Embed Size (px)
Citation preview
This article was downloaded by: [USC University of Southern California]On: 22 November 2014, At: 14:52Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Canadian Journal of Science, Mathematics andTechnology EducationPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/ucjs20
Examining teacher growth in professional learninggroups for in‐service teachers of mathematicsAnn Kajander a & Ralph Mason ba Lakehead University ,b University of Manitoba ,Published online: 26 Jan 2010.
To cite this article: Ann Kajander & Ralph Mason (2007) Examining teacher growth in professional learning groups forin‐service teachers of mathematics, Canadian Journal of Science, Mathematics and Technology Education, 7:4, 417-438, DOI:10.1080/14926150709556743
To link to this article: http://dx.doi.org/10.1080/14926150709556743
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in thepublications on our platform. However, Taylor & Francis, our agents, and our licensors make no representationsor warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of the authors, and are not theviews of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Examining Teacher Growth inProfessional Learning Groupsfor In-service Teachers ofMathematics
Ann KajanderLakehead University
Ralph MasonUniversity of Manitoba
Abstract: Teachers may face important challenges when encouraged to improvetheir mathematics teaching. Their personal beliefs, knowledge, confidence andpersonal intentions towards growth and change are all complex factors whichmay influence teachers' capacity, and their decisions about personal change intheir teaching. In this study, intermediate teachers and the conversations thattook place during their monthly Professional Learning Group meetings over aone-year period were examined in order to better understand issues teachers facein their growth and development as teachers of mathematics. We critically exam-ine the notion and meaning of success to different stakeholders.Résumé : Les enseignants sont susceptibles d'affronter des défis importantslorsqu'on les encourage à améliorer leur enseignemenl des mathématiques. Leursidées personnelles, leur niveau de connaissances, leur assurance et leurs intentionsdevant la perspective de changer et de s'améliorer constituent des facteurs com-plexes qui peuvent influencer leur capacité de changer leur façon d'enseigner et lesdecisions qu'ils peuvent prendre à l'égard de ces changements. Dans cette étude,nous avons observé des enseignants de niveau intermédiare et analysé les discoursqu'ils ont tenus pendant une année lors de rencontres mensuelles au sein d'ungroupe de formation professionnelle, afin de mieux cerner les questionsauxquelles font face les enseignants qui cherchent à améliorer leur enseignementdes mathématiques. Nous proposons une analyse critique de la notion de « succès »et de ce que celle-ci représente aux yeux des différentes parties prenantes.
IntroductionThis article describes a study investigating the behaviour and growth of
Grade 7-10 teachers of mathematics who participated in a professionaldevelopment project consisting of monthly professional learning group(PLG) meetings over the course of a year. Since 'relatively little researchexamines the specific interactions and dynamics by which professional
© 2007 Canadian Journal of Science, Mathematics and Technology Education
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
community constitutes a resource for teacher learning' (Little, 2003, p. 913),an exploratory study was carried out to listen to, examine, and attempt todescribe and characterize these conversations. Teachers' perceptions of theirparticipation and its effects on their beliefs and practices were investigatedthrough qualitative measures, including observation and audio recording ofmeetings of the professional learning groups and a process of narrativeinquiry or semi-structured interviews with selected participants. As well, aquantitative survey examined changes in knowledge and beliefs related tomathematics for teaching.
The learning processes of the professional learning groups, determinedby each group, varied widely. Although most participants felt their groupshad been successful, there were wide variations in intentions and approachesacross groups, which aligned with differences in the changes in teacherknowledge and beliefs identified by the survey. This article explores thesedifferent characterizations of successful professional learning, and relatesthe differences to the apparent advantages and disadvantages of PLGs asa form of professional learning. It concludes with a cautionary note aboutinherent biases in different conceptions of success for professional learningactivities.
Framework
Professional growth/development
The ineffectiveness of traditional professional development is well char-acterized in the literature (Ball, 1996; Darling-Hammond & McLaughlin,1995). 'The professional development currently available to teachers iswoefully inadequate In-service seminars and other forms of professionaldevelopment... are fragmented, intellectually superficial, and do not takeinto account what we know about how teachers learn' (Borko, 2004, p. 3).A variety of proponents suggest moving toward teacher-centred, ratherthan expert-centred, forms of professional growth. 'Even the apparentlysound principles of the teaching standards movement should not be imposedon teachers in a top-down manner. The reflection and initiative of teachersmust be respected and engaged in the ongoing enterprise of teacherdevelopment and school renewal' (Beck, Hart, & Kosnik, 2002, p. 191).Intentional efforts to question and reflect on personal teaching experiencesprovide an appropriate opportunity for growth (Manouchehri, 2002).
PLGs offer an alternative process for professional development: Smallgroups of teachers with common goals for improving practices hold regularmeetings and manage their own processes (Bissaker & Heath, 2005). It is amodel with no reliance on outside expertise, external management, orimposed direction. PLG approaches to teacher learning place considerablereliance on teachers to form effective communities of discourse and engage
418
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
in self-directed change processes (Darling-Hammond & McLaughlin, 1995).The PLG model is based on group members setting their goals collabora-tively and continuously negotiating how they will address those goals.As a consequence, PLGs are a model of professional development that islikely to be 'more sensitive to local conditions and teacher knowledge'(Windschitl, 2002, p. 161) than traditional workshop-based models.However, the success of PLG groups is clearly dependent on the choicesmade by the participants.
Mathematics knowledge for teaching
The content knowledge of teachers of mathematics has been the subjectof much recent study and considerable effort to reframe how we characterizeit (Franke, Carpenter, Levi, & Fennema, 2001; Hiebert, Gallimore, &Stigler, 2002). For the purposes of this article, we consider that a goodstarting point can be derived from recent efforts to view deep understandingof school mathematics content as a credible form of applied mathematicswith its own qualities and emphasis: a 'mathematics for teaching' (Ball& Bass, 2003; Ball, Hill, & Bass, 2005). On the other hand, when mathe-matics teachers' knowledge is considered to be a hybrid of traditional highermathematics and instructional methodology (Schulman, 1986, in Sternberg& Horvath, 1995), it is tempting to revert to old models of professionaldevelopment in which teachers are taught by experts in a traditionalmanner. A mathematics-for-teaching perspective opens the possibility thatteachers of mathematics could generate the understandings of mathematicsthat they need from where it is enacted—mathematical classrooms (Davis &Simmt, 2003)—if given opportunities to reflect on examples and evidence ofthe practices that embody it (Ball, 1997).
Criteria for success of teacher developmentinitiatives
While research suggests that the intentional self-questioning of practicecan support professional growth (Little, 2003; Manouchehri, 2002), otherrecent work (e.g., Balfanz, Mac Liver, & Bynres, 2006; Ross & Bruce, inpress) suggests that increases in student achievement can be achieved withboth formal training or the availability of external expertise, together withopportunities for reflection. Thus, an important question is whether PLGscan independently support teacher growth and, if so, how such growth canbe described or measured.
Characterizing and describing the quality of the PLG experienceremains a challenge. One question is: What aspects are to be measured ordescribed? An outcomes approach would presume that professional learningultimately would mean changes in student learning through changes inteaching practices. Improved achievement scores would add credibility to
419
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
claims of a professional program's effectiveness, but such an approach hassignificant tactical limitations, including the indirect and partial (at best)causal relationship between teaching approaches and student learning(Loucks-Horsley & Matsumoto, 1999). If the outcomes were consideredto be changes in teachers' practices, measurement, and observation,possibilities include self-reported descriptions of teachers' practices (Ross,McDougall, Hogaboam-Gray, & LeSage, 2003) or classroom observations(Boaler, 1997; Ma, 1999). The current study used a third kind of outcome asone of its foci by considering changes in teacher beliefs or knowledge as avaluable potential outcome of professional learning.
Alternatively, success for a professional learning approach could bejudged from the project's processes. The processes observed to be enactedwithin the approach could be compared to a list of desired elements andqualities determined by previous studies. Little (2003), for example, foundthat teacher learning communities that are conducive to improved practiceare ones in which groups 'reserve time to identify and examine problems ofpractice; they elaborate those problems in ways that open up new considera-tions and possibilities; they readily disclose their uncertainties and dilemmasand invite comment and advice from others; and artefacts of classroompractice (student work, lesson plans, and the like) are made accessible'(p. 938). Although this approach has the disadvantage of imposing anexternal set of criteria on PLGs that are designed to develop its own inten-tions and goals, it has the advantage for this study of requiring data that isaccessible through observations of the PLG process itself.
However, a more fundamental question as to how success will becharacterized is: Who is entitled to that decision? An institutional delineationof how success will be determined would, in the case of the professionallearning project reported here, mean that the school board's administrationwould make that determination. A school system may have explicit goals forfunded professional development such as changes in professional knowledgeor practice (Borko, 2004). Alternatively, as in the project described here,goal statements of the school board's intentions made by the project admin-istrator offer a framework for what could count as institutional evidence ofthe project's success.
The alternative to institutional parameters for success would be to letthe nature of success be determined by participants. Success could bedeclared for a professional learning process to the extent that those whoparticipated in the opportunity were satisfied, or according to the terms ofsuccess that were negotiated through their participation. Although thisapproach privileges the voices of the professionals who enact the PLGsand allows definitions of success to emerge through the actual process,the trustworthiness of a participant-based definition of success could havelimited credibility with external audiences.
420
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
Clearly, decisions made about how success will be characterized andwho will make that characterization are crucial to any evaluation of aprofessional learning project. Through its mixed methods design, thisstudy was organized so that decisions about the criteria for evaluating thePLG project's success could be a focus of the study itself. After providing adescription of the PLG process as it unfolded for participants, this articlewill address the relative viability of each of these approaches to characteriz-ing success.
Context and subjectsThe study took place within a single school board in a small city in
northwestern Ontario. Intermediate teachers of mathematics were invited bythe board to meet for a half day a month for the 2005-2006 school year todiscuss issues related to mathematics and its teaching. However, teacherswere not required to participate and many, especially in the upper half of thetargeted grade range, did not. The school board met the costs associatedwith the meetings, including the costs of substitute teachers.
Approximately 40 interested teachers were formed into groups by theschool board resource teacher, with each group comprising anywhere fromfour to eight teachers based mainly on geographic proximity. Four groupscontained only Grade 7 and 8 teachers, who are termed 'elementary'teachers in this jurisdiction. One group included teachers of Grade 7 to 10.Another group contained only teachers of Grade 9 and 10, considered tobe 'secondary' level mathematics teachers. All teachers in the groupstaught mathematics.
Although the teachers in the Grade 9-10 PLG decided as a group not toopen their group to the .research team, all other teachers agreed to partici-pate in the project. Some teachers whose teaching assignment or locationrequired them to withdraw from their PLG were also excluded fromthe research. The research cohort ultimately consisted of 31 teachers.Pseudonyms are used throughout this report when referring to individualteachers.
The PLG model was initially introduced by the board resource teacher(or 'consultant' in some jurisdictions), who hosted an introductory meetingin September for all participants. Recommendations for the PLGs were thenoffered by the resource teacher, such as suggesting the setting of a topic orspecific goals for each meeting, as well as the use of some time during eachmeeting to sustain a book study, with a book about teaching mathematicsselected from examples that were made available. However, these ideas wereoffered only as suggestions for the groups to consider, keeping in line withthe PLG model, which is based on the notion of professionals identifyingtheir own needs and deciding what choices are best for the group members.
421
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
For the rest of the year, all PLGs convened monthly at a location oftheir choice, during the scheduled half day of release time set aside for theprocess. The board resource teacher circulated to offer support and encour-agement, reinforcing the idea that the groups were to decide for themselveshow to develop their teaching through these meetings, but occasionallyoffered to supply resources as needed such as the Van deWalle and Lovin(2006) text. Generally, the professional learning groups followed the descrip-tion of the PLG model as described in the framework and functioned asautonomous groups, setting their own agendas and providing their ownleadership for processes they chose. For the purposes of writing, we haveused names of trees as labels for the different groups.
Design and methodologyA mixed methods design was used in the study in order to gather infor-
mation as broadly as possible. Surveys administered at the beginning andend of the PLG year generated quantitative measures of changes in teachers'values and understandings related to mathematics teaching. Transcribedaudio-recordings or detailed field notes of all PLG meetings were gatheredby research inquirers (the first author and three graduate students) assources for describing the PLG experiences of teachers. A narrative inquiryprocess was also initiated with several selected teachers (Clandinin &Connelly, 1996) as well as semi-structured interviews with a few participantsat the end of the year, which enabled selected teachers to validate andelaborate on the themes that the interviewers developed from the data.We will describe each element of this methodology, before sharing thedata and interpretations.
The survey instrument
The paper and pencil instrument, called the 'Perceptions ofMathematics' Survey (POM), was administered in the manner of a pre-test and post-test to participants in September and June. This survey is anexamination of beliefs about and understandings of mathematics, and hasbeen shown to be reliable as well as statistically valid both with pre-serviceteachers (Kajander, 2005) and in-service teachers (Kajander, Keene, Siddo,& Zerpa, 2006; Kajander, 2006). The POM instrument measures fourconstructs, namely the procedural knowledge of mathematics (correct useof standard methods), conceptual knowledge (understanding of mathemat-ical models, connections between methods, and justifications of methods),procedural values (beliefs in the importance of skills related to algorithmicand computational fluency), and conceptual values (beliefs in the impor-tance of understanding concepts and models).
The POM instrument measures values and beliefs with Likert-typequestions. Individual statements on the instrument portray a procedural
422
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
For questions 1 to 3 below on this page:
PART A):
PART B):diagrams.
1. 1.6x3
a)
2. 5 - ( -
a)
3. 1 « *
a)
Answer the questions, showing your steps as needed to illustrate the method you used.
Explain what you can about why and how the method you used in a) works, using explanations,models, and examples as appropriate. If possible, do the question another way.
b)
5)
b)
'A
b)
Figure 1: Part of the POM Instrument.
or a conceptual belief about the learning of mathematics in the teacher'sclassroom, and teachers rate each statement along a four point, Likert-typescale. For example, the item Everyone needs to deeply understand how andwhy math procedures work if they are going to make effective use of them isscored as Conceptual Values (CV), while the item It is important to be able torecall math facts such as addition facts or times tables quickly and accuratelyis scored as Procedural Values (PV). Procedural and conceptual valuesare considered to function independently of each other without inherentconflict: Teachers can express highly valuing either, both, or neither of thetwo kinds of beliefs.
Teachers' conceptual understanding of mathematics may be particularlyrelevant, and the Conceptual Knowledge (CK) construct has been shown tocorrelate strongly with other standard measures of knowledge of mathemat-ics for teaching (Kajander et al, 2006). To determine conceptual and proce-dural knowledge of mathematics, the POM instrument asks teachers toanswer mathematics questions relevant to intermediate-years mathematics,showing their steps to make their method clear, which then is scored asProcedural Knowledge (PK). Next, they are asked to explain and justifythe method used, in more than one style of representation and in more thanone way, which is scored as CK (see Figure 1 for a portion of this part of theinstrument).
In a previous use of the POM instrument, pre-service teachers have beenshown to change statistically in all four areas to a significant degree (p < .05)from the beginning to the end of a standard mathematics methods course,with all scores increasing except procedural values, which decreased(Kajander, 2005). Another study used this instrument with in-serviceGrade 7 teachers, who were receiving intensive professional training(Kajander et al, 2006). It also showed a statistically significant increasein their CV according to the POM instrument and a significant decrease
423
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTBRCESMT 7:4 October 2007
in their PV, when pre-training scores were compared to post-training(p < .05). As well, these Grade 7 teachers demonstrated a significant increasein their number and operations content knowledge of mathematics scores onan alternate instrument (CKT-M) for measuring middle school contentknowledge for teaching mathematics (Hill, Schilling, & Ball, 2005). Thisincrease correlated strongly with increases in CK as scored by POM, pro-viding validation for the use of the POM instrument in the current study.Increases in CV and decreases in PV were strongly related to improvedknowledge as measured by both the POM CK score and CKT-M scores.
In the context of this study, we conjectured that an increase in concep-tual understanding of mathematics for teaching might typically take placealong with a shift in beliefs, which may be characterized in part by anincrease in the value teachers place on conceptual learning, and a decreasein the value placed on procedural learning, as measured by POM. In otherwords, instead of emphasizing more traditional procedural fluency beforeinvesting time in problem solving activities, teachers may come to trust thatprocedural fluency on the part of students will develop along with deeperunderstandings. This is in contrast to a desire to focus more exclusively onprocedural practice as the main and first priority. Thus, it is argued thatteacher development may typically include increased knowledge and beliefsabout the need for conceptual understanding, and decreased beliefs in theimportance of procedural learning.
Qualitative aspects of data collection
Extensive field notes and/or transcripts from audiotapes were examinedfrom all meetings of four of the PLGs, and about half of the meetings of afifth and sixth PLG. Two of the PLG groups, chosen to exemplify the rangeof characteristics observed, were analysed in depth.
From the transcriptions and field notes, research inquirers constructednarrative texts and participants were able to respond to the narrative textthat described their participation, giving those participants an opportunityto add retrospectively to the narrative (Craig, 1995). 'Biographical storiesare collected as explanatory material, recovered as various narrative unitiesare traced, so that the teachers' stories can be told with new meaning'(Schulz, 1997, p. 2). The varied perspectives of participants are reflectedin the various forms of data.
The PLG experienceAs the monthly meetings progressed, we began to see differences in how
the groups functioned, as is also described in the literature (Little, 2003).In order to illustrate some aspects of this range, we chose two very differentgroups to describe in detail. These two case studies provide critical examplesagainst which we can test our initial concept of success as described in the
424
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
framework. We now turn to the case studies of these two selected PLGgroups.
The Pine PLG
The Pine group was a particularly large group: a Grade 7 teacher, aGrade 8 teacher, and six Grade 9/10 (secondary) teachers. Initially, Kevin,the Grade 7 teacher, made frequent comments illustrating his comfortlevel with classroom use of manipulatives, while a number of other second-ary teachers in the group frequently admitted they did not know how touse them. Kevin's offer to have other teachers drop into his classroomany time to watch his students working with concrete materials wasneither taken up by the other teachers in the group nor reciprocated.In another meeting Kevin led a demonstration using several different manip-ulatives. When the snap cubes were handed out someone asked, 'What arethese for?' The algebra tiles were also a mystery, with many questionssuch as 'Do you still have to teach them the method for FOIL?' and'Do you start with the outside left first or do you have to use one sidebefore the other?'
Kevin also made comments that suggested he was more comfortablewith alternative modes of assessment and grading than his other PLG mem-bers. While all other teachers in the group viewed assessment solely as unittests consisting of questions like the ones the students had done as practice,Kevin suggested the need for more extensive problems and rubrics. Severalother teachers then wanted to see an example of how rubrics could be usedto grade a test. However, the discussion did not continue; instead, someoneelse brought up using math journals to enhance literacy and the topic thenabruptly jumped to the role of literacy in mathematics.
Quick changes of topic such as this were typical for this PLG through-out the year. Often a suggestion was made or a need identified, but noattempt was made to investigate or resolve the problem, or to dig deeper.This type of interaction fits the general description of teacher discussions inthe United States provided by Wang, Strong, and Odell (2004): Ideas andsuggestions are typically left unelaborated and seldom "are probed deeply.
Discussion in this PLG was seldom centred on mathematics itself or onstudent learning, with the exception of the few initial attempts made byKevin. Discussion tended to be about teaching, but comprised mainly ofgeneral comments about difficulties and challenges of traditionalapproaches, with little or no consideration of alternate approaches thatare currently being discussed in reform-oriented literature and professionaldevelopment sessions. These qualities can be seen clearly in discussionsabout the difficulties related to 'teaching problem solving.' For example,during the second meeting, a discussion ensued about preparing studentsfor the provincial math test in Grade 9. One teacher commented that
425
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
'the biggest problem with EQAO [provincial test] is taking what the teacheris doing and putting it into real life examples. We struggle with having to dothis because we are used to having to show an algorithm method.'
There was a sense in these discussions that problem solving is a process'taught' after other ideas are 'taught,' so that students can 'take other ideasand extend it.' The sense was that the problem-solving task comes afterother more algorithmic learning and only if time allows. No discussionframed problem solving as a possible mode of learning, as discussed exten-sively in mathematics reform literature (National Council of Teachers ofMathematics, 2000).
Significant concern was voiced by many teachers in this group aboutresources and compatibility of textbooks to the new curriculum. One teacherin particular, Bernice, was particularly vocal at the first meeting about the'lack of black-line masters for the students who learn better that way.' Bernicemade many comments about the quality of texts and resources throughoutthe meetings. In her narrative inquiry response in March she wrote:
There should have been new Grade 9 and 10 text books and black-line masterspublished at the same time as the curriculum changed. We need textbooksthat match what we're teaching. We need to see some strategies for wordproblems too.
While she did speak of the need for problem solving, she seemed unwill-ing to tackle it independently. At the second meeting, she said, 'high schoolteachers need textbooks with multi-step problems instead of one-step rotelearning questions' to help students with problem solving. She dismissed thepossible consideration of other resources: 'We just don't have time to readthe stuff.' She wanted help with 'strategies for word problems. In a text-book, what does this look like?'
Bernice continued to be concerned about the skills students broughtwith them, but also showed a continued desire to understand more aboutproblem solving activities, writing:
Kids in Grade 9 really need to know their number sense and times tables. Theright answer matters and kids need to know how to get there. Manipulativescan help but most kids don't have a clue how to use them, and it's hard forteachers too. It's hard to find time to read all the new resources. I did try anactivity where students had to create their own word problems. Some of theproblems didn't have much problem solving involved, but some were prettycool. Next time I'll do it before the test.
Bernice was also concerned about students being appropriately placedand wanted to be sure that a Grade 9 applied class was always scheduled atthe same time as Grade 9 academic, so that students could move if they
426
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
found the academic too hard. In her narrative inquiry response in March,Bernice wrote:
It's really important to me that the right pathways are available for kids. If astudent can't handle 9 academic then there should be a 9 applied running at thesame time for them to move into. We need to track their progress especially ifwe think a student is placed wrong. For some of them the EQAO test [theprovince-wide final test for Grade 9 mathematics] is just like hitting a wall.
Overall, we view Bernice's priorities for change as being focused onmaterials, rather than on the teaching and learning processes of her mathe-matics classroom. Many of these priorities were external factors and did notrelate to an active desire to change her own knowledge or teaching skills.Because these priorities were shared by other group members, the teachers inthis PLG decided early on that they would focus a significant amount of theirPLG time on creating resources and diagnostics. During the third meeting,the teachers in this PLG split into two groups. One group had the task ofcreating a mid-year and end-of-year Grade 8 test, which Grade 8 teacherscould use to recommend proper Grade 9 placement for students. The secondgroup was charged with the task of identifying skills needed to succeed in theprovincial Grade 9 mathematics test; they were to find 'what's missing, andwhat's needed' in their current practice in terms of better preparing studentsfor the test. These subgroups operated for at least part of all remainingmeetings, building placement tests and classroom-ready sheets of practicequestions. At the end of the year, these resources were shared with theboard resource teacher and the other PLG members at a final joint meeting.
A keen interest of the research team was to listen for shifts in the dis-course that lead away from issues such as textbooks, resources, studentpreparation, placement issues, and other external factors, and towards ques-tioning ineffective teaching techniques and examining new conceptions ofteaching and learning (Little, 2003). Discussions of personal change in termsof teaching style, beliefs, or the unpacking of mathematical ideas did notemerge to any significant degree in this group, and if they did come up theywere not addressed in any detail by the group.
Although occasional, brief comments were made in connection with theneed for deeper mathematical understanding, again the ideas tended not tobe pursued or resolved. In the fourth meeting, for example, when workingon the Grade 8 test being created by one of the subgroups, a teacher asked,'Where do they learn why they flip the fraction to divide?' The Grade 8teacher responded, 'My kids learn to cross multiply so they know they cando this.' It was unclear to the researcher whether this teacher was referringto 'invert and multiply' or something else, and no one questioned the state-ment. Comments regarding difficulties with understanding the use of manip-ulatives were similarly not followed up with in-depth discussions.
427
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
In the last meeting of the group, the conversation continued to befragmented and unelaborated. For example, after one teacher mentionedhow about the incompatibility between the textbook with the revised cur-riculum, the conversation quickly turned to teaching tips, such as how to getstudents to remember whether quadratics open up or down. Another teacherexplained that she did this by telling students to 'think about [the change invalue of] your bank account, are you smiling or frowning ...,' indicating amemory-based approach, rather than the promotion of understanding. Thegroup finished off their last meeting by completing their resource package tobe shared at the final meeting of all group members.
We turn now to examining the progress of a selected elementary group.It was chosen for discussion because we felt it was the group most noticeablydifferent from The Pine group.
The Maple PLG
The Maple group was a small group of four Grade 7-8 teachers. Rightfrom the start, PLG members initiated their own shared goals related toimproving their understanding. Participants were very willing to share andprobe experiences from their own teaching. Cathy mentioned that 'they[my students] need the confidence to play, to try things... none of themwant to lead the way.' Laura sympathized, saying that 'kids are afraid tostep in, to try if not given the exact steps.' Cathy went on to share how shehad tried a strategy suggested at a recent workshop.
I gave them the handshakes problem to think about for two days with noinstructions, and then asked them to record their answer and strategy on chartpaper. Then I left it alone for two more days. Then we revised it and looked atother strategies.
The group expressed interest in such strategies for offering challengingproblems to students.
Sherry added that 'Sometimes students figure things out a totally dif-ferent way that I've never even thought of.' She continued to describe howimpressed she was with this. The discussion of multiple student solutionsseemed of particular interest to the group members. They also expressedinterest in deepening their own mathematical understanding in order tobetter be able to support different student interpretations.
During the second meeting the group decided to read the fractionschapter in the VandeWalle and Lovin (2006) text before the next meeting.By the third meeting these teachers were ready to begin an in-depth discus-sion of this content. Cathy shared some of her experiences teaching fractionswith manipulatives. The other teachers participated frequently in the frac-tions discussion, sharing classroom examples and making suggestions. Theywillingly and candidly shared their successes and failures with respect to
428
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
models, strategies, and their struggles related to student understanding(and their own understanding) with one another.
Related to fractions, Laura commented that 'dividing decimals was atotal flop' in her class and went on to share some specific teaching episodes.Others responded with their own experiences. Participants continued todiscuss how they could more effectively help students.
The discussion then moved to how learning in an investigative waywith hands-on materials helped the students, and who it benefited. Cathysaid 'it's different, it's cool'... but how do you assess [such learning]?' Markresponded, by asking them to explain things... not just on assessmentsbut all the time Keep them engaged rather than the teacher talking. Ido see some hugely positive shifts. I've gone from here to here [broadgesture] in being able to engage kids, especially for struggling kids Theones that used to be so far off, that used to pretend... now there's alight, they're a little bit more comfortable. If they're engaged, they're learn-ing a little bit.
The discussion then turned back to the related reading they had beendoing and a further in-depth discussion of models for learning fractionscontinued. Many moments of excitement and illumination emerged, inwhich participants expressed satisfaction with finally really understandingan idea, as the next few examples show.
Sherry was excited about the textbook definitions of the top and bottomnumbers in fractions and said, 'I wish I had read this before I taught theunit; this makes so much sense to me.' Cathy then initiated a rich discussionof multiplication and division models of fractions, and showed her previousattempts (before doing this reading) to have her students explain the ideasusing pictures, which she felt she had had only partial success with. Sheshared some of the varieties of solutions, but then described her subsequentstruggle modelling 5 -f- 2/3. In her narrative inquiry response, Cathy wrote:
We used fraction strips before I taught them the common denominator algo-rithm. I started out with no mental model of division in my own head, so atfirst I had no idea how to use them for division. I like them to be able to draw apicture to a solution before I teach them an algorithm. For example, I hadthem try to draw 5 -r- 2/3 by having them count the pieces, but then we gotstuck. The models in VandeWalle were a big help in understanding division. Ihad never seen these area grids for fractions before. Using these methods didseem to engage some kids more, but other kids were really confused by usingthe models and drawing diagrams to represent operations with fractions,especially one student who had been to a tutor at [a tutorial agency] formath and had learned algorithmic ways of doing operations with fractions.
The reading on division models was particularly exciting for her. She said,'I never thought of that... that there are two different kinds of divisionquestions.' With the researcher's encouragement, she then proceeded to
429
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
redraw her earlier diagrams for 5 -J- 2/3 in front of the PLG group, andtalked her way through it using her model to arrive at the correct solution.She said with excitement, 'I wouldn't have been able to draw a picture forthis... unless I read this [the VandeWalle book].' An in-depth discussionand unpacking of models of division (partitive and measurement) ensuedusing other examples, with the researcher used as a resource as needed.
Another, later PLG meeting for this group focused on the topic ofintegers, and a similar pattern of individual reading followed by shareddiscussion of teaching experiences, models that individuals had used withstudents, resolved misconceptions, and development of a new shared under-standing was observed. This particular meeting seemed to be especially help-ful to Laura, who made excited comments, such as 'this is good, this is whatI need,' and 'oh now I see—so now you CAN take away -3. I never knewwhat a zero pair was.' In this meeting, Mark, rather than Cathy, was themain sharer of his classroom experiences in using various models withstudents and, in a sense, drove and pushed the discussion to reach deepshared understanding. Mark freely invited the researcher to participateand questioned her in depth as to the accuracy of the classroom modelsand methods he showed. Again, the discussion was highly rich mathemat-ically for the participants and seemed to be very engaging and satisfying tothem. During the last meeting, the group viewed and discussed videos ofclassroom practice and talked excitedly about how they might use the ideasthey saw.
In this group, in contrast to the Pine group, teachers were observedapproaching discussions with the definite intent of achieving personalgrowth. They identified and unpacked problems of personal practice, readilyshared classroom artefacts, and solicited detailed advice from peers as wellas using the available external resources. They actively pursued new teachingpossibilities and shared their experiences. Discussions of the nature ofstudent learning that used examples of specific student products wereparticularly significant evidence of the teachers' shared commitment tomaking the PLG experience lead to changes in their teaching.
Results
Qualitative results
Generally, members of both groups felt positive about their experienceswith professional learning groups. The Pine group had developed a surveytest that they hoped would be useful for helping Grade 8 students to cate-gorize themselves appropriately when selecting Grade 9 courses, and theyhad created a list of skills they saw as needed for the provincial test. TheMaple group had explored new-to-them ways of teaching difficult content,and had encouraged each other to expand their repertoire of instructional
430
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
Table 1: Pretest and post-test mean scores on POM survey (all scoresshown out of 10)
PV
cvPKCK
Pretest Data
Mean5.84908.76329.03234.3226
N31313131
Sid. Deviation1.178011.041151.622422.50848
Post-test Data
Mean5.41978.49399.61295.3871
N31313131
Std. Deviation1.329551.165840.667202.94027
strategies for that content by sharing their students' products and addressingthe challenges such as assessment strategies together.
Yet, clearly, these two successes reflect very different kinds of profes-sional learning. The Pine group tended to reinforce their members' commit-ment to traditional teaching methods, and resistance to the consideration ofalternatives was persistent and effective. Discussion seldom or neverinvolved the analysis of classroom processes or student products.Discussion about students focused on achievement and deficits ratherthan processes of learning. Collaboration created two kinds of teachingresources: one that could affect student selection of courses, and one thatadded to the materials on hand for using a teacher-directed approach toteaching. If the two groups' processes and products were evaluated accord-ing to the goals of professional learning groups for reforming mathematicsteaching and learning outlined earlier in the article, one group would be seenas exemplary, and the other would be seen as having missed the point.We will return to considering how the success of the PLG process dependswholly on how and by whom success is characterized, but we turn ourattention now to one more kind of determinant of success: the quantitativedata.
Survey results
The results for the survey will be reported for the full set of 31 partici-pants in the PLGs and the study. Due to their population size, it would beinappropriate to apply statistical methods to survey data for individualgroups. Table 1 shows pre-test and post-test means for all four variablesof the Perceptions of Mathematics (POM) survey instrument.
At the pre-test of the current study, the 31 teachers were able to answercomputational questions (or PK) with a mean score of 90%; however theirability to explain or model how and why the method worked (CK) yielded amean score of only 43%. A significant increase in PK was noted (p < .05) bythe post-test, using a repeated measures t-test. While it might be assumedthat procedural skills should be very high for all teachers in the contentareas they were to teach, the reality was that, for some teachers, even
431
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
Table 2: Paired samples test for knowledge and beliefs
PairiPair 2Pair 3Pair 4
Pre_PV-Post_PVPre_CV-Post_CvPre_PK-Post_PKPre_CK-Post_Ck
PairedDifferencesMean
.4294
.2694-.5806-1.0645
Std.Deviation
.90376
.767751.336022.75603
Std.ErrorMean
.16232
.13789
.23996
.49500
95%Confid
Lower
.0979-.0123-1.070-2.075
Upper
.7609
.5510-.0906-.0536
t
2.6451.953-2.42-2.15
df
30303030
Sig.(2-tailed)
.013
.060
.022
.040
This work was funded by the NSERC University of Manitoba CRYSTAL grant Understandingthe Dynamics of Risk and Protective Factors in Promoting Success in Science andMathematics Education.
their procedural skills in these curriculum areas were incomplete. For exam-ple, two teachers, both new to teaching at the Grade 7-8 level, had pre-testPK scores of 50% or less. After the year of teaching at the intermediatedivision and participating in the PLG experience, improvement was shown,especially among those teachers whose scores were very low on the pre-test;all teachers scored 80% or higher on PK by the post-test.
Significant improvement was also evident overall in ConceptualKnowledge, indicating that teachers' classroom experiences, combinedwith their PLG activities and discussions, also supported significantchange (p <.05) in conceptual mathematical understanding for teachingas measured by this instrument (see Table 2).
A shift in teachers' values about mathematics and its teaching was alsofound. A significant decrease was seen in the variable PV {p < .05) from thepre-test to the post-test, and so it could be conjectured that by the end of theyear teachers prioritized procedural fluency as a learning goal less than atthe beginning of the year. However, the change in CV did not achievestatistical significance. This partial achievement of significant evidence ofchange in teachers' values aligns well with the qualitative results describedearlier for particular PLGs.
Interactions between qualitative and quantitative results
The narrative-inquiry approach to data gathering and analysis withindividual teachers meant providing teachers with opportunities to interactwith the data they had generated and the researchers' interpretations of thatdata. One result of this process was the participants' confirmation that thePOM surveys' pre- and post-scores contributed usefully and appropriatelyto the characterization of their personal professional growth for the year.Members of the Maple PLG were generally quite pleased that thescores on the instrument provided confirmation that, in most cases, theirmathematical understandings had advanced. For example, Mark and Cathy,
432
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
two members of the Maple group, offered strong linkages between theirscores on the POM instrument and their PLG experiences. IndividualPOM scores for Mark and Cathy showed a marked change in CK especially.The mean gain in CK scores for the full population of participants was 0.768on a 10-point scale (see Table 1). Cathy's CK score changed from 6 to 8,and Mark's from 1 to 10. Lesser changes were seen for members of thePine group, which also aligns with the lack of emphasis on such growthin that group.
In a follow-up interview, Mark described his PLG experience in theMaple group:
I was fortunate to be in a group that took it seriously... and was there tolearn My math classroom would not have evolved as it did without thePLG experience. People were very generous sharing ideas, I was alwaysamazed. The whole idea of sharing ideas about manipulatives—we helpedeach other Jearn... you need to be shown.. . . Planning what we were going to do at the next meeting... forces you to dosome thinking you may or may not have done otherwise, because you aremeeting with a group you have become quite close to.
This description speaks strongly not only to the effectiveness of their PLGfor affecting teachers' values and conceptual knowledge, but also confirmssome of the factors attributed to PLGs as effective professional learningformats described earlier in this paper.
DiscussionThe Pine and Maple groups enacted starkly different processes for their
professional learning group meetings. As just elaborated, the groups clearlyhad different senses of intention. One group developed a culture of profes-sional reform, supporting each other in taking the risks of trying unfamiliarteaching approaches and discussing the perceived deficiencies in their class-room practices. Their discussions frequently focused on student learning,and their engagement with textual resources was in support of their desire tosucceed in changing their teaching. The other group felt less inclined tochange their teaching. Their conversations stated the challenges they facedin their practices, but seldom included possibilities for addressing thosetaken-as-immutable problematics. Their discussions about students focusedon their deficiencies—their readiness to learn, their approaches to theirresponsibilities, their background knowledge—but rather than seeking tochange their teaching to match the students, the group pursued waysto ensure that students matched themselves better to the different coursesin Grades 9 and 10. Overall, according to all forms of data that werecollected, the teachers in one group advanced their pedagogic contentknowledge as they learned to teach in ways that were previously unfamiliar
433
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
to them. The teachers in the other group were more comfortable with theircurrent values and practices as mathematics teachers, and rather thanengage in changing these values and practices, they engaged in resourcedevelopment.
To what do we attribute these differences? Two categorical differencesand one experiential difference stand out when the two groups are com-pared. The group that engaged in professional reform (Maple) included onlyGrade 7 and 8 teachers; the group that engaged in resource development(Pine) included primarily Grade 9 and 10 teachers. This difference alignswith a difference in job description: The Maple group members were gen-eralists by background and by job description, whereas the Pine groupincluded members who taught mathematics as specialists.
In the end, it is not possible to attribute the differences in processes andoutcomes across the multiple groups to particular characteristics of thegroups' membership. Nor would such attributions in this one instance begeneralizable to other professional learning communities. -Perhaps the moststriking conclusion to be drawn from the striking contrasts in processes andoutcomes exemplified here is the flexibility that the PLG approach to pro-fessional learning allows. These two groups were each able to evolve indifferent ways, and the PLG process continued to provide a forum for thegroups to pursue their professional goals.
However, as Bateson (1989) suggests, when we are looking at two par-ticular cases in detail, we must be cautious not to be overly conscious of thedifferences between the cases. The reality is that both groups' stories havemuch in common that illuminate the nature of professional learning groupsas a structure to foster teacher learning. Both groups developed a commonsense of purpose. Both groups developed a conversational comfort zone inwhich members knew what fit the interests and intentions of the group as awhole. And, most importantly, both groups, despite very different concep-tions of what professional learning they valued, felt that the PLG processhad been worthwhile. This returns us to the underlying theme of this article:what the PLG experience described here can tell us about how we can andshould characterize success in professional learning.
Conclusion
Earlier in the article, different approaches to characterizing success forevaluating professional learning processes were outlined. The declaredsuccess of the PLG program and the two individual groups described inthis study would differ, depending on which conception of success was used.
An outcome approach to defining success would require first a decisionon what counts as an outcome. If changes in student learning were required,there is no data available within this study. In truth, defining a form ofstudent data that would generate differences attributable to the PLG process
434
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
would be problematic. If changes in teachers' instruction were required, theMaple group's discourse and interviews provide plentiful and rich descrip-tions of teachers changing how they taught, but the Pine group's does not.If the development of teaching materials was taken as a desirable outcome,then only the Pine group, with their development and sharing of twodifferent kinds of teaching resources, would be seen as having succeeded.Finally, if the desired outcomes were changes in teachers' beliefs and under-standings about teaching mathematics, then the POM instrument success-fully quantified aspects of successful outcomes for individuals and for theprogram.
Alternatively the professional learning groups in this study could beheld to account according to.a set of process criteria that have been foundto characterize other successful professional learning programs (e.g., Little,2003—see the framework section of this paper). Such a decision would againmean that one of the groups featured in this article succeeded, and onedid not.
So who should make that determination? If an institutional stance todetermine success were taken, the relative success would depend on thevalues of the school board, or the values of the project leader. In his intro-duction of the PLG process to potential participants, the project's leaderstated that instructional change was the ultimate goal. In the end, however,he felt that the outcomes across groups were at least satisfactory. The Maplegroup had achieved such changes, and the Pine group, he felt, had laid afoundation for such changes in the future. Separate from this study, theproject leader surveyed participants regarding their relative satisfactionwith the process. Apparently, teachers' feelings that they had been providedwith school board support for a legitimate and rewarding process wasan implicit institutional expectation. Ultimately, the institutional successof the project was reflected in the project's renewal the following year,with expanded participation.
But why not let the participants decide what should determine thesuccess of their professional learning? Because each group developed theirown intentions, participant success would need to be determined on a group-by-group basis. Both groups were pleased, even proud, of their work. Bothgroups had tangible evidence that demonstrated and characterized theirsuccess. Although the priorities of the participants may not match the prio-rities of interested parties (including the researcher-authors of this articleand perhaps its readers!), it is certainly a political act to impose an externaldelineation of success on a project that privileges the professionalism andindependence of its participants.
In fact, we, the researcher-authors, did have a predetermined set ofcriteria ready to be imposed through our research on the teachers' enact-ments of professional learning groups. As educators deeply committed to
435
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
the reform of mathematics in schools, we found ourselves significantly morepleased with groups such as Maple that intended and achieved the develop-ment of expanded repertoires of teaching. From that initial bias, we foundourselves drawn to the determinants of success that differentiated mostbetween the two featured groups. The process approach, which privilegescriteria drawn from previous studies that held the goals of professionallearning to be changes in teaching practices, and the outcomes approach,which relied on the POM survey that had been validated in part through itsability to pick up changes in teachers' beliefs and knowledge for teachingmathematics, both delineate by their design what will characterize success.In this instance, both approaches clearly favoured those groups like theMaple group, with its commitment to and demonstration of personalteacher growth. However, users of these approaches must be conscious oftheir orientation toward mathematics reform.
Ultimately, we remain committed to evaluating professional learning inways that value changes in teachers' values and content knowledge forteaching. At the same time, this article makes clear that other forms ofevaluation bring into view other forms of effective professional collabora-tion. In the particular case of evaluations of the PLG approach to teacherlearning, however, we have come to believe that evaluations that imposeconceptions of success—either the preconceptions of the evaluators/researchers or those of the sponsoring institution—are contrary to thenature of the approach. As described in the framework, the PLG approachis organized to value the autonomy of teachers and to trust in their capacityto be self-directed and purposeful. For compatibility with the approach thatit is evaluating, research on PLG process should grant to participants'conceptions of success a privileged position in its design.
Acknowledgements
We appreciate the assistance of the many teachers described in this study,who helped us not only to see some of the biases in play in the PLG projectand in our research on that project, but also helped us to see the validityin divergent views on teaching, learning, and professional development.In closing, we are grateful to the teachers in the study who opened theirprofessional communities to scrutiny.
ReferencesAdler, J., & Davis, Z. (2006). Opening another black box: Researching mathematics for teach-
ing in mathematics teacher education. Journal for Research in Mathematics Education,37(4), 270-296.
Balfanz, R., MacLiver, D.J., & Bynres, V. (2006). The implementation and impact of evidence-based mathematics reforms in high-poverty middle schools: A multi-site, multi-year study.Journal of Research in Mathematics Education. 37, 33-64.
436
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
Teacher Growth in Professional Learning Groups
Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we know andwhat we need to know. Phi Delta Kappan, 77(7), 500-509.
Bateson, M.C. (1989). Composing a Life. New York: Plume.Beck, C , Hart, D., & Kosnik, C. (2002). The teaching standards movement and current teach-
ing practices. Canadian Journal of Education, 27(2-3), 175-194.Bissaker, K., & Heath, J. (2005). Teachers learning in an innovative school. International
Journal of Education, 5(5), 178-185.Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting. London:
Open University Press.Borko, H. (2004). Professional development and teacher training: Mapping the terrain.
Educational Researcher. 33(8), 3-15.Clandinin, D.J., & Connelly, F.M. (1996). Teachers' professional knowledge landscapes:
Teacher stories—Stories of teachers—School stories—stories of schools. EducationalResearcher. 25(3), 24-30.
Craig, C.J. (1995). Knowledge communities: A way of making sense of how beginning teacherscome to know in their professional knowledge contexts. Curriculum Inquiry, 25(2),151-176.
Darling-Hammond, L., & McLaughlin, B.R. (1995). Policies that support professional devel-opment in an era of reform. Phi Delta Kappan, 76(8), 597-604.
Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education andcomplexity science. Journal for Research in Mathematics Education. 34(2), 137-167.
Franke, M.L., Carpenter, T.P., Levi, L., & Fennema, E. (2001). Capturing teachers' generativechange: A follow-up study of professional development in mathematics. AmericanEducalional Research Journal. 38, 653-690.
Hiebert, J., Gallimore, R., & Stigler, J.W. (2002). A knowledge base for the teaching profession:What would it look like and how can we get one? Educational Researcher. 31(5), 3-15.
Hill, H.C., Schilling, S.G., & Ball, D.L. (2005). Developing measures of teachers' mathematicsknowledge for teaching. Elementary School Journal, 105(1), 11-30.
Kajander, A. (2005). Moving towards conceptual understanding in the pre-service classroom: Astudy of evolving knowledge and values. In G.M. Lloyd, M.R. Wilson, L. Wilkins,S.L. Behm, (Eds.) Proceedings of the 27th Annual Meeting of the North AmericanChapter of the International Group for the Psychology of Mathematics Education.Roanoke, VA: All Academic.
Kajander, A., Keene, A., Siddo, R., & Zerpa, C. (2006). Effects of professional develop-ment on intermediate teachers' knowledge and beliefs related to mathematicsProgramming Remediation and Intervention in Mathematics Research Report.Ontario Ministry of Education. Available: http://www.curriculum.org/lms/files/prism/PRISMreportNWO.pdf(accessed October 28, 2007)
Kajander, A., & Zerpa, C. (2006). Effects of in-service for middle school teachers using multipleinstruments related to knowledge and beliefs. In A. Cortina, M. Saiz and Mendez A.,(Eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of theInternational Group for the Psychology of Mathematics Education, Merida, Mexico:Universidad Pedagogica Nacional.
Lieberman, A. (1995). Practices that support teacher development: Transforming conceptionsof professional learning. Phi Delta Kappan. 76(8), 591-596.
Little, J.W. (2003). Inside teacher community: Representations of classroom practice. TeachersCollege Record. 105(6), 913-945.
Loucks-Horsley, S., & Matsumoto, C. (1999). Research on professional development for tea-chers of mathematics and science: The state of the scene. School Science and Mathematics,99(5), 258-271.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding offundamental mathematics in China and the U.S. Mahwah, NJ: Lawrence Erlbaum.
437
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
CJSMTE/RCESMT 7:4 October 2007
Manouchehri, A. (2002). Developing teaching knowledge through peer discourse. Teaching andTeacher Education. 18(6), 715-737.
National Council of Teachers of Mathematics. (2000). Principles and Standards for SchoolMathematics. Reston, VA: Author.
Ross, J.A., & Bruce, C. (0000). Enhanced teacher efficacy as an outcome of professional devel-opment: The case of Grade 6 mathematics. Journal of Educational Research, (in press).
Ross, J.A., McDougall, D., Hogaboam-Gray, A., & LeSage, A. (2003). A survey.measuringelementary teachers' implementation of standards-based mathematics teaching. Journalfor Research in Mathematics Education, 34, 344-363.
Schulz, R. (1997). Interpreting teacher practice: Two continuing stories. New York: TeachersCollege Press.
Sternberg, R.J., & Horvath, J.A. (1995). A prototype view of expert teaching. EducationalResearcher, 24(6), 9-18.
VandeWalle, J., & Lovin, L. (2006). Teaching student centered mathematics grades 5-8. Boston:Pearson.
Wang, J., Strong, M., & Odell, S. (2004). Mentor-novice conversations about teaching: Acomparison of two U.S. and two Chinese cases. Teachers College Record. 106(4), 775-813.
Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: Ananalysis of the conceptual, pedagogical, cultural, and political challenges facing teachers.Review of Educational Research. 72(2), 131-170.
438
Dow
nloa
ded
by [
USC
Uni
vers
ity o
f So
uthe
rn C
alif
orni
a] a
t 14:
52 2
2 N
ovem
ber
2014
