies of the teachers’ written workwere also obtained. This work included problems they solved individuallyand in groups, notes they made about their thinking while solving the prob-lems, and summaries of their individual and group reflections required aspart of the program.


Data Analysis

The effect of the program on the participants was considered in terms ofchanges in their personal meanings and teaching of problem solving, andthe relationship between their post-PSI personal meanings and practice.Changes in personal meaning were determined by comparing the partici-pants’ thinking before and after the PSI experience. Interview transcriptswere used to identify recurring themes (Bogdon & Taylor, 1975) of howthe participants viewed self, content, teaching, and learning. To establishtriangulation, these themes were compared to transcripts of their groupdiscussions, to their written work during the program, and to transcripts oftheir actual classroom discourse.

Characteristics on teaching mathematics from a problem-solvingperspective (NCTM, 1991) facilitated identification of meaningful,common changes in the participants’ actual teaching. These charac-teristics were used to determine the level of occurrence of specificteaching behavior in each participant’s teaching of problem solving.Comparison of the participants’ teaching, in terms of the relationshipbetween their personal meanings and practice, was done more holistic-ally. Narrative summaries (Polkinghorne, 1988) were prepared for eachteacher’s classroom process. These summaries described the teachers’and students’ roles from the point of assigning a problem to students tobringing it to closure. All outcomes for each participant were shared withher to get her comments and feedback in terms of the appropriateness ofthe description of her behavior.

CHANGES IN PERSONAL MEANING

This section presents the outcome of the study in terms of the influence ofthe PSI activities on the participants’ personal meanings of self, problemsolving, and the teaching and learning of problem solving. The focus is onproviding common characteristics and themes reflected in the participants’thinking and teaching about their personal meanings before and after thePSI experience in order to illustrate the nature of the changes influencedby the experience.

Teachers’ Personal Meaning Prior to the PSI Program

Self.The participants often portrayed themselves negatively as mathema-tical problem solvers as indicated by the following statements:

So I am doing this [the PSI] because I am not a great problem solver myself. [Mary, Pre-PSI Interview]

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I found it really hard,. . . trying to understand the problems myself, let alone teach andexplain it to the kids. [Rose, Pre-PSI Interview]I had never really focused on real problems. I had no ideas, zip. It was whatever was in themath text.[Ann, Post-PSI Interview 1]

The teachers believed that they were deficient in the type of thinking oneshould have to be successful with mathematical problem solving.

I am not very analytical. [Ann, Pre-PSI Interview]I was never a great abstract thinker. [Mary, Pre-PSI Interview]

Fear was also a dominant theme in their thinking.

I am somebody who came out through school terrified for years in doing maths.. . . I stillhave that inside of me, panic attack, especially with problem solving. I get totally paranoidabout it. [Mary, Pre-PSI Interview]Like I was terrified [of problems],. . . I always had high anxiety over a problem. And forme to carry and to tell something different to my kids [students], they can sense that. [Ann,Pre-PSI Interview]Whenever I got stuck, I would panic and quit. [Pam, Pre-PSI Interview]

This fear allowed them to resonate with their students’ views of problemsolving.

When I assign word problems and get thegroooansand all those looks of terror from mystudents, I can totally relate to them. [Rose, Pre-PSI Interview]I can understand the kids’ griefs and problems when we deal with it [problem solving].[Wendy, Pre-PSI Interview]

This empathy for their students often influenced them to minimize theassignment of word problems so that they would not ruin the students’confidence.

Problems and problem solving.The teachers considered problems tobe word problems that usually contained numbers. In general, problemsolving was an open-ended process, albeit more controlled in the contextof mathematics.

Math is different because you always have to know theright method or formula to get theanswer. [Mary, Pre-PSI Interview]There are different strategies, or formulas, or whatever, for different problems.. . . So youhave to know how to match theright strategy with the problem. That is the hard part. [Ann,Pre-PSI Interview]

All of the teachers noted that one of their weaknesses as problem solversin mathematics was not knowing how to identify the correct algorithm orstrategy, particularly when it was not obvious through cue words.


Teaching and learning.Consistent with their view on problem solving,the teachers considered the teaching of problem solving to involve gettingthe students to identify the right strategy/algorithm and to execute it.Showing students how to identify and use cue words in a problem wasmost important. In general, the teachers’ description of their classroombehaviors emphasized a teacher-centered approach that focused on theteachers’ methods for solving the problems assigned to the students. Forexample,

It’s really me saying, ‘Here is a problem. Let’s work through it.’ And I am trying to get thekids to come up with the answer. [Pam, Pre-PSI Interview]I was accustomed to the teacher being the problem solver and the students follow along.So I find it difficult to be different. [Mary, Pre-PSI Interview]You do have the answer in your head, and you do want them to come up with that answer.[Ann, Pre-PSI Interview]

Thus, during intervention while students solved problems, the teacherswanted to see only their thinking being reflected in the students’ work.

[Prior to PSI] I always looked at the [student’s] answer. . . . I looked at the written part thatwas important to me. . . . I guess I really considered my way was the only way. [Mary, Post-PSI Interview 1]But it was like I never looked at the justification [of answers] before [the PSI]. It was justlike it [the answer] didn’t fit where I was intending it to be. Or I would say, “Well, that isnot what I was looking for.” [Pam, Post-PSI Interview 1][Prior to PSI] whenever they were getting close to the answer, I would get excited and jumpin and say “That’s right!” [Wendy, Post-PSI Interview 1]

Teacher intervention, then, was intended to make sure that students wereusing the teacher’s approach and to help or encourage them to do socorrectly.

Teachers’ Personal Meaning After the PSI Program

Self.The teachers now portrayed themselves more positively in relationto mathematical problem solving. In particular, they felt more confidentin their ability to solve both routine and non-routine problems withinthe context of their teaching. They were more comfortable dealing withproblem solving in the elementary school curriculum. They also believedthat they had a better understanding of problem solving. The followingexamples are representative of their thinking:

I am more confident with problem solving and with teaching it. I think that I have a betteridea where I want to go as opposed to last year. I think I am a little stronger this year in therespect that I feel more confident in what I am doing. So I am experimenting more withmy own ideas. [Wendy, Post-PSI Interview 2]Definitely, I am more confident. . . . But I also got more confident in myself as a problemsolver, even though, I know I still have a long way to go yet. But I think I could do it, you

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know. . . . So that way it has made me more confident as a problem solver and definitely inteaching it too. [Rose, Post-PSI Interview 1]I can talk more freely with the kids because my understanding is better and I have learnedthings [from the PSI] that I never learned in school.. . . This year I feel better. That I cansay. [Ann, Post-PSI Interview 3]

In general, the teachers seemed to have developed a better under-standing of themselves as problem solvers. They now saw themselves inways they had not experienced or considered before. Although severalfactors possibly contributed to this shift in their personal meanings, threewere highlighted by the teachers. First, their group interactions allowedthem to see each other struggle with similar issues in solving the problems.This allowed them to understand their personal struggles with problemsolving in a more objective way, i.e., it was not something unique to themas individuals. As Mary explained,

But to sit and work through these things and to see other people,. . . to see them strugglingtoo! And I remember one day when. . . Susan was asking some questions and . . . Susan is amuch more respected teacher than I am, and yet she is asking questions which I know theanswers to. That really I think was the turning point for me. She couldn’t see something andto me it was as plain as can be. But to realize that all people have these kinds of difficulties,that was a big thing for me. [Mary, Post-PSI Interview 1]

Second, the teachers experienced success in solving problems bydepending on their own thinking. This success conflicted with what theywere led to believe by their experiences as students and freed them totry problems they normally would have avoided. Third, a more realisticview of mathematical problem solving by the teachers allowed them tounderstand experiences such as barriers they encountered in the solutionprocess, and the resulting frustrations, as integral features of problemsolving and not as situations they created because something was wrongwith them.

Problems and problem solving.The participants now viewed a problem asany situation that had barriers to a solution. Such situations could occur ina variety of forms.

It doesn’t always have to do with numbers, it could be logic or anything that has achallenge. [Ann, Post-PSI Interview 1]

This shift in thinking allowed the teachers to establish more scope in theproblems they now used in their teaching.

Last year [pre-PSI] the problem solving that I did wasn’t great. No. . . . [The problems] weretaken from a little bank that I had put together. But they were all basically algorithmic.I know what an algorithmic problem is this year as opposed to real problems. So I am


forming a new bank with all kinds of problems including some the students make. [Ann,post-PSI interview 3]I kind of felt like last year I was following the problem solving book. But now I use someproblems from that and other sources and others I have made up on my own that are actuallymore fun and challenging for the kids. [Rose, post-PSI interview 3]

Consistent with this view of problems, the teachers’ thinking ofproblem solving reflected a more flexible process. The teachers nowviewed real mathematical problem solving as an open-ended process inwhich the problem solver had to be in control in terms of interpretingthe problem and deciding on how to overcome barriers to a solution. Theprocess could be unpredictable in terms of what might or might not work.Thus, problem solving was viewed not as linear but as involving a cycleof failure and success. Both positive and negative emotions were naturalconsequences of the process. In particular,

It has to do with thinking . . . and knowing what to do to get out of being stuck. . . . So therehas to be a lot of reasoning and brainstorming also. [Pam, Post-PSI Interview 1]I see it as getting stuck, then getting unstuck, which is the part that really makes you think.That is why there is frustration, but then it feels good when an idea works. [Susan, Post-PSIInterview 1]

The teachers’ thinking also reflected that problems and problem solvingshould be interesting and meaningful for the students.

Well it is interesting, I had a comment a couple of weeks ago from a girl who said, “Whenare we going to do problem solving?” . . . And I said, “Well, we have been doing problemsolving all along.” But, so in a way I guess that it is more fun for them, that they don’trealize what they are actually doing sometimes. [Rose, Post-PSI Interview 4]

Teaching and learning.There were definite shifts in the teachers’ viewsof teaching and learning problem solving. In particular, there were signifi-cant changes in their views of the teacher’s role and the students’ role inthe classroom process. Students were now viewed as active participantsin a personal process of thinking. Thus, the teacher had to be flexible inallowing the students to develop their abilities as problem solvers. Theparticipants’ thinking reflected specific features of these roles from thepoint of presenting a problem to the students to bringing it to closure.For example, in presenting the problem, students should be allowed todetermine and discuss their meanings of the problem before solving it, andthe teacher should facilitate the discussion of the meanings. In solving theproblem, students should use what they know individually and collectivelyin order to arrive at a solution. The teacher should intervene in a thoughtfuland strategic way. For example,

Don’t jump in too soon.. . . There is a fine line between jumping in too soon and stayingback too long.. . . The kids will need time to work things out for themselves. [Wendy, Post-PSI Interview 1]

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Questioning too is really important. . . . Through questioning you may be able to help themthrough it so they may feel better in the end. [Susan, Post-PSI Interview 1]

Finally, to achieve closure on the problem, students should share anddiscuss their solutions and reflect on whether the solutions made sense,and the teacher should facilitate the discussion of the solutions.

The participants’ thinking also reflected a passive and an active modeof intervention. Passive intervention required that the teacher only listento the students’ discussions in order to become aware of the students’thinking and to give the students time to think on their own. Active inter-vention required that the teacher communicate with the students when theywere stuck, off-track, or lost in order to stimulate their thinking to helpthem get pass barriers and allow them to make sense of their processes.

CHANGES IN TEACHING

This section presents the outcome of the study in terms of the influenceof the PSI activities, in general, and the participants’ post-PSI personalmeanings, in particular, on the participants’ teaching of problem solving.The focus is on how the teaching actions of the teachers changed followingthe PSI program.

The teachers’ classroom processes following the PSI experiencereflected significant changes in many key factors in their teachingof problem solving. For example, the teachers shifted from textbook-dominated teaching to teaching in which textbooks and other problem-solving resources were used to select problems. In their post-PSI teaching,teachers engaged students in problem solving more frequently and usedcooperative learning groups in all of their problem-solving classes.However, for some of the classes, the work in cooperative groups wassupplemented by individual work so students would gain experience inproblem solving on their own. Also, teachers shifted to emphasizingprocess over answer, to considering alternative solutions, and to askingstudents to share solutions and their thinking behind right and wronganswers. As they explained:

Before [PSI], I never really thought much about. . . questioning [their thinking] . . . That isa real eye-opener. [Pam, Post-PSI Interview 2]This year I have really concentrated on not jumping in for them and saying, “Yes, thatis right!” . . . Now, instead, I question them. . . . I have made them and myself more awareof other solutions. . . . I am making them more aware of looking at things from differentangles. [Wendy, Post-PSI Interview 3]And also the business of hearing how they did it. It is fun to see their ways and to see thatthey can be quite different, but still very acceptable. [Rose, Post-PSI Interview 3]


You have to let go of that [answer in your head] and. . . let them make their own discoveries.So I keep trying to do this, to ask the kids questions in the right way and not forcing themto come up with my answer. [Ann, Post-PSI Interview 2]

All of the teachers pointed out that their teaching after the PSI programwas more challenging but also more interesting and rewarding particularlybecause they were learning a lot from the students. They also noted that,as they progressed through the school year, the way they were teachingother areas of the mathematics curriculum was being influenced by theirproblem-solving approaches. For example,

I see math as everything is a problem to the kids and I am using manipulatives muchmore than I did last year and I am being turned on to different resources. [Ann, Post-PSIInterview 3]

Despite the general similarities in changes in their teaching, each partic-ipant lived out the effect of the PSI experience in her classroom in herown unique way. Thus, each teacher’s classroom actions reflected differentemphasis on some of the common changes. For example, some of theteachers used a larger ratio of non-routine to routine problems. Similarly,they all applied their post-PSI meanings of teachers’ and students’ roles,but each made modifications to fit her perceived classroom situation. Toillustrate how these roles unfolded with the youngest students involved,summaries of the teaching of the two Grade 3 teachers, Mary and Pam,are presented. These summaries highlight the teachers’ and students’ rolesfrom the point of presenting a problem to bringing it to closure.

Mary

Prior to class, Mary usually wrote the problem on a flip chart. Studentsgathered around the chart on the floor. Mary asked them to read theproblem quietly to themselves. She then asked one student to read it aloudfor the class. She asked if they saw the difference between the informationpart and the question part of the problem. The students were asked toquietly read the question part, then discuss it. She wanted to know whatthe question part was and what the problem was asking them to do. Shecalled on a few students to respond and checked on whether the othersagreed. If there was disagreement, she asked for an explanation. If studentsencountered difficulties identifying or interpreting the question part, sheoften used leading questions to help them. Next, Mary asked them to readthe information part and to talk about what was given in the information.She would end the discussion by asking: Does the information make sense?or, What do you think about the information? Once the discussions werecompleted, the students were asked to go into groups of three to solve theproblem.

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Mary circulated around the room to check on the progress of the groups.She intervened whenever they asked for help or she perceived them to beoff-track or lost. When they requested help or were off-track, Mary gener-ally asked them to explain what they were trying to do, then referred themto the information and/or question part of the problem. She suggested thatthey check some specific aspect of their processes. For those consideredlost, she reviewed the question part and information part of the problemand asked leading questions to help them see how to proceed. Once theygot going, she allowed them to continue in their own way, unless theyasked for help or got off track again.

After solving the problem, the students regrouped at the flip chart topresent and discuss their solutions. Mary sometimes asked them to sharesomething they liked or did not like about the problem. She then allowedthem to present their solutions regardless of whether they were right orwrong and to explain why their solutions made sense. She would remindthem to listen to the ideas of others to see if they made sense to them. Shewould encourage discussion of the solutions by asking questions such as:What do you think of . . . ? Which of the answers do you think are correctand why? Why does it make sense?

Pam

Pam sometimes wrote the problem on the blackboard and sometimesreferred the students to the textbook or a worksheet. She asked studentsto read the problem aloud. She gave them a few minutes to think aboutwhat the problem was saying. She asked them to work with one or twopartner(s), share what they thought with each other and come to someagreement. She called on several volunteers to share with the class whatthey thought about the problem, regardless of whether they were sayingsimilar things. She did not comment on any of the students’ responses, butshe allowed them to challenge each other and to justify their positions. Sheasked them to write their interpretation of the problem. She asked if theyhad any questions – other than how to solve the problem – or if they neededany clarification. If questions arose she solicited volunteers to respond tothe questions. Finally, she asked the students to continue working in theirgroups, to solve the problem, and to record their processes.

As students worked in groups, Pam circulated among them and inter-vened both actively and passively. She intervened actively wheneverstudents asked for help, were perceived to be off-track, or completed thesolution ahead of other groups. If students were stuck or off-track, sheintervened with questions such as: What have you tried? Why did youadd? What part of the problem asked you to do that? What else do you


think you can try? Why don’t you try drawing a picture or using a chart? Isthat what you really want to do? She gave them time to resolve difficultieson their own. If students were lost, she provided more direct guidance byasking leading questions or telling them what was wrong or how to getstarted. But in general, Pam allowed students to arrive at a solution in theirown way even if their solution was incorrect. At the end of the lessonshe dealt with the appropriateness of the proposed solutions. She tried toget students who finished early to reflect on their solutions and consideralternative ways of thinking about the problem.

After solving the problem, the students were required to present theirsolutions, regardless of whether they were right or wrong, and defend themto the class. In a common scenario, after a group presented their solution,Pam would ask the class questions such as: What do you think about theirmethod? Does it make sense? What doesn’t make sense? How can they fixit? The presenting group got to respond to the class whenever necessary.

Comparison

In general, Mary’s and Pam’s teaching reflected the same essence of theteachers’ post-PSI personal meaning on teaching and learning problemsolving. However, there were elements in Mary’s and Pam’s classroombehaviors that reflected different styles in conducting the lessons. Oneexample of this involved the presentation of problems. Mary alwayspresented a problem as consisting of a question part and an informa-tion part. Students were required to explicitly identify each part. Thesecategories then played an integral role in framing teacher-student discus-sions of the problem throughout the lesson. Mary also presented a problemin a communal context, i.e., she had students gather together to interpretthe problem. This context often led to one interpretation of the problemwhich was usually consistent with Mary’s interpretation. Thus, Mary’sintervention in order to provide help when requested was often influencedby her interpretation. Her goal, however, was not to discourage alterna-tive solutions to the common interpretation of the problem. She did notactively intervene when students were making good progress in solvingthe problem in different ways.

Unlike Mary, Pam did not use a communal context to present a problem,and she did not require students to explicitly identify the different parts ofthe problem. She required her students to first think about the problemindividually, then collaborate with a partner, then share with the wholeclass, and then record their interpretations of the problem. She did notattempt to establish a common interpretation when one did not emerge.Instead, all interpretations (correct or incorrect) were dealt with in a whole-

138 OLIVE CHAPMAN

class context after students were allowed to work on their solutions. Thus,Pam’s intervention appeared to be influenced by the students’ thinking;Mary’s interventions, on the other hand, tended to be driven by her owninterpretation of the problem and a particular conception of a solution.

Despite the differences in styles, Mary’s and Pam’s post-PSI classroombehaviors, like those of the other participants, reflected a similar qualityof teaching that was superior to their pre-PSI teaching in terms of facil-itating students’ problem-solving thinking. Their teaching had shiftedfrom a show-and-tell approach to active involvement of students inthinking through problems from beginning to end. In this shift, they alsoemphasized reasoning and communication as integral aspects of students’involvement and demonstrated higher expectations of students’ abilitiesto construct their own solutions to problems. In general, unlike their pre-PSI teaching, many aspects of their post-PSI teaching were consistent withcurrent reform recommendations like theProfessional Standards(NCTM,1991).

IMPORTANCE OF PERSONAL MEANINGIN FACILITATING CHANGE

The outcome of the study suggests that the PSI approach to teacher devel-opment can produce meaningful and fundamental changes to teachers’personal meanings and teaching. The humanistic context of this approachputs the focus on the teacher and not on a set of skills or teaching tech-niques. In the PSI project, this context was accomplished by focusingon personal meaning. Thus, as implied by the outcome of the study, theway personal meaning is recognized in professional development can beof significant importance in facilitating mathematics teacher change. Thefollowing discussion considers four ways of recognizing personal meaningin professional development based on theory and the PSI project. Thesefour ways, which are interrelated, are also being proposed as key contrib-utors in facilitating the changes in the PSI participants’ thinking andteaching.

First, personal meaning must be seen to have value in terms of Dewey’s(1938) theory of continuity in characterizing experience, and, of partic-ular importance, in terms of Polanyi’s (1958) theory of the personal. If,as Polanyi (1958) argued, personal meaning provides a basis of personalidentity, then to ignore or denounce it in inservice teachers’ development,directly or indirectly (e.g., by prescribing behavior), is demoralizing in thatwhat the teacher has become is being invalidated with no apparently soundreason from the teacher’s perspective. Also, because personal meaning is


self-constructed based on experience (Polanyi, 1958), ignoring it trivializesthe teacher’s ability to create his/her own classroom reality. Such debasingtreatment of personal meaning is likely to stimulate resistance to changeinstead of professional growth.

The PSI participants were considered to be very good teachers intheir school systems. This suggests that they were capable of constructingeffective personal meaning based on the understanding they had of the situ-ations involved. They also had useful knowledge about teaching on whichto build. The open-ended, non-prescriptive nature of the PSI programallowed them to use this knowledge. Thus, the program recognized andvalued them as professionals and as experts of their own classroom reality.With its focus on problem solving as an explicit topic, the PSI programdid not place the teachers in a situation in which they felt that all aspectsof their teaching of mathematics and their credibility as teachers of math-ematics were being challenged. This choice of topic also allowed them tostart rethinking their teaching of mathematics in general without creatinga feeling of having it imposed on them.

A second way of recognizing personal meaning in the professionaldevelopment process is by making teachers aware of it. According toPolanyi (1958), when one uses personal meaning to give meaning to exper-ience, one is aware of it only in a subsidiary manner, i.e., one acts from orwith it while focusing on something else. Thus, personal knowing involves“the pouring of [oneself] into the subsidiary awareness of particulars. . . which compose a whole” (pp. 64-65). The PSI participants demon-strated subsidiary awareness of their personal meanings when, prior to thePSI program, they tried to change their teaching by focusing on prescribedactivities and procedures. The outcome of their efforts did not reflectsignificant changes in their understanding of genuine problem solving,given that all problems were being algorithmitized in their teaching. Theprescribed situations were viewed in familiar ways that reflected their oldpersonal meanings. Thus, their efforts to change their actions did not havea significant effect on their personal meaning. Instead, the outcome of thePSI program supports the humanistic view that teachers’ awareness andunderstanding of their personal meaning are necessary before meaningfuland substantial changes in their behaviors are likely to occur.

A third way of recognizing personal meaning in the professional devel-opment process is through the nature of the experiences created for theteachers. Personal meaning, according to Polanyi, has a tacit component.Thus, “we know more than we can tell” (Polanyi, 1958, p. 55). However,as further explained by Polanyi, we can communicate this knowledgeindirectly if we are given adequate means for expressing ourselves. Thus,

140 OLIVE CHAPMAN

with adequate experiences, the tacit component of personal meaning couldbe modified. If professional development experiences are limited to theacquisition of predetermined concepts and techniques, they are unlikely toinfluence tacit knowledge. Experiential stories, however, implicitly containtacit knowledge (Bruner, 1990; Polkinghorne, 1988), and working withsuch stories would likely provide a more effective basis to modify thisknowledge. The outcome of the PSI program supports this use of stories.The PSI participants’ collaborative narratives seem to have helped themrecognize and modify that which, though active in their behaviors, waspreviously hidden from them.

The PSI program facilitated experiential stories by asking participantsto share and explore common interests, by preventing the participants fromcritiquing or analyzing each other or the stories, and by allowing the partic-ipants to focus on helping each other to articulate personal meaning. Moregenerally, the situations in the PSI program that provided relevant exper-iences in order to recognize personal meaning consisted of (a) concreteand experiential contexts for storytelling, (b) interactions with oneself andwith others of similar and different experiences, (c) observation of self andothers, and (d) a caring and encouraging environment for taking individualand collective risks.

A fourth way of recognizing personal meaning in teachers’ professionaldevelopment is in terms of its connection to the classroom context. Polanyi(1958) explained that we reshape our personal meanings in order to includethe lessons of a new experience. This reshaping occurs tacitly, originatingin our desire for greater clarity and coherence. The initial applicationof reconstructed personal meaning to the classroom would create newexperiences for students which in turn would create new experiences forthe teachers and new lessons from those experiences. Thus, reshaping ofpersonal meaning will likely occur. The PSI study suggests that out-of-classroom professional development activities that support reshaping ofpersonal meaning are likely to encourage further reshaping based on newclassroom experiences in meaningful ways.

The PSI participants encountered challenging and enlightening exper-iences in the classroom in the first term following the PSI program thatcould have had a negating influence on their personal meanings. Forexample, initially, there was opposition from students who wanted to betold “how to do it”, whether they were right or wrong, and to be givendirect responses to their queries. However, students’ initial oppositions didnot force the teachers back to their old teaching approaches. Instead, theteachers found meaningful ways to encourage the students such that, by theend of the first post-PSI school term, students seemed to be very comfort-


able with the problem-solving classes. In fact, during Susan’s post-PSIclassroom observation, her students were observed to model her approachto intervention in dealing with each other, although they had been givenno explicit suggestion from her to do so. For example, when one groupthat had determined a solution to a problem before the others was askedfor help, they responded by giving open-ended clues instead of telling howto get the answer. Thus, any further reshaping of the teachers’ personalmeaning seemed to have maintained the spirit of the PSI program.

CONCLUSION

In relation to the key questions of the study, the outcome has indicated thatthe PSI activities had a significant influence on the elementary teachers’personal meanings and teaching of problem solving. The outcome alsoimplied that explicit recognition of personal meaning in the PSI approachwas a significant factor in facilitating this change. The PSI activitiesallowed the teachers to reconstruct their personal meanings, to reflect amore positive view of themselves as mathematical problem solvers, andto develop an inquiry orientation, instead of the traditional view, of thenature of problem solving and its teaching and learning. The activities alsoallowed them to transform their teaching of problem solving from a show-and-tell model to one that captured the active, social, and constructivenature of the learning process. In general, there were meaningful andfundamental changes to the teachers’ personal meanings and teaching.Based on the nature of the PSI activities, such changes could be facili-tated by a professional development approach that explicitly recognizedpersonal meaning in at least four related ways: collaborative narrativeexperiences, self-understanding of personal meaning, valuing of personalmeaning, and allowing for adaptation of personal meaning to actualclassroom context. This professional development approach was a meansto help teachers understand themselves, rather than to judge them, and tohelp teachers understand practice, rather than to dictate practice to them.Future studies could investigate this approach as a way to facilitate changesin the teaching of mathematics from a problem-solving perspective.

REFERENCES

Bogdon, R., & Taylor, S. (1975).Introduction to qualitative research methods: Aphenomenological approach to social sciences. New York: Wiley.

Bruner, J. (1990).Acts of meaning. Cambridge, MA: Harvard University Press.Bruner, J. (1986).Actual minds, possible worlds. Cambridge, MA: Harvard University

Press.

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Buber, M. (1969).Between man and man. (R.G. Smith, Trans.). London: Collins Press.Carr, D. (1986).Time, narrative and history. Bloomington, IN: Indiana University Press.Carter, K. (1993). The place of story in research on teaching and teacher education.

Educational Researcher, 22(1), 5–12.Chapman, O. (1997). Metaphors in the teaching of mathematical problem solving.

Educational Studies in Mathematics, 32, 201–228.Connelly, M.F. & Clandinin, J.D. (1990). Stories of experience and narrative inquiry.

Educational Researcher, 14(5), 2–14.Cooney, T.J. (1985). A beginning teacher’s view of problem solving.Journal for Research

in Mathematical Education, 16, 324–336.da Ponte, J.P. (1994). Mathematics teachers’ professional knowledge. In J.P. da Ponte &

J.F. Matos (Eds.),Proceedings of PME 18. Lisbon: The University of Lisbon.Dewey, J. (1938).Experience and education. The Kappa Delta Pi Lecture series. New

York: Collier Books.Elbaz, F. (1990). Knowledge and discourse: The evolution of research on teacher thinking.

In C. Day, M. Pope & P. Denicolo (Eds.),Insight into teachers’ thinking and practice(pp. 15–42). New York: The Falmer Press.

Ernest, P. (1989) The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.),Mathematics teaching: The state of the art(pp. 249–254). London: The Falmer Press.

Even, R. & Tirosh, D. (1995). Subject-matter knowledge and knowledge about studentsas sources of teacher presentations of the subject-matter.Educational Studies in Mathe-matics29, 1–20.

Knapp, N. & Peterson, P. (1995). Teachers’ interpretations of “CGI” after four years:Meanings and practices.Journal for Research in Mathematics Education, 26, 40–65.

Lester, F.K. (1994). Musings about mathematical problem-solving research: 1970–1974.Journal for Research in Mathematics Education, 25, 660–675.

Lester, F.K. (1985). Methodological considerations in research in mathematical problemsolving instruction. In E.A. Silver (Ed.),Teaching and learning mathematical problemsolving: Multiple research perspective(pp. 41–69). London: Lawrence Erlbaum Associ-ates.

Mishler, E. (1986).Research interviewing. Cambridge, MA: Harvard University Press.National Council of Teachers of Mathematics (1989).Curriculum and evaluation stan-

dards for school mathematics. Reston, VA: Author.National Council of Teachers of Mathematics (1991).Professional standards for teaching

mathematics. Reston, VA: Author.Polkinghorne, D. (1988).Narrative knowing and the human sciences. Albany, NY: State

University of New York Press.Polanyi, M. (1958).Personal knowledge. Chicago: The University of Chicago Press.Polanyi, M. (1975).Meaning. Chicago: The University of Chicago Press.Simon, M. & Schifter, D. (1991). Towards a constructivist perspective: An intervention

study of mathematics teacher development.Educational Studies in Mathematics, 22,309–332.

Thompson, A. (1984). The relation of teachers’ conceptions of mathematics and teachingto instructional practice.Educational Studies in Mathematics, 15, 105–127.

Wood. T., Cobb, P. & Yackel. E. (1991). Change in teaching mathematics: A case study.American Educational Research Journal, 28, 578–616.

The University of Calgary

ALICE F. ARTZT

A STRUCTURE TO ENABLE PRESERVICE TEACHERS OFMATHEMATICS TO REFLECT ON THEIR TEACHING

ABSTRACT. This article presents a conceptual framework for studying the relation-ship between cognition and instructional practices of preservice secondary mathematicsteachers. It describes how the framework was used as a basis for activities in which pre-service teachers engaged in structured reflection on their teaching as a means towards theirprofessional growth. The approach required student teachers to engage in both prelessonand postlesson reflective activities. These activities are described, and details of two casesare given. This article demonstrates how this approach can facilitate the progression ofpreservice teachers’ pedagogical techniques and conceptions.

In his discussion of the current state and future directions of researchon teacher education, Cooney (1994) acknowledged the challenge ofpreparing future mathematics teachers. He raised the question of whattypes of experiences preservice teachers would need in order to becomeeffective teachers of mathematics. Research suggests that teacher reflectionis central for the improvement of mathematics teaching (Jaworski, 1994;Kemmis, 1985; Schön, 1983). Furthermore, within the last two decades,researchers have emphasized the importance of teacher cognition as theyhave recognized a well-defined link between teachers’ cognition and theirinstructional practices (Artzt & Armour-Thomas, 1998; Brown & Baird,1993; Ernest, 1988; Lappan & Theule-Lubienski, 1994; Shavelson, 1986;Shulman, 1986). In addition, projects that have resulted in change in bothteachers’ cognition and instructional practice have placed emphasis on theteachers’ experiences as a focus for reflection (Cooney & Shealy, 1997).Such research suggests that teacher education programs should includestrategies and activities that engage teachers in reflection on their ownthinking and on their instructional practice.

Researchers who have described changes in mathematics teachers’beliefs and practices (e.g., Cooney, Shealy, & Arvold, 1998; Fennema,Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Schifter & Simon,1992; Schram, Wilcox, Lappan, & Lanier, 1989; Thompson, 1991) agreethat there may be several developmental stages of teaching. For example,the initial stage can be characterized by traditional instruction: The teacheris driven by the belief that students learn best by receiving clear infor-


144 ALICE F. ARTZT

mation transmitted by a knowledgeable teacher. Subsequent stages ofinstruction are characterized by instruction that is more focused on helpingstudents build on what they understand and less focused on helping themsolely in the acquisition of facts. The instruction is grounded in theteacher’s belief that students should take greater responsibility for theirown learning. The final stage can be characterized by instruction in whichthe teacher arranges activities that involve both thehows and whys ofmathematical concepts and processes. The teacher is motivated by thebelief that, given appropriate settings, students are capable of constructingdeep and full mathematical understanding.

Goldsmith and Schifter (1997) suggested that motivational and indi-vidual dispositional factors may significantly affect the course of ateacher’s development. It is possible to examine the dispositions andmotivations of preservice teachers through writing assignments. Goldsmithand Schifter supported the use of writing, in that it is one possible socio-cultural transition mechanism that has the potential to enable the growthfrom one stage of teaching to the next. Writing, they suggested, helps one“to hold an idea or experience still for reflection” (p. 43). This form ofwriting played a significant role in the project presented in this article.

This article describes a model of structured reflection intended topromote reflective behavior of preservice secondary school mathematicsteachers. As part of this process the preservice teachers had to write aboutspecific elements of their thoughts and their practices and about the rela-tionship between both. I examine how this process of reflection served as ameans for facilitating their development from one stage of teaching to thenext.

RELATIONSHIP OF TEACHER COGNITION ANDINSTRUCTIONAL PRACTICE

Recent research that uses a conception of teaching as problem solving hasbegun to indicate that teachers’ cognition drives their instructional practice(Artzt & Armour-Thomas, 1993; Carpenter, 1989; Fennema, Carpenter, &Peterson, 1989). The method presented here for studying teacher cognitionis based on a framework that was developed by Artzt and Armour-Thomas(1996). This framework represents one way to view teaching as an inte-grated whole in which cognition plays a well-defined role in instruction(see Figure 1).

Specifically, the framework suggests that teachers’ knowledge, beliefs,and goals directly impact their instructional practice. The different factorsaffect the nature and quality of teachers’ thoughts and actions before,

A STRUCTURE FOR TEACHER REFLECTION 145

Figure 1. Framework for reflection on teacher cognition and instructional practice.

during, and after their lessons. They affect the way teachers design alesson, the way they monitor and regulate their instruction during theteaching process, and the way they analyze the lesson after it has beenconcluded. The framework served as a conceptual basis for the work withthe preservice teachers. A more detailed description of the frameworkfollows.

The framework presented in Figure 1 outlines the relationship betweenteachers’ mental activities and their lessons. Teachers’ knowledge, beliefs,and goals appear to drive their instructional practice.Teacher knowl-edge includes knowledge of subject matter, knowledge of pedagogicalstrategies, and knowledge of the students (Shulman, 1986). These compo-nents of teacher knowledge can affect instructional practice and studentlearning (Ball, 1991; Fennema & Franke, 1992; Peterson, 1988).Beliefsrefers to teachers’ integrated system of personalized assumptions regardingthe nature of mathematics, of students, and of ways of learning andteaching. Syntheses of the existing literature on beliefs by Ernest (1988),Kagan (1992), Pajares (1992) and Thompson (1992) suggest that teachers’

146 ALICE F. ARTZT

beliefs influence their instructional practice.Goals refers to teachers’expectations about the intellectual, social, and emotional outcomes forstudents as a result of their classroom experiences. TheStandards(NCTM,1989, 1991) have set forth goals for all students which are expected to bereflected in instructional practice: that students value mathematics, becomeconfident in their ability to do mathematics, become mathematical problemsolvers, learn to communicate mathematically, and learn to reason math-ematically (NCTM, 1989, p. 5). The goal of teaching for conceptual as wellas procedural knowledge has been addressed by Hiebert (1986), and Silver(1986), as well as in recent reform initiatives (MSEB & NRC, 1991).

Jackson’s (1968) distinctions of preactive, interactive, and postactivestages of teaching are useful for examining teacher cognition before,during and after teaching a lesson. Clark and Peterson (1986) andShavelson and Stern (1981) have done comprehensive reviews of researchon teacher thought processes. The major components that appear to impactinstructional practice are (a) planning during the preactive stage (Clark& Elmore, 1981; Clark & Yinger, 1979), (b) monitoring and regulatingduring the interactive stage (Clark & Peterson, 1981; Fogarty, Wang, &Creek, 1983), and (c) evaluating and revising during the postactive stage(Ross, 1989; Simmons, Sparks, Starko, Pasch, Colton, & Grinberg, 1989).As Shavelson (1986) pointed out, however, these aspects of thinking arenot conceptually distinct, but rather interconnected components of theteaching process.

Findings from these studies suggest that the cognitive components ofteaching play a critical role in shaping a teacher’s instructional prac-tice. There are many lenses through which instructional practice can bestudied. The perspective described below was developed by Artzt andArmour-Thomas (1996) and was used to examine the instructional prac-tice of secondary school mathematics teachers. It is grounded in theaforementioned literature.

A STRUCTURE FOR REFLECTION

The framework provided the conceptual basis for the approaches used withpreservice teachers during their entire year-long program of preparatorystudy. That is, based on the fact that preservice teachers have observedand participated in the teaching and learning process as students for mostof their lives, the program built on the teachers’ existing knowledge andbeliefs. The methods course focused largely on these existing systemsof knowledge and attitudes as well as on the current goals of secondarymathematics instruction as outlined in the Standards (NCTM, 1989, 1991).


This focus served as a basis for the teachers’ reflective activities duringthe following semester of student teaching. During the student teaching,students were required to teach at least one class per day in a secondaryschool. In addition to being supervised by cooperating teachers in thefield who worked with them on a daily basis, the student teachers wereobserved by their college supervisors four times during the course of thesemester. The supervisors analyzed and assessed the student teachers usinga structure based on the framework. Prelesson and postlesson reflectiveactivities were used to facilitate preservice teachers’ analysis of theircognition and instructional practice before, during, and after their lessons.Specifically, before teaching their lessons, in addition to writing lessonplans, the student teachers were required to submit a paper in which theydescribed their prelesson thoughts and concerns. After the lesson, thestudent teachers engaged in a conference with the supervisor and cooper-ating teacher in which the student teacher gave his or her impressions andanalysis of the lesson. During the latter part of the conference, the cooper-ating teacher and supervisor shared their impressions of the lesson andgave suggestions for further thought and improvement. After the confer-ence, the student teachers were required to write a paper describing theirpostlesson thoughts. This paper was submitted to the supervisor at the nextclass meeting. In addition, student teachers documented their thoughtsand experiences through weekly entries in their journals. Each of theseactivities was structured in a way that was consistent with the conceptualframework previously described. A detailed description of these reflectiveactivities follows.

Prelesson Reflections

In order to help student teachers formulate their broad goals and activatetheir knowledge they were required to provide a written account of theirprelesson thoughts. This reflective writing assignment was in addition tothe actual lesson plan. The following instructions served as a guide for theprelesson reflection:

As you begin to think about constructing your lesson plan, write down all of your concernsand the steps you are taking to account for these concerns. Use the list below as a guide:

1. Goals for students2. Knowledge of students (e.g., ability levels, interests)3. Knowledge of content (e.g., its place in the curriculum)4. Knowledge of pedagogy (e.g., alternate ways of teaching the lesson)5. The teacher’s role in the lesson6. The students’ role in the lesson7. Anticipated difficulties8. Sources used to get ideas and criteria for selection

148 ALICE F. ARTZT

The purpose of this list was to make explicit to the student teacher andthe supervisor the knowledge, beliefs, and goals that drive the design andimplementation of a lesson. The hope was that the suggested points of theprelesson reflection could help the student teachers advance more quicklyfrom the initial stage of teaching with content as the main focus to a morestudent-focused stage. In the first item on the list, the student teachers wereasked to consider their goals in terms of students only. Because my experi-ence suggests that beginning teachers’ overriding concern is often contentcoverage, my intention here was to convey the message that contentcoverage has value only in relation to student understanding. The nextthree items on the list (2–4) were designed to encourage student teachersto activate the knowledge they had about the students, the content, andpedagogical strategies as a means for informing their lesson design. Thenext three items on the list (5–7) were meant to help student teachers envi-sion the engagement and interaction of both the teacher and the studentsduring the lesson and to anticipate and plan for difficulties that might arise.Item 8 was designed to encourage the student teachers to use differentresources as a means for increasing their knowledge and expanding theiralternatives with respect to the design of the lesson.

Although beliefs are one of the key components in the model, they werenot directly addressed in the student teachers’ assignment for prelessonthoughts. There was an important and interesting reason for this. Afterextensive study of the NCTM Standards and other contemporary mate-rials in the methods course that preceded student teaching, the studentshad a clear idea of the latest philosophies that were valued for mathe-matics instruction. By this time in the year, if asked about their beliefsregarding mathematics instruction, the student teachers knew what theirinstructors wanted to hear and were very adept in providing it. Artztand Armour-Thomas (1998) suggested that actions are a more reliablereflection of teachers’ beliefs than are words. Specifically, they found thatseveral teachers gave lip service to their beliefs about student-centeredinstruction, yet contradicted these ideas during their instructional practice.In fact, when interviewed while they viewed the videotape of their lesson,they often contradicted their prelesson espoused beliefs in order to justifytheir action during the lesson. Therefore, the supervisor placed a greatervalue on the beliefs expressed by student teachersafter their lessons asthey reflected on their practice than on beliefs they would describebeforethey taught their lesson. Thus, the discussion about beliefs was postponeduntil the lessons were concluded and the teachers were asked to justifytheir actions.


The descriptions of their prelesson thoughts provided opportunities forthe student teachers to activate and make explicit their knowledge andgoals that drove their lessons. In addition, the supervisor, who observedthe lesson, gained insight into the decision making and reasoning that wasbehind the lesson. The knowledge of the student teachers’ reasons for whatthey did made it easier for the supervisors to understand and assess theinstruction they observed.

Postlesson Reflection

During the postlesson conference, student teachers reflected on theirinstructional practice as well as on their thoughts and decision-makingrelated to that practice. Following the conference, the student teacherswere required to submit both a written analysis of their lesson and adescription of their thinking about the lesson. Details of this method ofpostlesson reflection follow.

Self assessment.After the student teacher had completed his or her lesson,time was set aside for a conference with the supervisor. At that time,the student teacher was called upon to reflect on and assess the lesson.The supervisor encouraged the student teacher to do all of the talking,but helped structure the student teacher’s thoughts by asking questions.Specifically, the student teachers were first asked to examine the relation-ship of their prelesson thoughts, their lesson plans, and their teaching. Forexample, they were asked to recall their original goals for the lesson, howthese goals were addressed in the lesson plan, and the extent to which theybelieved they had accomplished the goals. They were asked to comparewhat they planned to do in the lesson with what they actually did. Second,the supervisor asked the student teachers to explain the decisions that influ-enced them to deviate or not deviate from their original plans. That is,they were asked to explain their monitoring and regulating actions duringthe lesson. For example, they were asked to describe what feedback theyreceived from students and how this feedback informed their practice.

In some cases, student teachers were then asked to further assess theirlesson with a focus on the three elements of tasks, learning environment,and discourse. They were asked to think about, describe, and evaluate thenature of these elements as they unfolded in their classrooms. They wereasked to consider factors that might have influenced their teaching. Finally,they were asked to consider suggestions they might have had for improvingthe lesson if they were to teach it again.

150 ALICE F. ARTZT

TABLE I

Sources for data collection

Time Activity

Before the lesson Prelesson activities:

Written prelesson thoughts

Written lesson plan

During the lesson Enactment of the lesson

After the lesson Postlesson activities:

Self assessment at conference

Exchange of Ideas with supervisor at conference

Written postlesson thoughts

Ongoing Journal entries

Supervisor feedback.After the student teachers completed their ownpostlesson reflections, the supervisor shared her thoughts regarding thelesson. It was at this time that the supervisor could request clarificationof issues and help the student teachers think about what new knowledgethey had gained or beliefs that may have changed as a result of teachingthe lesson. The supervisor asked such questions as: As a result of teachingthis lesson what have you learned? What new ideas did you learn aboutthe content, the students, or best ways of teaching the lesson? What beliefsdid you have about the content, the students, or best ways of teaching thelesson that changed in some way?

Postlesson thoughts.After the conference with the supervisor the studentteachers were required to write about their postlesson thoughts regardingthe lesson and the ideas discussed during the postlesson conference. Theywere asked to write about the strong points and the weak points of theirlesson. They were asked to suggest how the lesson might have beenimproved. This written evaluation served at least two purposes. First, itgave the student teachers the opportunity to reflect on the lesson once againin light of all they and the supervisor had said. Second, it gave the super-visor the opportunity to assess what the student teachers had learned fromtheir experience and whether they had internalized any of the ideas thatwere discussed. Table I summarizes the above data collection activities.

Examples will be given of the prelesson reflections, the instructionalpractice, the postlesson reflections, and the journal entries of two preser-vice teachers, Ms. Carol and Mr. Wong. These two students were selected


to exemplify how the structure was used in contrasting situations. Althoughboth were novice teachers, each began student teaching with a differentdisposition which seemed to account for their different progressionsthrough the course of the semester. With each example a discussion willfollow regarding how these reflective activities informed both the teachereducator and the preservice teachers about aspects of professional growth.

THE REFLECTIVE STRUCTURE IN ACTION

Ms. Carol

Ms. Carol had been an accountant who had decided to make a careerchange into teaching. Unfortunately, during the methods course hercooperating teacher would ask her to teach his class five minutes beforethe class started. She felt obligated to say yes, despite the fact that shecertainly was not required to do so. This on-the-spot teaching resultedin several upsetting experiences. She found herself completely “flusteredand unable to teach.” Therefore, at the beginning of the student teachingsemester she revealed that she was insecure about her teaching abilitiesand was almost ready to change her mind about entering the teachingprofession. She questioned her knowledge about mathematics and her ownbeliefs about whether she really wanted to be a teacher. She wrote:

I spent the entire Christmas break stressing about student teaching. All I know is that Ireally don’t want to student teach, and I am having second thoughts about teaching ingeneral. My self-esteem is at an all-time low, and I am more depressed than I can begin toexpress. I dread coming to school every day. I feel I have no strengths as a teacher. I don’tknow the progression of the curriculum. I don’t know the precise definitions of things.(February, Journal Entry)

Through this writing assignment the supervisor was able to see Ms.Carol’s low self-esteem. This low self-esteem may have created the moti-vation that Goldsmith and Schifter (1997) suggest is an important factorfor progressing through the stages of teaching. In fact, as will be shownfrom the excerpts of her work, Ms. Carol did progress rapidly.

During the course of the semester, Ms. Carol learned that by explicitlyformulating her goals and giving written explanations of what she knewabout the content, the students, and methods of pedagogy, she could planand implement better lessons. It appeared that by reflecting on her lessonsin a structured manner she learned how to improve on her own teaching.In the following lesson, which she taught toward the end of the student-teaching semester, we can see how Ms. Carol had progressed well beyondthe initial stage of teaching. Her student-centered focus is evident in her

152 ALICE F. ARTZT

written expressions as well as in her instructional practice as evidenced inher planning, the tasks she designed, the learning environment she created,and the discourse she orchestrated.

Prelesson reflection.Ms. Carol’s lesson involved a review of graphingsystems of linear equations and systems of inequalities. In her prelessonthoughts Ms. Carol was asked to state the broad goals she had for herstudents. She wrote that her goals were to get her students to “really thinkabout what they’re doing and understand the differences between graphinga system of equations and [a] system of inequalities.”

The prelesson activity also prompted Ms. Carol to activate and makeexplicit her knowledge regarding the nature of her students’ level ofunderstanding. She wrote:

The students seem to have a lot of trouble with graphing in general. They don’t seemto understand the underlying concepts of what the graph of an inequality, equation, orline really represents mathematically. I had noticed the day before that when the studentswere asked to label the solution set to a system of inequalities, they marked the point ofintersection, as if they were asked to find the solution to a system of equations. They woulddo the shading, but it meant nothing to them. They don’t understand why the solution to asystem of equations is (generally) one point, and the solution to a system of inequalities is(generally) a region consisting of many points. I wanted them to have a little more practicewith both types of graph systems, and then be able to sum up the differences at the end.(May, Prelesson Thoughts)

Unlike an initial stage teacher, Ms. Carol was not satisfied withstudents’ procedural knowledge alone. She was not satisfied that studentscould complete the shading, because she sensed underlying conceptualweaknesses. The lesson she designed was student-centered. She createda learning environment in which students were encouraged to interact withone another as they worked in small groups to complete four problems(see Appendix 1). Two of the problems involved systems of inequal-ities; one problem involved a system of equations; and a fourth problemasked students to highlight the differences between graphing a system ofequations and a system of inequalities.

Because Ms. Carol had recorded her anticipated difficulties, therebyactivating her knowledge about the students, she arrived at a plan for anappropriate instructional strategy. She said:

They [the students] have the tendency not to think, and the tendency to question them-selves to the point where they’re constantly raising their hands to ask, ‘Is this right?’ I’mhoping that if they work together, they’ll askeach otherif they’re ‘right.’ (May, PrelessonThoughts)

Because she used a small-group learning environment it appeared thatMs. Carol believed in the value of students sharing ideas and relying on


each other for feedback. She was also willing to try new techniques, asneither she nor her cooperating teacher had ever used small groups.

Instructional practice.Ms. Carol began her class with an interestingproblem in which students were shown a graph of a system of inequalitiesand were asked to discover the inequalities that comprised the system.This was the reverse of most problems, in which students are given thesystem of inequalities and are asked to draw the graphs. As the studentsworked on the problem, Ms. Carol circulated around the room encouragingsome students to begin working and observing how other students wereapproaching the problem. One student agreed to come to the front of theroom to show his solution and to answer any questions posed by otherstudents. Most students appeared to be on task and involved in the work.When students made incorrect comments, Ms. Carol did not interfere,but rather allowed the students to debate the issue until they resolved themisunderstandings.

Students were then given a worksheet that consisted of the four prob-lems. In her directions to the class, Ms. Carol explained that a member ofeach group would be randomly selected to explain his or her work to therest of the class. As the groups began working, Ms. Carol noticed that therewas not enough time for each group to complete all four problems anddecided to let each group work on one problem only. Some confusion aroseas she changed the directions and assigned a single problem to each group.Because the students had already begun drawing their graphs on graphpaper, further confusion was created when she distributed one transparencyto each group on which they were to draw their graphs for presentation tothe class on the overhead projector. As the students settled into their groupwork, Ms. Carol walked around overseeing what the students were doing.Occasionally the students would ask her a question, but rather than answerit herself, she referred the student to the other members of the group.

With only ten minutes remaining in the class, one student was randomlyselected to describe her group’s work and display the transparency onthe overhead projector. Ms. Carol allowed that student, as well as othermembers of her group, to field questions from the class. With only oneminute remaining, Ms. Carol engaged the students in a brief summarydiscussion of the differences between a system of inequalities and a systemof equations.

Although far from perfect, the nature of this instructional practicewas well beyond that of a teacher in his or her initial stage of teaching.The student focus, evident in the prelesson reflection, was evident in thelesson as well. By allowing students the opportunity to work in groups

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and requiring them to answer each others’ questions, she encouraged themto take responsibility for their own learning.

Postlesson reflection.In her postlesson conference, Ms. Carol was askedto reflect on and assess her lesson. Specifically, she was asked to reviewher prelesson comments to see how they might have accounted for thenature of her lesson enactment. For example, Ms. Carol was asked todetermine whether she had accomplished her goals: that students arriveat an understanding of the concepts and rely on each other for feedback.She noted that she had allowed students to explain their group’s work atthe board and that, when they did or said something incorrect, she hadwaited for other students to make the correction rather than commentingon it herself. She noted how she had walked around the room and listenedto what students were saying to one another and that when they asked hera question, rather than answering it herself, she referred the question backto the group. She also pointed out the conceptual questions she asked thestudents: “Where would you find a point that satisfies the first inequalitybut not the second? How could you change the system of equations so thatthe solution set is empty? How could you change the system of inequalitiesso that the solution set is empty?” However, she acknowledged that shewas unsure as to how much conceptual understanding her students haddeveloped because only a few students had responded. She also ran outof time and never reviewed the last problem that asked for a synthesis offindings.

Ms. Carol was then asked to account for the difference between whatshe planned to accomplish in the lesson with what actually took place. Ms.Carol’s immediate reply was that she ran out of time. When asked why thathappened she explained how both the tasks she designed and the instruc-tional strategy she used prevented the class from getting to a discussion ofthe last, most important problem. When asked what she learned as a resultof teaching this lesson, Ms. Carol replied:

By wandering the classroom I got to see the things that really trouble students; aspectsof mathematics that are completely obvious to teachers are not necessarily comprehendedso easily by the students. I learned that instructions that are clear to me might not beclear to them. For one thing, I said “work together,” but my definition of “work together”isn’t necessarily theirs. I also thought that saying something once was enough, but theysometimes need constant reminders. Also, their tasks not only have to be meaningful, theyhave to be more structured. (May, Postlesson Conference)

In her written postlesson analysis she wrote:

The strong point of the lesson was that most of the students eventually got to workingtogether and arguing their ideas. They got the chance to explain their ideas at the overhead,


using the transparencies enabled the students to get an accurate description of what wasgoing on. I asked questions that really made the students think. But since that was a goal ofmine, I should have put fewer graphing examples on the worksheet so the students coulddo all of the problems and then spend the majority of the time working on the thoughtquestions. I think that with a few minor adjustments, this lesson could have been a powerfulone for the students. I really like the cooperative learning aspect. The period goes by muchfaster, I talk less, and the students get more out of it. (May, Postlesson Thoughts)

The postlesson reflection prompted Ms. Carol to analyze the lessonin light of her original goals for students and her beliefs about the roleher students should play in their own learning. From her statements, itis clear that she was able to make the connection between the elementsof her lesson and their role in helping or impeding her efforts to accom-plish her goals for students. It is also clear that she used the studentsas her barometer for determining her level of goal accomplishment. Herthoughtful, coherent, and logical comments suggest that Ms. Carol hasthe ability to assess her lesson carefully and critically and to generateconstructive ideas for revision. These self-reflective abilities suggest thatMs. Carol has the tools for continuing her professional growth after thesupervisor has left the scene.

More importantly, Ms. Carol was able to give an insightful evaluation ofher own competence as a teacher. This was evident in Ms. Carol’s final self-evaluation at the completion of the student teaching semester. As suggestedbelow, Ms. Carol had learned that the ingredients for success include bothher ability to reflect and her underlying beliefs about students and how theylearn. She wrote:

As far as evaluating my ability to teach at the present time, I can only say that I’m betternow than I was when I started. The only thing that makes me think that I will be a goodteacher is that I can see what it is that I’m doing wrong, and I am really concerned if thestudents are gaining conceptual understanding of the underlying mathematics of a problem.I respect their thoughts, and I wait and expect them to explain themselves fully. I insist thatthey work together and listen to each other. I value the importance of getting them touse reason and speak like mathematicians. I put a lot of work into my lessons so that thestudents will benefit and get the most out of it. Most importantly, I really care about them.I want to see them all do well and appreciate mathematics. (June, Journal Entry)

Mr. Wong

Mr. Wong had been a computer programmer who, like Ms. Carol, hadmade a career change into teaching. His beliefs at the beginning of thesemester were characteristic of a first stage teacher and quite contrary towhat was being promoted in class. In his journal at the end of the studentteaching semester he wrote:

156 ALICE F. ARTZT

At the beginning of the student-teaching semester, my perception of teaching mathematicswas rather naïve. I thought that all you needed to do was to think of an interesting problemto get the students motivated in the beginning. Then, just present the lesson pretty much asthe textbook has it written. I was imagining that the students were attentive and ask ques-tions. I love to deal with children and I love to be the one to provide the correct answers.I had the image of me just teaching and talking for the entire period, like a preacher, withthe students absorbing all the knowledge that I impart to them. (June, Journal Entry)

In contrast to Ms. Carol, Mr. Wong’s strong beliefs about the “right”way to teach left him inflexible and unmotivated to learn new instructionalapproaches. According to Goldsmith and Schifter (1997) someone whosedisposition is so contrary to the kind of teaching envisioned by the reformmovement may need extra support in changing teaching practice. Inthis regard, the structure for reflection was particularly useful since itchallenged Mr. Wong to reexamine many of his beliefs about his role as ateacher and his students’ roles as learners. In the following example wewill see how Mr. Wong was encouraged to take on a more student-centeredfocus by gaining more knowledge about his students and the content ofhis lesson.

Prelesson reflection.Mr. Wong’s lesson involved probabilities ofcompound events. Contrary to Ms. Carol’s detailed comments about whatshe wanted her students to understand and how she wanted them to beactively involved in that process, Mr. Wong tended to speak in generalterms, reflecting a lack of knowledge regarding both the content and hisstudents. He wrote:

The goal in this lesson is to help the students to understand the concept of compoundevents, independent events, the counting principle and finally how to use the aforemen-tioned knowledge to compute the probability of compound events. The students hadreviewed some of the basics of probability during the previous lesson. In that lesson, thestudents reviewed the definition of outcome, sample space, and event. They also reviewedthe basic rules of probability.

Based on the experience of the previous class, there is, in general, a fair level of interestamong the students for the subject matter. Maybe because they were exposed to this matterin their last course, the students seem to be able to pick up on the material without toomuch difficulty. (March, Prelesson Thoughts)

Although Mr. Wong made mention of wanting his students to under-stand the material, he gave no indication of what aspects they might finddifficult nor how he would render it understandable to them. He suggestedthat students did not find the material particularly difficult. Given that prob-ability is generally considered a very difficult topic for students, suspicionarises that Mr. Wong has not been monitoring his students’ understandingvery well, if at all. Furthermore, although Mr. Wong mentioned several


topics, he never discussed them in any detail. His lack of student focusand apparent lack of awareness of the trouble spots entailed in teachingsuggested that he was still at the first stage of teaching. The supervisor usedMr. Wong’s prelesson thoughts to point out to him in a tangible way that heneeded to increase his knowledge of the content, of his students’ intuitiveknowledge of the content, and of ways of helping students understand thecontent.

Mr. Wong’s lesson plan consisted of an interesting and potentiallymotivating problem which required students to take a multiple choice quizcontaining the following questions: What is Neil Armstrong’s Birthday?How long is the Nile River?, and What is the diameter of Jupiter? Thequestions had 3, 4, and 2 choices, respectively, and the choices were closein value. In his postlesson thoughts Mr. Wong admitted that he had outlineda tight script for his lesson. He wrote:

I was going to give them the quiz and then prod them to recognize the compound eventsby asking them if they noticed any difference between what they did yesterday (rolling adice and spinning the spinner) and what they are doing now. I was expecting the studentsto come up with the correct answer – compound events. I would then write the “AIM.”Following the AIM, I then planned to ask the class what was needed to find the probabilityof getting all three questions correct in the quiz. I was expecting the class to reply that weneeded to know the sample space. This would have provided the opening to investigate thesample spaces of this activity. The sample space investigation was to be a tree diagram toprovide the necessary clarity. This was to be followed by asking the students for an easierway of getting the sample space – the counting principle. After establishing the countingprinciple, the plan was to then compute the probability of getting all three quiz questionscorrect by knowing the number of elements of the compound event or the number of waysthat the compound event can take place. As a finale to the lesson, I was going to cover theCounting Principle with Probabilities by asking the students if there is an easier way tocompute the probability of the above compound event. (March, Postlesson Thoughts)

Based on what he wrote, it was not surprising that the lesson enactmentwas problematic. That is, Mr. Wong’s focus was mostly on whathe wasgoing to do, with specific expectations for what the students were goingto say. This rigid disposition precluded the flexibility and explorationcharacteristic of later stages of teaching.

Instructional practice.The lesson did not proceed as Mr. Wong had envi-sioned. The first problem he encountered was with the discourse. Afterthe students took the quiz, one boy yelled out the correct answer to theprobability question, “What is the probability that you will get all of themultiple choice questions correct?” which Mr. Wong had not yet asked. Mr.Wong told the student that he was correct, and then he was at a loss for whatto do. For the remainder of the lesson he lectured, giving the definition ofa compound event and writing out a tree diagram of the sample space for

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the students. The students were passively taking notes and the potentialmotivation of the interesting task was all but lost.

Mr. Wong’s inflexible disposition and limited knowledge impeded hisefforts to salvage the lesson and presented a challenge for the supervisor tohelp him understand the nature of his difficulties. Fortunately, Mr. Wong’spostlesson reflection activities seemed to help him focus in a systematicand objective way on how and whether he was indeed facilitating hisstudents’ understanding of mathematics.

Postlesson reflection.In his postlesson conference, Mr. Wong was askedto review his original goals for the lesson and determine whether they hadbeen realized. When he stated that he covered all of the material he hadplanned, he was asked whether he “helped the students understand” theconcepts as he had stated in his original goals. He said he had tried butwasn’t sure whether the students understood or not. He was asked why hedidn’t know, and he said, “The class was quiet. It was just my voice.” Hewas then asked about his beliefs regarding the best ways to help studentsunderstand. Did he think it was by explaining everything to them as he haddone? Through this questioning Mr. Wong began to recognize that to reallyhelp students understand and be able to know if they understood therewould have to be more informative discourse in the class. He admittedthat he would have to “ask more probing questions,” that he would haveto “hear students explain their answers,” and that students would need to“respond to other students’ questions.” Mr. Wong was then asked to recon-sider his knowledge regarding students. He was asked if he still thoughtthe students did not have much difficulty with the topic, as he had statedin his prelesson thoughts. He stated again that the class was very quiet. Hewas asked to consider the reason for their silence. Mr. Wong was askedto consider the following questions: Did the fact that one student knew theanswer to the unasked question at the beginning of the class mean that eachof the students in the class knew the answer as well? Even if they knew theanswer, did it mean that they understood why that was the answer? Perhapsit was a lucky guess? How would the teacher know if it was a lucky guess?

Mr. Wong was also asked to examine the nature of the task he haddesigned. He thought the quiz was very motivational but realized that themotivation ended when the student gave the probability. He was asked howhe thought the motivation might have been maintained. How could the taskbe sustained in such a way that the other concepts such as sample spacecould be developed? Because the class had been so quiet, he realized thatthe learning environment had been awkward. He was asked how he mighthave actively engaged the students. After a lengthy discussion, Mr. Wong


suggested that he might have recorded each of the students’ answers on thetest. This would have helped in developing the idea of sample space and atthe same time engaged each of the students.

After considerable discussion, Mr. Wong was asked what he hadlearned as a result of teaching this lesson. Had any of his beliefs changed?He said that he realized that “a good problem is not enough to motivatestudents.” He expressed the remainder of his thoughts in his postlessonreflective activity. The following comments suggest that Mr. Wong hadlearned a great deal.

One of the major problems with my lesson was my inability to capitalize on the make-believe quiz to motivate the class to learn the subject matter. My approach was too dry andtoo inflexible. The class was not motivated. Some students were heard asking what wasgoing on. I was too anxious to get the word “compound event” out of the students’ mouthso I could write the AIM. Why not stir up enthusiasm by asking the class what is theprobability that anyone in the class has answered all three questions correctly? Challengethe class to determine how many students answered all three questions correctly. Thisshould lead naturally into the examination of sample space of the activity. When I did askthat question on sample space, one student answered correctly (not understanding why,however), and I applauded him for the “correct” answer and then went on to obtain thesample space by doing the tree diagram. I DID ALL THE EXPLANATION, AND ALLTHE TALKING. Big mistake! I should have used this opportunity to assess the studentsas to their understanding of sample space. Instead of applauding the student’s answer, Ishould have asked why and how he arrived at the answer. (March, Postlesson Thoughts)

This reflective activity appeared to enable him to systematicallyexamine all elements of his lesson: the tasks, the learning environment,and the discourse.

Throughout the semester, Mr. Wang was called on to constantlyexamine his teacher-centered approach but it seemed difficult for him toleave the security of lecture-driven lessons. He spent the entire semestergrappling with his beliefs about how students learn best. He was affectedby his own positive experiences in China with learning through listening.Only after extensive experience with examining the learning environmentand discourse that took place in his class did he begin to get the feelingthat his students were not paying attention, much less learning, while heengaged in “chalk and talk.” Through the reflective activities he beganto realize that he was not monitoring student understanding sufficiently,which was one of the reasons that he had rarely regulated his instructionalpractice to fit the needs of his students. In his final self-evaluation at thecompletion of the student teaching semester he wrote:

I must get rid of my old habit of tightly following the script of my lesson plan. It is a habitthat I still follow in the classroom setting although I am much more aware of it now thanbefore. In a classroom setting, I have got to learn to use the lesson plan only as a guide. I seethat I am not assessing the understanding of the students continuously. Interestingly, being

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aware of my bad habits, I tried to observe my tutoring technique for the past few days.What I have noticed is that I do not make many of the important mistakes as I have donein the classroom setting. During tutoring, I asked a lot of questions to assess the learninglevel of the students, and I tended to allow the students to learn by prodding them withquestions instead of just “teach” by talking away. (June, Journal Entry)

Judging from Mr. Wong’s comments about his tutoring, the super-visor got an indication that his comments were authentic and that thereflective activities facilitated his efforts to examine and question the valueof his teacher-centered approach to instruction. The structure for reflec-tion helped serve as a mechanism through which he could dislodge someof his inflexible attitudes and approaches and begin to progress in hisdevelopment.

FINAL REMARKS

Working with student teachers is a challenging and often mysterioustask. I have often found myself frustrated by my inability to help preser-vice teachers develop as I wished. I have often sat in the back of theirclassrooms bewildered by the things I saw. I used to ask myself whetherthese students had really attended my classes for the previous six months.When interacting with them I focused on their instructional practice andaddressed the issues that I felt needed attention. I tried to hide my frustra-tions to protect their fragile egos and often left the conference disturbed.Since I used the approach described herein, much has changed. I do notmean to claim that my students are now all final stage teachers. What I doclaim is that, by getting into the minds of my students in a structured way,I am better able to make sense of what they do and am therefore betterable to help them progress. Additionally, I have the impression that bybeing required to probe, express, examine, and question their own thinkingprocesses, my students are better able to understand and improve theirinstructional practices.

Goldsmith and Shifter (1997) suggested that teacher change is charac-terized by (a) qualitative reorganizations of understanding, (b) orderlyprogressions of stages, (c) transition mechanisms, and (d) motivationaland dispositional factors. The approach presented in this article facilitatesteacher change by engaging students in activities that served as transitionmechanisms for professional growth. It encouraged preservice teachersto reveal their motivations and dispositions and to organize their under-standings of the relationship between their thoughts and their instructionalpractice. When students engaged in thinking and writing about their goals,


their knowledge, and their beliefs in relation to their instructional prac-tice, their motivational and dispositional factors became apparent. Ms.Carol initially revealed that she felt insecure about her knowledge, herabilities, her ideas about teaching. Her feelings of discomfort and dissat-isfaction served as motivation for her to be open-minded about learningnew approaches. In contrast, Mr. Wong’s writing revealed that he enteredteaching with fixed beliefs that the best way to teach was through themedium of lecture. He initially resisted learning new approaches. Ashis supervisor, I realized that the only way to help him progress was toencourage him to focus on his students and challenge him to examine theextent of their learning. By knowing a preservice teacher’s motivationsand dispositions the supervisor is in a better position to facilitate change.The reflective activities served as the vehicle through which change waspromoted.

According to Goldsmith and Schifter (1997) a developmental modelfor mathematics teaching needs to account for a qualitative change in thebelief-behavior complex. The framework used in this study served as theconceptual basis for the structure that enabled preservice teachers to relatetheir thoughts and actions. Without the reflective activities, which requiredMr. Wong to confront his beliefs and focus on his students, it is possiblethat Mr. Wong might never have changed his teaching style. In fact, inhis written prelesson thoughts, he did not focus on student understandingand ways of learning; rather, he focused on whathe was going to do inthe lesson. It was only after the conference, in which he was continu-ally asked to examine student learning, that he was able to make someimportant connections. According to his postlesson thoughts, he came tounderstand that his rigid approach to teaching (using a script and transmit-ting knowledge) did not allow him the flexibility to monitor his students’understanding or to adapt his instruction as it became necessary.

Since I used this structured approach I have learned much about howmy students think, and have thereby changed many of my ideas and myapproaches. I am now less rigid in my views and therefore less frustratedabout what I see as I sit in the back of a classroom. I have come to realize,more than ever, that there are reasons for how preservice teachers teach.These reasons are a reflection of the knowledge and beliefs they haveabout the mathematical content, the particular students they teach, and theway students learn. Their reasons also reflect the goals they have for theirstudents, which are often influenced by the cooperating teachers’ goals.Preservice teachers need help and support in order to construct new mean-ings about what it means to be an effective mathematics teacher. Theymust continually be called upon to share, reexamine, and question their

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knowledge, their beliefs, and their goals for students. They must be madeaware of the need to monitor student understanding during the lesson asa means for reconstructing their own meanings about what is going onin their classrooms. I have found that when preservice teachers are mademore aware of the monitoring they do of their students, they become moreconscious of the need to change their instruction accordingly. Rather thanlooking at these interactive changes as a negative aspect of their lesson(“I couldn’t do what I planned”), they realize that they are indeed actingon behalf of their students and are engaging in more effective teaching.Finally, when they reflect on their lessons in a structured way I have theassurance that they have attended to the most critical facets of classroominstruction: tasks, learning environment, and discourse. Whereas initiallythey tended to focus on more efficient ways to cover the content and main-tain control, they now tend to focus on better ways to involve all of thestudents.

For preservice teachers the structure for reflection appears to be apowerful tool for facilitating their continual professional development.They are encouraged to be more analytical about their teaching. Theyare encouraged to examine and attend to the underlying assumptions andbeliefs that drive their practice. They are encouraged to think about whythey make the decisions they do in light of their goals for students. Theyare encouraged to think about how their knowledge and beliefs regardingthe content, their students, and methods of teaching impact the design oftheir lessons. Hopefully, these experiences will enable them to develop ahabit of reflective thinking about their teaching. Ms. Carol’s comment inher evaluation of the course suggests that this habit of reflection is possible.

Most of the work done as a student teacher was essential for self awareness. I think bybeing focused in on prelesson and postlesson thoughts I became able to focus on themsubconsciously. These thoughts and my awareness of them were brought constantly to myattention, and now they always will be. I wasn’t always thrilled to write them, but I nowrealize that it will be to my benefit always.

Most importantly, I hope that the prospective teachers will view them-selves as authorities who can evaluate their own classroom instruction interms of their own knowledge, beliefs and goals for students and be flex-ible enough to modify their beliefs as the evidence indicates. According toCooney and Shealy (1997) such autonomous behavior is critical for teacherdevelopment. In this article I have attempted to show how an approachfor examining teachers’ thought processes and instructional practice canbe used as a tool for the professional growth of preservice mathematicsteachers. Teacher educators can use this approach as a vehicle throughwhich preservice teachers are made aware of their underlying thinking and


how it impacts their instructional practice. By being provided a compre-hensive structure for self-reflection, preservice teachers can be empoweredto assess and improve their teaching.

ACKNOWLEDGEMENTS

The author is grateful to the many people who have contributed to thedevelopment of this work. Thanks go to Naomi Weinman who assisted mein supervising the student teachers, to Eleanor Armour-Thomas who gaveextensive feedback in the writing of this article, and to my students whoworked so hard.

APPENDIX A

Class Work Assignment to be done in Groups

1. a. On the same set of coordinate axes, graph the following system ofinequalities:

y + x < 5y ≥ 2x + 3

b. Based on the graphs drawn in part a, write the coordinates of:

i. A point in the solution set of the system of inequalities.ii. A point that satisfies the first inequality, but not the second.

c. Explain in a few brief sentences how you would check your answers topart b algebraically.

2. a. On the same set of coordinate axes, graph the following system ofinequalities:

y < 12x + 2

y ≥ 12x − 1

b. Label the solution set of the system of inequalities S.c. Change the inequalitites so that the solution set is empty. Explain your

answer.3. a. On the same set of coordinate axes, graph the following system of

equations:

y − 2x = 7x + y =−2

b. What is the solution of the system of equations? Check your answer.4. In your group discuss the differences between graphing a system of equations

and a system of inequalities.

164 ALICE F. ARTZT

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Cooney, T.J. & Shealy B. (1997). On understanding the structure of teachers’ beliefs andtheir relationship to change. In E. Fennema & B.N. Nelson (Eds.),Mathematics teachersin transition(pp. 87–110). Mahwah, NJ: Lawrence Erlbaum Associates.

Cooney, T.J., Shealy, B.E. & Arvold, B. (1998). Conceptualizing belief structures of preser-vice secondary mathematics teachers.Journal for Research in Mathematics Education,29, 306–333.

Ernest, P. (1988, July).The impact of beliefs on the teaching of mathematics. Paperprepared for ICME VI, Budapest, Hungary.

Fennema, E., Carpenter, T.P., Franke, M.L., Levi, L., Jacobs, V.R. & Empson, S.B. (1996).A longitudinal study of learning to use children’s thinking in mathematics instruction.Journal for Research in Mathematics Education, 27, 403–434.

Fennema, E., Carpenter, T.P. & Peterson, P.L. (1989). Teachers’ decision making andcognitively guided instruction: A new paradigm for curriculum development. In N.F.Ellerton & M.A. (Ken) Clements (Eds.),School mathematics: The challenge to change(pp. 174–187). Geelong, Victoria, Australia: Deakin University Press.


Fennema, E. & Franke, M.L. (1992). Teachers’ knowledge and its impact. In D. Grouws(Ed.),Handbook of research on mathematics teaching and learning(pp. 147–164). NewYork: Macmillan Publishing Company.

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Queens College of the City University of New YorkDepartment of Secondary Education and Youth ServicesFlushing, NY 11367-1597Email: [email protected]

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ENHANCING AND ASSESSING PRESERVICE TEACHERS’INTEGRATION AND EXPRESSION OF MATHEMATICAL

KNOWLEDGE

ABSTRACT. The construction of concept maps and the writing of interpretive essaysin mathematics courses for preservice and continuing teachers provide students with richlearning experiences and yield substantial insights into the degree of connectedness oftheir knowledge with respect to a given topic. With this dual approach students are giventhe opportunity to express their knowledge in different ways, which allows for individualdifferences in learning styles and verifies the relationships illustrated. As students activelyparticipate in the task of developing connections between related concepts, reflect on theirthinking, and become engaged in mathematical discourse, students are provided withan opportunity to mature mathematically and to experience an alternative approach toinstruction and assessment.

KEY WORDS: concept maps, writing in mathematics, mathematics education, assessment

The development of an integrated knowledge-base, the communicationof mathematical knowledge, and the use of assessment as a means toguide and improve learning are three aspects of a meaningful mathematicalexperience strongly emphasized in the current literature (see, for example,Bishop, Clements, Keitel, Kilpatrick & Laborde, 1996; National Councilof Teachers of Mathematics [NCTM], 1989, 1995). Accordingly, studentsmust be encouraged to investigate the connections among various mathe-matical concepts and topics, to reflect on and clarify their own thinkingabout mathematical ideas and situations, and to express mathematicalideas in writing. Consistent with these goals, the education of teachersof mathematics must not only promote the development of this type ofcontent knowledge and discourse, but also model the assessment of suchknowledge (Clarke, 1996; NCTM, 1991).

The combined use of concept maps and interpretive essays is onemeans by which these goals can be promoted for preservice and inser-vice teachers. Concept maps provide a graphical representation of themathematical connections perceived by the student; the interpretive essayexpands upon these relationships and focuses on communicating mathe-matical ideas through writing. These two avenues of expression, oneschematic and the other verbal, can enable students to explicitly commu-


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nicate aspects of their knowledge of mathematical concepts and/or topics,and to identify the strengths and weaknesses of their understanding.

CONCEPT MAPS AND INTERPRETATIVE ESSAYS

In a concept map, related concepts are represented as nodes and thespecific relationship between two concepts is indicated by the linkingwords written along the line that connects the nodes (see Figures 1 and2 for examples of unedited student concept maps).

Based primarily on the work of Novak (1984) in science education,concept maps were “developed specifically to tap into a learner’s cognitivestructure and to externalize, for both the learner and the teacher to see,what the learner already knows” (p. 40). The construction of conceptmaps as an indication of the connectedness of mathematical knowledge issupported by Hasemann and Mansfield (1995) and is closely tied to Hiebertand Carpenter’s (1992) model for analyzing the learning and teaching ofmathematics. Based on the assumption that mathematical representationsare connected in some useful way, concept maps characterize networks ofknowledge as vertical hierarchies in which some representations subsumeother representations or as web-like non-hierarchical arrangements oflinear chains or complex networks.

The construction of concept maps has been used effectively as a way ofteaching mathematics, as a means of identifying student misconceptions,and as an assessment instrument (Bartels, 1995; Beyerbach, 1988; Kouba,1994; Mansfield & Happs, 1991; Novak, 1984, 1991). Their use can “stim-ulate students to create, and allow faculty to assess, original intellectualproducts that result from a synthesis of the course content and the students’intelligence, judgment, knowledge, and skills” (Angelo & Cross, 1993,p. 181).

The use of writing to enhance and to assess student understanding isconsistent with the current trend to incorporate writing in mathematicsclasses (Connolly & Vilardi, 1989; Ellerton & Clarkson, 1996; Sterrett,1990). Various researchers have emphasized the use of journals or writingprompts to promote mathematical learning (Borasi & Rose, 1989; Clarke,1993; Miller & England, 1989; Powell & Lopez, 1989; Swinson, 1992)and the use of writing to identify students’ misconceptions (Birken, 1989;Miller, 1992).

If one accepts Webb’s (1993) contention that “assessment be integral toinstruction; multiple methods should be used; and that all aspects of mathe-matical knowledge and its connections to other branches of knowledgebe assessed” (p. 2), concept maps and interpretive essays can be viable

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Figure 1. Concept map #1.

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Figure 2. Concept map #2.

assessment tools in courses for preservice teachers. The combined useof both tools can enable educators to better monitor student progress andguide instruction. Their use can encourage students to explore a variety ofrelationships among mathematical concepts and topics, to develop missingconnections, and to clarify misconceptions. Moreover, emphasis can beplaced on several valuable, but frequently neglected, learning objectives,such as the illustration of structures in mathematics or the clear andcreative communication of ideas (Clarke, Clarke, & Lovitt, 1990). Theconstruction of concept maps can be used as an assessment tool; prop-erly used concept maps can reflect the kind of assessment advocated by

INTEGRATION AND EXPRESSION OF KNOWLEDGE 171

Cooney, Badger, and Wilson (1993) because they involve significant math-ematics, can be completed in a variety of ways, elicit a range of responses,require students to communicate, and stimulate the best possible perfor-mance on the part of the student. Experience with this type of activityencourages prospective teachers to “begin to think about alternative meansof assessing students’ learning in mathematics classrooms” (Chappell &Thompson, 1994, p. 187).

CONCEPT MAPS AND INTERPRETIVE ESSAYSIN TEACHER EDUCATION

Concept maps and interpretive essays can be used in a variety of instruc-tional ways – at the beginning of a lesson as a measure or review ofpast learning, during instruction to develop understanding, or at the endof a lesson as a summative activity. For example, early in a beginningcalculus course students can construct a concept map utilizing terms suchasmapping, graph, domain, range, inverse, andone-to-one. This reviewof the concept of function is a very effective way for education majors toreflect on and connect various aspects of their prior knowledge of functionsand to build a strong foundation for future use in the secondary classroom.Prospective elementary teachers can construct maps using terms associatedwith sets and functions, number theory, or polygons as they encounterthese areas of mathematics in content courses. These maps promotestudents’ mathematical understanding and can also serve as an ongoingassessment of that understanding. In a Survey of Geometries course forsecondary education students, a map that incorporates terms related tofinite, Euclidean, and non-Euclidean geometries can be constructed as afinal review of the course material. Figure 3 provides a list of terms thathave been used in each case.

Introducing the Activity

Students can be introduced to the construction of concept maps and thewriting of the accompanying interpretive essays during a regular classmeeting. They can be given a list of words related to a familiar topicand then led through the multi-step process as a class, with each stepillustrated on an overhead transparency. Students read the list of termsto become familiar with the general topic being explored, and then sortthe terms into clusters according to the extent to which they are related.Once sorted, the terms are arranged in clusters around the central conceptor topic. Because an arrangement may be either hierarchical or web-like,

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Figure 3. Terms for several concept maps.

depending on how one views the relationships between the terms, studentscan be shown examples of both types of maps and encouraged to exploreseveral arrangements of the terms until they are satisfied with the organi-zation of their map. Next, students draw linking lines between terms andindicate the relationships being illustrated by adding linking words to thecorresponding lines. Although the inclusion of linking words is critical tomap interpretation, students have the tendency to omit these labels becauseit is difficult for them to specify some of the relationships. Consequently,this aspect of construction should be discussed at some length. The class


can generate possible linking words (e.g.,has the property, is an exampleof, involves) and discuss the need to draw directional arrows on the linkinglines to indicate the direction of the relationship that is expressed. It iscritical to emphasize the individualized nature of the task to students. Thereis no one correct map to be constructed; students may omit any terms theyare unable to use or may add additional terms. The goal is for students toconstruct a map that makes sense to them as a means of constructing theirown knowledge.

Once students are familiar with the process of constructing a map, theycan be given a list of approximately 20 words related to the selected topicand asked to construct a concept map based on these terms. Depending onthe course and purpose of the assignment, the instructor may restrict termsstrictly to concepts, such as function or limit, or expand them to includetopics, such as finite or Euclidean geometry. Students are encouraged touse outside resources, as needed, to clarify the meaning of terms. A vari-ation of this method is to have students construct a concept map as a group,in which the group generates their own terms and reaches a consensus asto the arrangement; in this case, groups share their maps with other groupsand discuss the similarities and differences. A class discussion of the finalmaps can broaden students’ perspectives about the open-ended nature ofthe activity and expose them to relationships they may not have previouslyrecognized. The discussion also allows students who are less proficientwith writing to communicate their knowledge orally.

After completing a draft of their concept map, students should writean accompanying interpretive essay in which they clarify and expand onthe relationships expressed in the maps. These essays are meant to givestudents the opportunity to reflect on the relationships illustrated in theirconcept map, refine their thoughts, and explain how the map is orga-nized. Because students are allowed the freedom to set the essay in anycontext, the results are frequently more personal and informal than tech-nical writing. (Appendix A contains the unedited essays corresponding toconcept maps shown in Figures 1 and 2.) Students who are not comfortablewriting about mathematics should be given a few broad guidelines, such as

(a) make the paper reader-friendly,

(b) think of a classmate as the reader rather than the instructor,

(c) be as thorough in the discussion as possible,

(d) include additional information that might be relevant or that repre-sents personal insights, and

(e) consider personalizing the essay by putting it in a story or some othercreative context.

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Figure 4. Holistic scoring criteria for concept maps.

If used as an assessment tool and a quantitative score is desired, eachconcept map and interpretive essay can be evaluated with a set of holisticscoring criteria (see Figures 4 and 5). The concept map criteria focuson organization and accuracy; the interpretive essay criteria focus oncommunication, organization, and mechanics (grammar and punctuation).If used strictly as a learning tool, the descriptive portion of criteria can becombined with written and/or oral comments to provide qualitative feed-back to the student. In either case, students should be given the scoringcriteria prior to beginning the activity to allow them the opportunity toevaluate their work.

Scoring Concept Maps and Interpretive Essays

Two concept maps on number theory from a mathematics content courserequired for all elementary education majors are shown in Figures 1 and2; the accompanying essays are included in Appendix A. Initially, eachmap was assigned a preliminary score based on a scale from 0 to 10,with up to 6 points for organization and up to 4 points for accuracy. Toensure consistency in scoring, maps were then sorted according to theirpreliminary scores for organization and compared with the other mapsthat had received the same score; scores were adjusted as necessary. Asecond reading resulted in the final organization score. The same procedure


Figure 5. Holistic scoring criteria for interpretive essays.

was used for determining the accuracy score and for scoring the essays;comments and corrections were written on each map and essay.

The concept map shown in Figure 1 received a score of 9. The organi-zation was excellent (6), based on the following features: the relationshipbetweendivisor, factor, anddivisibility was clearly indicated; correctnotation to show divisibility and the Euclidean Algorithm was utilized;the term fundamental theorem of arithmetichad been added; clusterswere cohesive and well organized; and the linking words indicated an in-depth understanding of relationships between the terms. Because the mapindicated all prime numbers are odd, accuracy was rated as fluent (fewminor errors) rather than excellent (no errors). The concept map shownin Figure 2 received a score of 4. The organization of the map (weak)indicated minimal understanding of the relationships between the terms.There were no links constructed between any of the termsdivisor, factor,andmultiple; the linking words indicated a superficial understanding of themeaning of the terms (e.g., “divisor is part of greatest common divisor”);the termdivisible was not used. The accuracy score (good) was based on

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the connections shown betweenrelatively primeandprime and betweenfactor, factorization, andproduct.

The essay that accompanies Figure 1 received a score of 10. The mathe-matical relationships were discussed in a clear and systematic manner. Itshould be noted that the student wrote in her essay that “prime numbersare always odd,” which supports what was indicated on her map; thiswas evidence of a misconception rather than an error in constructing themap. The transitions between paragraphs were smooth, given the rangeof terms incorporated in the map and the essay; there were no errorsin grammar, punctuation, or capitalization. The essay that accompaniesFigure 2 received a score of 6. Based on the explicit statement abouthow the map was organized and the student’s interpretation of the termsinvolved, communication was rated as fair. Communicated understandingswere somewhat superficial or lacking at times (e.g., connecting relativelyprime with prime because the GCD is one and one is prime), which indi-cates only a partial understanding of the content involved. Note that thesentence referring to relatively prime clarifies the connections shown onthe map. The student knew the definition of relatively prime but not whythis term was used; he did not have a full understanding of the concept ofrelatively prime. Although the connections and transitions between topicswere appropriate, the essay appears somewhat disjointed and difficult toread which reflects an adequate (2 points) organization. The mechanicswas acceptable and received one point.

BENEFITS TO TEACHERS AND STUDENTS

Instructional Method

Numerous educators have noted the benefits of using concept maps andinterpretative essays as part of mathematics instruction and assessment.According to Bartels (1995),

The real value in constructing concept maps is the visual representation of mathema-tical connections that is produced. With this tool, the connections are explicitly depictedand are visible to the person constructing the map as well as to anyone who observesthe map. . . . Since the connections are visually depicted with the concept map, theperson constructing the map has the opportunity to evaluate perceived connections. Thisevaluation allows the map creator to deal with the effectiveness and meaning of theconnections. In addition, the creator can modify connections while constructing the map.(p. 548)

With respect to writing, Rose (1989) contended, “changing deeplyrooted beliefs about teaching and learning mathematics involves trans-


forming the classroom into a place where students may experience mathe-matics as a creative activity, particularly through written language” (p. 15).Birken (1989) maintained that:

Students see topics in mathematics as unrelated steps rather than as building blocks ofknowledge. Little opportunity exists in the fast pace of college courses for professors toreflect on past learning, to tie together concepts, or more importantly, to ask students tomake these connections on their own. This is where I see the greatest and most lastingvalue for writing in mathematics classes. (p. 38)

Comments from students who constructed concept maps in an algebracourse for preservice elementary teachers supported these contentions.When asked what, if any, advantages they saw for this activity, one studentwrote:

It helps you to connect all your learning. It made me think about all the connectionsbetween the concepts. It also helped me to organize my thoughts. Doing a storyreallyhelped me because I put the concepts into my own words. It was also great because weput the concepts into real life situations that helped make it more meaningful! Workingtogether also helped to bring the learning I have done into meaning. This way I had toexplain myself so that another person understood.

Another student responded,

I feel that the biggest advantage of this assignment was making the connections betweenthe different types of functions. I found that all the different functions had pretty much thesame parts. For example, a quadratic function is still a relationship betweenx-values andy-values. So is a linear function. I was glad to make this connection.

The two activities reinforced students’ basic mathematical under-standing and encouraged flexibility in the use of mathematical terms, asindicated in the opening and closing paragraphs of the second studentessay (Appendix A). Students initially had to have a basic understanding ofwhat each term means, yet when striving to show the relationships betweenthe terms be prepared to look at the meaning of the terms from differentperspectives. For example, on a polygon map, the termsacute, obtuse, andright can refer to a classification of triangles or the measure of individualangles, andregularcan be connected as a cross-link toequilateral triangleandsquare. Omission of these linking lines and no reference to these rela-tionships in the essay indicate a narrow or limited understanding of theseterms.

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Assessment Tool

The construction of concept maps and the writing assignment have bene-fits beyond those of using either method individually. The combinationof both visual and verbal communication of mathematical knowledgeyields a more complete representation of students’ mathematical knowl-edge. Frequently students will omit certain relationships in their conceptmap because they feel the map will become too cluttered; however, theydiscuss these relationships in their essay. Other times students discuss arelationship in the essay, perhaps vaguely, but are unable to illustrate therelationship on their map because they do not have a clear understanding ofthe relationship. Each representation provides valuable information aboutthe student’s understanding. A second benefit of the combined approach isthat the creation of the concept map gives an underlying structure to theessay without being overly rigid. By the same token, the essay justifiesthe organization of the concept map and clarifies possible technical inac-curacies in the construction of the map, such as inappropriate or missinglinking lines. It allows the students to expand on the connections they seebetween the terms, to explain the connections they perceive in a specificcontext, and to refine their thinking. If no hierarchical structure is imposedon the organization of the maps, as Novak (1984) proposes, the students areallowed more flexibility and opportunity for creativity than is sometimesthe case when concept maps are used. Such flexibility provides studentsconsiderable latitude in communicating their understanding of the topicinvolved.

Some misconceptions are readily identified in a concept map if linkingwords are missing or inappropriate, if terms or linking lines are omitted,or if the organization is weak. For example, a common error shown onfunction concept maps is the identification of all functions as one-to-one.However, many times in the context of the essay, it becomes obvious thestudent can distinguish between the defining characteristic of a function,single-valuedness, and the one-to-one property. Subsequent instruction canfocus on remediation of possible misconceptions. As patterns of miscon-ceptions arise for given topics, teachers can alter their future instruction toincorporate additional examples and activities that focus on these issues.Assessment, so conceived, can facilitate instruction.

Students’ View of Mathematics

Due to their open-ended nature, concept maps and interpretative writingcan help dispel some dysfunctional mathematical beliefs. For example,in many students’ conception of mathematics, the scope of mathema-tical activity is limited to problems that are well defined and have an


exact and predetermined solution; mathematical activity is characterizedas recalling and applying learned procedures. Students often see mathe-matics as a subject in which answers to questions are either right or wrong(Borasi, 1990). But concept maps involve mathematical activity that defiesa simplistic right or wrong orientation. Comments from preservice teachersindicate an awareness of this issue. One student commented, “I also under-stand now that math concepts are broad universals that hold true, not juststeps to solve to get an answer.”

PRESERVICE TEACHERS’ EVALUATION OF THE ACTIVITY

Enhancement of Learning

Although students find the construction of the maps time-consuming,sometimes spending up to four hours constructing one, they generallyreport that they enjoy the activities and find the maps and the inter-pretive writing to be worthwhile. One secondary education student wrote:“Concept maps, what a concept! This was a very fun thought provokingactivity. At times I thought that I would never be able to make it work,but perseverance prevailed!” Many students felt that the construction ofthe concept map and the writing of the corresponding interpretive essayencouraged them to reflect on their knowledge and enhanced their abilityto make mathematical connections. Others indicated that they enjoyed theopportunity to demonstrate their knowledge in a non-numerical, creativeway. For example, several students in the Survey of Geometries coursefor secondary mathematics education students utilized a road map analogyas the basis of their concept map and essay. One student drew a map ofthe City of Geometry in the County of Axiomatic Systems; several othersrelied on color coding to differentiate paths and destinations on their roadmap.

Concept maps and essays dealing with functions can result in storiesin a variety of settings. One preservice elementary mathematics minordeveloped an analogy to parts of the human body, where a function is likea brain and numbers and variables are the heart. Another created “TheLand of Function” that is populated by two tribes, where the “X’s areextremely independent and only capable of providing input. They formedrelationships with the Y’s, who are incapable of independent thought, andwho have become very dependent upon the X’s.” A preservice secondaryteacher in calculus wrote a more complex story called “Fred the Function.”

The domain is a group of numbers that Fred likes. These are numbers that work with himand do not do bad things like make him not exist. . . . Imagine Fred pulling a number out

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of his right-hand pocket (domain), squishing it between the palms of his hands, and thenputting the result into his left-hand pocket (range). This is what Fred does for fun.. . . WhenFred gets together with his friend Gary, they can play together quite interestingly. FirstGary takes a number out of their collective pocket, does some math to it, and then gives itto Fred. He then does his own math to it and puts it in the collective pocket they use as therange. This is called composition. . . . Well, now that you know Fred and what he does, youcan use his knowledge to help you in the world of mathematics. But do not stop there, alsolook for him in physics, chemistry, biology, and even economics!

Some students were uncomfortable with the open-ended nature of thetask; they were concerned about whether they were doing it correctlyand whether there is a prescribed length for the essay. These studentswere encouraged to feel comfortable with creating their own map, onethat means something to them. No page limit was imposed on the essay;the only requirement was that it effectively communicate connections tothe reader. Merely listing textbook definitions or a series of disjointedstatements was strongly discouraged.

Students’ written comments indicated that the activity contributed totheir understanding of mathematics and motivated them to develop a betterunderstanding of mathematics. For example, one student noted that

To be able to explain something, we must understand it. This project made us do just that.We have been habitually bombarded with the mechanics of math and not the ideas behindit. The relationships of all the terms to each other in different circumstances made megrasp the ideas and concepts. . . . Mathematics has always been fun for me. This quarter Iwas exposed to a different way.. . . I was motivated to learn more and see if I could findother relationships not mentioned in class.

Students frequently indicated that they valued the opportunity topersonalize the content. The student who used the human body analogystated that she could now relate the mathematical terminology to heranalogy; another indicated “I will take this knowledge with me becauseit is mine. I didn’t just regurgitate it through practicing equations.” Twostudents who worked cooperatively felt the “project made us think aboutwhat we had learned and how we would explain it to someone else. . . . Ithink it will help us more in the next math class because I will go into theclass thinking how would I connect the ideas or make a concept map outof the material being presented.”

Effect on Future Teaching

Preservice teachers readily see the benefits of using concept maps invarious contexts with students at the middle- and high-school level.The majority of students believed a deeper understanding of importanttopics and concepts will enhance their teaching, regardless of the grade


level they planned to teach. One student indicated that the construc-tion of concept maps motivated her to examine the mathematical contentshe will be teaching more carefully. She foresaw using concept mapsas a curriculum-planning tool to help her organize class activities andpresentations.

When asked whether they would use concept maps and essays in theirclassroom, preservice teachers responded in a variety of ways: “If they’redrawing webs that would make a spider proud, then I feel pretty goodabout where they are as a class. It’s something that I, as a teacher, couldget pretty good mileage out of.” Another student stated “concept mapswouldn’t be the only thing I would use to assess my students’ work, butI would definitely use them to assess understanding.” Some commentedthey would use writing because they found writing enhanced their ownunderstanding; others indicated they would use concept maps as a learningtool with younger students, but not for assessment. One student suggestedthat the use of concept maps and essays could be helpful when teachingfractions; another suggested the idea of walking younger students througha concept map to help them link familiar concepts.

A number of preservice teachers have reported using concept mapsand essays with students during their field experiences. One secondarymathematics education minor who used concept maps with junior highschool students found it difficult to select the terms students would usein their maps; as a result, this became a valuable learning experiencefor the student teacher and his students. Several elementary mathematicseducation majors adapted the list for the polygon map for use with middlegrade students during a practicum; they reported students found the activitychallenging and enjoyable.

A teacher must have a very thorough understanding of the mathematicsbeing explored in order to effectively use this approach with students. Theselection of terms and the evaluation of maps can be demanding, and maybe difficult for some novice teachers. However, beginning teachers can useconcept maps and writing to aid their lesson planning. As their knowledgeand confidence grow, they can implement this technique in the classroom.

SUMMARY

The use of concept maps and interpretive essays provides students witha rich learning experience which can yield substantial insight into theconnectedness of mathematical topics. Each method can be used effec-tively individually; however, when used in combination, students are giventhe opportunity to express their knowledge in different ways, which allows

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for individual differences in learning styles. In the process, they activelyparticipate in the worthwhile task of developing connections betweenrelated concepts, reflect on their thinking, and become engaged in mathe-matical discourse. Although this approach does not encompass all typesof mathematical knowledge, it does provide preservice and continuingteachers with an opportunity to mature mathematically and to experiencean alternative approach to instruction and assessment. As a result, theirprofessional development is enhanced as these experiences become partof the “threads that are woven as the fabric of successful mathematicsteaching.”(NCTM, 1991, p. 126).

APPENDIX A

Unedited Essay for Concept Map #1, Figure 1(Italics added to indicate termsused on map.)

Positiveintegerscontaining the relationship (b) divides (a), states that (a) isdivisible by (b) and (a) is amultiple of (b). (b) in the relationship is adivisoror factor (which are the same thing) of (a). A factor of a number means thatthat factor can be multiplied by another or other factors which will be equal to amultiple of those factors. There is never aremainderin this relationship.

Integers containing exactly two factors areprime. Namely, these factorsinclude one and itself. Prime numbers are alwaysodd. Integers with factors otherthan one or itself are calledcompositenumbers, and can be eitherevenor odd.Composite numbers can be expressed byfactorizations. Factors are multiplied byone another, so that theproductof these factors equals the composite number.Prime factorizationoccurs when all the factors in a factorization of a compositenumber are prime. Ways of obtaining prime factorization are by thefundamentaltheorem of arithmeticor by using afactor tree.

Thegreatest common divisor(GCD) of two or more natural numbers can befound by three methods: 1)intersection of sets, 2)prime factorization, and 3)Euclidean Algorithm, which is used when numbers are difficult to factor. If theGCD of two numbers equal one, they are said to berelatively prime.

Following the map in the other direction, (a) in the relationship, (b) divides(a) is amultipleof (b). Finding theleast common multipleof two numbers maybe obtained by three ways: 1)intersection of sets, 2)prime factorization, and 3)division by primes, which is used for determining the LCM of more than twonumbers.

Unedited Essay for Concept Map #2, Figure 2(Italics added to indicate termsused on map.)

I found that this map was a lot harder to construct than our previous map. Iused thegreatest common divisorand theleast common multipleas my startingpoints.Prime was also a very important term that related to many of the otherterms. I used the term’s definitions, and relationships to decide the similarities.


I also included the term’s purposes to make my connection between words. Iwish we could do the maps before we begin a section because it is so helpful inclarifying the material.

The first thing I did was connect all the various methods of finding theGCDand theLCM to the GCD and LCM. For example, I connected theintersection ofsetsto the GCD and the LCM. TheEuclidean Algorithmmethod and theprimefactorizationmethod were the only methods that I connected to anything elseto besides the GCD or the LCM. I included theremainderterm with EuclideanAlgorithm because without the use of the remainder it would be impossible tomake the algorithm work.Prime factorizationhas a lot of terms that relate it tobesides the GCD and the LCM.

A factor tree is a way of finding the prime factorization so I connected it.Moving down from factor tree is the termfactor which are the components ofa factor tree.Productandfactorizationwere also included because factors are aproduct of factorization. Moving directly up from prime factorization is the termprime because prime factorization uses only prime numbers.Relatively primejoins with prime because relatively prime numbers have a GCD of one which isprime. I also putintegerwith prime because in order for a number to be prime thenumber has to be a positive integer. A integer consists ofevenandoddnumbers,that is why those terms are with the term integer. Lastlycompositeandprimearejoined because they make up all the factors.

The map helped me make these concepts more concrete in my mind. Like theprevious map this map was very helpful for making similarities and differencesbetween all of the terms. I wish we could make a map before every test that wayall the terms would be truly understood.

REFERENCES

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Bartels, B. (1995). Promoting mathematics connections with concept mapping.Mathe-matics Teaching in the Middle School, 1, 542–549.

Beyerbach, B. (1988). Developing a technical vocabulary on teacher planning: Preserviceteachers’ concept maps.Teaching and Teacher Education, 4, 339–347.

Birken, M. (1989). Using writing to assist learning in college mathematics classes. In P.Connolly & T. Vilardi (Eds.),Writing to learn mathematics and science(pp. 33–47).New York: Teachers College Press.

Bishop, A., Clements, K., Keitel, C., Kilpatrick, J. & Laborde, C. (Eds.) (1996).Interna-tional handbook of mathematics education. Boston: Kluwer Academic Publishers.

Borasi, R. (1990). The invisible hand operating in mathematics instruction: Students’conceptions and expectations. In T. Cooney (Ed.),Teaching and learning mathematicsin the 1990s(pp. 174–182). Reston, VA: NCTM.

Borasi, R. & Rose, B.J. (1989). Journal writing and mathematics instruction.EducationalStudies in Mathematics, 20, 347–365.

184 LINDA A. BOLTE

Chappell, M. & Thompson, D. (1994). Modeling the NCTM Standards: Ideas forinitial teacher preparation programs. In D. Aichele (Ed.),Professional development forteachers of mathematics(pp. 186–199). Reston, VA: NCTM.

Clarke, D. (1996). Assessment. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick &C. Laborde (Eds.),International handbook of mathematics education(pp. 327–370).Boston: Kluwer Academic Publishers.

Clarke, D. (1993). Probing the structure of mathematical writing.Educational Studies inMathematics, 25, 235–250.

Clarke, D.J., Clarke, D.M. & Lovitt, C. (1990). Changes in mathematics teaching call forassessment alternatives. In T. Cooney (Ed.),Teaching and learning mathematics in the1990s(pp. 118–129). Reston, VA: NCTM.

Connolly, P. & Vilardi, T. (Eds.) (1989).Writing to learn mathematics and science. NewYork: Teachers College Press.

Cooney, T., Badger, E. & Wilson, M. (1993). Assessment, understanding mathematics, anddistinguishing visions from mirages. In N. Webb (Ed.),Assessment in the mathematicsclassroom, 1993 Yearbook(pp. 239–247). Reston, VA: NCTM.

Ellerton, N. & Clarkson, P. (1996). Language factors in mathematics teaching and learning.In A. Bishop, K. Clements, C. Keitel, J. Kilpartick & C. Laborde (Eds.),Interna-tional handbook of mathematics education(pp. 987–1033). Boston: Kluwer AcademicPublishers.

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Novak, J. (1984).Learning how to learn.New York: Cambridge University Press.Novak, J. (1991). Clarify with concept maps.Science Teacher, 58(7), 44–49.Powell, A. & Lopez, J. (1989). Writing as a vehicle to learn mathematics: A case study. In

P. Connolly & T. Vilardi (Eds.),Writing to learn mathematics and science(pp. 157–177).New York: Teachers College Press.

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Sterrett, (Ed.) (1990).Using writing to teach mathematics. Washington, DC: MathematicalAssociation of America.

Swinson, K. (1992). Writing activities as strategies for knowledge construction andidentification of misconceptions in mathematics.Journal of Science and MathematicsEducation in Southeast Asia, 15(2), 7–14.

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Department of Mathematics, MS-32Eastern Washington University526 5th StreetCheney, WA 99004-2431

THOMAS G. EDWARDS and SARAH M. HENSIEN

CHANGING INSTRUCTIONAL PRACTICETHROUGH ACTION RESEARCH

ABSTRACT. The vision of mathematics learning and teaching captured in the NCTMStandardsdocuments holds clear implications for needed changes in mathematics teachers’instructional practices. This article describes an action research collaboration between amiddle school mathematics teacher and a mathematics teacher educator that focused onthe teacher’s attempts to change her instructional practice in the direction of the NCTMStandardsvision. In particular, the middle school teacher wrote a narrative description ofthe collaboration and of the changes she made in her instructional practice as a result ofthe collaboration. An interpretive analysis of the teacher’s narrative by the teacher educatorreveals that the collaboration itself, the support for change inherent in the collaboration, andthe teacher’s regular reflections on her own beliefs and practices which derived from thecollaboration were important to her process of change.

KEY WORDS: action research, collaboration, teacher change, teacher learning, reflectivepractice, teacher beliefs, support, middle school

The vision of school mathematics education captured in theStandardsdocuments published by the National Council of Teachers of Mathematics(NCTM, 1989, 1991, 1995) “represents a radical departure from traditionalmathematics classes” (Simon, 1994, p. 72).The Professional Standards forTeaching Mathematics(NCTM, 1991, p. 3) articulated five major areas ofchange in instructional practice that are necessary to bring this vision tolife in classrooms, noting that each of these areas of change will requirea shift in teachers’ world views about mathematics and its learning andteaching.

In attempting to foster such changes, a number of researchers havereached the conclusion that successful implementation of educationalreform demands an understanding of the processes by which teacherschange their instructional practices (Hart, 1993; Richardson, 1990;Schifter, 1996; Shaw & Jakubowski, 1991; Wood, Cobb & Yackel, 1991).Schifter and Fosnot (1992) captured the essence of this thinking when theystated: “Given the vision of mathematics instruction animating the reformeffort, teacher development – teachers constructing for themselves the newmathematics pedagogy – is at the heart of that vision” (p. 16). What followsis a description and interpretation of a middle school teacher’s efforts


188 THOMAS G. EDWARDS AND SARAH M. HENSIEN

to construct the kind of reform-oriented pedagogy of which Schifter andFosnot speak. Her efforts to change are placed in a context of a collabora-tive framework in which she and a university mathematics teacher educatorcome to grips with a variety of instructional issues. We begin by consid-ering what has been previously learned about teacher change and factorsrelated to teacher change.

TEACHER CHANGE AND LEARNING TO TEACH

A review of the voluminous research on teacher change reveals severalpervasive themes. Among those often associated with teachers’ abilityto change their instructional practices are peer collaboration and support(Clarke, 1994; Fullan, 1991; Lortie, 1975; Nelson, 1993; Schön, 1983),teachers’ beliefs (Ball & McDiarmid, 1988; Nelson, 1993; Richardson,1990; Thompson, 1992), and the development of a reflective practice(Clarke, 1994; Doyle, 1990; Nelson, 1993; Richardson, 1990; Schön,1983; Shulman, 1986).

Richardson (1990) suggested that research in the intersection of theteacher change literature and the learning-to-teach literature might beuseful for developing standards of warranted practice. In her formulation,“research becomes one basis for the development of warranted practiceswith which teachers may experiment in their classrooms” (p. 16). In thisspirit, we will consider collaboration and support, reflect on teachers’beliefs, and finally introduce the notion of reflective practice.

Collaboration and Support

In reviewing and updating the findings of the Rand Change Agent studyof the mid-1970s, McLaughlin (1990) noted the importance of support forteachers who are attempting to change instructional practices. Accordingto McLaughlin, the Rand study indicates that successful implementationrequires active and on-going support, and the level of support neededis “sometimes unpredictable” (p. 12). Support and its partner, collabo-ration, also constitute two of the four guiding principles for the reform-oriented Summer Math for Teachers project as described by Schifterand Fosnot (1992). These authors not only emphasize support but alsoargue that collaboration is essential to the process of reform. They write,“regular classroom consultation provides support for continued reflectionas changes are introduced into the classroom” (p. 17).

Fullan (1991) also mentioned the strong influence of efforts to supportteachers who are attempting to change. Moreover, collaboration which

CHANGE IN INSTRUCTIONAL PRACTICE 189

provides teachers with regular feedback and a voice in curricular decisionshas often been mentioned as crucial to the enhancement of teaching (Fullanet al., 1989; cited in McLaughlin, 1990, p. 15). Schön (1983) furthersuggested that this sort of interactive, collaborative support is “congenialto reflective practice” (p. 335).

Teachers’ Beliefs

The research on mathematics teachers’ beliefs and conceptions indicates acomplex relationship between a teacher’s beliefs and her or his instruc-tional practices. Thompson (1992) suggested that this relationship is adialectic one when she noted that

there is support in the literature for the claim that beliefs influence classroom practice;teachers’ beliefs appear to act as filters through which teachers interpret and ascribe mean-ings to their experiences as they interact with children and the subject matter. But, at thesame time, many of a teacher’s beliefs and views seem to originate in and be shaped byexperiences in the classroom. (pp. 138–139)

If this analysis is correct, the interplay between teachers’ beliefs and theirinstructional practices seems to be more dynamic, interactive, and cyclicthan a simple linear cause-and-effect relationship.

In reviewing the literature on the process of change in instructionalpractices in the direction of the NCTMStandardsvision, Nelson (1993)noted that “changing. . . . teaching to better facilitate students’ mathema-tical thinking appears to require several interconnected changes in beliefabout the nature of learning” (p. 5). Nelson also traced three approaches inorder to promote change in teachers’ beliefs:

• introducing disequilibration (Schifter & Simon, 1992; Schifter &Fosnot, 1992), an approach in which teachers are stimulated toreconsider their ideas about the nature of mathematics teaching andlearning and reconstruct more powerful ones;

• restructuring knowledge about mathematics learning (Carpenter,Fennema, Peterson & Carey, 1988; Peterson, Fennema, Carpenter &Loef, 1989), an approach which strives to help teachers integrate new,research-based knowledge about the process of learning mathematicsinto their prior knowledge base; and

• renegotiating beliefs about learning, teaching, and mathematics(Wood, Cobb & Yackel 1991), an approach in which teachers resolvea series of conflicts between their prior beliefs and their observationsof what happens in their classrooms as they renegotiate classroomnorms with their students to allow for students’ construction ofknowledge and rich mathematical discussions.


A unifying theme in these approaches is that each appears to be based ona reflective, constructive view of the process of teacher change.

Teachers’ beliefs are a critical factor not only for changes in mathe-matics teachers’ instructional practice but also in other areas of instruction.For example, Richardson, Anders, Tidwell, and Lloyd (1991) reported ademonstrable relationship between teachers’ beliefs and their classroompractices when teaching reading comprehension in Grades 4, 5, and 6.

Reflective Practice

Schön’s reflection-in-action model (1983) is frequently cited when atheoretical basis for studies of teacher change is established. Schön usedhis model to analyze and explain the MIT Teachers’ Project, suggestingthat “reflection-in-action is essential to the process by which individualsfunction as agents of significant organizational learning” (p. 338). TheRand Change Agent study found a number of strategies to be effectiveways to foster teacher change. Among those strategies are:

• concrete, teacher-specific, and extended training• teacher observation of similar projects in other classrooms, schools,

or districts• regular project meetings that focused on practical issues• teacher participation in project decisions• local development of project materials (McLaughlin, 1990, p. 12)

Although McLaughlin did not use the term reflection in his analysis, onecan argue that individual teacher reflections are at the heart of each of theabove strategies.

Thompson (1992) also put reflective activity at the heart of the processof change in teacher beliefs when she observed that “teachers appear toevaluate and reorganize their beliefs through reflective acts” (p. 139). Thisformulation links teachers’ reflections with their experiences, which, ashas been suggested, is the only way in which experience is educative(Richardson, 1990; Schön, 1983; Shulman, 1986). Cooney (1994) viewedattempts to reform mathematics teaching as “exercises in adaptation fromwhat we areable to doto what wewant to do” (p. 9, emphasis in original).He saw the realization of the NCTMStandardsvision of mathematicsteaching and learning as the result of a process of individual adaptation,with teacher reflectivity central to the process.

The importance of teacher reflection to the process of change in instruc-tional practice is not limited to teachers of mathematics. Anders andRichardson (1991) reported a similar effect for teachers of reading compre-hension. Their school-based staff development program was intended to


Figure 1. A model for teacher change (adapted from Edwards, 1996).

help teachers critically examine their beliefs and practices in teachingreading. Anders and Richardson found that as teachers in the programreflected on the reasoning related to their practices, instruction wasenhanced. What seems readily apparent is that reflection and change gohand in hand. The process of change necessitates that teachers reflect ontheir practice, compare their practice with some form of idealized practice,and begin to change as they move toward that idealized practice. Further,that reflective practice is enhanced when the teacher’s efforts to reflect andreform are based on a foundation of collaborative support, as was the caseof the teacher described herein.

A reflective model of the process of teacher change (adapted fromEdwards, 1996) may be used to undergird the study of mathematicsteachers’ attempts to reform their teaching. In this model, a teacher’sbeliefs underlie and interact with a cyclic process of change that progressesfrom interaction to perturbation to change and back again to repeat thecycle. The teacher’s reflective activity is posited as the context whichconnects the points in the cycle with each other as well as with theteacher’s belief structure. This cyclical process, embedded in a reflectiveenvironment, characterizes the change process described in this article.


COLLABORATIVE ACTION RESEARCH

Schön (1983) argued that instantiations of his reflection-in-action modelwill require some degree of collaboration, noting that:

The development of action science cannot be achieved by researchers who keep themselvesremoved from contexts of action, nor by practitioners who have limited time, inclination, orcompetence for systematic reflection. Its development will require new ways of integratingreflective research and practice. (p. 320)

McLaughlin (1990) built on this premise, suggesting that engagingteachers in professional collaboratives holds promise as a means ofeffecting change in classroom practices.

Miller and Pine (1990) defined action research as a procedure bywhich teachers examine the process of teaching and learning in theirown classrooms “through descriptive reporting, purposeful conversation,collegial sharing, and critical reflection” (p. 57). Clift, Veal, Johnson, andHolland (1990) characterizedcollaborativeaction research as focusing onpractical problems of individual teachers as they interact with universitystaff. In addition to providing a context for collaboration and supportfor a teacher’s attempts to change her teaching practices, such collabo-ratives also hold rich potential for inducing a reflective teaching practice(Raymond, 1996).

Perhaps the greatest strength of collaborative action research is itsfocus on “issues that hold interest to both university and school partners”(Raymond, 1996, p. 2). This focus then provides a context within whichthe practical knowledge of the teacher and empirical premises derivedfrom research might interact. Such interactions between the partners in anaction research collaborative may very well provide an arena in which “thepublic criteria of discursive reasoning can be used to facilitate the growthof knowledge about teaching” (Orton, 1991, p. 23).

Collaborative action research also seems a good fit with at least twoprerequisites to success widely recognized in studies of innovation: sitelevel emphasis and continuing staff development focused on classroom-level implications (Levine & Cooper, 1991). The power of these twofactors might well be multiplied in the case of collaborative actionresearch, because a site level emphasis and classroom focus are inherent tothe collaboration.


A MIDDLE SCHOOL MATHEMATICS COLLABORATIVE

The authors, a middle school mathematics teacher and a mathematicsteacher educator, have been engaged in a collaboration which has beenongoing for three years. They originally met during a six-week summerprogram for middle school teachers aimed at increasing teachers’ use ofinquiry methods during mathematics instruction. The collaboration hastaken on a number of forms, including:

• collaborative planning of term activities, as well as individual lessons,• lessons taught by the teacher educator and observed by the middle

school teacher,• lessons taught by the middle school teacher and observed by the

teacher educator,• lessons co-taught by the collaborators, and• debriefing following each lesson taught as part of the collaborative.

The benefits that each perceived herself or himself receiving from thecollaboration have been reported elsewhere (Edwards & Hensien, 1997).

This action research collaborative has focused on the teacher’s effortsto change her practice in the direction of the NCTM Standards vision. Theproject has included all of the elements cited by Miller and Pine (1990) intheir definition of action research:

• descriptive reporting, which includes the teacher’s own writtennarrative in which she describes changes in her instructional practicethat have occurred over the course of the collaboration, as well as theteacher educator’s reactions to it;

• purposeful conversation, including identifying big ideas in thecurriculum, as well as planning individual lessons and activities;

• collegial sharing, in which each partner actively seeks the other’sinput when planning or debriefing; and

• critical reflection, in which each analyzes and evaluates the other’swork, as well as her or his own.

Because the collaborators’ first association with each other was asstudent and teacher, the teacher educator has taken care to develop a senseof collegiality throughout their collaboration. For example, because themiddle school teacher perceived the teacher educator as the mathematicsexpert, the teacher educator has relied on the middle school teacher’sknowledge of the ways in which middle school students learn in anattempt to cast her as the middle school expert. In fact, all of the teachereducator’s prior classroom experience was in grades 7–12, and most of thatin grades 11 and 12, so he was somewhat apprehensive about the prospect


of teaching fifth-grade students. Therefore, it was very natural for him todraw on the middle school teacher’s experience with fifth-grade students.

Thus, the collaborators usually acted on an equal footing as colleagues,rather than as university professor and middle school teacher. The resultingcollegiality provided an atmosphere of mutual trust and comfort in whichthe collaborators often shared their beliefs about mathematics and itslearning and teaching. This sharing was also present during the frequentsessions in which they engaged in collaborative planning, as well as duringdebriefing.

This report focuses on the teacher’s narrative and descriptive report ofher work and the changes she made, or attempted to make, in her prac-tice of teaching mathematics. Although the formal writing of a narrativedescription of one’s own work clearly provides a context for reflectionon that work, this activity also provides a context for ongoing support ofthe teacher’s efforts to change (Schifter, 1996). In this way, the collabo-rative included three variables critical to successful change initiatives byproviding the teacher with:

• collaboration and support,• a non-threatening context within which to examine her beliefs, and• regular opportunities for reflecting on her practice.

The next section contains a descriptive narrative of the collaborationwritten by the middle school teacher, Sarah. It is followed by reflections ofthe teacher educator, Tom, on the narrative from the perspective of changein the teacher’s beliefs and practices during the collaboration.

SARAH’S STORY OF CHANGE

The first few years of teaching can be somewhat intimidating and lonelyregardless of the particular circumstances. Although the freedom of havingyour own classroom can be an exhilarating experience, it is still easy tolose your educational focus during those first few years in the classroom.It is easy to become so caught up in simply teaching objectives, that youneglect to focus on the “how” and “why” of your lessons. The culprit maybe that very few educators challenge themselves to work collaborativelyon a regular basis to share teaching practices with others. With even oneother person’s help on a regular basis in preparing lesson plans or puttingtheories into practice, some amazing things can happen.

Two years ago, I began a collaborative partnership with Tom, aprofessor of mathematics education whom I had met when I was a partic-ipant in an inquiry math program two summers earlier. As a third year


teacher at that time, I was eager to try new techniques in the classroom,yet nervous to expose the truth about how little I knew about teachingand my curriculum. In these two short years, the incredible possibilitiesof collaborative teaching have become very clear to me. Not only has mypreparation for a typical school year changed dramatically, but my tech-niques and my view of the teacher’s and the students’ roles in a successfulclassroom have also changed.

During the summer months before my first two years of teaching ina public school setting, a typical way I prepared for the year was tolook through my mathematics and science textbooks, chapter by chapter,to familiarize myself with the material. I never questioned the scope orsequence of the material, and I rarely conversed with other teachers asto the importance of teaching certain objectives a certain way. I viewedthe sequence almost as something written in stone. As a result, I regardedall of my educational objectives as separate components. At the sametime, I wondered why my students did not grasp any connections betweensubjects, or even between topics within the same subject.

Since the beginning of our collaboration, it is customary for Tomand I to lay out a sequence of our educational objectives before eachquarter throughout the school year. More often than not, that layout hasnot mimicked the sequence proposed in the textbook. Together, we havecreated new sequences within a given topic, as well as across topics inthe curriculum. Each quarter, we have become more adventurous in ourplanning and the activities we use. Our scope and sequence even changesfrom year to year, because we have attempted to accommodate individualand class differences. The amount of material my students are retainingseems to be increasing. I believe this is due to my own much greaterunderstanding and sense of the purpose of our educational objectives.

I have discovered that I am even further challenged by our discussionsof how we are going to accomplish our goals and objectives. I often findmyself reflecting on my prior teaching practices. Earlier in my career, Iconsidered myself a thorough teacher if I simply wrote down my objec-tives for the year. Now I am beginning to see the value of determininghow we will accomplish our goals and objectives. This is often done bybrainstorming activities that students may complete in groups in order todevelop and reinforce the objectives that we are teaching. These groupactivities have proven to be rewarding for the students and for us, becausethey promote the idea of teamwork and provide an interactive learningenvironment that is fun for students.

As an extension to these group lessons, I have practiced posing “How?”and “Why?” questions such as “How does this concept relate to the real


world?” or “Why are we learning this?” I find the students’ ideas usuallygo well beyond thoughts I had considered.

Very soon after our collaboration began, I discovered that there weregaps not only in the layout of my lesson plans, but also in my ability tohelp my students make connections between one idea and another. I wasunable to step back and see the reality that I was not making connectionswithin the subject matter. Consequently, neither were my students. Tomhas become the catalyst in my classroom for connecting concrete andabstract representations of mathematical concepts. He has used illustra-tions, manipulatives, and technology to help make these connections. Forexample, his use of the TI-92 overhead calculator exposed this year’s fifthgrade class to the angle-sum theorem for triangles in a way they may neveragain see, not even in a high school geometry course. Without Tom’s will-ingness to share his knowledge of technology, I believe my students wouldbe less able to link the concrete with the abstract.

In the years prior to our collaboration, I did not regularly use manipula-tives. By observing Tom as he incorporates manipulatives into almost anygiven lesson, my comfort level with manipulatives has increased tremend-ously. For example, I now use base 10 blocks to teach place-value, boardsand ball-bearings to experiment with inclined planes, dice and spinners forprobability experiments, and square tiles to explore concepts of perimeterand area.

Since I have increased my use of manipulatives, I have noticed a consid-erable increase in the amount of material my students retain. The recitationof formulas in my classroom has decreased, and I am pleased to see that mystudents’ manipulation of objects has helped them to reason deductively ontheir own and come up with their own formulas and theories.

The teaching techniques that I am currently using in my classroom havea much different flavor than those I used before Tom and I began workingtogether. At one time, I placed a lot of emphasis on steering my students tothe one answer I was looking for. I asked questions such as, “How do youfind the area of a rectangle with a length of 7 feet and a width of 4 feet?”This technique, I discovered, was not effective in telling me which studentstruly understood the concept of area. Instead, it set up an immediate barrierbetween myself and those students who may not have known the exactformula for finding the area of a rectangle. Furthermore, it led me to falselybelieve that the students who actually told me the correct formula had acomplete understanding of the concept of area.

From watching the techniques that Tom uses to check for understand-ing, I am now learning to use more open-ended questions. For example,if students have a number of square tiles arranged in a rectangle, I might


ask if anyone can figure out the area by pointing at the tiles. This tech-nique allows everyone to be an active participant, whether individuals arefamiliar with the area formula or not. While one student explains her wayof finding the area, there may be others who can find a different way. Later,when the correct formula is given, I ask why the formula makes sense.Doing so helps students connect their more concrete counting strategy withthe more abstract formula. This method encourages all types of learnersto get involved. At the same time, students are better able to retain theinformation, because they have been able to both visualize and explain aformula they have themselves discovered.

Having had more experience with asking open-ended questions, I noticethat I am now focusing more of the activity in my classroom on mystudents’ knowledge than on my own. The more I attempt to focus onfollowing the students’ lead in the discussion, the better I can learn what itis they know and understand and what remains for them to learn. By givingstudents the responsibility to lead the class discussions, I am realizing howmuch more challenging, yet rewarding, my job becomes. My studentshave begun to accept the reality that when they have a question, theymay find another question coming from me, in return, rather than a quickanswer. Gradually, I have witnessed a small evolution in my classroomas the children change from needy, answer-driven students to thoughtful,problem-solving mathematicians. I have learned that asking open-endedquestions requires more effort from both the teacher and the students,but the rewards are so much greater for both. The paths that my studentshave carried us down during our discussion times have caused me to workharder to familiarize myself with the “bigger picture” of the specific objec-tives of my lessons. At the same time, my students’ understandings havemoved well beyond what they would have expected of themselves.

When Tom and I exchange roles in the classroom, it has made me awareof how beneficial the collaboration is for all of the parties involved. Forexample, when Tom is teaching, I take notes and write down suggestionsabout other things he could have included in the lesson. Similarly, heobserves my lessons and gives me feedback. Tom has been very helpfulby filling in gaps in my own knowledge base. This constructive criticismhas helped me to see how far I have come and how far I have to go. Thequestions Tom writes down for me at the end of a lesson I have taughtare as open-ended and helpful to me as the ones he asks our students. Hisquestions have challenged me to go one step further than I thought I could.Reflection has become a daily routine as I plan for the next lesson. Tomand I often discuss the types of questions that would be appropriate, and


the students benefit from a very well-rounded lesson that has been shapedby two different perspectives.

Working with someone who has a much greater knowledge of math-ematics and science, I have witnessed a much deeper understandingfifth-grade students can have than I ever expected. Tom has been an excel-lent model of how actively teachers might challenge higher-level thinkersby not answering their questions, but rather by allowing them to ponderquestions for a while. Ironically, as I have watched him challenge mystudents, I have realized he uses the same techniques with me during ourcollaboration. As a result, I am more aware of shortcomings in my teachingthan I ever was before we began our partnership. Although some mightview this as a negative sign, I interpret it as a positive sign of progress inimproving my practice of teaching. I have learned to set goals for myselfthat two years ago would have been unimaginable.

There are so many benefits from collaborative partnerships that theyare almost immeasurable. An important lesson I have learned is that beinga successful teacher is like taking a journey. There are many paths thatcan be taken to achieve the same goal, and the involvement of manypeople makes the journey all the richer. Sometimes having that one otherperspective allows you to observe things that previously you would havemissed completely. Most of all, though, I have learned that a teacher shouldnever feel completely settled, for a teacher’s work is never done.

TOM’S INTERPRETATION OF SARAH’S STORY OF CHANGE

Sarah’s narrative provides a window through which we might view herprocess of change in beliefs and practices. Doing so can help us understandher process of change, as well as how the change process might unfold inothers. Aspects of the narrative can be interpreted as instances of changein both her beliefs and practices. Moreover, her narrative can be inter-preted as evidence of the importance of the collaboration itself, with itsinherent support, to Sarah’s process of change. Finally, throughout Sarah’sdescription of her own process of change, one senses the importance of herreflective thoughts to the change process. What follows is my interpretationof Sarah’s change in beliefs and practice.

Change in Beliefs and Practices

When Sarah writes, “Not only has my preparation for a typical school yearchanged dramatically, but my techniques and my view of the teacher’s andthe students’ roles in a successful classroom have also changed,” her state-


ment may be interpreted as evidence of change in practices (preparation,techniques), as well as change in beliefs (teacher’s and students’ roles).Sarah elaborates the changes in practice she believes she has made in anumber of ways, citing changes in planning, questioning patterns, the useof manipulatives, class discussions, emphasis, and locus of activity.

Sarah’s narrative highlights the changes that I, the collaborating teachereducator, have indeed observed. At the beginning of the collaboration,Sarah was a teacher in transition in the sense that her practice of teachinghad been perturbed. She knew of the reform movement in mathematicseducation and aspired to move her practice in that direction. However, sheseemed uncertain of how to do so and apprehensive to “expose the truthabout how little I knew about teaching.” As the collaborative progressed,I witnessed movement in Sarah’s mathematics teaching toward a morereform-based vision. There also appeared to be an increase in Sarah’sconfidence regarding her own teaching.

For example, one of the most difficult aspects of inquiry-based instruc-tion in mathematics is the tension between providing information tostudents and allowing students to grapple with an idea. As most teacherswould, Sarah often struggled to resolve this tension. About halfwaythrough the second year of the collaboration, Sarah was teaching a lessonon area and perimeter using plastic tiles. Students had worked in groupsto form rectangles from 24 tiles and, without recourse to any formula,determined the perimeter of the rectangles they had formed. In the classdiscussion which followed, Sarah called on students from the variousgroups to come to the overhead projector and share their responses tothe activity. During this discussion, a question arose: Is a 4x6 rectangledifferent from a 6×4 rectangle?

Sarah certainly could have answered the question herself, and a yearand a half earlier she probably would have. But here she turned the ques-tion back to the students, asking, “Well, what do you think?” The studentsshared their ideas, and one student came to the overhead projector to phys-ically turn a 4x6 rectangle to show how she saw the two rectangles tobe the same. The class, not just Sarah, eventually decided in favor of theequivalence of all such pairs of rectangles.

Throughout the activity, the students, for the most part, counted thenumber of tiles around the outside edges of the rectangle to arrive at theperimeter. Some used the short-cut of counting halfway around and doub-ling. After several students had used this short-cut, it would have been easyfor Sarah to move the class in the direction of a formula for the perimeterof a rectangle. However, she did not. Rather, it was one of the students


who did so, articulating the formula as, “you could just add the length andwidth together and then double that.’

Both of us believe that the courage and self-confidence that allowedthese class discussions to unfold as they did developed over some timeand as a result of our collaboration. From Sarah’s story we can see thatthe modeling of instruction was important to her development and change.What seemed to be particularly important was not only the modeling ofteaching during the six-week professional development program, but moreso the modeling that occurred inher classroom withher students. HereSarah had the opportunity to observe not only reform-oriented teaching,but also the students’ reactions to such instruction.

Sarah frequently mentioned the power of these observations. Shebelieved that, freed of the responsibility to lead the instruction, she wasable to focus more of her attention on her students and the ways in whichtheir understandings developed. Thus, Sarah saw first-hand what studentsare capable of. On the basis of this new knowledge, she then began toconstruct a vision of what her instructional practice might become.

Although it can be argued that changes in practice must be accom-panied, perhaps even preceded, by changes in beliefs, most of the changein beliefs that Sarah articulates seems to be related to her view of how herstudents learn mathematics and what they are capable of. Nowhere is thismore apparent than in her description of a metamorphosis in her studentsfrom “needy” and “answer-driven” to “thoughtful” and “problem-solving.”

However, I regard the metamorphosis Sarah describes as evidence ofchange in her beliefs as much as any change in her students. In fact, Ihave taught lessons in Sarah’s classroom in three different school yearsand at various times of the year, from September through May. At notime did I perceive Sarah’s students to be particularly “needy” or “answer-driven,” and I was frequently impressed by the thoughtfulness of theirresponses to my teaching. From my perspective, Sarah has progressed fromviewing her students as “needy” and “answer-driven” to seeing more oftheir potential as thoughtful problem-solvers. If my analysis is correct, itmay be that a conceptual shift such as that observed in Sarah’s case maytrigger a transition mechanism between levels of teachers’ understandingsof reform-based mathematics teaching (Goldsmith & Schifter, 1997).

This interpretation of Sarah’s changes in beliefs and practices iscompatible with Nelson’s (1993) observation that “changes in teachers’beliefs about teaching and learning appear to have taken place in inter-action with changes in their classroom teaching practice” (pp. 7–8).Moreover, the suggestion that the relationship between beliefs and prac-tices is a dialectic one (Nelson, 1993; Thompson, 1992) seems borne out


by Sarah’s narrative. For example, on the one hand, her change in beliefabout evidence of student understanding seems to have led to her morefrequent use of open-ended questions. On the other hand, her increaseduse of open-ended questions seems to have changed her belief about whatthe focus of classroom activity ought to be: her students’ knowledge ratherthan her own knowledge. This dialectic example emphasizes the impor-tance of understanding the cyclic, interactive nature of the change process,as suggested by Edwards (1996).

Collaboration and Support

The importance of collaboration and support is a persistent theme inSarah’s narrative. Right from the start, she writes about the opportunity towork collaboratively and share teaching practices with others. She continu-ally returns to the theme of collaboration in discussing changes in herbeliefs and practices. Although she does not use the termsupportwhenshe describes the collaboration, she does describe what may be inter-preted as aspects of support. For example, she mentions the feedback,the constructive criticism, and the help she received from her collaborator.Sarah summarizes the importance of the collaboration to her process ofchange when she writes, “Sometimes having that one other perspectiveallows you to observe things that previously you would have missedcompletely.”

I believe that the mutual feedback that occurred throughout our collabo-ration provided Sarah a means by which she could enact change. Feedbackseemed to perturb Sarah’s view of her own teaching and provided her witha context for critically reflecting on her teaching. On the other hand, thegiven and received feedback also provided me with a context to reflect onmy own activity.

At the beginning of the collaboration, my view of what I might modelfor Sarah involved only my work with the students. Thus, I consciouslymodeled the use of manipulatives, cooperative learning groups, open-ended questioning techniques, allowing student ideas to frame the classdiscussions, and the like. I did not at first realize that the modeling couldalso extend to the debriefings which followed every lesson where one ofus observed the other teaching, as well as to our collaborative planningsessions. As the collaboration unfolded, I became aware of the opportunityto model reflective activity during the debriefing sessions, particularlyduring those which followed the lessons I had taught. Consequently, Ibegan to consciously model the turns of my own reflections by beginningto focus on student understanding and the reasons underlying apparentlysuccessful lessons or activities. I started to ask Sarah things like, “What


do you think the kids are understanding about. . . ,” or, “Why do you thinkthat worked so well?”

I became aware of the modeling potential of the collaborative plan-ning sessions only after reading Sarah’s narrative. I was intrigued by theimportance she seemed to place on those planning sessions, especially inthe beginning of the cooperation. In retrospect, I see two possible ways inwhich I might have unconsciously been modeling aspects of my practiceof teaching for Sarah during the planning sessions.

The first of these is related to the mathematical connections whichSarah laments having not been made prior to the collaboration by eitherherself or her students. Throughout the collaborative planning sessions, Ialways attended to such connections, because this is an aspect of my math-ematics teaching practice. Apparently, this activity was not lost on Sarah.She writes, for example, “Tom has become the catalyst in my classroomfor connecting concrete and abstract representations of mathematicalconcepts.”

The second way in which I might have been modeling aspects of mypractice for Sarah during the planning sessions is related to Sarah’s appar-ently new-found focus on students’ knowledge. Although in the narrativeSarah attributes this to her expanding experience with asking open-endedquestions, I believe that the collaborative planning may also have playeda role. As a habit of practice, I always attempted to connect instructionto students’ prior knowledge. Thus, when I asked Sarah if students wouldknow or be familiar with certain mathematical concepts or representations,she seemed to increasingly focus her attention on her students’ knowledge.

Reflective Activity

Toward the end of her narrative, Sarah writes, “Reflection has becomea daily routine.” Throughout the narrative, she describes some of herreflective activities. She discusses reflecting on the scope and sequenceof her mathematics curriculum, her educational objectives, and her lessonplanning and preparation. She reflects on student understanding and reten-tion of skills and concepts, her use of manipulative materials, her increaseduse of open-ended questions, and the connections her students madebetween the concrete and the abstract. She envisioned not only the chal-lenge of understanding her students’mathematicalthinking but also thechallenge of considering her ownpedagogicalthinking. Thus, she hascome to recognize the importance of reflection to her own learning process.

At first, Sarah focused her reflections on comparisons between herteaching prior to the collaboration and that during our collaborative effort.Reflection seemed to mean noting differences between what she and I


were doing now with what she had done previously. During the secondyear of the collaboration, there were shifts in both the content and thenature of Sarah’s reflections. She began to critically examine what she, orI, was doing in the classroom, often asking why a particular lesson hadbeen effective. At this point, reflection for Sarah seemed to involve ananalytic/synthetic consideration of teaching. Finally, near the end of thesecond year, Sarah began to seriously reflect on her students’ knowledge,and during the third year of the collaboration, she had begun to ask herselfwhat her students understood. In these instances, Sarah seems to regardreflection as a critical analysis of student learning.

Sarah now regularly measures her work against a vision of mathematicsteaching that she has shaped from her interpretation of the NCTMStan-dardsand our collaborative efforts. The power of these regular reflectionsas agents of change lies in the content of those reflections, for “if weare interested in change that is significant and worthwhile, the contentof reflection should relate to standards of appropriate classroom practice”(Richardson, 1990, p. 13).

CONCLUSION

We have outlined ways in which we believe our collaborative actionresearch project has served as a catalyst for change in a middle schoolmathematics teacher’s beliefs and instructional practices. Our collabora-tion involved three features McLaughlin (1990) deemed necessary forthe enhancement of classroom practices, viz., regular feedback to theteacher concerning her work with students, a place for her voice incurricular decision-making, and a high level of collegial interaction. Thesefeatures permeated our collaborative effort; their importance is evidencedthroughout Sarah’s narrative. Further, we believe that our action researchcollaborative demonstrates at least six of Clarke’s (1994) ten key principlesfor professional development, specifically,

• addressing issues of concern and interest identified by the teacherherself, thus involving a level of individual choice for the teacher;

• using teachersand real students engaged in classroom activities tomodel and project a clear vision of the proposed change;

• recognizing that changes in a teacher’s beliefs about teaching andlearning are intimately entwined with her classroom practice;

• allowing sufficient time for planning, reflection, and feedback;• enabling the participant teacher to gain substantial ownership of the

project by allowing for shared decision-making and recognizing heras an equal partner in the collaboration; and


• providing ongoing and critical support, because “change is a gradual,difficult, and often painful process” (Clarke, 1994, p. 45).

As important as we consider these features, we believe that the mostsignificant factor for change was the long-term and mutually reinforcingnature of our relationship. This relationship provided a context in whichthe relevant features identified by Clarke promoted professional develop-ment. Change is gradual and difficult and having a collaborative friendthroughout the change process helps the aforementioned features take holdand numbs some of the inevitable pain associated with reforming one’steaching.

Throughout the project, the teacher provided most of the directionby identifying the broad areas within which we would collaborate. Forexample, her first request was to do something with probability and statis-tics with her students. Even after such broad areas had been identified,Sarah continued to be involved in the decision-making. Her advice wassought on proposed lesson alternatives, and the decision as to whichone(s) to use was shared. Thus, the teacher was not only involved inthe decision-making process, but was recognized as an equal partner inthe collaboration. Although it was often the case, especially early in thecollaboration, that the teacher educator generated lesson alternatives, itwas the teacher who refined the alternatives to better meet the needs ofher students.

The teacher’s change process is consistent with the reflective, construc-tivist model for understanding teacher change that was posited by Edwards(1996). Her narrative provides testimony to the nature of a cyclicalinteraction-perturbation-change learning process when that process iscouched in a reflective and collaborative environment. We surmise thatthis learning process is not limited to one teacher’s professional develop-ment but could characterize other teachers’ learning as well. Sarah’s case,along with others, can produce a coherent mosaic that can illuminate ourmore general understanding of the process of teacher change. Our story isintended to contribute to this general aim.

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THOMAS G. EDWARDS SARA M. HENSIEN

Wayne State University Eagle Elementary SchoolDetroit, Michigan, USA W. Bloomfield, Michigan, USA


EDUCATING PRIMARY SCHOOL MATHEMATICS TEACHERS INTHE NETHERLANDS: BACK TO THE CLASSROOM

FRED GOFFREE and WIL OONK

EARLY APPROACHES TO TEACHER EDUCATION

Before 1800, Dutch primary teachers were not specifically trained forteaching as such. When applying for a teaching post, it was sufficient forapplicants to demonstrate a sound understanding of the subjects they wereexpected to teach. In those days the best pupil from the graduating class ofthe primary school would be chosen to assist the head teacher on a regularbasis. Sitting in the back of the classroom, the pupil would observe theteacher and learn the art of teaching in an informal way. In order to learnmore about the subject he or she was going to teach, the pupil would takeadditional lessons at home from the head teacher. These private lessonslater became more systematic or normalized and became known as normallessons. The normal schools that evolved from this practice later becamethe teacher training colleges. The first such colleges in the Netherlandswere established by the government around 1800. Many more teachercolleges quickly followed, including colleges affiliated with Protestant orRoman Catholic churches. Most towns had at least one teacher college,general and/or church affiliated.

Until 1952, the curricula of these training institutes were essentiallythe same as those for higher secondary schools, albeit with the addi-tion of pedagogy and teaching methodologies and with half a day aweek allocated for working in the schools. Teacher training was radi-cally changed, however, as a result of the New Training College Act of1952. The school subjects of secondary education were replaced withthe teaching methods for the subjects taught in primary schools. Forexample, mathematics was replaced with teaching methods for arith-metic. However, because the teacher educators remained the same, littlechanged in practice. The teaching methods for arithmetic were frequentlyaugmented with tough calculations for the student teachers, supplementedwith tips for working in the classroom. Most of these tips were of a


208 FRED GOFFREE AND WIL OONK

general educational nature, e.g., they referred to teaching with visual aids,free activity, and elements of educational psychology such as differentlevels of thinking. In 1968 teacher-training colleges were renamed, by law,as Pedagogical Academies. Behind the scenes, government commissionswere busy thinking up a new programme and a new approach to teachertraining for primary education. The discussions were about issues such asextending the programme from 3 to 4 years, placing a central focus onpractice teaching, embracing competency based and humanistic curricula,and considering the consequences of bringing together teacher educationfor primary schools (ages 6–12) and infant schools (ages 4–6).

THE WISKOBAS MODEL FOR TEACHER EDUCATION

With the impetus of the world-wide New Math movement behind him,Hans Freudenthal established the Institute for the Development of Math-ematics Education (IOWO) and set into motion the Wiskobas project(Freudenthal, 1978; Treffers, 1978). The Wiskobas (which can be trans-lated as Mathematics in the Elementary School) project was intended toinfluence mathematics education at the national level through the trainingof teachers. Shortly after the first Wiskobas curriculum publications onprimary education went to press, an educational experiment was startedat a pedagogical academy in Gorinchem (Goffree, 1977). Every week,Freudenthal and two members of the Wiskobas project attended thelectures given to the student teachers and went to the school where theteachers acquired practical experience. Freudenthal worked with childrento show the student teachers how it would be possible to initiate andobserve mathematical learning processes. His observations and analyseswere intended to impress on the student teachers the idea of the teacheras a researcher and give them the feeling that there was much thatcould be learned from the children themselves. He also tried to bring anarrative element into the teaching with stories such as “Walking withBastiaan” (Freudenthal, 1977). Goffree (1979) has provided a descriptionof the impact of Freudenthal’s approach on the pedagogical academy inGorinchem.

Materials for the student teachers were developed and tested. Freud-enthal made theoretical contributions to these materials as evidenced in hischapters “The phenomenon of ratio” and “Measurement as phenomenon”(see Freudenthal, 1983). Both of these chapters represented Freudenthal’sapproach to educational phenomenology of mathematical structures andwere made available to his IOWO colleagues for comment years inadvance of their publication.

TEACHER EDUCATION AROUND THE WORLD 209

Figure 1. Model for primary mathematics teacher education.

A MODEL FOR LEARNING TO TEACH

The learning model depicted in Figure 1, sometimes referred to as theAeroplane Model, shows that mathematics education, both for studentteachers and pupils, takes as its starting point concrete situations andfamiliar contexts. The model is based on the concrete and the familiarand the activation of pupil’s subjective structures (the informal mathe-matics) which then allows the mathematical learning process to start withpupils’ intuitive notions and informal procedures. In a sense, this is whatFreudenthal meant with his educational phenomenological analyses: math-ematics comes into being in reality, mathematics provides the structuresfor this reality, reality gives meaning to mathematics and, in well-chosenreal-world situations, pupils get the opportunity to rediscover mathematicsunder the guidance of the teacher who knows the objective structures ofmathematics (formal mathematics) (see Freudenthal, 1983).

The subjective structure of a student teacher is affected by his or herearlier experiences with learning mathematics and teaching mathematics.While mathematization plays an important part in the learning processesof children, for the student teachers it is a process of both mathematisingand didactising (Freudenthal, 1991, p. 174). Student teachers carry outmathematical activities at pupils’ own levels and then reflect on and discussin small groups with their peers the results of those activities, often guidedby questions such as, “How do children learn?” and “How do we teach


children?” These reflective discussions create a foundation for learninghow to work with children.

Reflections on children’s learning processes combined with the studentteacher’s own experiences in learning mathematics contributes to thecreation of an educational basis for teaching mathematics to youngchildren. Sometimes big ideas from general educational theory, rooted ineither pedagogy or formal mathematics, can also make a contribution. Inthis way, the student teacher gets into a cyclical process in which mathema-tical problems, mathematization, reflective problem solving, and masteringteaching methods follow naturally from each other and, in each successivedomain, become more sophisticated. During this process, student teacherswork with children and study their learning processes while continu-ally referring back to their own learning process. This succession ofprocesses strongly resembles the cyclical processes of action research usedin projects in which the “teacher as researcher” is the object of study(Jaworski, 1998).

During the 1980s, the aeroplane model (Figure 1) was developedinto a three-volume text, collectively known as theMathematics andDidactics for Primary Mathematics Students. At that time, some devel-opment research was being carried out at the University of Utrecht asa continuation of the IOWO Project. This work led to a pilot studyfor a national programme for realistic mathematics education in primaryschools (Treffers, De Moor & Feys, 1989). It also provided the authors ofMathematics and Didacticswith useful material for student teachers whoincreasingly recognized the Wiskobas approach in the new mathematicsschoolbooks then coming onto the market. Newly-developed courses inthe areas of counting, calculations up to a hundred, doing sums, frac-tions, decimals, ratios, mensuration, and geometry were also developedfor student teachers along the lines of the aeroplane model.

Teacher anecdotes from the research project also fit well into teachereducation and often enlivened the working material for student teachers.After some years, the so-called reflective solutions of mathematical/didactical exercises were collected and distributed to encourage studentteachers to be self-motivated and to develop independent activities. Theresulting books were used in more than 80% of the teacher trainingcolleges.

THE NEED FOR A NEW PARADIGM

In 1984, Pedagogical Academies were again renamed as PABOs (Collegesof Education), but it was not until the early 1990s that any significant


changes occurred. The cause of these changes lay in a large-scale inspec-tion of all PABOs in 1991 – the first inspection of its kind. The judgmentof the inspection was damming. The criticism was mainly directed towardthe lack of a good academic background for primary school teachers and ofa clear training concept involving teaching methods. Further, the colleges’ability to change was called into question and the management got a “fail.”Under the threat of subsidy cuts, the PABOs were put under pressure, andchange became the new credo. At the insistence of the Ministry of Educa-tion, a large number of PABOs were incorporated into Colleges of HigherEducation (where the emphasis was on vocational education in general).Now it was a matter of creating independent business management withinthe existing frameworks.

In the middle of all of this turbulence, the curriculum for teacher educa-tion also had to be revised. This time the curriculum had to be basedon a well-defined training concept and fit within the framework of aspecific teacher education pedagogy that still needed refining. Problem-based learning and thematic education were espoused, and teachers fromall disciplines were expected to develop their own materials according tothese two concepts. Again, the colleges were required to leave behind theparadigm of a programme dominated by the school subjects and to lookfor themes, case studies, and problems that would have obvious validity tothe study of teaching per se.

A NATIONAL APPROACH TO TRAINING PRIMARYTEACHERS

While all of this was going on, a group of ten mathematics and didacticseducators began to work on a pilot study for a national programme forthe PABO. This pilot work was based on and coordinated with the “PilotStudy for Primary Mathematics Education.” This new work became knownas the PUIK Project (PUIK: Dutch acronym for Programming, Outlining,Designing, and Quality) and was carried out under the auspices of theNational Institute for Curriculum Development (SLO). The results of thiswork were published in 1995. The core of the study consisted of eighteenstandards for primary school mathematics.

These standards as well as other parts of the report were translatedinto English for a discussion between the developers and fellow teachereducators from the United States (Goffree & Dolk, 1995). This visithad major consequences for the training of Dutch primary school math-ematics teachers. In particular, the Dutch trainers became acquainted withMagdelena Lampert and Deborah Ball’s MATH project (Lampert & Ball,


1998). The student learning environment developed in their project coin-cided with ideas developed in the Netherlands about the training of studentteachers and the way they learn. It was decided to develop an environmentfor student teachers in the Netherlands similar to that developed in Lampertand Ball’s project. The first prototype, called Petit-MILE, consisted mainlyof recordings of grade 3 lessons (daily for 5 weeks) and grade 7 lessons(daily for one week) which were subsequently arranged into video frag-ments with an average length of 1 to 2 minutes complete with transcriptsand a search engine. Each fragment was considered a separate narrativeand had its own title and abstract that summarized the essence of the story.The fragments could be accessed with the search engine (full text retrieval)or by using the archive.

The Minister of Education, Culture, and Welfare was pleased withthe progress made in the development of the PABO curriculum andmade his approval clear in two ways. First, he made funds availablefor extending Petit-MILE into MILE – a project that involved a greaterrange of grades. Secondly, a task force was established for developing acommon curriculum for all PABOs. In putting this curriculum together,the task force had to take into account a very recent and finely workedout publication, “Skills Required of Beginning Primary School Teachers”(SLO/VSLPC, 1997). This paper paid particular attention to the teachingof language and mathematics. After lengthy considerations and discus-sions, the task force decided that the present fear of the discipline-orientated learning plan should not be translated into discarding the schoolsubjects. Instead, a school subject was seen not purely as subject matter,but also as an area of knowledge and experience of learning and teaching,both within and outside school and with a distinct history and developmentof its own. This view of school subjects, which already existed for mathe-matics, made it possible to establish the working field of student teachersin two dimensions – skills required for beginning teachers and knowledgeof school subjects. The school subjects thus formed the working materialthat enabled student teachers to learn the trade of being a teacher.

The addition of reflections from a theory-based educational, psycho-logical, or pedagogical point of view enriches this work of the studentteachers. In this way, the student teachers also become involved inthe creation of practical knowledge, as has already been described formathematising and didactising.


Figure 2. A more recent model for primary mathematics teacher education (1998).

BACK TO THE CLASSROOM, BUT DIFFERENTLY

The MILE project fits very well with the model depicted in Figure 2.But the viability of MILE’s digital representation of school practice stillrequired an entirely different way of looking at things for the teachereducator. With MILE we are back in the classroom, and this means thata student teacher’s own classroom observations form a starting point foranalysis, reflection, and discussion and also form the basis for his/her ownquestions about teaching. The impetus for learning to teach is, therefore,initiated outside MILE. But MILE provides an enriched orientation and


provides direction to the subsequent analyses of teaching. Moreover, thestudent teacher can, if it helps, use theory as a vehicle for further exploringthe possibilities of using MILE. It should be emphasized that MILE hasbrought theory and practice closer together to an unprecedented extent.As the millennium approaches, back to the classroom will have majorconsequences for how student teachers are taught to teach.

REFERENCES

Freudenthal, H. (1973).Mathematics as a educational task. Dordrecht, The Netherlands:Reidel.

Freudenthal, H. (1977). Bastiaan’s experiments on Archimes.Educational Studies inMathematics, 8, 3–16.

Freudenthal, H. (1978).Weeding and sowing: Preface to a science of mathematicaleducation. Dordrecht, The Netherlands: Reidel.

Freudenthal, H. (1983).Didactical phenomenology of mathematical structures. Dordrecht,The Netherlands: Reidel.

Freudenthal, H. (1991).Revisiting mathematics education. China lectures. Dordrecht, TheNetherlands: Kluwer Academic Publishers.

Goffree, F. (1977). Johan, a teacher training freshman studying mathematics and didactics.Educational Studies in Mathematics, 8, 117–152.

Goffree, F. (1979). Learning to teach with Wiskobas (Doctoral dissertation, in Dutch).Utrecht, The Netherlands: IOWO.

Goffree, F. & Dolk, M. (1995).Standards for primary mathematics teacher education.Utrecht, The Netherlands: Freudenthal Institute.

Jaworski, B. (1998). Mathematics teacher research: Process, practice and the developmentof teaching.Journal of Mathematics Teacher Education, 1, 3–31.

Lampert, M., & Ball, D.L. (1998).Teaching, multimedia and mathematics: Investigationsof real practice.New York: Teachers College.

Treffers, A. (1978).Three dimensions: A model of goal and theory description inmathematics instruction – The Wiskobas Project. Dordrecht, The Netherlands: Reidel.

Treffers, A., De Moor, E. & Feys, E. (1989).Proeve van een nationaal programma voor hetreken-wiskundeonderwijs op de basisschool[Model of a national curriculum for primarymathematics education]. Tilburg, The Netherlands: Zwijsen.

SLO/VSLPC (1997).Startbekwaamheden leraar primair onderwijs. Deel 1: Startbek-waamheden en situaties[Skills Required of Beginning Primary School Teachers. Volume1: Skills and situations]. Utrecht, The Netherlands: APS.

Streefland, L. (Ed.) (1991).Realistic mathematics education in primary school. Onthe occasion of the opening of the Freudenthal Institute. Utrecht, The Netherlands:Freudenthal Institute.

FRED GOFFREE WIL OONK

Bremlaan 16 Orionlaan 963735 KJ Bosch en Duin 1223 AK HilversumEmail: [email protected] Email: [email protected]


THE MISSOURI MIDDLE MATHEMATICS (M3) PROJECT:STIMULATING STANDARDS-BASED REFORM?

BARBARA REYS, ROBERT REYS, JOHN BEEM and IRA PAPICK

The Missouri Middle Mathematics (M3) Project, a three-year teacherenhancement project funded by the National Science Foundation (NSF),utilized collaborative curriculum investigations as a vehicle for profes-sional development. The investigations were based on standards developedby the National Council of Teachers of Mathematics (NCTM, 1989, 1991,1995). Project activities, organization, and emphases were informed byvarious professional development models including those advocated byCarpenter et al. (1989), Wood et al. (1991), Ball (1996), Silver and Stein(1996), and Sowder et al. (1998). The major emphasis of M3 was onsupporting teachers as they considered and implemented new curriculummaterials and instructional strategies. Specifically, the M3 Project wasdeveloped based on the following assumptions:

• Professional development activities should engage teachers inlearning mathematics in ways that will encourage them to reflect onhow their students learn mathematics.

• Successful professional development activities are generally long-term and focus on specific issues of practice of concern to teachers.

• Curriculum materials are strong determinants of what mathematics istaught and how it is taught and therefore should be considered withinprofessional development activities.

• Innovative curriculum materials serve as a tangible focal point fordiscussion and consideration and facilitate teachers making initialchanges to practice.

• Significant curricular change requires collegial support as well assupport from school and district administrators.

What follows is a description of M3 Project goals and activities anda summary of what was learned regarding the use of collaborative

? Research reported herein was supported by a grant from the National Science Founda-tion (#ESI 9453932). The findings and opinions expressed are those of the authors and donot necessarily reflect either the position or policy of the NSF.


216 ROBERT REYS ET AL.

curriculum investigation as a vehicle for professional development andStandards-based reform.

PROJECT GOALS

The Missouri Middle Mathematics Project provided a forum for middle-grade teachers and administrators to investigate Standards-based mathe-matics curricula. Specifically, project staff organized a series of activitiesto support teams of teachers and administrators to carefully review,try out, discuss, and exchange ideas regarding innovative, middle-school mathematics curriculum. Four sets of curricula –Mathematicsin Context, Connected Mathematics, MathScape, andMath Thematics–each developed with support from the National Science Foundation, wereshowcased throughout the project. One hundred sixty teachers and admin-istrators from 23 Missouri school districts studied the materials on theirown and at summer institutes, at curriculum-review sessions during theacademic year, at regional meetings, and through e-mail conversations.They explored not only the materials but also strategies for implementinginstructional practices consistent with the philosophy of these Standards-based curricula. Project staff included a cadre of regional associates(university professors and mentor teachers) from various locations aroundthe state. The regional associates facilitated continued dialogue amongproject participants within a particular region. This local network wascrucial to maintaining the commitment and ongoing conversation amongparticipants. The primary goals of the project were to:

1. facilitate investigation and consideration of four NSF-sponsoredStandards-based middle-grade curricula;

2. provide a forum for exchanging stories and engaging in substantivediscussions regarding innovation in teaching practices and curriculummaterials;

3. develop a cadre of mentors (middle-school teachers and adminis-trators, mathematics coordinators, and university faculty) who willprovide statewide leadership and assistance to middle schools explor-ing the possibility of curriculum change; and

4. support demonstration sites where Standards-based mathematicscurricula are utilized and school personnel able and willing to modelcurricular and instructional changes for other middle-grade facultyconsidering change.

The M3 Project was developed based on the premise that significantchange in middle school mathematics programs does not just happen (Reys


et al., 1997). Neither can it be legislated or forced upon unwilling partici-pants. The project began with a nucleus of participants from throughoutthe State of Missouri and established a support system to stimulate, foster,and monitor instructional and curricular change with the goal of improvingthe learning of mathematics by all middle-grade students.

PROJECT ACTIVITIES

The 23 participating school districts were selected based on an expresseddesire and commitment to improve their middle-school mathematicsprogram. Districts agreed to provide resources (e.g., release days, field-test materials, e-mail network) for project teachers and administrators asthey studied and tried out materials and instructional strategies. Projectparticipants met together five times during the first year (one week-long summer conference and four curriculum-review conferences) andin smaller groups at various regional sites. The conferences providedopportunities to study theStandards(NCTM, 1989, 1991, 1995), considerissues such as the influence of technology on teaching and learningmathematics, examine particular themes in mathematics such as problemsolving, communication, and number sense, and investigate innovativemiddle-grades curriculum.

The four curriculum-review conferences, each two-and-a-half dayslong, were led by curriculum developers and included a focus on:Mathe-matics in Context(MiC), Connected Mathematics(CMP), Six ThroughEight Mathematics Project(STEM, now calledMath Thematics), andSeeing and Thinking Mathematically(STM, now called MathScape).Curriculum review conferences were held every other month so thatteachers could continue their study and try the materials (at least oneunit) with students. Regional meetings were held between conferencesto debrief teachers on the successes and challenges associated with usingthese materials.

During the second and third years of the project, participants continuedto meet regularly either regionally or as a statewide group. They wereencouraged to select a particular curriculum to study in more depth duringthe summer institutes. Statewide and regional conferences were held tofacilitate discussion and to focus on instructional changes prompted bythe curricula. Participants were also encouraged by the project staff andtheir own district administrators to visit other Missouri school sites toobserve other project teachers who used the curriculum materials. ByYear 3 of the project, several districts had chosen to adopt a particularcurriculum, whereas others were still reviewing materials. Project partici-


pants developed and presented sessions on the curricula to their districtcolleagues and to teachers from neighboring districts. In addition, M3

participants and staff organized and facilitated a statewide DisseminationConference open to teachers throughout Missouri who had not been apart of the M3 Project but wanted to learn about the NSF middle-schoolmathematics curricula. Throughout the project, some participants werevery active e-mail users, interacting with fellow project participants or withproject staff on a regular basis.

Another aspect of the model involved the M3 Project’s use of regionalassociates. This approach provided the first opportunity in the state’shistory for college mathematics faculty from several different campusesto collaborate and work directly with teachers and schools for a sustainedperiod of time. This model worked exceedingly well and provided directcontact with districts across the state that would have been impossible toreplicate from a single campus.

IMPACT OF THE M3 PROJECT

In addition to planning and facilitating activities and conversations, thestaff regularly collected information from individual participants, groupsof participants (district teams), and regional associates. Project staff metregularly to plan activities, review what participants were talking aboutand doing in their classrooms, and monitor the model of collaborativecurriculum investigation as a vehicle for professional development. Inaddition, each staff member reflected regularly via e-mail conversationsprompted by the project director.

Data were collected from individual participants via a pre- and apost-survey and through regular journal entries kept by participants on aregular basis. District teams were also asked to record their decisions andconversations. These data provide evidence of the impact of the model ofprofessional development. Some of the outcomes of the project were anti-cipated and others were not. Here we summarize the outcomes documentedto date.

Anticipated Results

Professional growth of participants.The M3 Project has a positive impacton participants’ perceptions about teaching, learning, and the implemen-tation of new teaching strategies. The process of reviewing and usingStandards-based middle-school curricula enabled them to deal with avariety of questions including the following:


− What mathematics should be emphasized in middle grades?− What impact does continual and extensive review have on students?− What impact does exposure to challenging tasks have on students?− How do opportunities to engage in reflective writing affect student

thinking?− What impact does the utilization of multiple strategies have on

students’ problem-solving ability?

The extent to which particular participants became more reflective andflexible in their approach to teaching varied. However, most agreed thatthe mathematics included in their current textbooks was not enabling mostof their students to succeed and to develop motivation to continue theirstudy of mathematics. More than 80% of the participants indicated thatthe focus of collaborative curriculum review of Standards-based materialshelped them gain a better awareness and understanding of the vision ofthe Standards. They agreed that the comprehensive sets of curriculummaterials offered an “existence proof” that the philosophy and contentoutlined in the Standards could be translated into useable materials.

Effectiveness of model.The use of a collaborative curriculum investigationas a vehicle for teacher enhancement provided an opportunity for manyissues on assessment, teaching practices, and content coverage to emerge.This model provided continuous opportunities for teachers to make smallchanges as they tried various units of the curriculum materials within asupport structure that involved dialogue with other teachers who wereimplementing similar changes.

Each element – collaboration, curriculum, and investigation – wasimportant in the model.Collaborationwithin and across districts providedeach participant an opportunity to hear different voices and viewpointsregarding new curriculum materials and instructional approaches. Thiscollaboration provided a rare opportunity for sustained discussion andinteraction and validated the importance of considering Standards-basedcurriculum. The Standards-basedcurriculum was quite different thanmaterials they had been using. Most participants came to the projectbelieving that the materials they were using were not meeting the needsof most of their students. Consequently, the teachers were “hungry” forbetter materials. They came to believe that curriculum that emphasizedimportant mathematics (including attention to algebra, geometry, statistics,and number) was important for all middle-school students. Because thefour Standards-based curricula differed in several ways from what theywere using, investigation over a period of time was essential. Thisinvestigationincluded hearing about the materials from an expert (e.g.,


curriculum developer or teacher user), studying the materials by doingmany of the activities, trying them out with their own students, andengaging in discussion with colleagues who had had similar experiences.

Impact on students.The focus of M3 Project was on middle-schoolteachers. However, as the participants used these Standards-based middle-grade mathematics curricula with their students, the impact on middle-grade students was immediate. Initially these curricula challengedstudents’ perceptions of “What is mathematics?” Many middle-schoolstudents expected mathematics to be computational, and therefore did notassociate problem solving, reading, and writing with mathematics. Instruc-tional approaches required more student participation and less teacherdirected activities. Group work and projects became commonplace inmathematics in classes. These changes were not consistent with moststudents’ previous encounters with mathematics and did not go unnoticed.Some students were excited and said, “Doing mathematics is better thanhearing about it,” whereas others said they were “more comfortable withthe old mathematics” (Reys et al., 1998). It became clear that even a fewlessons from these Standards-based curricula had an immediate impact onstudents.

Unanticipated Results

Algebra – a powder key.We never anticipated the explosiveness of thepowder keg associated with algebra in terms of the questions, “For whomis algebra intended?” and “What kind of algebra and when should it betaught?” These issues emerged quickly and continue to linger among themiddle-grades faculty. All participating districts allowed, even encour-aged, their best eighth graders to take algebra. They wondered if theeighth grade units from the new curricula were better for these studentsthan a traditional first-year algebra course. Participants were reluctant tochallenge the existing school structure (as well as the anticipated parentalpressure) to have algebra as an option in the eighth grade.

In addition to placement, the amount of algebra in the middle gradereform materials was also an issue. If students successfully completedgrades 6–8 using a reform-based curriculum, would they have experi-enced the equivalent of a year’s study of algebra? Research on this issueis desperately needed before the algebra issue can be laid to rest, ifindeed it can be. The issue of articulation between these Standards-basedmiddle-school mathematics and the integrated 9–12 mathematics curriculafor secondary schools provides additional concerns about the merits forcontinuing algebra as a separate course.


Value of professional dialogue.It was anticipated that the conferences andregional meetings would provide opportunities for professional growthamong all participants. The project provided regular opportunities forteachers to talk about the challenges they faced in implementing reform.What we did not expect was the need for teachers from the same school,or school district, to have time to share their experiences in trying toimplement reform (Reys & Reys, 1997). Teachers indicated that theirregular school calendar provided virtually no opportunities for them todiscuss their mathematics teaching. In order to capitalize on the power ofteachers talking with teachers, adjustments were made in the conferences,meetings, and regional discussion groups to promote more of theseexchanges and to share ideas. Everyone agreed that providing time forteachers to talk with teachers was time well spent.

Impact beyond the middle grades.By the second year of the project theteachers were using more and more of the Standards-based mathematicscurricula. But they also began to wonder about the transition frommiddle-school to secondary-school mathematics. The middle-schoolteachers began to talk with secondary mathematics teachers in theirdistrict about Standards-based mathematics. This stimulated importantreflection on current practices and encouraged a much needed dialogue. Italso encouraged more people within the district to consider mathematicspreparation from elementary through secondary as a whole and exploreways to insure smooth transition.

Impact on preservice teacher education.Direct contact with Standards-based mathematics curricula and interactions with participating teacherswho used these materials made a significant impact on the college facultyinvolved in the M3 Project. This experience led to considerable changein the methods and materials used with undergraduates who prepared tobecome middle-school mathematics teachers. Perhaps the most obviousevidence of this is the use of selected lessons, units or modules fromthe Standards-based middle-school mathematics curricula with preser-vice teachers. The mathematics embedded in these materials is rich andprovided an ideal context for significant mathematical exploration anddiscussion.

REFERENCES

Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we knowand what we need to learn.Phi Delta Kappan, 77(7), 500–508.


Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C.P. & Loef, M. (1989). Using knowl-edge of children’s mathematics thinking in classroom teaching. An experimental study.American Educational Research Journal, 26, 499–531.




Reys, R.E. & Reys, B.J. (1997). Standards-based mathematics curriculum reform: Impedi-ments and supportive structures.Journal of Mathematics Education Leadership(July),3–8.

Reys, B.J., Reys, R.E., Barnes, D., Beem, J. & Papick, I. (1997). Collaborative curriculumreview as a vehicle for teacher enhancement and mathematics curriculum reform.SchoolScience and Mathematics, 97(5), 253–259.

Reys, R.E., Reys, B.J., Barnes, D., Beem, J.K., Lapan, R. & Papick, I. (1998). Standards-based middle school mathematics curricula: What do students think? In L. Leutzinger(Ed.),Mathematics in the middle, 153–157. Reston, VA: National Council of Teachersof Mathematics.

Silver, E.A. & Stein, M.K. (1996). The Quasar Project: The “revolution of the possible”in mathematics instructional reform in urban middle schools.Urban Education, 20(4),476–521.

Sowder, J.T., Philipp, R.A., Armstrong, B.E. & Schappelle, B.P. (1998).Middle-gradeteachers’ knowledge and its relationship to instruction. Albany, NY: State Universityof New York Press.

Wood, T., Cobb, P. & Yackel, E. (1991). Change in teaching mathematics: A case study.American Educational Research Journal, 28(3), 587–616.

104 Stewart HallUniversity of MissouriColumbia, MO [email protected]

GUEST EDITORIAL

TEACHER GROWTH AND SCHOOL DEVELOPMENT

In the last decades, we have made considerable progress in describing andunderstanding individual mathematics teachers’ growth and development.We have learned, for example, how long-term professional developmentprograms and research projects support mathematics teachers’ efforts tobring about change, and how these programs can rescue them from burnoutor from leaving the teaching profession (see, for example, Borasi, Fonzi,Smith, & Rose, 1999; Even, 1999; Halai, 1998; Jaworski, 1998; Krainer,1999). However, it is often argued that teachers who participate in suchintensive programs and projects are “always the same,” and those whoreally do need improvement do not come. Furthermore, the growth of asingle teacher does not necessarily have an impact on other colleagues athis or her school.

Experience shows that it is not easy for individual teachers to findcolleagues who really want to join in their efforts to improve teaching.Nor is it easy to sustain enough motivation and perseverance to realisechanges in the short or long run. Furthermore, innovative action at a schoolis often regarded rather critically by non-participating teachers, a situationthat can lead to open or hidden resistance and opposition. Even a pairof colleagues co-operating successfully might not be enough to have animpact on other teachers. Similar experiences were reported in Borasiet al. (1999, p. 75) who pointed out that their professional developmentprogram had additional benefits when not just pairs but a critical mass ofteachers from the same school participated in the program. If professionalcommunication among teachers is not an important feature of the schoolculture, innovations by individual teachers remain limited to them and theirclassrooms.

Individual-oriented professional development programs – althoughvery important – cannot be the only strategic intervention in order toimprove the teaching of a subject in our schools. The culture of a school,that is, the context in which the teachers live and work, has an essentialimpact on teachers’ actions and beliefs. Schools as organisations have their


224 GUEST EDITORIAL

own rules, habits, general conditions, and power relations which greatlyinfluence the role that mathematics teaching can play at a school.

These are arguments for alternative approaches to promoting mathe-matics teachers’ growth which take into account the organisational aspectsthat determine teachers’ work. One possibility is to support whole depart-ments of mathematics in schools. Experience shows (e.g. Krainer, 1999)that successful professional development programs take time and consid-erable support. Advantages that might be gained from working with entiredepartments or schools include the following:

• Professional communication among teachers might be promoted.• Teachers could have others (who work next door to them) to encour-

age their efforts, or even colleagues who were ready to join theirefforts to improve their mathematics teaching.

• Collaboration among the teachers might generate a transition from anassembly of lone fighters to a network of critical friends(see also thetermcritical colleagueshipin Lord, 1994, andprofessional cultureinLoucks-Horsley, Hewson, Love, & Stiles, 1998, pp. 194–199).

• It is possible to take the culture of the school into consideration andto contribute to its enhancement.

• Innovations would be more likely to become a relevant component ofmathematics teaching.

• Activities of mathematics teachers can give birth to whole schooldevelopment activities.

• Mathematics teaching could be more visible and could play a greaterrole at the school.

As all leaders of organisations, principals have a great influence oninnovations. Their willingness to support initiatives at their school dependson a variety of factors. For example, a principal’s view of mathematics andof the professionalism of mathematics teachers might influence his or herattitude toward professional development. For this and other reasons, manymathematics teacher educators (e.g., Peter, 1996) highlight the importanceof principals with respect to the professional development of teachers.Even if principals play a more indirect role, there are still opportunities forthem to actively and directly influence mathematics teachers’ work. Halai(1998, pp. 298–299), for example, describes how a principal in Pakistanin tandem with a teacher educator worked out the strategy for implement-ing a professional development program for mathematics teachers at thatparticular school.

Principals and other important stakeholders such as department heads,regional subject co-ordinators, or superintendents, who hold different rolesand functions in the school system, have their own ideas and beliefs

GUEST EDITORIAL 225

about the nature of learning, teaching, mathematical knowledge, andreform (e.g., Nelson, 1998). Because they have considerable influenceon decisions concerning general conditions of the quality of teaching,it is essential to pay more attention to their roles in the professionaldevelopment of teachers, both practically and theoretically.

Mathematics teacher education needs good stories (e.g., Cooney, 1999),and fortunately we have an increasing number of them about individualteachers. Maybe we should begin to tell good stories about groups ofteachers, departments of mathematics, and even whole schools.

REFERENCES

Borasi, R., Fonzi, J., Smith, C. F., & Rose, B. (1999). Beginning the process of rethinkingmathematics instruction: A professional development program.Journal of MathematicsTeacher Education, 2, 49–78.

Cooney, T. J. (1999). Stories and the challenge forJMTE. Journal of Mathematics TeacherEducation, 2, 1–2.

Even, R. (1999). The development of teacher leaders and inservice teacher educators.Journal of Mathematics Teacher Education, 2, 49–78.

Halai, A. (1998). Mentor, mentee, and mathematics: A story of professional development.Journal of Mathematics Teacher Education, 1, 295–315.

Jaworski, B. (1998). Mathematics teacher research: Process, practice and the developmentof teaching.Journal of Mathematics Teacher Education, 1, 3–31.

Krainer, K. (1999, May). Learning from Gisela – or: Finding a bridge between classroomdevelopment, school development, and the development of educational systems. In F. L.Lin (Ed.), Proceedings of the 1999 International Conference on Mathematics TeacherEducation(pp. 76–95). Taipei, Taiwan: National Taiwan Normal University.

Lord, B. T. (1994). Teachers’ professional development: Critical colleagueship and the roleof professional communities. In N. Cobb (Ed.),The future of education: Perspectives onnational standards in America(pp. 175–204). San Diego, CA: Academic Press.

Loucks-Horsley, S., Hewson, P. W., Love, N., & Stiles, K. E. (1998).Designing pro-fessional development for teachers of science and mathematics. Thousand Oaks, CA:Corwin Press.

Nelson, B. S. (1998). Lenses on learning: Administrators’ views on reform and theprofessional development of teachers.Journal of Mathematics Teacher Education, 1,191–215.

Peter, A. (1996).Aktion und Reflexion. Lehrerfortbildung aus international vergleichenderPerspektive[Action and reflection. Teacher inservice education from an internationalcomparative perspective]. Weinheim, Germany: Deutscher Studien Verlag.

Konrad KrainerAssociate Editor,JMTE

GWENDOLYN M. LLOYD

TWO TEACHERS’ CONCEPTIONS OF A REFORM-ORIENTEDCURRICULUM: IMPLICATIONS FOR MATHEMATICS

TEACHER DEVELOPMENT?

ABSTRACT. This paper describes two high school teachers’ conceptions of the coopera-tion and exploration components of a reform-oriented mathematics curriculum. Althoughthe teachers appreciated the themes of cooperation and exploration in theory, their concep-tions of these themes with respect to their implementations of the curriculum differed. Oneteacher viewed the curriculum’s problems as open-ended and challenging for students,whereas the other teacher claimed that the problems were overly structured. Each teacherattributed difficulties with students’ cooperative work to the amount of structure and direc-tion (too little or too much) offered by the problems. Discussion of such similaritiesand differences in the teachers’ conceptions emphasizes the dynamic, humanistic natureof curriculum implementation and gives rise to important implications for mathematicsteacher development in the context of reform.

By demanding changes in both the content and activity of mathematicsinstruction, recent reform recommendations in the United States challengea lasting tradition (Gregg, 1995; Richards, 1991). In light of the impressivedurability of traditional teacher-centered and procedure-oriented mathe-matics instruction, how do veteran teachers deal with calls for reform? Thispaper describes two secondary teachers’ experiences with reform recom-mendations in the context of their implementations of a set of innovativecurriculum materials. Focus is on the teachers’ conceptions of the meaningand importance of certain mathematics classroom activities, in particularcooperation and exploratory problem-solving.

Cooperation and Exploration in Mathematics Teaching and Learning

Cooperation and exploration are prominent themes both in the curriculummaterials implemented by the teachers (Hirsch, Coxford, Fey & Schoen,1995) and in more general documents that promote mathematics educationreforms (Mathematical Sciences Education Board [MSEB] & National

? The research reported in this study was supported in part by the National ScienceFoundation (MDR-9255257). The views herein are those of the author and do notnecessarily reflect those of the National Science Foundation.


228 GWENDOLYN M. LLOYD

Research Council [NRC], 1989; National Council of Teachers of Mathe-matics [NCTM], 1989). Teachers are urged to establish mathematicsclassrooms in which students engage actively in exploration and cooper-ative work in order to help students develop rich understandings ofmathematics as a vibrant and useful subject.

Reform-oriented models of teaching and learning are supported bya broad base of empirical and theoretical literature about how studentsunderstand and learn mathematics. This literature makes a strong casefor classroom activities that give rise to genuine mathematical problemsfor students to resolve (Hiebert et al., 1996; Lampert, 1990; Schoenfeld,1992; P.W. Thompson, 1985; Yackel, Cobb, Wood, Wheatley & Merkel,1990). In contrast to traditional classroom activities that emphasize correctanswers and routinized solution methods, problem-centered instructioncapitalizes on opportunities for students to learn as they cooperate inthe solution process. This process can “include accounting for surprisingoutcomes, such as when two alternative methods lead to the same result,justifying a solution method, or explaining why an apparently erroneousmethod leads to a contradiction” (Yackel et al., 1990, p. 15). When studentswork in groups to communicate their ideas and questions, agree anddisagree among themselves, and negotiate joint theories and ideas, richmathematical learning can occur (Richards, 1991; Slavin, 1990; Voigt,1996).

Teachers’ Conceptions and Reform

How do teachers make sense of the themes of cooperation and explorationas they implement innovative curriculum materials? The role of teachers’conceptions and classroom experiences in mathematics reform cannotbe overemphasized. An extensive body of research provides consistentevidence that teachers’ conceptions strongly impact instructional prac-tice (Brophy, 1991; Fennema & Franke, 1992; A.G. Thompson, 1992).Moreover, teachers’ conceptions have profound effects on their inter-pretations and implementations of reform recommendations and reform-oriented mathematics curricula (Cohen, 1990; Lloyd & Wilson, 1998;Romberg, 1997; M. Wilson & Goldenberg, 1998; S.M. Wilson, 1990).For example, in S.M. Wilson’s (1990) case study of a teacher imple-menting a new curriculum as part of the California mathematics reforms,the teacher’s conceptions of appropriate instruction interfered with hisability to foster student inquiry in the ways intended by the reformcurriculum. Because reform-oriented pedagogies require that teachersreconceive their roles in mathematical activity and student learning,implementation of an innovative curriculum can pose significant chal-

TEACHERS’ CONCEPTIONS OF CURRICULUM 229

lenges even to the most committed teachers (Clarke, 1997; Smith,1996).

Reform documents and curriculum materials do not prescribe or definepractice for teachers, but rather offer visions “orienting individuals andinstitutions toward collectively valued goals” (Shulman, 1983, p. 501). Aricher understanding is needed of the relationships between teachers’ ownconceptions of mathematics teaching and the recommendations for changeoutlined in curriculum materials. The study reported in this paper inves-tigates how and why two teachers encountered particular successes anddifficulties as each implemented a set of novel curriculum materials for thefirst time. How did the teachers conceive of cooperation and exploration asthey implemented the curriculum materials in their classrooms?

RESEARCH METHODS

Curriculum Materials

The curriculum of the Core-Plus Mathematics Project attempts to supportteachers in enacting many recommendations of theStandards(NCTM,1989). Each year of the high school curriculum (Core-Plus Courses1–4) features algebra and functions, geometry and trigonometry, statis-tics and probability, and discrete mathematics. The curriculum’s units,each designed to guide approximately four weeks of student work, areorganized into several multi-day investigations that emphasize modelingreal-world situations, experimenting in order to develop and test theories,and debating with classmates. The materials direct teachers to organizethe classroom so that students can work cooperatively as they exploremathematical problems and ideas. Cooperative learning is complementedby whole-class discussions in which activities are introduced and summa-rized.

An example of a Core-Plus lesson may be helpful. The first lessonin a unit about probability,Simulating Chance Situations, begins witha description of Chinese government policies that restrict families toone child. Students are first asked to think of some alternative plans forcontrolling population growth. Then students are instructed to flip coinsto simulate an alternative plan, that each family may have two children,and to record their results in a frequency table. Students must decide howcoin flips can be used to simulate the births and to describe the possibleoutcomes for this particular plan. A variety of problems and questionsfollow, including: “Use your frequency table to estimate the probabilitythat a family of two children will haveat least oneboy” and “Estimate the


probability that a family of two children will have at least one boy usinga mathematical method other than simulation.” Next, students are askedto consider a different plan in which families continue having childrenuntil a boy is born, and to record data in frequency tables and histograms.This time, students predict responses to a set of questions prior to theirsimulation, and then compare predictions to actual results. Finally, studentsanalyze one of their own alternative plans proposed at the beginning of theinvestigation.

Although most Core-Plus activities are based in real-world contexts,such as the probability simulation described above, some engage studentsin exploring more abstract mathematical situations. For example, in a unitabout functions, an investigation titledThe Shape of Rulesasks studentsto generate tables and graphs associated with four sets of equations thatrepresent different types of relationships (linear, quadratic, exponential,etc.). Students are instructed to look for patterns across the three repre-sentations and summarize their ideas in statements such as “If we see arule like . . . , we expect to get a table [or graph] like . . . .” Further examplesand discussion of the Core-Plus curriculum can be found in a report by thematerials’ designers (Hirsch et al., 1995).

Participants, Site, and Context for the Study

This study investigated the conceptions of two veteran high school mathe-matics teachers, Mr. Allen and Ms. Fay, in a public school district in theNortheastern U.S. where the Core-Plus materials were being field-tested.Empirical study of Mr. Allen and Ms. Fay began in 1994 and 1996, respec-tively, when each teacher implemented the Core-Plus materials for the firsttime. These two teachers were selected for study because (1) in contrastto most other teachers at the same school, Mr. Allen and Ms. Fay imple-mented the materials voluntarily, (2) each teacher communicated a desireto integrate more cooperation and exploration into his or her instruction,and (3) the teachers’ individual experiences appeared to have importantcontrasting elements.

At the beginning of the study, Mr. Allen had been teaching mathematicsfor 14 years and, by his own description, had largely adhered to traditionalclassroom practices. In the spring of 1994, he was invited by the chair ofhis mathematics department to participate in the field-testing of the Core-Plus materials. He said, “If this is the way we’re going to go, I want tomake sure I have experience in it.” During the 1994–95 school year, Mr.Allen implemented the Core-Plus Course 1 materials for the first time inone class of 32 ninth grade students. In the 1995–96 school year, he againtaught one Core-Plus Course 1 class of 32 ninth grade students. In 1996–


97, he taught one ninth grade Core-Plus Course 1 class and two tenth gradeCore-Plus Course 2 classes.

In the Fall of 1996, due largely to her interest in the Core-Plusprogram, Ms. Fay joined the mathematics faculty at the high schoolwhere Mr. Allen taught. Her previous employment included both 10years of classroom teaching and, most recently, a state government posi-tion in which she visited schools with innovative mathematics educationprograms. According to Ms. Fay, in her previous teaching she had “alwaysused groups and. . . always had a project focus,” and she wished for theCore-Plus materials to support her in developing and extending thesecomponents of her practice. Her commitment to the Core-Plus innovationwas one of the primary reasons that she was selected as a participant in thisstudy. During the 1996–97 school year, Ms. Fay taught with the Core-PlusCourse 2 materials in two classes of tenth-grade students.

Data Collection and Analysis

Data sources consisted of teacher interviews, classroom observations, andfieldnotes. Because the two teachers joined the project at different times,the number and dates of interviews and observations varied considerably.Over a 3-year period between September 1994 and January 1997, Mr.Allen participated in 17 interviews (8 in Year 1, 7 in Year 2, and 2 inYear 3) and was observed 73 times teaching with the Core-Plus materials.This paper focuses primarily on data collected during his first two yearsof curriculum implementation. During the 1996–97 school year, Ms. Fayparticipated in five interviews and was observed 10 times in her Core-Plusclasses. Most interviews lasted approximately 1 hour, and all were audio-recorded and transcribed. All classroom observations were audio-recordedand approximately half were video-recorded. Fieldnotes were taken duringobservations, and written artifacts such as student work were collected andphotocopied.

In the interviews, teachers were invited to reflect on recent classroomevents, suggest goals for upcoming classroom activities, and respond toquestions about more general emerging themes. Data were analyzed duringand after collection (LeCompte & Preissle, 1993). Careful readings of tran-scripts and fieldnotes and creation of interview and observation summariesresulted in the identification of preliminary themes for subsequent focus ininterviews and observations. After the observation periods ended, morethorough review of data took place. The development of major themeswas aided by the use of taxonomic and thematic analytic strategies(Spradley, 1979). In the final stage of analysis, major themes were furthersynthesized within and across the data sources to illustrate important, and


often contrasting, aspects of the ways in which the teachers viewed theirimplementations of the Core-Plus curriculum.

THE CASE OF MR. ALLEN

Conceptions of Exploration

From the start of his teaching with the new curriculum, Mr. Allen posi-tively differentiated the Core-Plus problems from those found in traditionaltextbooks by pointing out ways in which they engaged students in sense-making activities. He described the Core-Plus activities as having anapproach that required students to develop “informal ideas based on theirexperiences” in contrast to “the teacher or book feeding it to them” (Int. 3,Yr. 1, 11/14/94). More specifically, in his second year using the materials,he indicated that the Core-Plus questions themselves required extensiveinterpretation by students:

The Core-Plus questions are a little more general – not vague, but not as specific and sosometimes the students have to figure out exactly what it is they want them to do. That’swhat they need to be able to work at and practice because that’s what they’ll use eventually– the problem solving skills and attacking a problem, reading it, and struggling with it. (Int.2, Yr. 2, 11/10/95)

One characteristic of the Core-Plus questions that required more analysisthan traditional exercises was that they were “different each time and theydon’t do the same thing over and over again.” He contrasted problemsthat challenged students to think actively about mathematics with a moretraditional approach that enticed students to have the attitude “Just tell methe steps I need to do and I’ll do those, but I don’t want to think about it toomuch” (Int. 5, Yr. 2, 12/8/95). Further, Mr. Allen suggested that with tradi-tional problems and instructional methods, student “understanding isn’tnearly what it is in Core-Plus.”

Another reason that Mr. Allen thought the Core-Plus materials engagedstudents in sense-making was that the activities centered on “realistic prob-lems and situations that drive the mathematics rather than having a bunchof made-up book-type problems” (Int. 1, Yr. 2, 10/17/95). For instance,he reflected on how the Core-Plus materials introduced the concept ofvariablewhen students explored the relationship between weight and cordstretch in a bungee jumping situation: “The kids understand what a variablecan represent based on some practical applications – give them situationswhere things are related and then they get a sense for what ‘variable’means” (Int. 5, Yr. 2, 12/8/95). In Mr. Allen’s view the overall benefit


of more open-ended, contextualized problem exploration was that Core-Plus students develop “deeper understanding of the material rather thanjust at the surface and knowing some algorithm to solve something and notreally know the background of it” (Int. 2, Yr. 1, 10/3/94). He suggestedthat, because Core-Plus students will “have a background in lots of mathe-matics andthinking,” when they need to develop procedures or algorithmsthey will be capable of deriving them: “If factoring is important, they’regoing to be able to do the algorithmic factor” (Int. 8, Yr. 1, 6/1/95).These comments illustrate Mr. Allen’s strong valuation of the Core-Plusemphasis on “thinking about situations but not necessarily having theanswer be the major outcome – it’s the process that’s important” (Int. 1,Yr. 2, 10/17/95).

His valuation of the “more investigative” approach of Core-Plus wasevidenced on numerous occasions in his classroom when he suggestedways that students could use their different solution strategies to yielddiscussion and further learning. For example, when some students labeledthe axes of coordinate graphs in the reverse order from the conventionalapproach, Mr. Allen invited groups to compare how the different axislabelings might affect the representation of the data in the graphs. Forinstance, when students switched variables on the horizontal and verticalaxes of a graph showing adistanceversustime relationship, he instructedseveral groups of students: “Compare the graphs withtime anddistanceswitched. See if they are the same.” He indicated in an interview (Int. 5,Yr. 1, 12/2/94) that he was hoping to elicit more discussion by havingstudents analyze these differences. Similarly, Mr. Allen often remindedstudents that their personal preferences (for instance, among differentrepresentations of data) would help them to solve problems effectively.As these examples show, Mr. Allen encouraged students to personalizetheir solution strategies and make sense of the problems and ideas forthemselves.

Although Mr. Allen valued the Core-Plus approach, he often worriedabout how to help students deal with the problems and questions in thecurriculum materials. He reported that while working on Core-Plus prob-lems, students repeatedly asked him questions about “what they needto do.” Because the Core-Plus questions engage students an analysisof complex mathematical issues with less guidance than is offered intraditional mathematics texts, students often “think that they have to dosomething completely different or they may read more into the questionthan what’s really there” (Int. 6, Yr. 1, 12/12/94). Mr. Allen attributedmuch of the students’ frustration to the novelty of these types of problemsand activities for them:


The students haven’t had a lot of practice with reading and understanding. . . a problem.They are used to reading a five- or six-word sentence and then simplifying the expressions.It doesn’t take a genius to figure out what you should be doing. While here, you have awhole story to read and in that story is some critical information. (Int. 1, Yr. 3, 9/20/96)

However, even when students understood the questions, they were unac-customed to attaining more than one correct final answer: “They want tomull over these questions like they’re right and wrong rather than thinkingabout them and giving a basis as to why they answer a question oneway or another” (Int. 6, Yr. 1, 12/12/94). Mr. Allen repeatedly remindedstudents to “just say what [they] think” as they worked on the Core-Plus activities. For example, in one lesson when students were asked topredict the relationship between the time and height of a ball thrown inthe air prior to modeling the situation, students expressed concerns suchas, “How are we supposed to figure out how high it goes?” In response,Mr. Allen encouraged students to “give a guess of how highyou thinkitwent.” In a subsequent interview, as he reflected on the students’ reac-tion to this question, Mr. Allen described, “Rather than ‘What do youthink about it?’ they wanted to know exactly how to do it” (Int. 6, Yr.1, 12/12/94). Because “getting them more into using their own thoughtsas being valid has been difficult,” Mr. Allen identified the need to buildstudents’ confidence in offering personal theories and opinions in responseto the Core-Plus problems.

Conceptions of Cooperation

As he began to implement the Core-Plus materials, Mr. Allen identifiedhis appreciation for “the philosophy of working in groups to get thingsdone” (Int. 2, Yr. 1, 10/3/94). Because “the teacher wasn’t going to be thefocus anymore,” the group work would give students “more ownership” ofthe mathematics under consideration. He communicated disappointmentabout the failure of traditional problems to “lend themselves to cooperativework”:

The traditional exercises are something that you can do by yourself. . . there’s not a lot ofdiscussion asked or describing asked. It’s more or less just get an answer or do some taskand get it done. You certainly can do that just by yourself. You might sit next to somebodyand ask them a question or watch how they did it or have them help you, but it’s not likeyou sit in a group and let one person do this part and one person do the other part. (Int. 5,Yr. 1, 12/2/94)

As this statement suggests, Mr. Allen wanted cooperative activities torequire explicit collaboration, a theme that is further illustrated by hisalteration of some Core-Plus activities to emphasize greater interde-pendence among group members. For example, in a Year 2 lesson that


involved students performing experiments with their graphing calculators,he explained to the class:

There is a lot of work to do so we are going to split up the work. I’m going to give onegroup Experiment #2 and there are six graphs to do in that group. We don’t have time to doall six ourselves so you want to split up the work . . . The idea is to use everybody in yourgroup.

To Mr. Allen, this lesson was “a perfect example” of how cooperationmakes most sense when the work involves more than could be reason-ably accomplished by one person. In other words, students must “relyon each other” while solving the problems (Int. 5, Yr. 2, 12/8/95). Inthis example, students analyzed and developed patterns based not only ontheir groupmates’ work, but also on the ideas of other groups. Because hechanged a recommended activity to encourage more extensive discussionand joint analysis by students, this example also illustrates Mr. Allen’ssincere interest in creating significant opportunities for students to worktogether, independently of him, on the Core-Plus activities.

However, Mr. Allen’s implementation of group work was not withoutconcerns or problems. In his view, struggles with tackling the more open-ended and complex problems of the curriculum particularly impactedstudents’ abilities to work cooperatively in their small groups. Because thematerial required more analysis and discussion on the part of the studentsthan did the traditional materials to which they were accustomed, Mr.Allen sensed that the students were “anxious about being out there a littlebit free and not really quite sure where this is all going” (Int. 6, Yr. 1,12/12/94). Unless students had “some perseverance and some stick-to-it-iveness,” they frequently stopped working on the mathematics and beganto socialize. During his second year implementing the materials, Mr. Allenexplained,

We have a ways to go with the groups really working at trying to understand things. If thematerial isn’t too difficult, the flow seems to go pretty well. As the material gets a littlemore difficult or more abstract, it isn’t in front of them real well, then they struggle withtrying to work at trying different things or . . . trying to figure something out as a groupwhen it’s not so obvious that it might take an error or two and not get frustrated and stop.(Int. 6, Yr. 2, 1/12/96)

As this statement reflects, Mr. Allen felt strongly that the success ofcooperative activities depended not only on student perseverance, but alsoon whether students understood what they needed to do and how to accom-plish it. Because, in Mr. Allen’s view, most Core-plus activities demandedthat students take the initiative in interpreting questions and situations,these factors frequently came into play in shaping his sense of the successof classroom activities.


Attempts to Address Concerns

Due to his concerns about students’ inclination to work productively andcooperatively on the Core-Plus activities, Mr. Allen often felt compelledto provide more direction than the curriculum materials recommend. Oneway that Mr. Allen added more direction to the Core-Plus lessons was thathe gave students review problems to complete at the end of each set ofinvestigations. As he explained when he developed the first set of reviewproblems, “It seems like a good way of having a limited task for all ofthem to get done. The task, more or less, summarizes the material. . . then Iknow that I have exposed them to the general ideas that I think they need toknow” (Int. 4, Yr. 1, 11/28/94). The review problems established “guidingposts of how and where we are supposed to be” (Int. 5, Yr. 1, 12/2/94).As these comments suggest, the review problems allowed Mr. Allen tocompensate for any weaknesses he perceived in students’ previous workwith the Core-Plus activities.

Notably, Mr. Allen made no changes to the problems printed in theCore-Plus curriculum materials. In other words, despite his concerns aboutthe open-ended problems, he did not re-write the problems or activities toinclude more structure or direction. This is consistent with Mr. Allen’ssense that to a greater extent than traditional textbooks, the Core-Plusmaterials restricted the teacher’s ability to personalize the curriculum:

The traditional textbook had this content that was just algorithmic type things and youcould then pick things to come into that class to make it come alive. You had that freedomto pull in an article or an experiment to liven up this idea we were going to do out ofthis algebra book. This [Core-Plus] is more –they are putting together the real-worldapplication that they want you to use so you are tied to that. It is hard to get away from it.(Int. 1, Yr. 3, 9/20/96)

Feeling tied to the mathematical situations presented in the materials mayhave contributed to Mr. Allen’s reluctance to modify the provided activitiesand to insteadappendreview problems.

The most prominent way that Mr. Allen’s concerns impacted hisinstruction was in the nature of his interactions with students and the wayshe chose to organize his classroom. Mr. Allen viewed teacher interventionas a way to improve students’ work on challenging problems and ques-tions: “To keep the flow of the class going I want to get in there and helpthem a little bit” (Int. 6, Yr. 1, 12/12/94). He indicated that providing direc-tion about certain concepts or activities was more efficient than havingstudents explore and develop ideas themselves, for example in the caseof solving algebraic equations: “Eventually they may come up with someunderstanding but I don’t know if it would be as clear that way as it is withme trying to make sure I give them some direction” (Int. 6, Yr. 2, 1/12/96).


In Mr. Allen’s first year, during a typical class session, he spent lessthan 5 minutes on whole-class instruction in order to introduce and reviewproblems. In his interactions with small groups, he attempted to “getthem going in the right direction.” As he circulated among the groupsprompting and directing students’ work, he often asked the same questionor made the same comment to each of the groups. Recognizing that hewas providing significant direction to students, he explained in an inter-view: “It takes spoonfeeding it a little bit at this point to let them knowwhat they really need to put down” (Int. 2, Yr. 1, 10/3/94). In his secondand third years of using the materials, Mr. Allen provided much lessdirection to groups than in his first year. Instead he decided to providemore whole-class instruction to help students acquire “a decent sense ofwhat they’re doing” and “the perception that they can be successful” inthe cooperative activities (Int. 3, Yr. 2, 11/21/95). Whole-class instruc-tion included an invitation to students “to be the teacher” so they wouldpresent problems to peers. It also included discussions of problems inwhich Mr. Allen attempted to make his classroom “more interactive.”However, more commonly, during his whole-class discussion he modeledproblems so that students would be prepared to work in a group withouthis help – otherwise, “You just turn them all loose and two minuteslater everyone’s got their hand up and it becomes chaos” (Int. 2, Yr. 2,11/10/95).

These comments and examples show Mr. Allen’s conception thatteacher-directed instruction can provide a sort of scaffolding or supportfor students’ small-group explorations. Although he believed that studentsshould discuss and learn important ideas in their small groups, he claimedthat teacher-directed instruction, particularly in whole-class formats, wasnecessary to insure that students had the opportunity to consider all of theimportant material. When he provided students with more structure anddirection, Mr. Allen felt better about his work in the classroom: “Eventhough I’m giving them more direction . . . it’s helpful for me to feel likeI’m covering the bases of what I think they should know” (Int. 6, Yr. 2,12/12/94). But he was also conscious of the potentially negative impact ofhis direction on students’ opportunities for mathematical exploration andcreativity. He suggested that his concerns about keeping students on taskand moving through the materials caused him “to be the opposite of what[Core-Plus is] really trying to emphasize” (Int. 5, Yr. 2, 12/2/94). In hissecond year, he became even more concerned that he “tended to do mostof it [him]self” (Int. 5, Yr. 2, 12/8/95). He pointed out that “You’ve got tobe able to balance it and not go overboard and start doing every problemfor them either” (Int. 2, Yr. 2, 11/10/95). The attempt to find this balance


was central to Mr. Allen’s goal of eventually developing a practice moreconsistent with “the Core way.” His sincere enthusiasm and interest inreforming his instruction is evidenced by his comment: “I am learning a lotabout myself as a teacher so I just enjoy it a lot more from my experienceof fifteen years . . . and I am finding it to be a lot more fun” (Int. 7, Yr. 2,5/17/96).

THE CASE OF MS. FAY

Conceptions of Exploration

As Ms. Fay began to teach with the Core-Plus materials, she commu-nicated her appreciation that the students were working on “really richproblems.” She identified real-life contexts such as distributions of studentshoe sizes or fast food nutrition as important problem features, and sheclaimed, “I never hear ‘When are we going to use this?’ because they allsee that it’s relevant. When the students saw the pure math, it didn’t makeany sense to them. It was just a neat strategy” (Int. 3, 11/5/96). The realisticcontexts of the Core-Plus problems seemed to Ms. Fay to be very effectivein engaging students in explorations of important underlying mathematics.

Despite her positive descriptions, she communicated disappointmentwith the opportunities for student exploration within the lessons. Herconcerns related primarily to the structure of the problems:

The kids are really led through the problem in Core-Plus. It’s like using the problem asa way to show them how to do this work instead of the kids saying “OK, let’s organizeourselves. What do we need?” Even though they are really good problems, it’s handed tothem how to solve it. (Int. 3, 11/5/96)

Ms. Fay viewed the activities in the Core-Plus materials as overly definedsets of questions that led students to particular answers or ideas throughpredetermined solution paths. She feared that the structured nature of theinvestigations might contribute to students attending more to the proce-dural aspects of problems than to the central concepts. For instance, sheexpressed that “Many times, students focus more on the steps for the bigideas than on the reasons behind them,” and she wished that students wouldask more “Why?” questions (Int. 1, 9/20/96). In her view, having studentswork on fewer problems with less guidance from the materials would allowstudents to take greater leadership in the solution process and, as a result,to develop more conceptually rich understandings.


Conceptions of Cooperation

Ms. Fay expressed many positive thoughts about students working cooper-atively. She believed that cooperative work allows students to articu-late and extend their understandings through give-and-take among groupmembers:

They are each articulating their own thoughts. If I just got up there and explained it, thenthey are only going to hear it one way. Instead they hear it from the kids . . . from othergroups, and it becomes a classroom of everybody teaching and everybody learning. (Int. 1,9/20/96)

But because the structured nature of the typical Core-Plus problemgave students few opportunities to “take responsibility for organizingand solving problems in their own ways” (Int. 2, 10/25/96), Ms. Fayalso expressed dissatisfaction with the cooperation component of thecurriculum.

Three weeks into the semester, she was pleased with her Core-Plusstudents’ abilities to work together, indicating that “they truly do listenand I think they have a lot better communication skills than some adultsI know” (Int. 1, 9/20/96). She attributed the students’ communicationstrengths to “their past experiences in the Core 1 classes” (the previousyear). However, at the end of October, she suggested that “the groups wereactually working better at the beginning of the year than they are now,”a phenomenon she related to the structure of the Core-Plus problems. Inparticular, she observed:

The groups are not discussing as much as they should be. They work apart until they havea question and then they hit their buddy and ask how he did it but then they go back toworking . . . or somebody says “I think this is the answer” and everybody says “Okay” andthey write it down. (Int. 2, 10/25/96)

This quote alludes to one of her main concerns – that many of the Core-Plus activities can be completed individually: “There is not a lot of workin these books that require a shared interest like a jigsaw puzzle whereeach person takes a piece and we will work together. Instead each personcan work alone” (Int. 4, 12/16/96). Although some investigations providedopportunities for Ms. Fay to have the whole class share the work, suchas when each student contributed to a large number of trials in a prob-ability experiment about Chinese birth policies, in general, she did notfeel that breaking up this work used everybody’s talents to contribute to asubstantial group product.


Attempts to Address Concerns

Despite her complaints about the over-structured problems, during myobservations, Ms. Fay did not change the Core-Plus investigations tomake them more open-ended by eliminating, altering, or adding prob-lems. Instead, during her whole-class instruction, which typically followedstudents’ group work, she adjusted the focus of some problems to attendto what she viewed as the two main negative implications of over-structured problems: They did not permit a conceptual emphasis, andthey did not encourage students to develop their own solution paths. Forexample, one Core-Plus investigation stresses several rules associated withgeometric transformations. In the whole-class discussions after studentshad completed the group activities, Ms. Fay emphasized repeatedly thatstudents should visually consider the graphs in order to perform therequested transformations, rather than rely solely on the rules. Whilemodeling exercise solutions, for instance, she stated to the class, “I findit easier to look at it visually” and “It doesn’t make sense to memorizeformulas.” Another way that Ms. Fay altered her review of problems thatfocused on procedures (such as multiplying matrices) was to ask studentsto explain verbally the steps involved, reminding them that “more than justthe answer is important.”

Ms. Fay’s whole-class instruction was primarily devoted to givingdetailed but very brief explanations of problem solutions. She empha-sized that the solutions were her own and that students might havedifferent approaches. For instance, she often prefaced her explanationswith comments such as “Here is howI did it . . . yours might be different.”These types of comments evidence that Ms. Fay wanted students to beaware that there could be multiple solution methods even within the limitedstructure of the Core-Plus problems. However, she did not capitalize onor elicit these potential differences from students. Her review of prob-lems typically consisted of writing her solutions on an overhead andexplaining them. When she asked students questions, she rarely pressedthem to justify their responses. Although she asked the class questions,such as “Do you agree with that?” or “Is that right?,”shegenerally decidedwhen answers were correct. Ms. Fay’s efforts to have students view prob-lems conceptually and to be aware of the possibility of multiple solutionmethods were restricted to discussion following, not during, students’group work. Most of her interactions with students in their groups involveddirect responses to their questions (e.g., “Is this right?” or “Am I doing thisright?”) and administrative tasks such as checking homework or discussinggrades. In fact, she expressed uncertainty about what to do while students


Figure 1. Two corresponding pieces of Ms. Fay’s dominoes game.

worked in groups, wondering, “Why do they need me here?” (Int. 1,9/20/96).

Ms. Fay did make some changes to the ways students worked on theCore-Plus problems in their groups. To address the individual nature ofproblems, she occasionally required students to work in smaller groupson the problems. Although not representative of a general approach to herconcern, one noteworthy example of an attempt to deal with her dissatis-faction with students’ cooperative work comes from her teaching of theinvestigation about transformation rules described above. At the end ofone class session, Ms. Fay divided the students into pairs for a game thatwas not part of the Core-Plus materials. She distributed sets of 14 domino-like pieces, each of which contained one exercise and one solution, andinstructed the teams to match up their pieces as quickly as possible. Todo so, students placed the domino containing the correct solution next tothe one with the corresponding exercise, as illustrated in Figure 1. Duringa subsequent interview, Ms. Fay pointed out that the students could “seethat if they work together then they will have a better chance of winningthan if we do it individually” (Int. 2, 10/25/96). In other words, successat this activityrequiredstudents to work jointly. This example illustrateshow Ms. Fay made an addition to compensate for what she viewed asa weakness of the curriculum materials. However, this type of additionwas not repeated during my observations, and Ms. Fay’s more typicalefforts to improve students’ group work involved verbally encouragingthem to work together, discuss problems in their groups, and value others’opinions.

Ms. Fay admitted that she did not develop many problems or projectsto supplement the suggested Core-Plus activities, and she explained thatshe felt very constrained in her ability to do so. One reason for notmaking more changes was that she did not know the materials wellenough to decide “which investigations and problems are crucial” (Int.5, 12/17/96) and moreover, her mathematical background was insuffi-cient. For example, she explained, “I don’t know if I had to come upwith a project on matrices that I could have . . . . I haven’t had matricessince Linear Algebra” (Int. 1, 9/20/96). In addition, her knowledge ofmeaningful mathematical problem contexts was lacking: “Many of theapplications my students are seeing for the first time, I am also seeing forthe first time.” She predicted that in her second year of implementation, she


would be more prepared to “eliminate some of the problems” and, basedon those problems, create “one big problem that students could organizeand solve” (Int. 4, 12/16/98).

Ms. Fay suggested that her inclination to develop and implementmore extensive projects than the Core-Plus materials provided was furtherrestricted by working in a department “where everybody tries to be at thesame place at the same time” (Int. 4, 12/16/96). Ms. Fay believed that“if everybody did their own thing then [she] would be teaching probablyvery differently” (Int. 5, 12/17/96). Her sense of obligation to colleagueswho preferred to maintain uniform movement through the materials wasmost problematic because she greatly valued giving students ample timeto explore and interact: “The more time that they have to work with thismaterial, the more time they have to interact with each other – time that isnot just taking notes and trying to figure out what the teacher wants” (Int.2, 10/25/96). During an interview, she referred to an occasion in which shesuggested to the other Core-Plus teachers that the students should work onone of the end-of-unit projects:

One of the Core-Plus activities. . . was to have students look through newspapers and findarticles or tables with cause-effect relationships. I said “Great, let’s break up the monotonygive them newspapers and each group do a presentation.” Everybody was like “Well wecan’t take a whole class hour to do that.” I’m like, “Why not?” (Int. 4, 12/16/96)

This example illustrates a struggle between Ms. Fay’s desire for moreexploratory activities and her colleagues’ concern for more efficientcontent coverage. This struggle contributed to Ms. Fay’s sense that shedid not fully own her Core-Plus classroom: “Inmyclassroom, we wouldbe doing some experiments or projects right now” (Int. 1, 9/20/96).Although the Core-Plus materials do suggest numerous in-depth projectsas unit summary activities, feeling rushed and behind some of the otherteachers contributed to Ms. Fay’s decisions at times to skip these morelengthy activities. Her preference was for these extended projects to bethe main components of the curriculum’s activities so that explorationand collaboration among students would be more integral to the program.Her disappointment with some aspects of her Core-Plus instruction aside,Ms. Fay felt that she was learning from the many challenges she facedwhile implementing the curriculum: “It’s really made me think and I willcontinue to think about what I feel is important in mathematics and what Iwant to stress in mathematics class” (Int. 4, 12/16/96).


DISCUSSION

The teachers in this study saw similar benefits of exploration and cooper-ation, but they interpreted the curriculum’s activities in terms of thosebenefits differently. Mr. Allen viewed the Core-Plus problems as chal-lenging and open to student interpretation – at timestoo open. On theother hand, Ms. Fay claimed that the problems were overly structured anddid not permit students to solve problems in their own ways or exploreconcepts sufficiently. Ms. Fay’s comments about Core-Plus activities, suchas “Students are led through the problem,” bear remarkable similarity toMr. Allen’s comments about the traditional curriculum’s exercises.

Both teachers expressed disappointment with the quality of theirstudents’ efforts to discuss and debate with one another as they workedin small groups. Moreover, both teachers attributed difficulties with groupwork in their classrooms to the nature of the mathematics problemsin the curriculum materials. However, the teachers’ contrasting viewsof the Core-Plus problems gave rise to different descriptions of theirspecific difficulties with students’ group work: Ms. Fay suggested thatthe overly structured problems could be completed individually and thuslimited students’ opportunities for discussion and cooperation, whereasMr. Allen believed weak group interactions resulted from open-endedactivities that sometimes placed too heavy demands on students to developtheir own solution strategies. These explanations illustrate the teachers’identification of different connections between the underlying structureof mathematics problems and the quality of students’ work with thoseproblems.

During their instruction with the materials, each teacher added occa-sional activities and experimented with group formats. For instance, eachteacher changed from groups to pairs at times and discussed group prob-lems as a whole class. However, neither teacher explicitly altered thewording, number, or sequence of problems on which students worked intheir groups. The primary way in which both teachers addressed concernswas through their own interactions with students. In Mr. Allen’s firstyear, these interactions aimed to communicate to students the import-ance of developing and exploring multiple approaches to mathematicalsituations, and occurred while students worked on problems in theirgroups. Like Mr. Allen in his second year, Ms. Fay typically savedher comments and questions for students until whole-class instruction,at which time she encouraged students to rely on their own under-standings rather than on rules. In other words, both teachers used theirinteractions with students to communicate messages about their concerns


with students’ work on the mathematics problems in the curriculummaterials.

As these results illustrate, curriculum implementation consists of adynamic relation between teachers and particular curricular features. Thisnotion is consistent with warnings that reform recommendations andassociated curriculum materials cannot and do not bring about changealone – educational change is a complex human endeavor (Cooney, 1988;Freudenthal, 1983). Although the Core-Plus materials were designed withthe intent of supporting student collaboration and exploration and theteachers were motivated to enact these themes, for the teachers to do sowas neither a straightforward process, nor always possible. Concurrentconsideration of Mr. Allen and Ms. Fay’s classroom experiences gives riseto many important themes and associated questions related to curriculumimplementation and classroom reform. Three of these themes are discussedbelow.

Teacher Control of Student Learning

One of the most prominent themes of Mr. Allen’s Core-Plus instruc-tion was his interest in structuring and controlling students’ engagementwith mathematics. Despite his belief that students benefit from exploringand discussing mathematics in their groups, he struggled with whetherstudents would learn appropriate mathematics without more direction fromeither himself or the curriculum materials and review sheets. Mr. Allen iscertainly not alone in his desire to structure or direct students’ learning.When experienced teachers, particularly those with a history of learningand teaching in traditional classroom settings, attempt to reform practicethrough the implementation of novel curricula, concerns about classroomauthority are prevalent (Cooney & Shealy, 1997; Romberg, 1997; M.Wilson & Goldenberg, 1998). Mr. Allen seemed most comfortable when heor the curriculum materials explicitly outlined and thus held the authorityfor what students should be learning.

Though not as immediately apparent in her case, Ms. Fay’s instruc-tion illustrates the issue of authority as well. Her view of the Core-Plusproblems indicates that to a large degree she believed students could learnsignificant mathematics without explicit direction from the curriculum orthe teacher. However, when students asked her questions or requestedher direction, she typically responded directly rather than press studentsto make decisions for themselves. In this sense, she seemed satisfied toallow students to view her as an authority for determining right and wronganswers. One factor related to this tendency might be Ms. Fay’s uncertaintyabout what to do when students worked in their groups. This uncertainty


highlights the need for teachers to rethink how their jobs in the classroommust change to coincide with novel types of student activities.

These issues suggest the value of continued investigation of howteachers involved with reform might renew their sense of efficacy. That is,as teachers change classroom practices, what changes occur in their senseof the ability to affect student learning? Teachers’ sense of efficacy canpowerfully impact student learning, but the conceptions that support manyteachers’ sense of efficacy may be rooted in models of learning that are notconsistent with current reforms. As Smith (1996) explained, teachers havetraditionally felt most self-efficacious when they tell students what theyneed to know. How then can teachers come to feel efficacious as they adoptnew forms of mathematics instruction? Smith identified four potential sitesfor re-establishing self-efficacy: choosing problems, predicting studentreasoning, generating and directing discourse, and judicious telling. Thefocus of the current study raises the question of the self-efficacy of teacherswho are implementing innovative curriculum materials: Can teachers moortheir sense of efficacy in the areas suggested by Smith when they imple-ment curriculum materials, and if so, how? Because the structure ofcurriculum materials may both support and constrain teachers’ efforts tochange instruction, as discussed below, the potential role of curriculumimplementation in teachers’ redevelopment of self-efficacy remains acritical question.

Opportunities to Personalize Instruction

Both Mr. Allen and Ms. Fay faced struggles with cooperation and explor-ation in their Core-Plus instruction, and both attributed their struggles tofeatures of the mathematics problems in the curriculum materials. Despitetheir criticisms of the materials, neither teacher significantly changedthe nature of the problems on which students worked in their groups.Rather, both teachers used their interactions with students to address theirconcerns. Why did the teachers not change the Core-Plus problems andactivities to better suit their personal goals? Ms. Fay frequently expressedthat she was notable to personalize her instruction to a greater extent.Although Mr. Allen never expressed this sentiment, he too may havebeen confined in his ability to make further changes to the curriculum’sactivities.

One constraint to altering the materials was suggested by Ms. Fay. Shelamented that the culture of the mathematics department, which encour-aged uniform implementation of the materials, confined teachers’ abilitiesto develop individual teaching styles. School culture shapes how teachersmake sense of their decisions and actions in relation to perceived shared


understandings of the faculty (Feiman-Nemser & Floden, 1986; Gregg,1995). Like many of Mr. Allen’s mathematics department colleagues, heviewed the curriculum as material to cover. Though he expressed littleconcern, he admitted that his decisions to teach in certain ways were ofteninfluenced by departmental pressures. When teachers feel restricted in theirattempts to make adjustments to the curriculum, they may face difficultchallenges in adapting the curriculum to suit the needs of their studentsand to best fit their own goals and strengths.

It may also be the case that the curriculum materials themselvespresented a form of constraint to the teachers. Recall that Mr. Allensuggested that the Core-Plus materials had accomplished the developmentof mathematical situations for students to explore, leaving the teacher tiedto those situations. He highlighted an interesting notion: When a reform-minded teacher uses traditional materials in the classroom, he or she maybe afforded more room for personalization because the goals of the mate-rials are so different from his or her own goals. Because reform-orientedcurriculum designers accomplish much of the alteration of mathematicalcontent and activity in their production of materials, teachers with strongand innovative visions may experience a profound loss of previously-heldopportunities to personalize their instruction.

The issue of constraints upon teachers who implement curriculummaterials is particularly interesting when related to Prawat’s (1992) iden-tification of one impediment to teacher change:

Instead of viewing students and curriculum interactively . . . teachers tend to regard themas similar factors that somehow must be reconciled . . . . Teachers focus on the packagingand delivery of content, instead of on more substantive issues of knowledge selection andconstruction. (p. 389)

This imperative begs the question of whether the detailed design ofcurriculum materials encourages teachers to focus primarily on pack-aging and delivery. That is, to what extent do curriculum materials permitteachers to view the learner and curriculum as interactive?

If teachers are to view students and curriculum dynamically, they needto learn to make classroom-based developments within the curriculumimplementation process. After all, curriculum developers may wish tocreate certain learning experiences for students, but they cannot fullyanticipate how particular students will interact with the mathematicalactivities. If teachers are to extend their focus beyond delivery, thedesign of curriculum materials may require substantial changes. Balland Cohen (1996) have suggested that curriculum materials designedwith the intention to engage teachers as well as students “could helpteachers to learn how to listen to and interpret what students say, and to


anticipate what learners may think about or do in response to instruc-tional activities” (p. 7). Given this potential for curriculum materials todraw teachers’ attention to details of student learning, curriculum imple-mentation may indeed offer first-hand experiences that can help teachersdevelop constructivist views of mathematics learning and enact associatedpractices.

Tensions Between a Curriculum’s Philosophy and Teachers’ Visions

Past research reports have described teachers whose traditional concep-tions interfered with their abilities to implement novel curricula (e.g.,Cohen, 1990; S.M. Wilson, 1990). These teachers’ struggles have beenattributed to their misunderstanding or lack of appreciation for the philo-sophies underlying the curriculum’s proposed activities. The present studyoffers an example of a distinctly different type of conflict betweenteachers’ conceptions and curriculum design. The two teachers in thisstudy seemed to desire problems that required more extensive cooperationamong students than they believed was provided in the materials. Duringtheir work on the Core-Plus investigations, some groups in both teachers’classes were observed working on problems individually even though theywere seated in groups. In addition, in his second year, Mr. Allen sometimesinstructedstudents to work individually on the investigations that had beenintended for groups. These observations support Ms. Fay’s concern thatwork on the Core-Plus problems did not always require substantial collab-oration among students. These results emphasize how tensions betweenteachers and curriculum may not necessarily be based on deficiencies inteachers’ conceptions, but may instead develop through complex interac-tions between teachers’ goals and specific characteristics of the curriculummaterials.

The dynamic interaction between teachers and curriculum has potentialto be highly educative for teachers. What conceptions of mathematics,teaching, and students are most helpful to teachers as they attempt touse curriculum materials to teach in reformed ways? What qualities ofcurriculum materials, and what underlying philosophies, most produc-tively help teachers change their practices? These questions suggestfruitful areas for continued research activity that will extend our under-standing of how teachers and curriculum materials interact in the processof reform.


IMPLICATIONS FOR MATHEMATICS TEACHERDEVELOPMENT

As illustrated by the findings of this study, teachers’ process of changedraws on their conceptions of mathematical content, students, pedagogy,and the department contexts in which they work. For curriculumdevelopers, these results have important implications: In the continueddevelopment of reform documents and materials, the visions and meaningsthat teachers attach to themmust be carefully considered. To better supportteachers in facilitating classroom activities that include student explorationand cooperation, curriculum materials must not only recognize but alsoexploit and profit by the teacher aspect of the teacher-curriculum dynamic.

This study’s results also present considerable challenges to teachereducators and others concerned with the professional development ofteachers. We bear responsibility to create opportunities for teachers, bothbeginning and experienced, to develop conceptions that will help them todeal effectively with reform themes in their mathematics classrooms. Likemany inservice teachers, preservice teachers often possess weak know-ledge and narrow views of mathematics and mathematics pedagogy –conceptions that are bolstered by years spent as students in traditionalclassrooms (Brown, Cooney & Jones, 1990; Lortie, 1975; A.G. Thompson,1992; Zeichner & Gore, 1990). Professional development activities muststrive to assist inservice and preservice teachers in making sense of themany discrepancies between current reform recommendations and theirown classroom experiences as students and teachers.

One area of mathematics teacher education that warrants increasedattention is the preparation of teachers to enter into a more dynamic rela-tionship with the curriculum. It is notable that in this study both teacherswere dissatisfied with the structure of activities presented in the Core-Plusmaterials, but neither teacher changed the problems on which studentsworked during class sessions. Ms. Fay and Mr. Allen’s treatment of thecurriculum as fixed suggests that teachers may struggle to conceive ofcurriculum as an adaptable guide that permits and encourages alterationswith respect to the demands of particular students. Teachers likely requiresupport not only in coming to recognize the need to adapt curriculum, butalso in learninghow to adapt it.

Of course, even if Ms. Fay and Mr. Allen had viewed curriculum moredynamically, it is not necessarily the case that student learning would havebeen more central to their thinking. In fact, neither teacher displayed evid-ence of recognizing student understanding as what Goldsmith and Schifter(1997) termed “both the guide and the goal of their practice” (p. 29). Both


Ms. Fay and Mr. Allen seemed to be more concerned with specific problemand lesson designs than with details of their students’ thinking. Ironically,had Mr. Allen made changes to the curriculum’s printed problems, hisinstruction may have less successfully enacted reform themes. Based onhis conception that students need to follow more structured paths towardmathematical understandings, his tailoring of the materials would mostlikely have involved more direction to the mathematics problems, limitingstudents’ opportunities for exploration. This example also suggests thatteacher education activities must aim to encourage teachers to conceive ofstudents as independent learners who are capable of and can benefit fromtheir own development of powerful mathematical strategies and theories.

There are likely many useful materials and contexts for extendingteachers’ conceptions in the areas for professional growth to which Mr.Allen and Ms. Fay’s experiences draw our attention. One potentially richcontext for extending teachers’ conceptions involves the use of reform-oriented K-12 curriculum materials in preservice teacher education andother professional development activities (Lloyd & Frykholm, 1998).First-hand experiences workingas studentson the investigations presentedin curriculum materials may prompt teachers to expand their conceptionsof teaching and learning. Although Mr. Allen and Ms. Fay were able toparticipate in some of these experiences second-hand by observing theirstudents’ work with the Core-Plus materials, they may have benefitedalso by engaging with the materials more personally. Such experiencesmay also allow teachers to develop new understandings of mathematicalconcepts and real-world contexts that they did not encounter in their ownmathematics education. Furthermore, use of teachers’ guides associatedwith both innovative and traditional materials to plan and facilitate lessonsmight engage teachers in recognizing and making pedagogical choices,thus explicitly initiating a more dynamic relationship with curriculum.

Innovative curriculum materials are certainly not the only source ofreform-oriented pedagogical themes. Videos and cases are particularlyappealing teacher education tools because they offer detailed images ofwhat reformed mathematics teaching and student learning can look like.Davenport and Sassi (1995) have suggested that images of classroomdiscourse, for instance, are highly valued by teachers at early stages in thechange process. Moreover, video explorations and case-based discussionsallow focused analysis of very specific issues, concepts, and practices.For instance, videos and cases can provide vivid representations of theclassroom successes and difficulties faced by teachers like Ms. Fay andMr. Allen. Through collaborative analysis of classroom vignettes, teacherscan consider and develop empathy for multiple perspectives on teaching


and understanding (Barnett, 1991). In doing so, teachers may learn tomore carefully observe and listen to students, and as a result, expand theirconceptions of students and how they learn mathematics.

Challenges to teachers’ conceptions about mathematics teaching mayalso occur in school settings, as was the case in this study. Implementationof the Core-Plus curriculum materials compelled both teachers to reflecton their past and present classroom instruction. Mr. Allen’s instructionwith the curriculum materials permitted him to view his traditional prac-tices more critically and to begin to identify and articulate his own desirefor new approaches to mathematics teaching. Despite her dissatisfactionwith some aspects of her implementation of the curriculum materials,Ms. Fay, too, valued the opportunity to contemplate different mathe-matics teaching practices. Because reform-oriented experiences demandthat teachers rethink what it means to teach, curriculum implementationoffers a promising way to initiate and support significant professionalgrowth in many areas related to mathematics, teaching, and learning.

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Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburg, VA 24061-0123USAe-mail: [email protected]

ANGEL GUTIÉRREZ and ADELA JAIME

PRESERVICE PRIMARY TEACHERS’ UNDERSTANDING OF THECONCEPT OF ALTITUDE OF A TRIANGLE

ABSTRACT. The way teachers understand mathematics strongly influences their teachingand what their pupils learn. Using Vinner’s model of acquisition of mathematical conceptswith its distinction of concept image and concept definition as a framework, we analyzeconcept images, difficulties, and errors related to the concept of altitude of a triangle exhib-ited by 190 preservice primary teachers in a written test. We describe the influence of twovariables on the preservice teachers’ performances: (a) the presence of a formal definitionand (b) previous classroom activities that dealt with the concept of altitude. We categorizeand analyze some common errors and identify the concept images that may lead to thoseerrors. Finally, we present some implications of our results for teacher education.

One of the main foci of mathematics educators with respect to thegeometry taught in primary and secondary school (in Spain, Grades 1–6 and 7–12, respectively) is to understand how geometrical concepts arelearned by students at different educational levels. Strongly related tothis focus is the interest of mathematics educators in understanding thelearning processes of preservice and inservice teachers with respect to thegeometric concepts and properties they are supposed to teach. In researchon teachers’ understanding of basic geometrical concepts, we find differentapproaches according to the context, the teaching methodology, the organ-ization of the mathematical knowledge, and the kind of activities. Thisdiversity is a consequence of the complexity of the problem, even for thoseconcepts considered basic. The literature on preservice teacher educationis quite extensive. In particular, several handbooks of research includeoverviews of this literature from varied viewpoints: Brown, Cooney, andJones (1990) and Shulman (1990) discuss different research perspectives,or paradigms, to approach preservice and inservice teachers’ training andbeliefs. Clark and Peterson (1990) review results on teachers’ thinkingprocesses. Fennema and Franke (1992) review research on the relationshipamong teachers’ knowledge, their teaching strategies, and their pupils’learning. Lanier and Little (1990) and Brown and Borko (1992) discussissues related to teacher training processes.


254 ANGEL GUTIERREZ AND ADELA JAIME

In this paper we present a study aimed at providing information aboutpreservice primary teachers’ conceptions of altitude of a triangle. Relatedto the objective of this paper, Brown, Cooney, and Jones (1990) suggestedthat often preservice teachers do not have the knowledge and under-standing of mathematical content necessary to apply the methodologicalinnovations proposed by new curricula. In addition, the limited mathe-matics knowledge of preservice teachers is an obstacle for their training ondidactical and pedagogical knowledge (Brown & Borko, 1992). The resultsin Mason and Schell (1988) corroborate these findings for preserviceprimary teachers with respect to geometrical concepts.

Within the USA and Portuguese contexts, respectively, Mayberry(1983) and Matos (1985) studied the van Hiele levels of reasoning ofpreservice primary teachers on several geometric concepts. In Mayberry’sstudy, based on clinical interviews, 48% of the American teachers showedreasoning at Level 2 or lower, 16.5% at Level 3, and 19% at Level 4.In Matos’ study, based on a written multiple-choice test, 59% of thePortuguese teachers showed reasoning at Level 2 or lower, 18% at Level3, and 4.5% at Level 4 or 5. In both studies, more than half of thepreservice teachers were unable to understand deductive arguments, eveninformal ones, or to analyze definitions in order to classify families ofpolygons. Both researchers found a positive correlation between the vanHiele level assigned to teachers and their mathematical backgrounds insecondary school. In our experiments with Spanish preservice primaryteachers (Gutiérrez & Jaime, 1987; Gutiérrez, Jaime, & Fortuny, 1991),we have obtained results that are similar to those in Mayberry’s (1983)and Matos’ (1985) studies. It seems that weak mathematical knowledgeand reasoning skills of preservice primary teachers are independent ofcountries, educational systems, and techniques of assessment.

Linchevsky, Vinner, and Karsenty (1992) and Vinner, Linchevski,and Karsenty (1993) have examined in detail the conceptions preserviceteachers have of mathematical definitions. Preservice high school teacherswere asked to complete a questionnaire in which they were asked (a) todefine an equilateral triangle and a rectangle, (b) to evaluate the correctnessof several definitions of these concepts written by high school students,and (c) to reflect on a dialogue among a teacher and a student on theminimality of mathematical definitions. After the preservice teachers hadcompleted the questionnaire, the researchers held a group discussion on theminimality of definitions, that is, the sufficiency of properties included inthe definition, and on their arbitrariness, that is, the existence ofcanon-ical definitions and alternative equivalent definitions. The analysis ofthe answers showed that 65% of the preservice teachers believed that adefinition could include non-necessary conditions, and only 33% of them

UNDERSTANDING OF THE CONCEPT OF ALTITUDE 255

understood the minimality principle. The students in the first categorymatched van Hiele Levels 2 or lower, and those in the second categorymatched Levels 3 or higher.

Based on their research, S. Vinner and R. Hershkowitz developed amodel to explain the cognitive processes individuals use when they facenew mathematical concepts, in particular, geometrical concepts. One ofthe core components of this model is the difference between a mathe-matical concept, the concept image created in a student’s mind, andthe concept definition verbalized by the student (Vinner, 1991; Vinner& Hershkowitz, 1980, 1983). According to this model, a good learningprocess results in the two elements, concept image and concept defini-tion, being merged which then allows students to correctly discriminateexamples of the concept. Unfortunately, however, for many students thereis no link between the two elements, and students use one or the otheraccording to the given task. In the next section, we present the Vinnermodel, which we used as a framework for our research. We then describea study in which we analyzed preservice primary teachers’ knowledge andunderstanding of the concept of altitude of a triangle and the influence ofdifferent variables on their understanding of the concept. In conclusion, weidentify implications from our research for teacher education.

ON THE FORMATION OF GEOMETRICAL CONCEPTS

The van Hiele model of levels of geometric understanding is recognizedas one of the most comprehensive models with respect to the learning ofgeometry. The Vinner model (Vinner, 1991; Vinner & Hershkowitz, 1980,1983) is another useful framework to guide teachers and researchers intheir activity of understanding students’ mental processes. According tothe latter model, when we listen to or read the name of a known conceptor when we solve some task, our memory is stimulated and something isevoked. However, what is evoked is rarely only the formal definition ofthe concept, but rather a set of visual representations, images, properties,or experiences. This set of elements that can be recalled constitutes theconcept image. For geometrical concepts, a student’s concept image mayinclude various figures the student remembers as examples of the concept,and the set of properties the student considers belonging to the concept.A student’s concept image is viable when it allows the student to discrim-inate, without error, any example of the concept and when the associatedproperties are all necessary properties of the concept. On the other hand,as a result of the teaching methods, students may memorize a definition,which they may repeat when they are asked for it or if they are asked to


identify an example. This verbal definition, which can be memorized andrepeated by a student, is called the student’sconcept definition.

The properties included in a concept image are not always mathe-matical properties; irrelevant physical properties may also be considered,in particular by students who reason at the first van Hiele level (Burger &Shaughnessy, 1986; Clements & Battista, 1992; Fuys, Geddes, & Tischler,1988; Gutiérrez et al., 1991). For instance, for many primary schoolchildren, the concept image of a triangle consists of a set of specifictriangles in standard position and several properties of these figures,such as a triangle having a vertical (right) angle or slanted sides ofequal length (Hershkowitz, 1989). The presence of irrelevant propertiesin students’ concept images has also been reported by Wilson (1986) whostudied the influence of irrelevant features of figures on the usefulness ofcounterexamples.

The concept definition expressed by a student may or may not coincidewith the definition of the corresponding mathematical concept. On theother hand, the concept definition is not necessarily operationally linked toa student’s concept image when the student is solving a task. For instance,when asked to define rectangles, many students include the condition thatnot all the sides are of the same length, although these students identifysquares as rectangles when they are presented with figures. In contrast,other students state the definition of a rectangle as a parallelogram withright angles, but they do not accept squares as rectangles because allsides are of the same length (Wilson, 1990). Both behaviors evidencethe differences that exist for students between concept image and conceptdefinition.

The Vinner model emphasizes that a student’s experiences and theexamples of a concept encountered, either in school or in other contexts,play an important role in the formation of a concept image. Very oftenstudents are given only a few examples of a geometrical concept, allof which have a common specific visual characteristic; these examplesthen become prototypes (Hershkowitz, 1989), and they are the only refer-ences available when the student is judging new cases. Therefore, a wayto improve the quality of concept images is to try to detect the failingsof students’ concept images, to offer them a wider variety of examples,and to take into account especially those examples directly related tothese errors. Unfortunately, many teachers suffer the same errors as theirstudents, and so they cannot help them. An objective of this research is toprovide specific directions on how to analyze teachers’ concept images ofa specific geometrical concept, the altitude of a triangle, as a first step inthe improvement of their knowledge of this concept.


The Vinner model has been applied in several studies. For example,Vinner (1983, 1991) observed students who solved tasks based on conceptsfrom calculus, such as function, tangent, continuity, and limit of asequence. Vinner and Hershkowitz (Hershkowitz & Vinner, 1982, 1983;Vinner & Hershkowitz, 1980, 1983) studied primary and secondary schoolstudents’ geometrical concepts. In some studies (Hershkowitz, 1989;Hershkowitz & Vinner, 1984), students as well as preservice and inser-vice teachers were included in the samples, and all solved the same tasks.Two types of tasks were used: (a) given sets of figures, participants wereasked to identify examples of concepts such as angle, isosceles triangle,right triangle, diagonal, or quadrilateral, and (b) participants constructedexamples of concepts, e.g., altitude of a triangle. In some experiments, apart of the sample was provided with a test that included the definitions ofthe concepts, whereas the other part of the sample received the same testwithout the definitions.

The main conclusions relevant to our study are:

• When teachers, preservice teachers, and students were asked to solvethe same tasks, the same misconceptions were found in the threesamples; the preservice and inservice teachers’ concept images wereonly slightly better than those of the students (Hershkowitz, 1989;Hershkowitz & Vinner, 1984).

• The concept images of many teachers, prospective teachers, andstudents are based on a few prototypical figures and are inde-pendent of their concept definitions. For instance, a very commonconcept image of a diagonal of a quadrilateral includes only internaldiagonals; consequently, only one diagonal is commonly drawn inconcave quadrilaterals (Hershkowitz, 1989; Hershkowitz & Vinner,1984; Vinner & Hershkowitz, 1980).

• With respect to identification tasks, the presence of the definition ofa concept has almost no influence on the responses. However, in atask of construction of altitudes of triangles, the answers of subjectswho were provided with the definition were significantly better(Hershkowitz & Vinner, 1982, 1983, 1984; Vinner & Hershkowitz,1983).

Wilson (1988, 1990), in her studies of students in Grade 6 andabove, got results similar to those of Vinner and Hershkowitz. Wilsonasked students to draw examples of given geometric concepts, definethe concepts, select examples among several figures, and answer ques-tions about the concepts. The concepts selected were triangle, rectangle,and square. Wilson reported that students based their answers on inflex-ible prototypical instances. She also observed the presence of many


Figure 1. Typical (a) and atypical (b) textbook examples of a right triangle.

inconsistencies among students’ answers to the different parts of the test.Follow-up interviews showed that some students were aware of suchinconsistencies, but they did not feel uncomfortable with them. AlthoughWilson did not use the Vinner model as the conceptual framework for herstudies, her results can be interpreted in terms of that model. In particular,students used their concept images in some tasks and their concept defini-tions in other tasks. They were aware of the contradictions among imagesand definitions, but they were comfortable because they saw conceptdefinitions and concept images as two independent tools.

In light of the Vinner model, three types of behaviors can be identified,according to the quality of the associated concept images (Hershkowitz,1990):

• Individuals with the poorest concept images, developed from a fewprototypical examples and visual properties, base their judgments onthe visual aspect of these prototypes, compare them to the figureswith which they have to work, and reject as examples those figuresthat do not coincide with the prototypes of their concept image.For instance, the concept images associated with a right triangleoften include only those triangles with a horizontal and a verticalside (Figure 1a). Consequently, many students, preservice, and inser-vice teachers have great difficulty identifying right triangles withouthorizontal and vertical sides (Figure 1b) (Hershkowitz, 1989).

• Individuals with somewhat richer concept images make judgmentsbased on a few prototypical examples plus some mathematical proper-ties of those examples. They try to apply these properties to the figureswith which they have to work, and they reject those figures that donot fit these properties. For instance, for many students, the conceptimage of a triangle includes the property that the triangle is an acutetriangle and, as a consequence, has only internal altitudes; therefore,these students tend to say that there is no altitude to side a of thetriangle in Figure 2, or they draw an internal segment (Hershkowitz,1989).


Figure 2. An obtuse triangle-frequently not part of a student’s concept image.

• The third type of behavior described by the Vinner model corres-ponds to those individuals who have complete concept images, that is,images that include a wide variety of examples and all the importantproperties of these examples. The specific examples now play acomplementary role to provide ideas or to verify conjectures whichare later corroborated or formalized by the use of the mathematicalproperties. These students are able to make correct judgments basedon the use and analysis of the critical properties of the concepts.

Finally, Vinner and Hershkowitz also showed that many teachers-atdifferent educational levels-wrongly believe that their pupils’ reasoningis based predominantly on the formal verbal definitions of the concepts,and that their concept images play a secondary role, subordinated, both inits formation and its use, to the concept definition. Probably these teacherstake for granted as well that the concept definition used by their pupilsis the mathematically correct one the students have been taught. However,the student activities are based, in most cases, only on their concept imagesbecause for many students the concept definition is inactive, except whenthey are explicitly asked to repeat it, or does not exist; they forgot or neverlearned the definition that they were taught by their teacher (Vinner, 1991).

PRESERVICE TEACHERS’ CONCEPT IMAGES OFALTITUDE OF A TRIANGLE

Based on Vinner and Hershkowitz’s experiments on the concept of altitudeof a triangle, we address and extend their conclusions. We present theresults of a test designed to analyze preservice primary teachers’ under-standing of the concept of altitude of a triangle. We identify their reasoningprocesses and the influence of some variables, such as students’ previousknowledge, the presence of a formal definition in the test, or the influenceof learning activities that dealt with altitudes of triangles as part of thecontent of a course on mathematics education. The conclusions of thispart refer to the identification of the preservice teachers’ concept imagesand the way they make use of their concept images and the mathematical


definition of altitude of a triangle in the resolution of specific tasks, taskssimilar to those they will find in the textbooks when they begin theirprofessional life.

Method

Instrument. The instrument for the study was a test made by rearrangingthe items used in Hershkowitz and Vinner (1982) and Vinner andHershkowitz (1983). Originally, the test had been designed for studentsin Grades 6 to 8 (ages 11 to 14). One of the researchers’ aims had been toinvestigate whether or not the presence of the formal definition of alti-tude influenced the students’ answers. Vinner and Hershkowitz (1983)found that the presence of a definition in a construction task significantlyimproved results. In addition, irrelevant attributes of the triangles, suchas position or shape, induced a significant number of wrong answersboth when the definition was provided and when it was not provided(Hershkowitz, 1989). Incorrect responses were classified according to thecritical attributes of the concept of altitude used by the students, but thereasons for the errors were not analyzed.

Figure 3 shows one version of the test used in our study. The alternateversion differed from the one depicted only by the omission of the defini-tion of the altitude. The definition of altitude used in the test was the oneincluded in Spanish textbooks and taught by primary teachers. There wereno time limits to complete the test.

Participants and procedure. The test was administered to four groupsof students in the Primary Teacher Training School of the Universityof Valencia (190 students). Like all Spanish university students, thesestudents had studied mathematics from Grade 1 to 10. Most students in thesample had also studied mathematics, as an optional subject, in Grades 11and/or 12 of secondary school, although we do not have information aboutthe exact amount. Euclidean geometry, however, is usually studied only inprimary and lower secondary school (Grades 1 to 10). About half of thestudents in Group A (A2) were administered the version of the test thatcontained the definition of the altitude of a triangle. The other studentsin Group A (A1) took the alternate version of the test. All students inGroups B and C first took the test without the definition of the altitude.Immediately after completion, they took the test again, and this time thetest did contain the definition. Students in Group D were administeredthe test twice without the definition, once before and once after theyhad worked on activities involving altitudes of triangles. Table I summa-rizes the information about the distribution of the students in the differentgroups.


Figure 3. Test items including the definition of the altitude of a triangle.

Coding. To code the answers, we considered correct answers to be thosethat satisfied the mathematical requirements of the question. We allowed,however, a small error in the perpendicularity of the altitudes because thestudents were working without drawing tools. We coded as wrong answersthose in which the segments did not meet the mathematical requirements


TABLE I

Distribution of the Preservice Teachers Participating in the Study

Number of students taking the test

Group College year without / with definition pre-test / post-test Total

A1 3rd 28 / 0 — 28

A2 3rd 0 / 31 — 31

B 1st 33 / 33 — 33

C 3rd 34 / 34 — 34

D 2nd 64 / 0 64∗ / 49 64

Total 159 / 98 64 / 49 190

∗15 students answered only the pre-test.

of the question or those in which the segment was clearly not perpendicularto the base. We analyze the various kinds of errors later in the section.

Influence of the Presence of the Definition of Altitude

Because the concept ofaltitude of a triangleis considered important byprimary school teachers, we can assume that all students in the sample hadstudied the concept at some point during their primary school years. On theother hand, because Euclidean Geometry is not taught in upper secondaryschool, we can also assume that students had not formally dealt with theconcept for several years. The administration of the test without definitionof the concept of altitude allowed us to obtain information about whatthe students remembered several years after encountering the concept, andthus gain insights into the nature of their present concept images. If thedefinition was included in the test, it possibly served as a reminder for thestudents. That is, the definition might have activated the students’ conceptimage, although this concept image might not have matched the givendefinition. In other cases, the presence of the definition might have allowedstudents to analyze the definition and thus enabled them to answer an item.Therefore, the version of the test with the definition allowed us to gaininformation about the change in the students’ concept image fostered bythe definition.

The results of the tests for Groups A1 and A2 (28 and 31 students,respectively), B (33 students), and C (34 students) are summarized inFigure 4 and Table II. The design used for Groups B and C proved tobe richer than the one for Groups A1 and A2, because the twofold admin-istration of the test, first without and then with the definition, provided us


Figure 4. Percentages of correct responses in Groups A1, A2, B, and C.

with information about the changes in the ways students responded to theitems before and after they had read the definition.

Correct responses. Overall, there were more correct responses in the testcontaining the definition of altitude than in the test without the defini-tion, except for Item 9 in Group B. The definition seemed to provide thestudents with information that helped them improve their understanding ofthe concept of altitude. Although Groups B and C were administered thesame test twice, we believe that the influence of the practice effect wasminimal because students had unlimited time to check all responses andcorrect those they thought wrong before they returned the first test to theteacher.


TABLE II

Responses (%) to the Test Without Definition (NoD) and With Definition (Def) of Altitude

Correct responses (%) Incorrect responses (%) No response (%)

Item Cond. A1 A2 B C A1 A2 B C A1 A2 B C

1 NoD 85.7 93.9 88.2 14.3 6.1 11.8

Def 93.5 97.0 91.2 6.5 3.0 8.8

2 NoD 85.7 87.9 82.4 14.3 12.2 17.6

Def 96.8 97.0 91.2 3.2 3.0 8.8

3 NoD 89.3 93.9 82.4 10.7 6.1 14.7 2.9

Def 96.8 97.0 91.2 3.3 3.0 8.8

4 NoD 82.1 84.8 79.4 17.9 15.1 20.6

Def 96.8 97.0 85.3 3.3 3.0 14.7

5 NoD 82.1 69.7 58.8 17.9 24.2 35.3 6.1 5.9

Def 87.1 78.8 67.7 12.9 15.1 26.5 6.1 5.9

6 NoD 75.0 93.9 82.4 25.0 6.1 17.6

Def 90.3 97.0 88.2 9.7 3.0 11.8

7 NoD 57.1 69.7 55.9 42.8 27.3 38.2 3.0 5.9

Def 71.0 84.8 73.5 29.1 15.1 26.5

8 NoD 42.9 54.5 35.3 57.2 45.4 61.8 2.9

Def 58.1 69.7 61.8 42.0 30.3 35.3 2.9

9 NoD 64.3 66.7 44.1 35.7 27.2 47.1 6.1 8.8

Def 67.7 63.6 61.8 25.8 30.3 35.3 6.5 6.1 2.9

10 NoD 64.3 75.8 73.5 32.1 21.2 11.8 3.6 3.0 14.7

Def 67.7 84.8 85.3 25.8 15.1 11.8 6.5 2.9

11 NoD 82.1 75.8 79.4 17.9 21.2 17.6 3.0 2.9

Def 90.3 93.9 88.2 6.5 6.0 8.8 3.2 2.9

12 NoD 60.7 69.7 61.8 35.7 27.3 38.2 3.6 3.0

Def 77.4 90.9 73.5 22.6 9.1 23.6 2.9

13 NoD 57.1 60.6 44.1 39.3 36.4 52.9 3.6 3.0 2.9

Def 67.7 66.7 55.9 29.1 30.3 41.2 3.2 3.0 2.9

14 NoD 60.7 69.7 58.8 39.3 27.3 41.2 3.0

Def 71.0 81.8 73.5 25.8 18.2 26.5 3.2


Difficulty of items. About the same items seemed to be easy and difficult,respectively, for all groups. Based on the percentage of correct answers ineach group, the easiest items (3, 1, 2, 6, 4, and 11, ordered from easy toless easy) are, except one, the triangles with internal altitudes. The mostdifficult items (8, 13, 9, 7, 14, 5, and 12, ordered from more to less diffi-cult) are the triangles with non-internal altitudes, that is, obtuse or righttriangles.

Influence of position. Items 10 and 11 are the same triangles, but rotated.The graphs show a clear difference between the percentages of correctanswers for these two items from all groups of students. This can be inter-preted as a consequence of the different rotation of the figure, the easieritem (11) appearing in the prototypical position with a vertical altitude. Thesame effect can be seen for Items 7, 8, and 12, where the easiest item (12)is a triangle in the prototypical position, whereas the base of the triangle inItem 8, the most difficult one, is slanted.

Items with no responses. The number of items with no responses was lowin each group, even when the test did not provide the definition. Most ofthe blank answers corresponded to Items 5, 9, 10, and 13. Three of thosefour items involved right triangles. The fourth, Item 13, was a triangle withan internal altitude, but the altitude had to be drawn in an unusual position.From the low number of items without a response, we can conclude that allthe students had at least some concept image of altitude which they wereable to recall.

Influence of Instruction on Students’ Concept Images

Group D was administered the test without the definition of altitude bothtimes, but before and after the students did some work with altitudesof triangles as part of the second year course of their teacher educationprogram. The aim was to observe the influence of instruction that includedattention to the altitude of a triangle.

The intervention. As a part of a course on didactic of mathematics, GroupD worked one hour weekly during the year on problem-solving tasks basedon Cabri (Version 1.7 for PC). The aims of this course were twofold:(a) to improve the students’ knowledge and understanding of the mathe-matics taught in primary schools, and (b) to teach elements of didactic ofmathematics students would need for their professional activity. Duringthe instructional time, students solved geometric construction problems,made and proved or disproved conjectures, and designed simulations of


Figure 5. Percentages of correct responses in Group D.

geometrical properties studied in primary school. All tasks were relatedto basic geometrical concepts, such as triangles, quadrilaterals, paral-lelism, or perpendicularity. Computers were operated by groups of twoor three students. For each task, students first completed the task ontheir own. In the whole-group discussion that followed the group activ-ities, students discussed various ways of solving the problem, as well astheir findings, conjectures, and difficulties. Finally, there was a phase ofinstitutionalization by the teacher (Douady & Perrin-Glorian, 1989).

For three consecutive sessions, students worked on tasks involvingtriangles. In particular, students studied the construction of special linesegments, such as altitude and median, and points (e.g., orthocenter,barycenter). They observed, conjectured, and verified the properties ofsegments and points. One of the properties studied was the position,internal or external, of the segments and centers as relative to variouskinds of triangles. Although the concept of altitude of a triangle was used,its definition or characteristic properties were not explicitly stated by theteacher. Students drew altitudes by using the perpendicular line commandin the Cabri menu.

Pre- and post-test. The pre-test was administered one week before studentsbegan the three classes devoted to triangles, and the post-test was admin-istered two weeks after these classes. Figure 5 and Table III summarize theresponses obtained from the 49 students who participated in both the pre-test and the post-test. Figure 5 shows the improvement in the results of thepost-test in relation to the pre-test. It may be appreciated that the greatestdifferences between the correct answers on the pre- and post-test happen inItems 7, 8, 12, and 14, which are among the most difficult items. These datasuggest that, in the courses for preservice teachers, instruction and directedactivities influence students’ concept images, as would be expected.


TABLE III

Responses (%) to Pre- and Post-Test in Group D (n = 49)

Correct responses Incorrect responses No response

Item Pre-test Post-test Pre-test Post-test Pre-test Post-test

1 98.0 100 2.0 0

2 91.8 91.8 8.2 8.2

3 98.0 100 2.0 0

4 89.8 93.9 8.2 6.1 2.0

5 81.6 85.7 16.3 14.3 2.0

6 93.9 95.9 6.1 4.1

7 57.1 75.5 42.9 20.4 4.1

8 40.8 71.4 57.1 26.5 2.0 2.0

9 67.3 71.4 30.6 26.5 2.0 2.0

10 73.5 81.6 26.5 18.4

11 87.8 89.8 12.2 10.2

12 67.3 79.6 32.7 20.4

13 71.4 75.5 28.6 22.4 2.0

14 61.2 77.6 38.8 22.4

Error Analysis

In this section we analyze the answers of the 159 students from groupsA1, B, C, and D (pre-test) who answered the testwithout the definition ofaltitude (refer to Table I). Our aim is to present a catalogue of responsescharacteristic of different kinds of partial or poor concept images. Thestudent drawings were coded as follows:

1. No answer.2. The correct altitude on the specified base. A small error in the

perpendicularity of the segment drawn was accepted.3. The median to the specified base.4. The perpendicular bisector of a side.5. The correct altitude to a side different from the specified base.6. A segment perpendicular to the specified base, but with the wrong

length.7. A segment internal to the triangle, from the opposite vertex to the

specified base, but not perpendicular to it nor the median to the base.The difference between Codes 2 and 7 is that the sloping of thesegment drawn is big enough to discard imprecision in the drawing.

8. Other incorrect responses.


TABLE IV

Incorrect Responses (%) With Respect to Error Type

Item Code 3 Code 4 Code 5 Code 6 Code 7 Code 8

1 0 0.6 0 6.3 0 0

2 3.1 1.3 0 4.4 0 3.8

3 0 0.6 0 5.0 0 1.3

4 4.4 1.9 0 5.0 0.6 2.5

5 6.3 1.3 4.4 7.5 0.6 1.9

6 0 0.6 3.1 6.9 0 1.3

7 18.9 1.9 5.0 4.4 3.1 5.0

8 28.3 1.9 11.3 2.5 9.4 2.5

9 10.1 1.3 7.5 6.3 1.3 7.5

10 9.4 1.3 1.3 2.5 4.4 4.4

11 5.0 1.9 0 6.3 1.3 1.9

12 13.8 1.3 5.0 8.8 4.4 0

13 11.9 1.9 10.7 5.0 5.0 3.1

14 17.0 1.9 6.9 7.5 3.8 0

Table IV shows the frequency (in percent of the set of 159 responses)of each kind of Codes 3–8 observed. We attempt to analyze prototypes ofeach solution and to infer the concept image associated with it.

In the analysis of the errors for each item, it is necessary to realizethat some errors cannot be detected in some triangles. For instance, itemsinvolving isosceles triangles (Items 1, 3, and 6) did not allow us to detectwhether students confused altitude, median, and the perpendicular bisector(Code 3 and 4 errors) because these three segments coincide, except whenthe student considers the altitude on a side different from the specified one.

Next, we present and analyze examples that represent the most commontypes of responses. The answers allow us to recognize the consistency ofthe students’ concept image with respect to the altitude of a triangle. Somestudents’ responses are not included in the analysis because those studentsmade only a few mistakes, likely due to negligence or oversight, or theygave mostly wrong answers but without any apparent pattern.

In each case below, although only a few student responses arepresented, those responses allow us to identify the characteristics of thestudent’s concept image. The student’s other responses were consistentwith the ones provided.


Figure 6. Responses reflecting confusion between the concepts of altitude and median(Ana, Group D).

Figure 7. Responses reflecting a partial concept image influenced by the concept ofperpendicular bisector (Student 22, Group A).

Altitude vs. median. Figure 6 shows responses of a student whose conceptimage of altitude is a mixture of altitude and median (Code 3 error). As canbe seen in Table IV, this is the most frequent error. The confusion of thetwo concepts may be due, in part, to both concepts usually being studiedat the same time in the primary school. It also appears that the students,rather than using the definition of altitude when drawing, used only theirconcept image, a result that confirms Vinner’s (1991, p. 73) findings. Code3 was recorded in 26.9% of the tests without the definition of altitude andin 22.4% of the tests with the definition. It is interesting to observe that,in reality, most of these students, as the one featured in Figure 6, have apartial concept image; they correctly drew the altitude in triangles withinternal altitudes, but they were unable to draw an external altitude or analtitude that coincides with a side.

Altitude vs. perpendicular bisector. The examples in Figure 7 representresponses of a student who mixed the concepts of altitude and perpen-dicular bisector, an error that rarely occurred in our sample (Code 4).The student does not have a wrong concept image of altitude in that thedrawn segments are, in fact, altitudes on the required sides. The studenthas, however, only a partial image, which makes him/her draw the altitudewith an endpoint on the midpoint of the required base of the triangle.

Limitation to internal altitudes. Figure 8 shows the answers of a studentwho, in all cases, has drawn an internal altitude to a side of the triangle.


Figure 8. Responses reflecting a partial concept image that excludes external altitudes(Marta, Group D).

Figure 9. Responses reflecting a partial image which does not take into account the lengthof the altitude (Student 25, Group A).

The student considered a side different than the specified base when thecorrect solution would have not been internal to the triangle (Code 5 error).This student has a partial concept image, which does not include triangleswith non-internal altitudes. That is, the prototypical case is that of an acutetriangle, in which the altitude is internal to the triangle.

Disregard of length. Figure 9 presents the case of a student who did notconsider the length of the segment as relevant, but who took into accountall the other characteristics of the concept of altitude (Code 6 error). Itappears that this student associated the altitude with a ray or a segmentof an undetermined length, but having all the other characteristics of thealtitude. Again, one way of changing this partial concept image into acomplete one is to present problems that require students to use the char-acteristic that they had not considered, in this case, the length. A disregardof the length of the altitude was the second most frequent error in ourstudents.

Fixation on side. Figure 10 presents the work of a student (the only onein our sample who answered in this way) whose concept image of altitudeappears to include only isosceles triangles, because only Items 1, 3, and6 were answered correctly. The student had no image for other cases inwhich he/she highlighted some element of the triangle, in particular, oneside. More specifically, this student drew a side of the triangle or the dottedsegment (provided in the test to indicate the side on which the altitudeshould be drawn).


Figure 10. Responses representing fixation on a side of the triangle (Student 23, GroupA).

Figure 11. Responses in which the highlighted base served as distracter (Student 32,Group A).

Despite the apparent regularity of the answers, a closer look shows thatthe student had no criterion for choosing a particular side because the sidemarked was sometimes side a, and the other times a different side.

Marked base as distracter. Finally, the student whose answers are includedin Figure 11 presents a concept image that is quite confusing. The useof incorrect visual images prevailed over the use of the mathematicalproperties of the concept. For this student, the dotted segments that weresupposed to help identify the base might have served as distracters thatinduced the student to draw wrong segments. Once again, this conceptimage includes the prototype of an internal altitude, although the answersto the first and last triangles in Figure 11 (the same triangle in two differentpositions) are contradictory. Furthermore, for this student, the altitude mustbe a segment different from any of the sides (in the case of right triangles).This student did not consider perpendicularity when other visual charac-teristics made the drawings more like the figures that seemed to exist inhis/her memory.

We have shown several examples of preservice primary teachers’concept images that represent various incorrect and/or partial concep-tions of altitude of a triangle. Some errors occurred quite frequently,whereas others seem very unusual. The reason for certain errors is evident,although, as the two last examples show, interpretation of answers is some-times difficult. In light of the previous results, several consequences can beraised for pre- or inservice teacher training courses.


IMPLICATIONS FOR TEACHER EDUCATION

Certainly the concept of altitude of a triangle is not an easily graspedconcept by either pupils or preservice teachers. In light of our research andother similar studies (Hershkowitz, 1989; Hershkowitz & Vinner, 1984),we can begin to provide a basis for teachers of teachers to make informeddecisions when designing programs for preservice and inservice teachers.We found that many preservice teachers had the same poor concept imagesas primary or secondary students. This situation can provide a contextfor the teachers to examine their and their classmates concept images andconcomitantly learn what kinds of concept images pupils are likely to have.They could examine, for example, their understanding of the altitude andthe median, and subsequently investigate how they differentiate these twoconcepts and how their students might as well. This kind of an analysiscould provide them with insights when they become teachers with theresponsibility to teach these concepts to their students. The question thenbecomes: What kind of instruction could best enable teachers and studentsto develop a complete concept image?

We have based our research on the Vinner model. We found that thecore constructs of this model, the student’sconcept imageand conceptdefinition, and the description of the relationship among them providevaluable tools for interpreting the learning of basic geometric concepts.Our research has contributed to the particular context in which this modelcould be used with pre- or inservice teachers to enhance their under-standing and for them to develop a basis for understanding their students’concept images. Further, we recommend that future research focuses ontopics such as space, measurement, or isometries to provide an even richerresearch base for teacher education programs.

We see the following specific suggestions for teacher education asemanating from our research. We offer them in the hope they are of valueto other teacher educators.

• Any instructional program intended to enhance preservice teachers’concept images should take into account their previous conceptimages and possible misconceptions and should consider specificlearning situations useful for addressing those misconceptions.

• Teachers should be given opportunities to present, explain, and defendtheir particular conceptions of altitude and other basic geometricalconcepts. These discussions could provide the opportunity to considerand resolve any cognitive conflicts that arise from different conceptimages. These concept images should be compared with the formaldefinition.


• Teachers could conduct class discussions with their pupils to assesstheir concept images of the altitude of a triangle. Analysis of students’outcomes would also provide a context for teachers to reflect on theirown concept images.

• Teachers could be given pupils’ responses to questions like the onespresented in this study and asked to describe these pupils’ conceptimages. As they reflect on and discuss pupils’ concept images theymight at the same time reflect on their own concept images.

• Teachers could be presented with research similar to the one reportedin this article and use that research as a basis for reflecting on theirown concept images. They could, for example, study the natureof students’ understanding of subconcepts when difficulty occursregarding the primary concept. They could engage in designing activ-ities aimed at examining possible conflicts related to the differentsubconcepts, their relationships, and the logical nature of the defini-tion.

A final remark addresses the use of research such as ours and the educa-tion of teachers. There are many mathematical concepts that depend onlower order concepts (Skemp, 1971) or subconcepts. It often happens thathigh school or university teachers assume that their pupils understand theselower order concepts and subconcepts. Using the Vinner model with itsdistinction of concept image and concept definition, teachers and teachereducators can examine the nature of students’ poorly formed conceptimages and determine what geometric concepts, properties, and facts arelacking. This, then, could become a focal point for teacher educationprograms.

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Departamento de Didáctica de la Matemática Angel GutiérrezUniversidad de ValenciaApartado 2204546071 – ValenciaSpainE-mail: [email protected]

Departamento de Didáctica de la Matemática Adela JaimeUniversidad de ValenciaApartado 2204546071 – ValenciaSpainE-mail: [email protected]

KAREN LANGFORD and MARY ANN HUNTLEY

INTERNSHIPS AS COMMENCEMENT: MATHEMATICS ANDSCIENCE RESEARCH EXPERIENCES AS CATALYSTS FORPRESERVICE TEACHER PROFESSIONAL DEVELOPMENT?

I learned there must always be a nagging ignor-ance accompanied by accessible knowledge to achievelearning . . . . I can use my experience as a learner toenhance my teaching. I need to use ignorance to feedcuriosity. I want to model my mentor’s ability to let methink for myself before assisting. (Project participant)

ABSTRACT. Among the initiatives of the Maryland Collaborative for Teacher Prepara-tion (MCTP) is a summer research internship program that places preservice middlegrades teachers in extended collaboration with professional mathematicians, scientists,and educators engaged in research and curriculum development activities. We describethe MCTP internship program including the rationale for and structural features of theprogram. We also highlight the effects of the internship experience on preservice teachers’conceptions of and beliefs about the nature and processes of mathematics and science, andthe teaching of mathematics and science. Our findings suggest that the internship exper-ience is a fundamentally significant life experience for preservice teachers. Furthermore,internships have the potential for realizing reform in mathematics and science education;that is, preservice teachers who have participated in MCTP internships intend to bring aholistic, conceptually oriented view of mathematics and science to their classrooms. Thepaper concludes with a summary of ongoing programmatic and logistical challenges of theMCTP internship program.

This preservice teacher’s reflection followed the completion of a summerresearch internship in a program that places future teachers in full-time collaboration with professional mathematicians, scientists, museumpersonnel, environmental educators, and curriculum developers. Powerfulstatements like this compelled us to systematically study the effects ofthe internship program on its participants. The internship program isone component of the Maryland Collaborative for Teacher Preparation(MCTP) which, guided by current research on teaching and learning, ispursuing several novel approaches in the preparation of mathematics andscience teachers for Grades 4–8.

? Funding for writing this manuscript was provided by grants from The NationalScience Foundation (DUE # 9255745) and The University System of Maryland. Theauthors would like to acknowledge helpful comments on earlier versions of this manuscriptby James T. Fey, Amy Roth-McDuffie, Maureen Gardner, and Donna Ayres.


278 KAREN LANGFORD AND MARY ANN HUNTLEY

In this report we provide descriptive and analytical accounts ofthe MCTP internship program. First, we describe two contexts forunderstanding these internships – the MCTP program, and the MCTPperspective on preservice teacher internships. Second, we describe thestructural features of the internship program. Third, we present an empir-ical study of the effects of the internship program on preservice teachers’conceptions of the nature of mathematics and science, and on how theyplan to teach mathematics and science. We conclude with an examinationof the reform potential of the internship program, a summary of ongoingchallenges, and a discussion of future research.

CONTEXT

The summer research internship program is situated within the broadercontext of the Maryland Collaborative for Teacher Preparation, a projectfunded by the National Science Foundation (NSF). MCTP involves theteaching and research institutions of the University System of Marylandin collaboration with local community colleges and K-12 public schoolsystems. Its mission is to promote fundamental reform in the teachingand learning of mathematics and science by improving the preparation ofupper elementary and middle grades teachers of mathematics and science.(Upper elementary and middle grades students are approximately 9 to 14years of age). MCTP began its work in 1993 with NSF funding for a five-year project to design, develop, implement, and evaluate an innovativeinterdisciplinary program of courses, internships, and field experienceswith the goal to prepare teachers who will provide exemplary mathematicsand science instruction in Maryland. MCTP students either major in orintend to major in education, with mathematics and science for Grades4–8 as areas of concentration.

Five principles have guided the design and development of the courseand field experience components of MCTP (Fey, 1998):

1. In mathematics and science content and pedagogy courses, model thepractices that future teachers will be expected to employ when theyenter the profession.

2. Provide courses and field experiences in order to support thedevelopment of understanding and skill in both mathematics andscience, so that prospective teachers know and can take advantage ofthe important connections between the disciplines.

INTERNSHIPS AS COMMENCEMENT 279

3. Support the development of fluency with modern technologies asstandard tools for research and problem solving as well as for imagin-ative classroom instruction.

4. Prepare prospective teachers to deal with the broad range of studentswho are in public schools today; give special attention to the under-standings and skills needed to help students from diverse culturalbackgrounds.

5. Provide placement assistance and sustained support during the criticalfirst years of graduates’ induction into the teaching profession.

From these principles, each of the collaborating institutions designeda four-year preservice teacher preparation program which includes thefollowing components: new content courses, new methods courses, intern-ships, field experiences, and sustained professional support. Each of thesecomponents involves technologies and hands-on learning in an effort toproduce teachers who understand the connections between mathematicsand science and who can create interactive learning environments for allstudents. Among the fresh MCTP approaches for preservice teachers arethe scientific and informal education experiences that MCTP refers to asinternships, in which teacher candidates participate in mathematics andscience as it is really done, not solely as it appears in college lectures andstylized laboratory projects.

Internships and school-based field experiences differ with respect tosetting, collaborating mentor, and intended effects. Whereas internshipstake place in businesses, industry, scientific institutions, or in informaleducation settings such as museums, science centers, or zoos, school-basedfield experiences take place in elementary or middle schools. The mentorswith whom interns collaborate are mathematicians, scientists, and museumor environmental educators. In school-based settings, MCTP preserviceteachers work with experienced mathematics and science teachers whoare committed to an interdisciplinary, hands-on approach to teaching andlearning. While on site, interns typically do not engage in the designof instructional plans or materials based on a content-specific aspect oftheir experience; such activity, when it occurs, usuallyfollows the intern-ship experience. The primary focus of an MCTP internship experienceis to broaden the preservice teachers’ content knowledge; the devel-opment of their pedagogical content knowledge (Shulman, 1986) andtheir curricular knowledge are secondary goals. In contrast, school-basedexperiences focus primarily on the development of pedagogical contentknowledge and curricular knowledge. The distinctions between internshipsand school-based experiences are summarized in Figure 1.


Figure 1. Internships versus school-based field experiences.

The goals of the MCTP internship program are to offer preserviceteachers unique opportunities to: (a) understand the processes of mathe-matics and science, (b) enhance their content background in mathematicsand science, and (c) reflect on translating their experience into teachingpractice. Internships are preemptive measures to better ensure that thefuture teachers bring richer, broader conceptions of mathematics andscience to bear on their practice. Studies of the teaching and learning ofmathematics emphasize the urgent need to break a cycle that perpetuatesshort-sighted conceptions of mathematics and the teaching of mathe-matics. Such conceptions cast mathematics as a body of discrete rule-bound topics that is transmitted and preserved by virtue of teacherauthority (Grouws & Schultz, 1996). The cycle begins when studentsschooled in this conception of mathematics exit secondary school. Itcontinues when they complete teacher preparation programs that donot adequately challenge this conception. And the cycle becomes self-perpetuating when these teachers in turn pass on this legacy to the nextgeneration of students.

A countervailing cycle begins with the enhancement of content know-ledge and understanding, which then influences conceptions of mathe-matics, which in turn influences beliefs about how mathematics should


be taught and learned (Thompson, 1992). Little evidence supports thehypothesis that an increase in teachers’ content knowledge alone positivelyinfluences their students’ learning (Fennema & Loeff, 1992). Course-work alone – more courses, or courses taught in novel ways – may benecessary, but they seem to be insufficient to radically challenge andchange future teachers’ conceptions of mathematics (Brown & Borko,1992; Brown, Cooney, & Jones, 1990). Rather, “making an impact onprospective teachers requires powerful interventions” (Grouws & Schultz,1996, p. 449). The placement of preservice teachers in alternative out-of-school settings exemplifies what Brown, Collins and Duguid (1989)discuss as the situated nature of mathematical thinking, namely, thatthe learning of mathematics is embedded in the activities, contexts, andcultures in which it is developed and used.

The rationale for the traditional practice of reserving science andmathematics internships for content majors is that the experience preparesthe next generation of mathematicians and scientists. The rationale for thenovel practice of creating placements in science and mathematics settingsfor education majors is that the experience prepares the teachers who willeducate successive generations of mathematically and scientifically literatepeople. Most of the sites where MCTP interns are placed are science,rather than mathematics oriented, and thus, most interns work closelywith scientists, not mathematicians. Interns are nonetheless exposed to arich abundance of mathematics. The MCTP program rests on the premisethat mathematics and science are intrinsically connected, sharing commonassumptions, methodologies, and tools. Interns experience this relationshipfirsthand by routinely engaging in spatial and quantitative reasoning duringthe course of their activities.

Recent reform efforts in teacher preparation have supported expandedschool-basedfield experiences, cited by their proponents as being the mostimportant component of teacher education programs (McIntyre, Byrd &Foxx, 1996). Ifinternshipexperiences in the culture of mathematics andscience play a different but equally powerful role in teacher training, thenthese programs merit serious consideration.

STRUCTURAL FEATURES OF THE INTERNSHIP PROGRAM

The MCTP summer research internship program consists of a carefullyorchestrated sequence of events. A chronology of major features of theprogram appears in Figure 2.

Three of the collaborating MCTP institutions require that their studentscomplete an internship as part of their MCTP program. In the remaining


Figure 2. Chronology of major events.

institutions the internship is optional. Few internship applicants havecompleted teaching methods courses because such courses are usuallyreserved for the final year of their teacher preparation programs. Thenumber of mathematics and science content courses they have taken varieswith their academic year status. All MCTP students are invited to applyfor internships.

A committee consisting of representatives of MCTP and public educa-tion in Maryland reviews internship applications and makes selectionand placement recommendations. Internship site representatives reviewthe applications of individuals recommended to them and make finalplacement decisions. Internship sites span the entire Washington, D.C.metropolitan region, including college and university campuses, busi-nesses, government and industrial research laboratories, museums, andoutdoor education centers.

The key summer events are orientation, student forum, and presentationday. Orientation takes place before interns report to work at their sites. Atthis time we clarify expectations and stimulate networking ties within thecohort. At the student forum, which for most interns follows the conclu-sion of their work on site, interns informally share their research findingsand prepare for presentation day. On presentation day the interns formallypresent their research to MCTP faculty and students, site mentors, andrepresentatives of public education in Maryland.

The expectations for interns and site mentors are specific. Interns mustwork 40 hours per week for 8 to 10 weeks. The number of weeks is deter-mined jointly by the interns and their mentors. Interns are strongly urgedto avoid additional work or academic commitments during the internship.They must maintain a journal of their activities and reflections, contributeat least weekly to a listserve discussion, and participate in all programevents. Site mentors, who are critical to the success (or failure) of intern-ship experiences, must assign the interns’ summer projects and guide themthroughout all phases of the work. Mentors must be available to interns on


a consistent, frequent basis and ensure that interns engage in routine officeor laboratory chores only to the extent that all professionals at the siteengage in such tasks.

THE STUDY

One aspect of the evaluation of the MCTP summer research internshipprogram concerns the nature of outcomes of the program. Specifically,what effects does the internship program have on preservice teachers? In1997, we investigated two questions:

1. To what extent, if any, did preservice teachers’ conceptions ofthe nature and processes of mathematics and science change aftercompleting their internship?

2. To what extent, if any, did preservice teachers’ conceptions of teachingand learning mathematics and science change after completing theirinternship?

Participants

The participants in the study were 17 of the 18 interns who participatedin the 1997 program. We eliminated one intern who missed the orientationdue to illness. The sample represents five collaborating MCTP institutions.For 14 of the 17 participants, the internship was a program requirement.All participants were female. All but two participants were within an agerange of 18 to 21 years; one study participant was in her 30s, and anotherwas in her 40s. The types of work that the interns did and the numberof interns who engaged in each type of work were as follows: scienceresearch (3), mathematics research (1), science curriculum development(4), science resource development (1), integrated mathematics and sciencecurriculum development (1), science teaching (2), naturalist interpretation(4), and educational research (1). To ensure anonymity, all participantshave been assigned pseudonyms.

Sources and Methods of Data Collection

Our main source of data was the archived listserve discussions. Forthe summer months, the interns had dedicated use of an electronic list-serve. Participation on this listserve was limited to the Summer 1997interns and the internship staff. The moderated, unstructured listservediscussions were designed to build community and to document interns’summer experiences. We encouraged interns to share a range of matters


such as exciting experiences, areas of concern, top-of-the-mind issuesand questions, excerpts from their journals (what they had done on site,what they had learned, and how they felt about it), and their reflec-tions on one anothers’ postings. We electronically archived the listserveconversations.

For purposes of triangulation we administered a survey and collectedartifacts from orientation, student forum, and presentation day. The surveywas adapted from an instrument used to examine and track changes inteachers’ knowledge and beliefs (Kennedy, Ball & McDiarmid, 1993). Itcontained 65 questions separated into six sections: (a) views about mathe-matics and science, (b) importance of mathematics and science, (c) beinggood at mathematics and science, (d) learning mathematics and science,(e) prerequisites for teaching mathematics and science, and (f) strategiesfor teaching mathematics and science. The majority (61) of the questionswere evaluated on a five-point Likert scale:strongly agree, agree, neutralor not sure, disagree, and strongly disagree. We administered the pre-internship survey immediately prior to the opening activities on the firstday of orientation. We administered the post-internship survey, which wasidentical to the pre-internship survey, at the conclusion of activities on thelast day of the student forum. At both orientation and student forum, wecollected artifacts generated during group discussions. The final artifactwe collected was each intern’s research summary. Written for presenta-tion day, each summary was a 750–1000 word description of the natureand findings of the intern’s summer work and her perceptions of thesignificance of her research experience.

Data Analysis

For analysis of the listserve and artifact data, we used a variant of analyticalinduction (Bogdan & Biklen, 1992; LeCompte & Preissle, 1993; Miles &Huberman, 1994). We read the data several times, and each time we notedreflections and remarks that we deemed relevant to the questions guidingthe study. Next, we identified themes that emerged from the data, andchunked together instances from the data sources that fit the themes. Were-examined instances of data that conflicted with the developing general-izations and preliminary themes, and either generated explanations thatwere compatible with the emerging themes or revised the themes. Wecontinued this iterative process of identifying new themes and categorizingchunks of the data until we had exhausted the data, and the themes that hademerged were consistent.

For each survey question we compiled the changes in all interns’ pre-and post-internship responses. We weighted these changes based on two


criteria: (a) magnitude of change, and (b) whether the change was a move-ment away from either theagree or the disagreeside of the responsecontinuum. This process resulted in the identification of survey questionsfor which there was the most change across the interns. The compilationof interns’ responses for each item, in addition to specific changes in someinterns’ responses to particular questions, was used to triangulate emergingthemes from our analysis of the listserve and artifact data.

Findings

Three themes emerged from our analysis of the data: (a) the nature andprocesses of mathematics and science, (b) the teaching of mathematics andscience, and (c) the nature of the professional work place. The themes arediscussed in the context of the interns’ experiences over the course of thesummer. Summary information about each intern referred to in this sectionis provided in Figure 3.

Nature and processes of mathematics and science. As the interns plungedinto their summer work they exhibited a broad range of thoughts and feel-ings. Keesha, who developed instructional materials for an environmentalresearch facility, most succinctly expressed this range of emotions: “We’vebeen overwhelmed with information, frustrated, enlightened, delighted,and frustrated again” (listserve, Week 8 of 10). Interns found themselvesrunning into dead ends, facing ups and downs, realizing that new discov-eries may make old discoveries obsolete, and experiencing their work asbeing full of puzzles and unanswered questions.

The interns’ ups and downs were a natural response to the challengesposed by their work. For instance, Tina’s task was to analyze epidemi-ological data in order to describe historical trends in the incidence andmortality of selected cancers in the United States from 1950 to 1992. Asshe began to analyze daunting amounts of data, she was uncomfortablewith the dead ends she encountered. At the end of her third week (of 10)into this work she shared on the listserve, “I have run across a lot of deadends, but at least I am learning from them. I just can’t wait until this partof my research is over and I can begin to find some results.” By her sixthweek, Tina was overwhelmed with the results of her preliminary analyses,much of which still did not make sense to her. In fact, she found that hermentor, an experienced scientist, also struggled to discern the patterns inher data. Working alongside her mentor, Tina experienced science as ittruly is – full of unanswered questions and ambiguity – in stark contrastto the way science is often presented in stylized classroom laboratoryactivities. Tina articulated these feelings as follows:


Figure 3. Selected interns, research sites, and tasks.


I have so much work that I have done that [my mentor] really does not know what to makeof it. Researchers don’t know all the answers. I have so much of this stuff [data and graphs]that I have no idea how I will get it into a paper. Never mind how much stuff I have; wehave no idea what it means in the first place. . . . It’s just why particular trends are occurringthat is difficult. (listserve, Week 6 of 10)

Tina’s frustration with the analysis of her data was matched byBarbara’s consternation over her struggle to collect data. Barbara’s jobwas to collect and analyze biofilm specimens in a harbor that surroundsan industrial region. She created the racks on which the specimens wouldbe collected and then submersed them at various sites in the harbor. Incontrast to Tina’s prolonged struggle, Barbara managed within just twoweeks to overcome some of the obstacles she had faced; she was bothsurprised and delighted with the results. Excerpts of Barbara’s listservepostings during her first four (of eight) weeks on site chronicle her ups anddowns:

Week 1: I won’t bore you with the monotonous details of drilling andcutting the plexi-glass plates [for the racks] that will go into thewater. It’s not hard, just very boring and repetitive. Talk aboutbeing put to sleep . . .

Week 2: We put a rack out at [location] on Tuesday, with a diving weightand a bright yellow buoy and as of this morning, it is GONE!. . . that’s why I am sick of biofilm racks!!!

Week 3: We found the rack that disappeared from [location] – it had justdrifted VERY far away! We collected the disks from two sitesyesterday, and the growth on them is SPECTACULAR!

Week 4: Well, I have to run and examine some creatures that are onour disks (it’s so cool – barnacles, mussels, stentors, hydra,anemones, flatworms, round worms – just to name a few!).

Whereas both Tina and Barbara grappled with immediate tasks, Julia, incontrast, was disturbed as new scientific findings emerged that called intoquestion some of the data she had generated during an MCTP internshipthe previous year. Julia completed two internships, one in Summer 1996and another, at an entirely different site, in Summer 1997. During her 1996internship she had used photomosaics from the Viking Orbiter to cata-logue multi-ring impact basins on the Martian surface. During her 1997internship, she reflected on the findings that were then being generated byNASA’s Pathfinder mission and the rover, Sojourner:

Having studied the surface of Mars last summer, and hearing on the news that there aresome major differences in what they had thought about the surface and what actually is, Iwonder about the research that I did last summer. . . does it still hold true? It was unnervingfor me to think that scientists could have been wrong, or slightly off on something. . . . It’s


funny how when a scientist says or does something it’s like it has a certain magic likethey are too smart and all knowing to be incorrect on something, yet history shows this isdefinitely not always the case. Scientists are not always right, and do not always handletheir knowledge correctly. (listserve, Week 6 of 10)

Julia’s sense of security in scientific research was seriously compromisedas a result of new scientific data.

In contrast to Julia’s disquiet, Susan’s work with primates at a zoolo-gical park stimulated cognitive conflict, yet rewarded her with valuableinsights. Susan believed that her work had enriched her understanding ofher place in the world of living things:

It was amazing to see how much the animals acted like humans. I remember learningabout primates in school, but seeing first-hand the similarities between humans and otherprimates has changed my whole outlook on the nature of our species. (research summary)

As the interns worked with mathematicians and scientists, their percep-tions of their own capabilities changed. They recognized and prized theiraccomplishments. For example, Barbara noted, “I feel as if I am a real‘scientist’ as I have been struggling through this. . . stuff” (listserve, Week7 of 8). Carla, who worked with integrated mathematics and sciencelessons at an aeronautical and space research facility, became aware ofnewfound strengths:

Because of this internship, I have decided to stay in school for another year to get ascience concentration. My major is elementary/middle school education with a mathe-matics concentration, and I honestly never thought I was smart enough to “get” science. . . . I’ve been given a lot ofopportunities that have made me feel like I’m actually doingsomething – and I’m doing it by myself. (listserve, Week 8 of 10)

Kristen, who worked as a naturalist interpreter on a barrier island, believedthat her internship changed the way she looked at the world. In turn, sheexpressed a desire to be the kind of teacher who will help her studentspreserve the sense of wonder that she had recaptured over the summer:

I learned everything about the island: the important function of barrier islands, the dynamicchanges that take place over time on a moving island, the unique and fascinating creaturesthat live in such an unpredictable habitat, and the importance of protecting the island andits wildlife. Upon reflection, I realize it is the most wonderful experience I have ever had. Iuse the present tense in describing the past summer because I believe this experience willbe with me every day in my classroom. I learned from nature how to capture that sense ofwonder that is so important in understanding math and science. The curiosity that peaked inmy mind when I was a child, and was lost at some point in late adolescence, was retrievedthis summer at a very mystical place. (research summary)

Finally, Maureen, who studied chronic lymphocytic leukemia at a medicalresearch laboratory, expressed an unbridled confidence as she looked backover her summer’s accomplishments:


The most surprising insight that I have taken away after the completion of this project isthat I now realize that I am capable of doing anything. I came into this internship withoutany prior genetics knowledge and have come away knowing practically everything there isto know about this disease. (research summary)

By the end of the summer, despite – or, arguably, due to – having cycledthrough being “overwhelmed with information, frustrated, enlightened,delighted, and frustrated again,” the interns evidenced a willingness totolerate ambiguity that countered their pre-internship dispositions. Addi-tional support for this conclusion comes from the interns’ responses toselected items in the section of the survey concerning “what it means to begood at mathematics and science” (see Figure 4).

Just over half of the instances in which interns’ post-internshipresponses diverged from their pre-internship responses to these items,the shift was to the response,not sure. Regardless of whether theinterns shifted from agreement or disagreement with any given statement,when their thinking changed, interns appeared to be reconsidering theirpreviously-held beliefs and suspending judgment.

Teaching mathematics and science. Shifts in the interns’ beliefs about thenature and processes of mathematics and science appeared to be associatedwith similar shifts in their beliefs about teaching and learning topics andconcepts in these content areas. The interns came to believe that teachersare curious learners and that learning is a self-directed activity.

The interns’ pedagogical viewpoints seemed to reflect their experiencesof the realities of the research process. The way in which they experi-enced these realities was critically influenced by their mentors. Mentorsmodeled how to be lifelong learners by dealing openly and proactivelywith ambiguity. The interns appeared to project this message into the futureby envisioning themselves as teachers whose practice would mirror thatof their mentors. Their tolerance of ambiguity contradicted pre-internshipsurvey responses which suggested that they believed that teachers shouldknow the answers to any questions that arise in class, and that studentsshould always leave class enlightened and satisfied.

Tina expressed determination to be a teacher who can seek knowledgeand understanding along with her students. Her mentor most stronglymodeled the effort to wrestle sense out of confusion. Tina’s descriptionof the kind of teacher she is committed to becoming evoked images of hermentor’s behavior:

The number one thing that I have learned that I will definitely take into the classroom withme is not to act as an encyclopedia for students. They will know that I do not have all ofthe answers, but we can try to find them together. (listserve, Week 8 of 10)


Figure 4. Selected items from the survey.

Tina echoed this intention in her post-internship response to the surveystatement, “If a student asks a question in math or science class, theteacher should know the answer.” Tina’s post-internship response,stronglydisagree, was a radical departure from her pre-internship response ofagree.

Catherine, whose job was to set up and operate an air quality moni-toring station in an urban inner harbor environment, seemed remarkablycomfortable with the notion of learning alongside students. This is evidentin her reflections after an instructional episode:


When the kids were dissecting the squid they were asking me questions as if I knew whatI was doing and [I] was learning along with them–which was great because. . . everyone isstill learning and I didn’t know anything about squid. (listserve, Week 4 of 8)

Like Tina, Catherine also shifted her response to the survey statement, “Ifa student asks a question in math or science class, the teacher should knowthe answer.” However, Catherine’s shift fromagreeto disagreewas of alesser magnitude than Tina’s.

Ann, who developed an online interactive field guide to a dam site inAfrica, described a seemingly innate, long-standing pursuit of knowledgeand understanding:

The older and more educated I get, the more I find out how much others don’t know andhow it is ok not to know everything. I often find that I am in a “thinking bind” when itcomes to what we do and don’t know. I can make myself dizzy sometimes trying to makesense of things. (listserve, Week 7 of 10)

It is therefore not surprising that her pre- and post-internship responses,disagreeandstrongly disagree, respectively, to the same survey statement(“If a student asks a question in math or science class, the teacher shouldknow the answer”) indicates little change. However, her response to thesurvey statement, “Students should never leave class feeling confused orstuck,” shifted fromagreeto disagree, which suggests that her internshipexperience may have influenced her beliefs about her future students’need to tolerate ambiguity. The following are more of Ann’s musingsabout not knowing and the “dizziness” that accompanied her attempts atsense-making:

Perhaps that dizzy sensation is a good one to replicate in our students. Sometimes I learnmost from that dizziness. It makes me realize that we have only begun to think we under-stand what is going on around us, and even more, that it’s ok not to understand everything.We seem to have a natural desire to try to explain, so it is important to activate in yourstudents. And when they get dizzy, let them sit back for a minute, recuperate, and then diveback in. Kind of like recovery between sprinting workouts. (listserve, Week 7 of 10)

The internship experience appeared to support the interns’ developingsense that curiosity is a catalyst of lifelong learning. For instance, Joan,who worked in a zoological park facility that engages visitors in inter-action with scientists, seemed to leap beyond her own curiosity-drivenexplorations to consider what her experience presages for the classroom:

This week my curiosity about animals was running wild – and this feeling was driving meto ask questions and be a critical thinker about new ideas. Perhaps by creating environmentsin our classrooms that encourage and support curiosity, our students will be excited, andwe will create learners that are always asking questions. (listserve, Week 7 of 10)


Barbara, whose path from frustration to satisfaction has been tracedin preceding sections of this paper, here considered the pedagogicalramifications of her journey:

If we let our students discover things on their own, they will be more motivated to learn.I just think that the way to teach science (and every other subject for that matter) is byallowing students to teach themselves. They will gain more interest, and also feel like theyare in charge of their education. This will, in turn, provoke some excitement about learningfrom them. (listserve, Week 5 of 8)

Many of the interns talked about the role of technology in sparking,capturing, and feeding curiosity. Ann experienced this during her summerresearch in which she developed an interactive Web site. In reflecting onher experience she refined and extended her beliefs about teaching withtechnology:

By creating the first [online interactive feature for the work site] I learned the tools neces-sary to create a successful interactive feature for use in the classroom, and I have experienceusing these tools to create a product that I will be able to use directly in my classroom.As a result of producing this feature, I have also learned that computers are going toplay a greater role in our lives as new technologies evolve. Keeping up with these newtechnologies is going to be even more important as computers become a greater part of ourstudents’ worlds. Students see computers as a learning tool; I have learned to see computersas a teaching tool. Computers are more than word processors, fast calculators, and a wayto check e-mail. Computers are a tool that will add a new dimension to our world and ourstudents’ education. (research summary)

Nature of the professional work place. In addition to learning about thenature and teaching of mathematics and science, interns learned that theyhad to acclimate to the environment of the professional work place. Formost interns, this summer internship experience was their initial foray intothis arena.

Because many interns found their workplaces to be stressful and quick-paced, they struggled to quickly establish their place within the setting.Ann, a high-energy person herself, was nevertheless challenged:

It’s hard to really get settled in an environment that seems to be in a constant whirlwind.So much happens here in a day that I feel like I have 2 weeks everyday . . . . I have decidedto quit trying to navigate [the whirlwind] so much and go with the flow. (listserve, Week 4of 10)

Julia remarked that, “I sometimes think this is the hardest part . . . walkinginto an already established environment and routine, then having to figureout how you fit into the scheme of things” (listserve, Week 3 of 10).

Other interns spoke to the nature of the work itself, describing it asbeing both labor- and time-intensive. Contrasting her summer work withher experience as a student, one intern admitted that she had never had


to work so hard for courses. Another intern, Tina, came to appreciate thelabor involved in depicting data in a usable fashion:

[My project] is so time consuming and I am finally beginning to see some of my results,which relieves me a little bit. I was beginning to think that ten weeks would not do the job.It is incredible how much time goes into something that only yields one chart or one graph.(listserve, Week 4 of 10)

Barbara recognized that at the beginning of the research cycle, researchersmust commit tireless hours to sound data collection so that the results willbe usable data:

I have been sweating over biofilms racks; burning on the boat, as we try to anchor the boatand drop these stupid things in the water. . . . It’s just very difficult to get these thingsdone– CORRECTLY, so that the results will be usable. (listserve, Week 2 of 8)

All of the interns worked, not only with their mentors, but also withteams of individuals at their work sites. In their research summaries,some of them alluded to the collaborative nature of their work withcomments such as: “I learned the value of feedback and working as a team”(Julia), and “I learned the importance of teamwork and a supportive workenvironment” (Ann).

Interns were keen to note the work habits and dispositions of the profes-sionals in their work environments. They were likewise surprised to finda full range of human traits – hard-working and playful. Barbara wassurprised at the time commitments professionals make to their work: “Onevery shocking thing that I have noted is how ALMOST every single personhere puts in way more than 40 hours a week, and they LIKE it!” (listserve,Week 1 of 8). Julia and Ann, who worked at the same site and collaboratedto post the following message, described their surprise upon seeing a veryhuman side to people they had heretofore seen only in a professional role:

[We] were invited to a dinner cruise with the staff. It was nice to see the staff unwind, andit really made us feel like a part of the group. Before this evening neither one of us couldhave imagined the staff dancing around the room in a conga line or doing the limbo!!!(listserve, Week 2 of 10)

Lastly, Joan’s view of her mentor took into account both professionaland interpersonal attributes: “I have realized that scientists are real andwonderful people who are anxious to share their knowledge and also tolearn from you” (research summary).

Summary

The interns’ comments suggest that they viewed the knowledge that theyconstructed over the course of the summer to be a function of their inter-action with the tasks they were given and the people who mentored them.


They evidenced a strong, personal sense of ownership of and pride in theirknowledge and understanding. They recognized that they achieved thisknowledge and understanding amid a swirl of both positive and negativeexperiences and emotions. We highlighted three themes from our study ofthe MCTP summer research internship experience.

The first theme addressed the nature and processes of mathematics andscience. Interns met the challenges that often characterize the process ofdoing research. They sometimes agonized over collecting, analyzing, andmaking sense out of data. They experienced false starts and unexpectedreprieves. They learned that successful professionals must often wrestleorder out of confusion. They reconsidered their conceptions of their owncapabilities as a function of their summer research experiences.

The second theme concerns the teaching and learning of mathematicsand science. A commitment to teaching was a major component in eachintern’s desire to do an internship. It appears that the interns, on theheels of the challenge of their summer internships, had redirected theirthinking from what theydid during the summer to what theyhopedto do as teachers. They dared to take on challenging tasks; they envi-sioned themselves as risk-taking teachers who intended to question andpursue understanding alongside their students. They pursued countlessquestions and accomplished new levels of understanding; they envisionedthemselves as encouraging curiosity in their students. They attained theiraccomplishments by virtue of courage and persistence; they hoped toencourage their future students to take an active role in their own learning.

The third theme concerns the nature of the professional work place.The interns settled into their work environments by adapting to the quickpace and by learning to value the collaborative nature of their work. Whatinterns learned about researchers as people surprised them. Their mentors’dedication to hard work and grueling schedules were offset by playfulnessand a sense of humor.

QUESTIONS AND CHALLENGES

In this section we return to the questions that guided the study: Towhat extent, if any, did preservice teachers’ conceptions of the natureand processes of mathematics and science change after completing theirinternship? To what extent, if any, did preservice teachers’ conceptions ofteaching and learning mathematics and science change after completingtheir internship?


Evidence of Change

The internship experience seemed to be a fundamentally significant lifeexperience for the preservice teachers. At the culmination of their summerexperience, all of the interns reported a sense of pride and empowermentthat derived from their commitment to a challenging task, from tappinginner capacities that they had not heretofore recognized. They discoveredthat not onlycould they do the work, but theyhad done it, and they haddone it well. Moreover, their work was valued by their site mentors asbeing a substantive contribution to the ongoing work at their site. Overthe first three summers of the internship program, 11 interns were hiredto work part-time at their sites during the school year, and some werehired full-time the following summer. Ten have published; in two cases,the publications are in refereed science journals. Three have developedinteractive Web sites, and three have developed CD-ROMs.

The findings of this study are reminiscent of Wertime’s (1979) reflec-tions on the courage that taking on a problem, a “project for the future wecommit ourselves to by an act of the will” (p. 192), entails. The commit-ment is to invest some part of our personal resources (hence, self esteem)on a quest for what is unknown to us. Wertime suggested that incubationand inspiration fuel the quest:

The acknowledging of a problem entangles us deeply with ourselves and with the world. . . . If [our students] lack persistence, it is not because they are lazy, or cowardly, or docile;it is much rather because they have never had a knowledge of their persistence revealedto them . . . . Incubation and inspiration are the major mental powers least known by ourstudents, as well as the least trusted by those who do know of them. This fault is largelyours; we have not, as educators, taught our students very well how to activate these powers.(pp. 193–195)

If one views the construction of mathematical and scientific meaning asthe pursuit of a problem, and if the preservice teachers in this study areto teach their students how to activate their mental powers in the pursuitof understanding, then it is necessary that they themselves have engagedin that process. Interns’ reported observations and reflections suggest thatthe internship experience can be such an opportunity. The data suggest thatacclimating to a novel workplace and contributing to the research thereincalled upon substantive inner resources. As seen from the interns’ pointof view, they went into their work sites quite daunted by their tasks. Theyconcluded the summer with pride in their accomplishments.

Why do these internships have the potential for realizing reform?Perhaps it is precisely because interns are required to take the risks inherentin acclimating to and contributing to the work site. One of the interns,Julia, reflected on the risk-taking that the process entailed and projected


her experience into the future as she described her desire for her studentsto experience similar growth challenges.

I hold that we should teach students to be bold and take risks (not the life-threatening kind,but the kind where you risk failure or standing alone on your own ideas). . . . I want to bethe type of teacher that not only encourages my students to take risks, but takes risks alongwith them. I want to create a place where my students can feel comfortable taking risks,and will openly share their thoughts, questions, and ideas. (listserve, Week 6 of 10)

These goals radically depart from those of a teacher who simply transmitsand preserves a body of knowledge by virtue of authority.

We have learned from this research that preservice teachers who haveparticipated in internships intend to bring a holistic, conceptually orientedview of mathematics and science to their classrooms. We have set thesepreservice teachers on a trajectory, but we do not yet know where thattrajectory will take them, that is, how their experience will translate intotheir future teaching practice. However, we believe they are headed inthe direction indicated by the current national reform of mathematics andscience education (American Association for the Advancement of Science,1990, 1993; National Council of Teachers of Mathematics, 1989, 1991;National Research Council, 1996).

Challenges

The MCTP summer research internship program faces both programmaticand logistical challenges. One programmatic challenge concerns the trans-lation of the internship experience into classroom practice. Although theimmediate focus of the internship experience is on preservice teachers’acquisition of content knowledge, we have nevertheless capitalized onopportunities to help interns incorporate components of their research intoinstructional plans. Overall, however, the internship program lacks fundingto support a classroom implementation component. In an attempt toaddress this shortcoming, the MCTP internship program recently mergedwith a similar internship program for experienced teachers, the Univer-sity of Maryland Graduate Fellows Program. In every possible case, wenow pair a preservice teacher with an experienced teacher. It is ourexpectation that the experienced teachers’ perspectives will support thepreservice teachers’ thinking about translating their research experienceinto classroom practice.

Another programmatic challenge is to discern the influence of MCTPcourses on the internship experience. Interns did occasionally allude toMCTP coursework in the listserve discussions. For example, as Kristentalked about a preferred style for conducting the programs she wasresponsible for at her site, she made a veiled reference to her experiences


in MCTP classes: “I really like the Bay Explorers program because I canpractice conducting activities MCTP style. With the other programs, thereis not much freedom, they are formal lectures or demonstrations.” Cath-erine referred directly to an interdisciplinary viewpoint gleaned from anMCTP course:

Recently I took a science class which was chemistry and biology combined and the teacherswere constantly explaining to us that all science and math is interrelated and we need tolearn it that way and not as separated as it has been in the past. (listserve, Week 5 of 8)

These statements suggest that the ways in which interns respond to thechallenges inherent in their internship assignments are theoretically, atleast in part, attributable to MCTP course experiences.

In addition to programmatic challenges, the internship program faceschallenges of a logistical nature. We continue to struggle with the questionof whether all MCTP students should be required to complete an intern-ship. If internships are uniquely transforming experiences, then perhaps,yes, they should be required. However, financial considerations limit thenumber of students who are in a position to apply for internships. Althoughthe internship program provides a stipend, students who must earn theiracademic year living expenses during the summer find that the structure ofthe program presents difficulties. Also, for non-traditional students, mostof whom have children, child-care expenses present a potential obstacle toparticipation in a summer internship.

Lastly, we recognize the dearth of sites whose primary focus is mathe-matics. Such sites are difficult to find, and when found, involve the addeddifficulty of convincing the site representatives to take interns who aremajoring in education, rather than in mathematics or engineering. Atother potential sites, site representatives claim that the learning curves forlaboratory procedures at their sites preclude an 8 to 10 week internship.

Next Steps

Our findings suggest that the MCTP summer research internship experi-ence is a promising vehicle for effecting teacher reform. We believe it tobe a catalyst that launches interns on a trajectory to becoming the newkind of teacher MCTP envisions; that is, one who understands the connec-tions between mathematics and science and who creates an exciting andinteractive learning environment for all students. In this sense, we view theinternship experience as their commencement – from preservice teachersto teachers of mathematics and science.

Further investigation involves ongoing documentation of the summerresearch experience. A critical next step will be to study the teachingpractices of the MCTP interns after they graduate from college. It will be


important to note ways in which their teaching practices are characteristicof reform practices in teaching mathematics and science and to discernthe extent to which their adoption of these practices is a function of theirinternship experience. Equally important, for those practices that do notconform, we want to learn more about the nature of the barriers to suchreform practices.

REFERENCES

American Association for the Advancement of Science (1990).Science for all Americans.New York: Oxford University Press.

American Association for the Advancement of Science (1993).Benchmarks for scienceliteracy. New York: Oxford University Press.

Bogdan, R.C., & Biklen, S.K. (1992).Qualitative research for education: An introductionto theory and methods(2nd ed.). Boston, MA: Allyn and Bacon.

Brown, C.A., & Borko, H. (1992). Becoming a mathematics teacher. In D.A. Grouws (Ed.),Handbook of research on mathematics teaching and learning(209–239). Reston, VA:National Council of Teachers of Mathematics.

Brown, J.S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture oflearning.Educational Researcher, 18(1), 32–42.

Brown, S.I, Cooney, T.J., & Jones, D. (1990). Mathematics teacher education. In W.R.Houston, M. Haberman & J. Sikula (Eds.),Handbook of research on teacher education(639–656). New York: Macmillan.

Fennema, E., & Loeff, M. (1992). Teachers’ knowledge and its impact. In D.A. Grouws(Ed.),Handbook of research on mathematics teaching and learning(147–164). Reston,VA: National Council of Teachers of Mathematics.

Fey, J. (1998). Guiding principles: New thinking in mathematics and science teaching.In M. Gardner & D. Ayres (Eds.),Journeys of transformation: A statewide effort bymathematics and science professors to improve student understanding(11–21). CollegePark, MD: The Maryland Collaborative for Teacher Preparation.

Grouws, D.A., & Schultz, K.A. (1996). Mathematics teacher education. In J. Sikula, T.J.Buttery & E. Guyton (Eds.),Handbook of research on teacher education(2nd ed.) (442–458). New York: Macmillan.

Kennedy, M.M., Ball, D.L., & McDiarmid, G.W. (1993).A study package for examiningand tracking changes in teachers’ knowledge(Technical Series 93-1). East Lansing, MI:National Center for Research on Teacher Education.

LeCompte, M.D., & Preissle, J. (1993).Ethnography and qualitative design in educationalresearch(2nd ed.). San Diego, CA: Academic Press.

McIntyre, D.J., Byrd, D.M., & Foxx, S.M. (1996). Field and laboratory experiences. InJ. Sikula, T.J. Buttery & E. Guyton (Eds.),Handbook of research on teacher education(2nd ed.) (171–193). New York: Macmillan.

Miles, M.B., & Huberman, A.M. (1994).Qualitative data analysis(2nd ed.). ThousandOaks, CA: Sage.

National Council of Teachers of Mathematics (1989).Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.



National Research Council (1996).National science education standards. Washington,D.C.: National Academy Press.


Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research.In D.A. Grouws (Ed.),Handbook of research on mathematics teaching and learning(127–146). Reston, VA: National Council of Teachers of Mathematics.

Wertime, R. (1979). Students, problems and “courage spans.” In J. Lochhead & J. Clement(Eds.),Cognitive process instruction: Research on teaching thinking skills(191–199).Philadelphia, PA: Franklin Institute Press.

Success For All Foundation Karen Langford200 W. Towsontown Blvd.Baltimore, MD 21204-5200USA

Department of Curriculum & Instruction Mary Ann HuntleyCenter for Mathematics EducationUniversity of Maryland, College ParkCollege Park, MD 20742USA


IMPACT OF THE MISSOURI MIDDLE MATHEMATICS PROJECTON THE PREPARATION OF PROSPECTIVE MIDDLE SCHOOL

TEACHERS

IRA J. PAPICK, JOHN K. BEEM, BARBARA J. REYS and ROBERT E. REYS

In the previous issue ofJMTE, Reys, Reys, Beem and Papick (1999)described the Missouri Middle Mathematics (M3) Project1 (1995–1999).The goal of the project was to improve the teaching and learning ofmiddle school mathematics through collaborative Standards-based mathe-matics curriculum investigations. The project not only contributed to theimprovement of teaching mathematics in the participants’ classrooms, butit also significantly affected the program for prospective middle schoolmathematics teachers at the University of Missouri-Columbia. This teachereducation program is described in this article.

An essential ingredient in the successful transition to Standards-basedpractice at the middle school level is the development of teacher educationprograms that reflect the fundamental principles of the NCTMStand-ards (1989). In particular, the mathematical preparation of prospectivemiddle grade teachers, which traditionally had been integrated into theprograms for elementary teachers, needs careful consideration. The Mathe-matical Association of America, inA Call for Change(MAA, 1991),outlined recommendations for the mathematical preparation of middlegrade teachers. These recommendations differ significantly from recom-mendations for the preparation of elementary teachers and provide guid-ance to those developing new programs for middle grade mathematicsteachers.

Currently, as in the case of the M3 Project, much attention is beingfocused on the improvement of the knowledge and instructional skillsof inservice teachers. This is important in order to stimulate imme-diate and substantive change. However, it is imperative that preserviceeducation be simultaneously improved, so that new teachers entering thefield are knowledgeable and can collaborate with their more experiencedcolleagues in transforming the classrooms in which middle grade studentsstudy mathematics. In response to this urgency, college and universityfaculty throughout the nation are examining and restructuring their teacherpreparation courses and programs. These academic activities have created


302 IRA J. PAPICK ET AL.

a strong demand for resources and materials, and a desire to share ideasand successes.

At the University of Missouri-Columbia, faculty from the Departmentof Mathematics (within the College of Arts and Sciences) and the Depart-ment of Curriculum and Instruction (within the College of Education) havedesigned a comprehensive middle school mathematics teacher preparationprogram that implements Standards-based curricular materials and instruc-tional strategies. In this paper, we highlight some of the distinctive courseswithin this program that evolved from our work with inservice teachers.

BACKGROUND

The M3 Project employs collaborative curriculum investigations as avehicle for teacher enhancement and systemic reform (Reys, Reys, Barnes,Beem & Papick, 1997). For the past three years, we worked withteams of middle school mathematics teachers and administrators from23 Missouri school districts. Our collaboration involved extensive reviewand exploration of NSF-sponsored Standards-based middle school mathe-matics curricula and helped create and support a foundation necessary formaking informed decisions about middle school mathematics programs(Reys, Reys, Barnes, Beem & Papick, 1997; Reys, Reys, Beem & Papick,in press).

Although the M3 Project was an inservice teacher enhancementproject, the long-term and ongoing dialogue with practicing middle schoolteachers, as well as the involvement of university faculty in extensivecurriculum review and regular visits to middle school classrooms, influ-enced our thoughts about preservice teacher education. In fact, we foundthe curricula reviewed by the inservice teachers in the M3 Project (Mathe-matics in Context, Connected Mathematics Project, Math Thematics, andMathScape) rich in mathematics and pedagogical prompts. Consequently,these materials were often taken into the preservice classroom and usedto initiate and build important mathematical and pedagogical ideas. It isimportant to note that, true to the spirit of reform, the Standards-basedcurricula not only present mathematics differently, but they also intro-duce many important topics which have traditionally been reserved forthe secondary level. In particular, the new curricular materials containsignificant amounts of algebra, geometry, probability, and statistics. Ourexperiences with the M3 Project have made it clear that teachers need toknow quite a bit more mathematics than was required just a few years ago.These experiences convinced us of the real and immediate need to modifyboth content and pedagogy courses for preservice teachers.


The process of curriculum change at the university level is oftencomplicated and rather deliberate, even if there is a demonstrated needfor such change. If separate colleges within a university, such as theCollege of Education and the College of Arts and Science, are involvedin curricular changes, complexity increases. The creation of a core groupof determined faculty from each college, equipped with strong research tosupport change, is a necessary first stage of the procedure. The M3 Projectnot only provided compelling evidence for substantive curricular modi-fications, but also established a strong link that connected mathematicsfaculty and mathematics education faculty. This robust collaboration wascrucial for achieving the desired goals, and as inexplicable as it seems, itis somewhat unique within the academic walls of large U.S. universities.

In 1995, the Missouri Department of Elementary and Secondary Educa-tion announced a new certification area at the middle school level (Grades5–9). Previously, only elementary (Grades 1–6) and secondary (Grades 7–12) certificates had been available. With the addition of the middle schoolcertificate, the secondary certificate was changed to Grades 9–12. For thefirst time, teachers specifically trained for service at the middle school levelwould be certified by the state. This action provided a supporting frame-work for change at the university level. Utilizing insights and experiencesderived from the M3 Project, in conjunction with the new state certifica-tion platform, the core group of mathematics and mathematics educationfaculty began to develop and implement a middle school certificationprogram. This program was built upon the following Standards-basedprinciples:

• Preservice students will be engaged in extensive and supported fieldexperiences so they will gain a deeper understanding of schools,classrooms, and learners.

• Mathematics content preparation will be designed and delivered byestablished mathematicians who are recognized for their teachingexpertise and are knowledgeable of both the NCTMStandards(1989)and the recommendations of the MAA inA Call for Change(1991).

• Mathematics pedagogy preparation will begin in the junior year andwill be linked to the mathematics students study in the final two yearsof their program.

• Middle grades Standards-based curricula and instructional strategieswill be prominent in the mathematics and mathematics educationcoursework and in the field experiences.

• The program will be a four-year undergraduate certification programwith a minimum of 27 credit hours in mathematics and 6 credit hoursin mathematics education.


• Mathematics studied will be based on a foundation of algebra,geometry, and concepts of calculus. Additional courses, speciallycourses for middle school majors, will build on this foundation.

MIDDLE GRADES MATHEMATICS CERTIFICATION:A GENERAL VIEW

Students preparing to become middle school mathematics teacherscomplete the general education requirements of the University, the middleschool professional education requirements, and 21 semester hours ofsocial studies, English, or general science for a second-field teachingcertificate (minor). In addition, nine 3-credit-hour courses in mathematicsare taken along with two 3-credit hour courses in mathematics educa-tion. As part of the mathematics education courses, students participatein two linked field experiences in which they work with middle gradeteachers and students in classrooms throughout the two semesters prior tostudent teaching. The fundamental goal of the mathematics courses is thatthey thoroughly prepare middle school mathematics teachers to effectivelyutilize Standards-based curricula. (Course syllabi are available from theauthors.)

Mathematics Courses for Middle Grade Teachers

Pedagogically speaking, it may be advantageous to have a program inwhich each of the constituent mathematics courses services only educa-tion majors. In most cases, this is not practical, and alternate approachesare necessary. In particular, some of the required mathematics coursesin our program are not teacher preparation specific but are intendedfor more heterogeneous audiences. A useful strategy employed in ourmodel is the creation of a few distinctive mathematics courses that aredesigned specifically for preservice middle grade mathematics teachers.These courses introduce and explore significant mathematical conceptsand directly relate them to middle school mathematics. In fact, specificunits from the NSF-funded middle school curriculum projects serve asfocal points for classroom discussion and analysis. Important feedbackfrom the middle school teachers participating in the M3 project, as well as adetailed inspection of the materials, helped determine the nature of our newcourses. In particular, in-service teachers highlighted areas of mathematicsthat they felt unprepared to address. They also argued for the developmentof mathematical knowledge that would allow them to understand how the


mathematics they were teaching fit and connected to more sophisticatedmathematical ideas.

Algebraic and geometric concepts are central to the new Standards-based middle grade curricular materials, and the middle school teacherpreparation program has been significantly strengthened with the addi-tion of relevant algebra and geometry courses (Algebraic Structures andGeometric Axioms and Structures). These mathematics courses havebeen constructed in the spirit and philosophy of the NCTMStandards(1989, 1991, 1995), MAA’sA Call for Change(1991), and the jointNCTM/NCATE guidelines entitledInitial Programs Preparing Teachersto be 5–8 Mathematics Teachers(NCTM, 1999). They are designed toprepare middle grade teachers to meet the mathematical challenges oftheir future assignments. They primarily serve middle school mathematicsteacher education students but also are open to and attract other majors.

Algebraic structures for middle grade teachers. A few natural questionsserve as a prelude to our discussion:

1. Why should middle grade teachers study modern algebra?2. What modern algebra concepts are essential for middle grade teachers?

In traditional middle school curricula, eighth grade is usually the first timestudents are engaged in a formal course in algebra. The nature of thiscourse is generally abstract and skill driven and often does not connect thesubject matter to the real world or to other mathematics. Moreover, histor-ical and conceptual perspectives are rarely considered, and this contributesto the isolation of the subject. Given these deficits, resourceful teachersexpend a great deal of effort and time searching for meaningful and inter-esting applications. This task, although not elementary, can be achieved. Acarefully constructed and administered modern algebra course can providemiddle grade mathematics teachers with enormous insight and maturityinto algebraic thought. Furthermore, intensive exposure to inductive anddeductive reasoning in the context of selective axiomatic structures helpsprovide the middle grade teacher with important foundational perspectivesand the ability to understand and convey a wide spectrum of algebraicideas.

In contrast to traditional curricula, Standards-based materials introduceand study algebra throughout the entire middle grade program. Algebraoccurs in the context of other mathematics, in real world situations, and isprominent as a tool for modeling various problems. The requisite skills arecontinually reinforced, and necessary practice is appropriately included.If teachers feel more practice is warranted, it is not difficult to find addi-tional sources (e.g., any traditional textbook contains numerous practice


exercises). Consequently, much less time and effort are spent on findingand distributing supplementary materials, and more energy can be devotedto the teaching and learning of algebra.

Our reflections on Question 1 helped set the stage for Question 2, i.e.,the course content. In developing our modern algebra course, we care-fully chose topics that would adequately prepare teachers to effectivelyuse Standards-based curricular materials. For example, a detailed studyof the ring of integers provides an ideal setting for the analysis of suchfundamental topics as: the division algorithm, GCD, Euclid’s Algorithm,GCD Identity, prime numbers, The Fundamental Theorem of Arithmetic,modular arithmetic, and natural generalizations to polynomial rings. Also,a rigorous examination of arithmetic properties in various algebraic struc-tures deepens the understanding of traditional arithmetic and accentuatesthe importance of axiomatic mathematics. These concepts, as well as otherrelated ones, are essential to the middle grades mathematics teacher andare pervasive throughout the Standards-based curricula. It should be notedthat this course is not static and will continue to improve as it evolvesthrough additional iterations.

A most important tenet of Standards-based curricula is that content andpedagogy share equal importance. How students understand and employthe power and utility of mathematics is strongly dependent upon how thesubject is taught. New roles for teacher and student, including coach andco-investigator, are an essential part of the reform classroom, and multipleforms of assessment provide greater versatility in the measurement ofstudent achievement. In order to support and sustain the substantivechanges in the middle school mathematics classroom, it is quintessen-tial that professional development and teacher education jointly respondto the appropriate alterations needed for inservice and preservice teachertraining.

Modern algebra not only is a foundational imperative for mathe-matics teachers, but provides a natural setting for exploration, conjecture,modeling, problem solving, and the analysis and construction of rigorousarguments. Exposure to such mathematical endeavors contributes to theteachers’ content base, and, concomitantly, equally reinforces criticalStandards-based teaching strategies. In our modern algebra course, wehave structured the classroom environment to parallel the Standards-basedclassroom. Group learning combined with traditional lecture help guidethe way through the course topics. Consequently, individual and collectivepresentations, as well as written assignments, constitute the regular dailyactivities. Other forms of assessment include: take-home and in-classexams, historical assignments, critical analyses of Standards-based and


traditional curricula, and term papers. Student feedback has been quitepositive.

Geometry for middle grade teachers.The need for teaching more geometryin the middle grades is very clear. Teachers in the M3 Project found thegeometry content of the Standards-based curricula to be significantly moreintegrated and sophisticated than in traditional materials they had previ-ously used. They argued for more attention to the connection betweengeometric and algebraic concepts, and they desired to study more deeplythe ideas presented in middle school curricula.

We focus primarily on Euclidean geometry. However, we also covertaxicab geometry and non-Euclidean geometries in order to providecontrast. The introduction of alternative geometries serves to deepen theunderstanding of Euclidean geometry because one must see what non-Euclidean is about in order to better understand the special nature ofEuclidean geometry. Proof and rigor are emphasized to some degree, butintuition is also highlighted. Geometry is equally well suited for buildingintuition and for bringing home the need for precise definitions and state-ments. It is an excellent place to illustrate the power of logic and proof. Inaddition, geometry has a concreteness that people invariably see as relevantto the real world.

Standards-based mathematics curricula introduce more geometry tostudents at much earlier ages and investigate geometry through discoverytechniques, problem solving, and modeling strategies. We have used activ-ities from the middle school mathematics curriculum projects with thepreservice teachers. These activities are vivid reminders for the preser-vice teachers that middle school students are doing this mathematics.The preservice teachers are motivated to engage in the exploration ofthese activities and, in the process, become aware of the rich mathematicsavailable in these middle school curricula.

We find that dynamic software programs, such as the Geometer’sSketchpad, enable future teachers to see firsthand the power of technologyas an investigative tool. We also utilize a range of organizational strategies,including cooperative learning and group presentations of content. Thesepresentations provide evidence of the preservice teachers’ growing know-ledge as well as their understanding of how to present mathematicalcontent and how to engage learners (in this case, their peers).

Mathematics Education Courses for Middle Grade Teachers

The two 3-credit-hour pedagogy courses for middle grade mathematicsmajors are taken at the beginning of the junior year and concurrently


with the advanced mathematics courses (e.g., Algebraic Structures andGeometry courses described above) and with related field-based experi-ences. The first course immerses students into teaching mathematics byengaging them in the study of middle grades curricula, instructional tech-niques, assessment strategies, and resource materials. These in-class fociare supported and enriched by weekly visits to middle grade classrooms.Students also work one-on-one with individual middle grade students ina supervised after-school tutoring environment. This course is meant tostimulate their thinking about appropriate and effective teaching strategiesand challenge their beliefs about appropriate content for middle gradestudents. The second course builds on the first and provides supportedopportunities for developing lesson plans, teaching small groups ofstudents, and engaging in more in-depth study of particular units of studyor big ideas at the middle school level (e.g., proportional reasoning).Preservice teachers also explore ways to integrate content with othersubjects and complete several projects with preservice teachers who pursuea different major (e.g., middle school literacy). These activities are aimedat helping them develop collaborative working partnerships with teachersin other discipline areas.

SUMMARY

Our experiences with in-service teachers in the M3 Project prompted usto make considerable changes in the preservice middle grades teacherpreparation program. It also informed the changes through the ongoingand intense conversations between in-service middle grade teachers anduniversity faculty. In fact, it became clear that the changes in content andinstructional emphasis advocated by the M3 Project staff were also neededin our own university course experiences.

The first graduates of our new mathematics middle school certificationprogram graduate this year, so they are untested by traditional standards.Nevertheless, they have had a mathematics preparation uniquely focusedon middle school, along with extensive experiences in middle schoolclassrooms. Conversations with these students as they complete their 16-week student teaching internship, and with their cooperating teachersand administrators, confirm that they feel confident and well prepared toinitiate their careers.

An important goal of our program is to help graduates realize that theirundergraduate program prepares them for a good start, but that continuedgrowth and professional development must become an integral part of


their careers as middle school mathematics teachers. In that spirit, we willcontinue to monitor their development and transition to full-time teachers.

Our program is not offered as a utopian model. However, our approachillustrates how the experience and knowledge gained from workingwith inservice teachers as they investigate Standards-based mathematicscurriculum materials can help produce significant change in the under-graduate preparation of future teachers. It is described here in the hopethat it may be helpful to other faculty facing similar challenges in programdevelopment. The institutional collaboration of faculty in MathematicsEducation and the Department of Mathematics has been critical in estab-lishing this program. Although this collaboration has been instrumental,it has also been incremental. We succeeded in this initial effort basedon the close collaboration of two mathematicians and three mathematicseducators. For us, the challenge is to sustain the program by expanding thecadre of mathematicians and teacher educators committed to improvingteacher preparation.

NOTES

1 The Missouri Middle Mathematics Project had been supported by a grant from theNational Science Foundation (#ESI 9453932).

REFERENCES

Mathematical Association of America (1991).A call for change: Recommendations for themathematical preparation of teachers of mathematics. Committee on the MathematicalEducation of Teachers, Mathematical Association of America.

National Council of Teachers of Mathematics (1989).Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.



National Council of Teachers of Mathematics (1999).NCATE-approved curriculumoutcomes: Initial programs preparing teachers to be 5–8 mathematics teachers. Reston,VA: Author.

Reys, B. J., Reys, R. E., Barnes, D., Beem, J. K., & Papick, I. J. (1997). Collaborativecurriculum review as a vehicle for teacher enhancement and mathematics curriculumreform.School Science and Mathematics, 97(5), 253–259.

Reys, B. J., Reys, R. E., Beem, J. K., & Papick, I. J. (in press). The Missouri MiddleMathematics (M3) project: Stimulating standards-based reform.Journal of MathematicsTeacher Education.


Reys, B. J., Reys, R. E., Barnes, D., Beem, J. K., & Papick, I. J., National Science Founda-tion Teacher Enhancement Grant (ESI 9453932) (1995–1998).Reshaping mathematicsin the middle grades in Missouri: A model to improve mathematics curriculum, teaching,and learning(M3 Project).

Department of Mathematics Ira J. PapickUniversity of MissouriColumbia, Missouri [email protected]

Curriculum and Instruction Barbara J. ReysUniversity of MissouriColumbia, Missouri [email protected]

Department of Mathematics John K. BeemUniversity of MissouriColumbia, Missouri [email protected]

Curriculum and Instruction Robert E. ReysUniversity of MissouriColumbia, Missouri [email protected]

BOOK REVIEW

Lampert, Magdalena & Ball, Deborah Loewenberg (1998). Teaching,multimedia and mathematics: Investigations of real practice. New York:Teachers College, Columbia University. ISBN 0-8077-3757-7 (paper)0-8077-3758-5 (cloth)

It happened one day in the autumn of 1976. In the Juliana van StolbergCollege of Education in Gorinchem (Netherlands), seating was arrangedfor the next teacher educators’ meeting. It was almost three o’clock, andthe teacher educators were coming in one by one. At the centre of theclassroom, all by itself, stood an empty school desk complete with bench,which appeared, based on its size, to be from a first-grade classroom of aprimary school. There it stood, puny amid the many weighty gentlemenand the far fewer ladies, clearly intended as an eye-catcher. The principalof the College of Education surveyed the circle, turned to us, pointed tothe desk, and said, “Now that’s what it’s all about!”

Emphasis may sometimes shift, but apprenticeship has always been partof the prospective teacher’s learning environment. Problems arising fromattempts to combine teaching and learning theory with actual practice havenever really been solved successfully for all parties involved. The connec-tion between practice and theory has always been problematic, not only ina conceptual way but also in a practical sense. Teachers in training havedifficulty gaining access to what goes on in practice. Their knowledge ofwhat actually takes place in the classroom is superficial at best.

The advent of information and communication technology is bringingabout change. Lampert and Ball have developed theStudent LearningEnvironment [SLE], a digital representation of real practice, carefullyrecorded in real classrooms and organized into an accessible entity. Whatmakes SLE truly remarkable is its scope, which includes recordings in twoclasses over a full year, and the unaffected nature of that environment.

The titleTeaching, Multimedia, and Mathematicsnames three lines ofapproach, one of which examines the training of mathematics teachers atthe primary school level. The subtitleInvestigations of Real Practicehasmore connotations than first meet the eye: (1) mathematics education in


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primary schools, (2) teacher education students learning to teach at col-leges of education, and (3) the educational training of prospective teachers.In each of these fields investigations are carried out by K-12 students,teacher education students, and the authors of this book.

This book is multifunctional and can be used for teacher educationpurposes, as a manual for designers of multimedia environments, or as aresource for teacher educators who take practice as the point of departure.It also puts a new teacher education programme into perspective and showswhat abilities reflective practitioners can demonstrate. And last but notleast, it is a summary of an extensive developmental teaching project. Inshort, the book tells the story of an educational design from its first con-ception, goes on to describe the experiences of the first users, and evaluatestheir learning processes.

Lampert and Ball describe an educational design quite unrivalled inthe domain of teacher education. The design offers a substantial, well-reasoned answer to the question of how real practice can be taken as astarting point in teacher education, and how, against that background, theavailable and relevant theoretical knowledge about learning and teachingcan contribute. Lampert and Ball discuss how prospective teachers candevelop an investigative attitude and can continue to build on their ownlearning processes once they are actively involved in K-12 teaching. Theirdesign is destined to come as a culture shock in conventional teacher edu-cation circles. It will also make higher demands of teacher educators thanhas been the case so far.

TWO REFLECTIVE PRACTITIONERS JOIN HANDS

The first two chapters describe the prehistoric phase of the design, so tospeak, as Lampert and Ball are establishing themselves as reflective prac-titioners. Seen in the light of the following chapters, they each describetheir development from student to teacher educator. Step by step, the readercan track how their professional insights and personal viewpoints evolved.This is inside information that promotes a better understanding of laterdevelopments. At the end of both chapters their paths converge, and in theperiod after 1980 they join forces. It is in those early years that the seedsfor the futureMathematics and Teaching Through Hypermedia[M.A.T.H.]project were sown.

Having arrived at Chapter 3, the reader knows that both authors will berounding off the story as a team. So as not to overlook that fact, we shallcall them M&D (Magdalena & Deborah) from now on.

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A NEW APPROACH TO AN OLD EDUCATION CONCEPT

Criticism

M&D are deeply critical of teacher education as it stands. Disappointingresults call for a thorough analysis of the causes. Why is there hardly anytransfer from theory to practice and what effect do piecemeal, fragmentarytraining programmes have? Just look at teacher education students duringteaching practice. They observe K-12 teachers teaching their subjects, butthey have no opportunity to determine why they do what they do or howthey explain things that happen in the classroom. Teacher education stu-dents tend to regard teaching as a skill requiring little more than commonsense. “The more they feel that they can teach using common sense andprior experience, the less likely they are to appreciate the importance ofsome kind of professional knowledge” (p. 25). Strange things go on intraining as far as focus on knowledge is concerned as well. For a longtime, thinking on the content and the form of knowledge was rigid. So, too,were notions on determining what required knowledge was; how it shouldbe organized, presented, and acquired; and how its general and specificcharacter and its association with everyday practice could be established.Moreover – and here M&D add a surprising touch – teacher educationstudents have never personally experienced the fact that certain knowledgeis of essential importance to teaching.

New Insights

Fundamental criticism invites solutions, and new insights can provide thenecessary inspiration for coming up with those solutions. Practice, forinstance, is always talked about in the singular. Practice in the classroomdoes, however, take many guises. Sometimes the teacher is called uponto temporarily distance himself from his own conceptions to take up thatof the pupil, or to adapt his own philosophy to follow the view taken bya textbook author, or, quite the opposite, to put the book aside to look atmathematics from an everyday vantage point, or to descend from the aca-demic level of formal sums to working with concrete material. In training,teacher education students should at least be able to experience all thesefacets of actualpractice.

Teachers’ knowledge depends on the context and this knowledge isintensified and broadened in exchanges with others. M&D provide newinsights in the field ofsituated knowledgeand collaborative learningenvironmentswhich make a contribution to a new teacher education.

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A Vision, Fundamental Changes, and New Technologies

M&D dream of a teacher education course in which their ideas take shape.The dream is nourished by previous experiences: “In short, we wantedthem to be able to use our classrooms as sites for inquiry into practice”(p. 40).

But the impression given in the dream is not as easy as this statementsuggests. A fundamental transformation of teacher education is, in fact,what is suggested, an innovation that turns the subject matter (what isteacher education all about), the discourse (how does interaction aboutlearning and teaching in the teacher education take place), and the learningenvironment (how is the training organized in a didactical and techno-logical sense) completely upside down. Once M&D start thinking aboutactually realizing their dream they use Shulman’s thoughts on “strategicpractical knowledge” and Wittgenstein’s idea of “knowledge as a terrain tobe explored and discovered by way of multiple journeys” as backgroundconcepts (pp. 44–45). They do so at the end of the 1980s when the unpre-cedented opportunities of the new information technology and multimediaare promoted with great panache. Now they can give their fantasies freereign:

Flexible electronic cataloguing and linking, in our collection of records of practice, forexample, would mean that a teacher educator could prepare to engage teacher educationstudents in an investigation of classroom culture by choosing some relevant video clips,attaching them to reflections the teacher wrote on those moments in the lesson, writ-ing some commentary about ideas in the sociology of small groups that were related towhat was happening, putting this all together with children’s work to make it possiblefor novices to investigate how ideas travelled from one part of the room to another, andmaking everything accessible at a few mouse clicks for printing or projection. Then boththe teacher educator and the teacher education students could record their reflections onthese materials – questions, commentaries, interpretations, and the like – for real-time andnonsynchronous collaborative analyses. This activity would produce the sort of multiplejourneys across the same terrain that we imagined could represent the study of teachingfrom the perspective of practice. (p. 55)

In August 1989 the National Science Foundation awarded a three-yeargrant for development and research, thus launching the Mathematics andTeaching Through Hypermedia (M.A.T.H.) project. The aims were, inshort, (1) to record practice in Grade 3 (Ball) and Grade 5 (Lampert), (2)to create a digital environment with these records of practice for teachereducation students, and (3) to investigate the use of this learning environ-ment by teacher educators and teacher education students and the effects ithad.

BOOK REVIEW 315

PRACTICE, TEEMING WITH OPPORTUNITIES, ISMULTIMEDIALLY RECORDED

Before graduate students armed with video cameras went into theclassroom, a number of decisions were made that were important to theway in which the material as a whole would represent the practice ofmathematics education. There were three recording categories: normalimages of everyday life in the classroom, special recordings for investiga-tions relating to the MATH project, and additional information about thelessons.

Apart from that, a well-thought-out decision was made to take an entireschool year as a suitable unit for recording lessons and to record everythingthat happened during the lessons. No selection was to be made, no spe-cific theme chosen, and no particular educational implications were tobe included. The idea was to record practice as it truly was. To reduce,wherever possible, the influence of existing views and prejudices whilstrecording, a rotating system of graduate students was used.

Once everything was ready and the recordings could be assembled,ordered, and put away neatly, the project team saw this as yet anotherchallenge to be met. They judged this correctly, as the organization ofmaterial does determine what the learning environment looks like, and thusthe way in which teacher education students experience and comprehendactual practice. Unfortunately, in this study no specific attention could bepaid to the link between the organization of the records of practice and theways teacher education students learned about (actual) practice.

As soon as the first practical prototype of the SLE is available, workcan be started on designing and researching alearningenvironment. Thismeans

that the focus will have to be on teacher education students, becauseit is their activities that transform the digital environment into a learningenvironment.

THE DIGITAL ENVIRONMENT BECOMESA LEARNING ENVIRONMENT

To arrive at a befitting learning environment for the teacher educationstudents, M&D select an approach they callinteractive curriculum devel-opment. This means that they set to work with the teacher educationstudents in the given surroundings, using prudently chosen concepts frompractice. A fine example of a practical situation is the lesson given by

316 BOOK REVIEW

Deborah Ball on 18 December 1989. What she makes of the problem-of-the-day, the pupils, the discussion, the interaction, the reflection, and theatmosphere, contributes interesting material for working with the Grade3 students. That is the moment a learning environment is created. And asteacher educators and investigators, M&D provide a crystal-clear analysisof that environment.

In summary, the commentaries of the teacher education students onMs. Ball’s lesson reflect their own experiences as pupils, especially thoseencounters that left lasting impressions. They also reflect on what theyconsidered a good teacher to be, as well as on their earliest experiencesin dealing with (young) children. The picture of an exploratory base takesshape, built on personal experiences that have led to a sort of case-by-case knowledge of which they were never actually conscious, as such,and so could never have reflected on. Sometimes, snatches of theoreticalknowledge are recorded, cropping up anywhere as distinctive theoreticalconstructs (such asfeedback, reinforcement,context, andstrategy).

INVESTIGATING THE INVESTIGATIONS

In the years between 1991 and 1996, there were 68 investigative learningprojects initiated, supervised, monitored, and analysed by the M.A.T.H.team at Michigan State University (East Lansing) and the University ofMichigan (Ann Arbor). Close to 200 teacher education students took partin this Investigation Project, some alone, most in small groups. M&D’sresearch questions boil down to the following: What practical knowledgeis stored in the environment? How do teacher education students acquirethis knowledge? What can and must the teacher educator contribute to this?

Questions Put to Practice

The creation of learning environments for prospective teachers can beenhanced by knowledge of which questions novices want to ask aboutpractice. The questions indicate what practical knowledge they wish tohave. M&D set aside the first sessions for formulating and compilingquestions, after which the pros and cons for the investigation are discussed.

The list of investigation questions put together by the teacher educa-tion students shows what knowledge of practice they want and expect toacquire, but reveals nothing about the knowledge teacher education stu-dents gain through SLE, and certainly not what each individual student canget out of the investigation. That is perhaps the reason why M&D describesome investigations in more detail. We see the learning processes in those

BOOK REVIEW 317

descriptions where learning about mathematics education and learning toteach from practice go hand-in-glove.

An Example

Although the authors do not refer to the termpractice knowledgeas such,the reader encounters various manifestations of this kind of knowledge.In addition, it is impressive that this is presented in the context of teachereducation, and by M&D as teacher educators and investigators.

The reader is introduced to Patricia, a motivated student searching for aknowledge of fractions within a context, supervised by Lampert. The case(recorded in the autumn of 1994) is impressively presented in Chapter 7.To convey a glimpse, I would like to enumerate some points from thisextensive chapter. One point is the on-site construction of knowledge, thedevelopment that Patricia goes through that leads to a drastic change inher ideas of practical knowledge and that leads her to speak in terms ofsuppositions and conjectures. There is also the positive effect of having tokeep an electronic notebook as well as the fact that the teacher educator assupervisor can act as co-investigator because the student can cut the datashe has consulted from records and can paste it in the notebook. Anotherpoint is the importance of careful and expert supervision and a concep-tion of autonomous work in the SLE. Finally, this chapter discusses whathappens in connection with the question, “What does knowing fractionsentail?” which enables an integration of the student’s own mathematicalwork with the practice and theory surrounding it.

Patricia appears to have learned a thing or two. This is the way M&Dput it: She is no longer an outsider in Deborah’s class, she is no longeron the outside looking in, with no empathy. On the contrary, it seems thatshe is better able to project herself in the role of the teacher and the pupil.And she shows a different approach to practice, one which is now moreinvestigative.

Disappointed, the reader could wonder whether that is all there is andwhether that is all that one semester’s hard work has yielded. On furtherreflection it becomes clear that it would not be fair to use old standardsfor measuring the new pedagogy of teacher education. If we take the ideasand opinions presented above as criteria, then it becomes apparent howlarge the dividends are. Patricia has not learned the theory of rationalnumbers, nor did she analyse a well-structured learning strand of fractionsin primary school, or figure out the Rhind papyrus. Rather, from now onshe will confront the practice of teaching with an open mind, in a carefuland investigative manner. She will most certainly attempt to create a goodatmosphere in the classroom and may have eyes – and ears – for the indi-

318 BOOK REVIEW

vidual pupil. For her, mathematics is a subject on which discussions cantake place and where children can also learn from children.

M&D say the following about Patricia: “We can make no claims aboutwhether Patricia learned more with these new technologies in the contextof a new pedagogy of teacher education. She did learn differently than shewould have in a conventional methods course, and perhaps she learned ina way that will have an impact on her knowing K-12 teaching in practice”(p. 157).

THE RESULTS OF THIS STUDY

The crux is whether, by investigating K-12 teaching in practice, teachereducation students were able to acquire relevant knowledge and, if so, howthat took place. As to this study in particular, it is essential to know whetherit is possible – by observing specific episodes in a classroom – to be ableto discuss with teacher education students in an unconstrained way thebig ideas of pedagogical content knowledge and educational psychology.Finally, it is helpful to know what demands are to be made upon the teachereducators in this new modus operandi for teacher education.

This book gives an answer to all these questions. Sometimes new ques-tions come up; from time to time doubts prevail. As in the second question,the reader is regaled with a few fine examples that illustrate how suchan operation can proceed from a specific case (such as Ball’s lesson on18 September 1989) to a general principle (What role does the teacherhave if she wants to encourage pupils to participate in a mathematicaldiscussion?).

The important role of the teacher educator becomes apparent. Theprincipal task is to point teacher education students in the right directionwhen they investigate K-12 teaching practice, to allow the confronta-tion to be as meaningful as possible, and to help them see the acquired,context-based, circumstantial knowledge in a broader context. Even moreimportant is to have them experience how much there is still to learnabout practice and in practice and how they can extend their potentialas teacher-as-researcher. The implications of the study for the teachereducator should not be misconstrued. M&D show that in the US theseaptitudes are still rare in (prospective) teacher educators. Thus they con-clude with a cry from the heart: “What sort of policies and programs couldbe designed to support an appropriate education for teacher educators?”(p. 166).

Back in Gorinchem in 1976, an empty school desk stood to remindteacher educators (at Colleges of Education) of the portent of practice. In

BOOK REVIEW 319

the work done by M&D in 1999, this school desk can be replaced by anSLE workstation. Not only would this symbolize real practice, but teachereducators and teacher education students alike could, if they so desired,start investigations in a digital representation of real practice.

Bremlaan 16 FRED GOFFREE

3735 KJ Bosch en DuinThe NetherlandsFreudenthal Institute (MILE-project)Email: [email protected]

ACKNOWLEDGEMENT

The editors thank the following colleagues who reviewed manuscripts forJMTE in 1998. We appreciate our reviewers’ thoughtful critiques of themanuscripts and their contributions to the field of mathematics teachereducation.

Douglas AicheleBridget ArvoldDeborah L. BallMichael T. BattistaJoanne BeckerSarah BerensonCarol BohlinRaffaella BorasiGeorge W. BrightCatherine A. BrownLeone BurtonWilliam S. BushJinfa CaiThomas P. CarpenterVictor CifarelliDavid ClarkeDouglas H. ClementsBeatriz S.D’AmbrosioA. J. DawsonLinda DeGuireDonald J. DessartMichael De VilliersThomas DreyfusPaul ErnestMegan Loef FrankeToshiakira Fujii

Fulvia FuringhettiJoe GarofaloFred GoffreeKaren GrahamTheresa GrantDoug GrouwsLynn HancockRina HershkowitzDeAnn HuinkerBarbara JaworskiMartin JohnsonChristine KeitelHenry KepnerThomas KierenJeremy KilpatrickPeter KloostermanJohn KolbKonrad KrainerChronis KynigosColette LabordeCharles LambDiane LambdinMagdalene LampertGilah LederSteve LermanMary LindquistRomulo Campos Lins

Gwendolyn LloydJohn MasonJose Filipe MatosSue MauKatherine K. MersethDenise MewbornL. Diane MillerTatsuro MiwaBarbara NelsonNel NoddingsSandi NormanJohn OliveNicholas OppongBarbara PenceJoão Pedro da PonteThomas PostNorma PresmegEdward RathmellLeo RogersSusan RossJane SchielackDeborah SchifterAnna SfardMike ShaughnessyKen ShawAnna SierpinskaMartin A. Simon


322 ACKNOWLEDGEMENT

Larry SowderLynn StallingsLes SteffeHeinz SteinbringMarilyn StrutchensPeter SullivanJulianna Szendrei

William TatePatrick ThompsonCarol ThorntonDina TiroshTad WatanabeDiane WearneLinda D. Wilson

Patricia WilsonSkip WilsonTerry WoodOrit ZaslavskyRose Mary ZbiekLaura van Zoest

Documents

Journal of Mathematics Teacher Education_2