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Changing Beliefs: Teaching and LearningMathematics in Constructivist PreserviceClassroomsDianne S. Anderson a & Jenny A. Piazza aa Boise State University , USAPublished online: 06 Jan 2012.

To cite this article: Dianne S. Anderson & Jenny A. Piazza (1996) Changing Beliefs: Teaching andLearning Mathematics in Constructivist Preservice Classrooms, Action in Teacher Education, 18:2,51-62, DOI: 10.1080/01626620.1996.10462833

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Action in Teacher Education Summer 1996, Vol. XVIII, No. 2, pp. 51-62

Changing Beliefs: Teaching and Learning Mathematics in Constructivist Preservice Classrooms

Dianne S . Anderson Boise State University

Jenny A. Piazza Boise State University

Abstract

Constructivism is a learning theory which has emerged from Piagetian research. Reform in mathematics education frequently centers around constructivist principles. Many barriers prevent reform from occurring. The authors describe changes in instruction which have been implemented in their university mathematics education classrooms and relate these changes to their own constructivist philosophies. They examine the effects of change in instructional pedagogy and classroom environment on preservice teachers' beliefs about mathematics learning and teaching as well as the prospective teachers' feelings and attitudes toward mathematics.

The authors are engaged in changing the instructional pedagogy of their mathematics education courses. The courses involved are the one-year sequence in mathematics content taught within the mathematics department and the mathematics curriculum and instruction course taught within the elementary education program. The intent of the changes is to promote a constructivist view of teaching and learning mathematics in an elementary preservice classroom. This view is informed by: (a) the constructivist perspective on teaching and learning; (b) the current reform in mathematics; (c) what we know about the elementary constructivist environment; and (d) perceived barriers to change for preservice and inservice teachers. We include a summary of each of the four influences, the program changes made in the mathematics content and curriculum and instruction courses, and analysis on changes in preservice teachers' beliefs.

The Constructivist Perspective

The purpose of education is to assist the student in learning how to "obtain" knowledge, not the memorization of a body of facts (Bruner, 1963, 1966). Instruction, therefore, should take into account the capacity of humans for learning, the nature of the learning process, and what knowledge is (Bruner, 1971). Genetic epistemology, the study of the origin of knowledge, was initiated by Jean Piaget (1970). The process of reflective abstraction resulting in assimilation and accommodation that Piaget describes informs educators on how children learn (Piaget, 1970, 1952/1963). Piaget describes assimilation as consisting of the child taking in new information from hidher environment and organizing it into existing schema. Accommodation is the change in the child's understanding of reality which results from the construction of new schema necessary for the child to organize the new information (Piaget, 1970, 1952/1963). Children revise their thinking when new information brings about a discrepancy in their former view of the world (Rogoff, 1990). This revision then results in cognitive growth and development. Knowledge is the result of construction and invention, rather than discovery (Piaget, 1970).

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Because the Piagetian view of learning is a process of concept construction, rather than absorption and accumulation of information, the theory is termed constructivism (Schifter & Fosnot, 1993), and ". . . remains the most coherent account of how knowledge, especially logico- mathematical knowledge, develops in children" (Kamii & Livingston, 1994, p. vii). Constructivism is a theory about learning; it is not a prescription for instruction. The proponents of constructivism often have differing views on certain issues associated with this philosophy. The following ideas on constructivism, however, are generally agreed upon: (a) knowledge is constructed; (b) cognitive structures are activated in the process of construction through assimilation and accommodation; (c) these cognitive structures are constantly constructed and result in growth; (d) there is no external reality; and (e) acceptance of constructivist tenets leads to the adoption of constructivist pedagogy (Fosnot, 1989; Noddings, 1990).

Preservice teachers have limited personal experience with constructivist perspectives on learning. They need a clear understanding of these constructivist perspectives. They also need appropriate experiences and time to develop a philosophy on how children learn mathematics.

Current Changes in Mathematics Education

In 1989, the National Council of Teachers of Mathematics published Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). These standards are a set of recommendations for both content and pedagogy and energize current reform movements in teaching mathematics. The NCTM's rationale explains that knowing mathematics means much more than mastering number facts and algorithms.

. . . "knowing" mathematics is "doing" mathematics. A person gathers, discovers, or creates knowledge in the course of some activity having a purpose. This active process is different from mastering concepts and procedures. (p. 7)

Often in "traditional" mathematics classrooms, learning is defined as the passive acquisition of information, usually gained through repetitive, rote practice (NCTM, 1989). Instead, NCTM calls for instruction to include five aspects: "appropriate project work, group and individual assignments, discussion between teacher and students and among individual students, practice on mathematical methods, and exposition by the teacher" (NCTM, 1989, p. 10). The emphasis on active and purposive learning procedures outlined is congruent with a child-centered, developmentally appropriate philosophy. NCTM, however, falls short of a constructivist view in that it supports the "discovery," rather than construction, of knowledge.

Nevertheless, much of the current reform effort in mathematics education is based on what constructivism tells us about how people learn. In order to understand the need for changes in mathematics curriculum and instruction, we must first understand how and why traditional mathematics education is inappropriate for children. Brooks and Brooks (1993) offer several faults of the traditional mathematics classroom: (a) teachers talk too much; (b) teachers rely too much on textbooks; (c) most classrooms discourage cooperation and force students to work alone on lower-order skills; (d) student thinking is not valued; and (e) schooling has as an underlying premise the behaviorist idea that one external reality exists which all students should come to know. Many traditional mathematics practices are inappropriate because they lead to learners who are dependent on authority, are unable to solve multifaceted, higher-level mathematics problems, are competitive rather than collaborative, and engage students in limited knowledge construction.

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A Constructivist Classroom

In a constructivist classroom, autonomy is the goal of education (DeVries & Kohlberg, 1987; Kamii & Livingston, 1994). Autonomy is the ability to govern oneself and take responsibility for one's decisions (Kamii & Livingston, 1994). Autonomy involves the determination of right from wrong or truth from untruth using one's logical reasoning and sense- making. In promoting autonomy, educators promote the engagement of the child's mind through coordination of viewpoint which leads to the construction of knowledge. Autonomy and coordination of viewpoint go hand in hand--coordination of viewpoint is necessary for making a commitment to a position and exercising logical reasoning to defend the position. This defense leads to alterations in cognitive structures and develops the structures at a higher level than previously held (Fosnot, 1989).

Mathematics curriculum should be based on the posing of problems of emerging relevance to students and should take children's interests into account (Brooks & Brooks, 1993; Bruner, 1963, 1966). This premise does not mean, however, that only those things in which children are interested can be taught. Interest may be generated by the teacher through the use of good problems which promote cognitive dissonance as problems arise in context. A good problem is one which students view as relevant, is complex enough to have multiple solutions, and demands a group problem-solving effort.

Learning is structured around primary concepts or what is often termed "big ideas" taught in context. Traditional instruction frequently centers around learning many small skills, with the expectation that at some time students put all of the small pieces together to make a whole concept (Brooks & Brooks, 1993). In fact, students are frequently unable to make the part to whole transition. Large concepts provide a context in which the learning of individual skills takes place.

Learning takes place when students experience "disequilibrium" as a result of new information which elicits the equilibration process; ". . . the creation of new cognitive structures springs from the child's need to reach equilibrium when confronted with internally constructed contradictions; that is, when perception and 'reality' conflict" (Brooks & Brooks, 1993, p. 26). The teacher's role is to present situations to the students which promote disequilibrium. As students resolve their differing viewpoints, they achieve the desired equilibrium. This equilibration process results in the growth and development of knowledge.

Preservice environments need to be constructed to promote teacher and learner autonomy. Preservice teachers will, in turn, govern their future classrooms through logical decisions based on what they know about how children learn mathematics. The constructivist teacher education. environment is necessary if future classrooms are to look different from the classrooms of preservice teachers' experiential background. Time is needed for self-reflection in order for the relevance to emerge.

Barriers to Reform in Mathematics Education

Several barriers to reform exist and inhibit the change process (Schifter & Fosnot, 1993). Changing beliefs causes feelings of discomfort, disbelief, distrust and frustration.

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1. Inservice and preservice teachers, who must be the agents of change, are products of the system they are trying to change.

2. Not only is this pedagogy different, it is also much harder to learn and to develop.

3. Teachers need deeper content knowledge and understanding of the connections between mathematical topics and about the connections of mathematics to real life.

4. Teachers' reliance on textbooks undermines their professional judgment about what constitutes appropriate teaching.

5. The frequent use of standardized tests to assess speed and accuracy of computation ensures that teachers will be unlikely to change the emphasis of their teaching.

6. The idea that there is "one right answer" is in conflict with the nature of mathematical structures and demonstrates a limited knowledge of what mathematics involves.

7. Even when supervisors promote reform efforts and give teachers inservice education on what they perceive to be constructivist methods, they still evaluate teachers using the old paradigm.

8. Teachers' cognitive beliefs influence their instructional decisions and many of the feelings, beliefs, and values of some teachers are in direct conflict with those inherent to constructivism.

9. The goal of education today is heteronomy (Kamii & Livingston, 1994), in that education focuses on producing individuals who think alike and have like ideas and values.

In order to ensure reform in mathematics education, we must overcome these barriers. Assessing teachers' beliefs and moving slowly along the continuum of change are essential to success. Preservice and inservice teachers' traditional beliefs about mathematics and the learning and teaching of mathematics, however, must change if their practices are to change (Clark & Peterson, 1986; Cobb & Yackel, 1995; Peterson, Fennema, Carpenter, & Loef, 1989; Schifter & Fosnot, 1993). Several principles are necessary for those changes to take place in preservice education: (a) Preservice teacher education should be based on the same pedagogical principles as mathematics instruction; (b) If preservice teachers are expected to teach mathematics for understanding they must themselves become mathematics learners; (c) Regular classroom experience and consultation provides support, sustaining preservice teachers' learning in the context that matters most; (d) Collaboration among preservice and inservice teachers is essential to the process of reform; (e) Autonomous preservice and inservice teachers should be the goal of education (Cooney, 1994; Kamii & Livingston, 1994; Schifter & Fosnot, 1993).

Program Changes

In response to the NCTM Standards and the tenets of constructivist philosophy, changes have been made in the mathematics content and the curriculum and instruction courses offered at our university. Our years of teaching experience, our reflections on our observations of student learning, and ongoing study of learning theory led to our dissatisfaction with traditional

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mathematics instruction. These changes represent our attempt to implement reform in our own classrooms.

Mathematics Content

The mathematics content sequence addresses problem solving, the structure of arithmetic, geometry, probability and statistics. Students entering these courses frequently demonstrated a high level of mathematics anxiety and a low level of mathematical understanding (despite usually being skilled at computation). Students dreaded these courses and often postponed taking them until they had no other choice. While in the courses, the students complained that the material studied had no relevance to what they would be doing when they became classroom teachers. Although that relevance was clear to the instructor, it seemed difficult to communicate it to the students.

In response to these obvious problems, and assisted by research in learning and teaching, significant changes have been made in these content courses. The changes involve delivery of instruction, not changes in content. The purpose of the change was two-fold. The first goal was to develop real understanding of the content on the part of the students. The second purpose was to model for the students the kind of teaching we hoped they would practice when they became classroom teachers. We felt that by experiencing mathematical learning under a new paradigm, they would be more open to new methods of teaching.

The first major change is the elimination of lecture as the primary instructional device. Instead, the emphasis is on active learning through group problem solving. Students do not sit at individual desks but in groups of four or five at tables. Almost every class day, they work together to solve problems which are intended to promote connections among mathematical relationships and develop understanding on the part of the learners. Discussion among group members is a critical element of this process. In addition to the group activities, students also are assigned homework problems to be completed individually. These homework problems form the basis for whole class discussion the following day in which alternative methods of solution are presented and analyzed by students and teacher. The focus is on moving away from algorithms and toward solutions based on understanding, "making sense" of the mathematics. A standard textbook is used as a reference for student reading and as one source of problems. Many other sources are used also, including some commercial products and many problems and activities developed by the instructors.

The second major change involves the use of physical models to develop understanding. These models range from manipulatives such as counters, multi-base blocks, or centimeter rods, to pencil-and-paper activities such as pictures and diagrams. While these may be the same manipulatives used by elementary school teachers, they are not at this point used in the context of teaching methods. They are used to solve problems that these college students find challenging. In the process, students gain understanding of mathematical concepts. The same is true of the pictures and diagrams. Students are encouraged to use models rather than algorithms to solve problems.

Below is a typical problem. A traditional solution to this type of problem involved working backwards through several steps. Our students solved it in manner similar to that shown in Figure 1.

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Figure 1 Example of Student Problem Solving Using Models

$12 $24

$12 $12

$12 $12

~

$48 $12

Draw a rectangle to represent the money the twins had at the beginning. Divide it into three equal pieces, and label one of them as representing the money spent on the parents' gift.

l p l Now there are two remaining pieces, representing the remaining funds. If one fourth of this is to be spent on Granny's gift, divide the two pieces into four and label one of them as representing the amount spent on Granny's gift. This leaves three pieces unlabeled.

Since five sixths of the remaining money was spent buying gifts for siblings, divide the three pieces into six. Five of the six represent the amount spent per sibling, and the last piece then must be the $12 left to spend on lunch.

Jill and her twin brother Jack saved their money for Christmas. Last week, they took their savings and went shopping. They spent one third of what they had saved to buy a gift for their parents. Then they spent one fourth of the money that remained on a present for Granny. After buying her present, they spent five sixths of what was left on gifts for their siblings. Having finished their shopping, they spent the last $12 on lunch for themselves. How much money did they have when they started?

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Another difference in current instruction is the emphasis on writing. As illustrated in Figure 1, writing about the problem solving process is an intrinsic part of this new approach. It is not enough to just draw the picture and give the answer; the important thing is the description of the reasoning that resulted in the answer. These changes all are consistent with an emergent constructivist philosophy.

Mathematics Curriculum and Instruction

Changes in the mathematics curriculum and instruction course goals, projects, and assessments were guided by the notion that autonomy should be the aim of education if the potential for students' knowledge construction is to be met. The "content" of the course is not as affected by the change as is the style of teaching, the classroom community, and the assessment. The goal of the mathematics curriculum and instruction course is to enable prospective elementary mathematics teachers to make autonomous decisions regarding equitable practice. Equitable practice involves curriculum and instruction which offers the opportunity for all children to construct logico-mathematical knowledge. A constructivist philosophy informs the curricular and instructional decisions of this course, (i.e. students are provided opportunities in which to construct their ideas about mathematics learning and teaching). Some specific areas of inquiry include cognitive theories of development, methodology, curriculum, materials, content and equity.

An inquiry approach which involves investigation, active participation, and reflection is used to promote growth and development of equitable elementary mathematics constructs. Individual student goals and needs are met through the individualized experiences of reflection, critical thinking, coordination of viewpoint, reading, and writing on current issues in mathematics curriculum and instruction. Individual growth and development is also promoted through the students' examination of their personal philosophy of elementary mathematics education and the resulting implications for educational practice in their future classrooms. The teacher works as a facilitator of student-directed learning. Assessment practices emphasize the demonstration of assimilation and accommodation with evidence of conceptual connections and understanding of content.

Analysis of Changes in Thinking About Curriculum and Instruction

As we promoted a constructivist environment in our classrooms, our informal observations indicated that changes were taking place in student thinking. This paper is the result of an effort to gain a more formal understanding of the extent of those changes. We randomly selected 50 student journals from a group of 154 submitted in 1994. Based on our review of the literature on constructivist pedagogy and our own personal experiences, we analyzed the journals to determine a student's commitment to a constructivist philosophy. Our judgements were based on the self- described behaviors of the students as well as the language used in the journals. Four layers of learner commitment emerged which are described below.

Strong Constructivist Commitment

The principle factor in classifying students as possessing a strong constructivist commitment is evidence that the individual's language and behaviors focused on hidher efforts to understand from where children's understandings derived. The language and behaviors of these preservice teachers centered around inquiry into the source of children's mathematical knowledge

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and they were engaged in trying to determine how children learn. For example, after reading a case study on constructivism, one student wrote,

As I read about [the teachers in the case study], I see them working hard to implement constructivist principles, and it doesn't happen overnight! Shoot, it has taken years to "train" kids into the old paradigm, and now we've got to "untrain" them, or "retrain" them back. Back to what? Back to the way they (we) naturally learn. . . through inquiry and experimentation. It took some time and hard work, but she [the case study teacher] made progress. . .We're only as strong as our philosophy, and our commitment to it.

The belief that students construct their own knowledge, that knowledge cannot be transmitted from teacher to student, guided their curricular and instructional language and behaviors as well.

Valuing Some Elements of Constructivism

Students valuing some elements of constructivism indicated an appreciation of the pedagogical value of many of the practices or behaviors commonly associated with constructivist classrooms. The most commonly mentioned practices included the use of manipulatives and the use of cooperative learning groups. Some of these students also recognized the value of active learning and the need for problems to be real and relevant to the learner. The language and behavior of these students, however, is not always consistent; i.e. the language is often that of constructivist philosophy, but the behaviors observed are very controlling and behaviorist in nature. Occurring not as often is the flip side to this. Sometimes the preservice teacher rejected the language of constructivism, yet was observed as a highly intuitive facilitator with few behaviorist intents.

Lacking Evidence of Commitment to Constructivism

No or little evidence of commitment to constructivist principles is indicated by students who simply mention constructivism and constructivist ideas but make no reference to incorporating these principles into their own teaching and/or learning. They are still in a state of disequilibrium, struggling to determine who they are going to be when they enter the classroom. These students often use language and behaviors that imply a need for external forces for personal definition, particularly the cooperating teacher. These students focus on what was done in class as opposed to the educational relevance of the class in terms of how children learn. This is demonstrated in the following journal entry.

I enjoyed doing the place value game. At first it was a little confusing and I felt that [the student presenting] could have given us more directions before turning us loose, but after a while it became easy. . This was a very fun activity. Children/students learn by manipulating objects and practicing.

Antagonistic

This group of students is most difficult to characterize. The language and behaviors of these individuals demonstrate definite antagonism. The antagonism may be directed toward the philosophy itself, the instructor, or both, and their peers. They see little need to change the way we currently teach mathematics, regardless of their personal successes or failures with the content

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of mathematics in their past educational experiences. In discussing group work, one student writes,

. . . it was hard to keep everyone working together. Some people don't want to stay behind waiting for someone to understand. . . I found it interesting to . . . watch how everyone went in different directions.

For many of these students, this was their first introduction to constructivist ideas. Changes that were made in the content courses after these students were enrolled imply future students will have greater exposure to learning in a constructivist environment.

Analysis of Changes in Thinking About Content

With the changes in the content courses well established, we gathered additional data on preservice changes in thinking regarding mathematics content. In the fall of 1995, near the end of the first semester, students in the mathematics content course wrote an essay about their learning experience in the course. They were invited to comment on whatever they found most interesting or notable about the course. The respondents consisted of 45 female and 3 male students, and slightly more than half of them were traditional students. Two types of student responses emerged from the initial review of the essays. One type of response demonstrated students' beliefs about learning mathematics (cognitive factors) and the other reflected students' feelings about their learning experience (affective factors). Although responses varied in terms of the words students chose to use, recurring themes developed. Ten different themes emerged in the category of cognitive beliefs and another five themes emerged from the affective factors (see Tables 1 and 2).

Table 1 shows cognitive responses to student learning experiences. The most notable result was the number of students (30) who specifically mentioned their belief that the use of manipulatives promoted mathematical understanding. Comments included statements such as "Manipulatives gave me a better perspective," "The use of manipulatives introduced concrete evidence as to why and how math works," and "Working with manipulatives helps me to reach a deeper level of understanding for the concept being taught. We believe that as a result of their own learning experiences, these preservice teachers are much more likely to use manipulatives when teaching mathematics.

About 40% of the students addressed the value of their cooperative learning groups. One said, "Today, I see the value of assigning students to groups in order to learn, communicate ideas, and learn the value of others' ideas." Even though so many students mentioned the value of working together in groups to learn the mathematics, only 3 individuals mentioned student discussion and explanation as important. The lack of value placed on student discussion implies that the instructor is still the dominant voice in the classroom or that students perceive the instructor's talk as more valuable than student talk. This may be due to their old belief system in which the teacher is the ultimate authority on right/wrong answers. Of the 7 students rating repetition as important, 2 qualified their statement by saying that they no longer felt it was the most important factor in learning mathematics.

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Table 1 Cognitive Responses to Student Content Learning Experiences

Type of Response Number of Responses

Use of manipulatives enhanced learning Use of group work enhanced learning Use of pictures and diagrams was valuable It is important that mathematics make sense People learn mathematics in many different ways It is important to recognize that there are many

Repetition is important Student discussion and explanation is important Memorization is important Careful modeling by the teacher is important

correct ways to solve problems

30 20 11 11 11 10

Table 2 Affective Responses to Student Content Learning Experiences

Type of Response Number of Responses

Student feels less anxiety about learningheaching mathematics Student gained a better/deeper understanding of mathematics Student enjoyed learning mathematics and feels a greater sense of

21 18 10

confidence Student learned to value others' ideas Student feels s/he has something to contribute

1 1

Many of the 48 students stated that these learning activities had positive outcomes for them. These kinds of positive experiences have the potential to change the teaching methods which these individuals view as appropriate in their own future classrooms.

Table 2 lists affective responses. Affective statements exhibited more variation in word choice, which required making judgments about similar meanings. For example, such comments as "I'm not so nervous," "I'm less fearful," "I don't find math so intimidating," were all grouped into a category called Student feels less anxiety about learningheaching mathematics. Twenty-one students made statements of this sort, showing that this type of mathematics instruction had helped ease the mathematics anxiety for many of these preservice teachers. Lessened anxiety may lead to more exploration of teaching strategies in this particular content area.

Comments placed in the second category, Student gained a betterldeeper understanding of mathematics, included such statements as "I've expanded my problem-solving ability, " "I achieved a higher level of thinking," "I have grasped things I couldn't grasp before," and "I began to understand what I've been doing for the last 18 years." The most interesting aspect of the second category was that many of these responses came from students with strong backgrounds in

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mathematics. Some of them felt they progressed from thinking about which algorithm to use to trying to make sense of the problems.

The third category, Student enjoyed learning mathematics and feels a greater sense of confidence, included responses such as "It was the most enjoyable math experience of my life," "My confidence has been raised," "I feel much more confident," and "I actually like math now." While some students view themselves as being less anxious about mathematics, and others see themselves as having more confidence, the result is an indication of a substantial decrease in negative feelings on the part of students who have traditionally been fearful of mathematics and avoided it whenever possible.

The results of this inquiry are encouraging. These prospective teachers appear to have a more positive attitude toward mathematics in general. Several commented that they are looking forward to teaching mathematics and realize that it is possible to make learning mathematics interesting and exciting.

Conclusion

Constructivist beliefs emerge from experience and reflection. Our own beliefs have evolved as a result of experience with preservice teachers and reflection upon those experiences. Our classroom environments are reconstructed as a result of these changes. Changing preservice teacher beliefs about the teaching and learning of mathematics involves the same process. Ultimately, reform in mathematics education rests on the shoulders of the classroom teacher. Professional teacher educators are in a position to influence the course that reform takes through promoting changes in the thinking of preservice and inservice teachers. As teacher educators implement reform in their own classrooms, preservice teachers gain both the academic knowledge and the personal experiences that make it possible for these changes to occur.

References

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Handbook of research on teaching (pp. 255-296). New York: Macmillan Publishing Company.

Cobb, P. & Yackel, E. (1995). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the seventeenth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3-29). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Cooney, T. J . (1994). Teacher education as an exercise in adaptation. In D. B. Aichele, & A. F. Coxford (Eds.), Professional development for teachers of mathematics (pp. 9-22). Reston, VA: National Council of Teachers of Mathematics.

DeVries, R., & Kohlberg, L. (1987). Constructivist early education: Overview and comparison with other programs. Washington, DC: National Association for the Education of Young Children.

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Noddings, N. (1990). Constructivism in mathematics education. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 7-18). Reston, VA: National Council of Teachers of Mathematics.

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Peterson, P. L., Fennema, E. , Carpenter, T. P, & Loef, M. (1989). Teachers' pedagogical

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Piaget, J . (1970). Genetic epistemology. New York: W. W. Norton and Company. Rogoff, B. (1990). Apprenticeship in thinking. New York: Oxford University Press. Schifter , D . , & Fosnot , C . T. ( 1993). Reconstructing mathematics education: Stories of teachers

meeting the challenge of reform. New York: Teachers College Press.

Dianne S. Anderson is a Special Lecturer in Mathematics and Computer Science, and a doctoral candidate in Curriculum and Instruction. preservice elementary education emphasis, at BSU. Currently teaching math content for elementary preservice teachers.

Jenny A. Piazza holds an Ed.D. in Curriculum and Instruction, elementary mathematics educatiodlearning theory emphasis, Oklahoma State University, Stillwater. Currently Assistant Professor at BSU teaching graduate and undergraduate math curriculum and instruction courses.

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