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Teaching and Teacher Education 21 (2005) 439–456 Autonomy, beliefs and the learning of elementary mathematics teachers $ Janet Warfield a, , Terry Wood b , James D. Lehman b a Department of Mathematics, Illinois State University, Campus Box 4520, Normal, IL 61790-4520, USA b Department of Curriculum and Instruction, Purdue University, West Lafayette, IN 47907-2067, USA Abstract Seven elementary teachers participated in a project designed to help them learn to teach mathematics according to reform recommendations. Teachers were provided opportunities to learn through both private reflection and public inquiry about their teaching and children’s learning. The teachers’ instruction, reflection, and beliefs were studied. All of the teachers adopted some reform-based procedures including having children report problem-solving strategies. However, only three of them developed more complex practice in which children were involved in inquiry into one another’s strategies. The groups had different beliefs about the autonomy of children to construct mathematics and their own autonomy to make instructional decisions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Mathematics teacher education; Teacher beliefs; Teacher learning 1. Introduction A goal of education during the last quarter century has been to enable students to become independent learners (Darling-Hammond, 2000; Feiman-Nemser, 2001). This is particularly true for mathematics education where the goal is to enable students to think creatively and flexibly about mathematical concepts and solve mathema- tical problems with understanding (National Council of Teachers of Mathematics, 1989, 2000). To do this, teachers need to create situa- tions in which students are involved in exploring mathematical ideas, making conjectures about those ideas, and justifying their mathematical reasoning. This involves both reflecting on math- ematical situations oneself and communicating about one’s thinking with others (Hiebert et al., 1997). It is believed that students will develop independence or autonomy only through such activities (Kamii, 1982; National Council of ARTICLE IN PRESS www.elsevier.com/locate/tate 0742-051X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tate.2005.01.011 $ A shorter version of this paper was presented at the Annual Meeting of the American Educational Research Association, April 2003, Chicago, IL. Corresponding author. Tel.: +1 309 664 0409. E-mail addresses: jwarfi[email protected] (J. Warfield), [email protected] (T. Wood), [email protected] (J.D. Lehman).

Autonomy, beliefs and the learning of elementary mathematics teachers

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ARTICLE IN PRESS

0742-051X/$ - se

doi:10.1016/j.ta

$A shorter v

Meeting of the

April 2003, Chi�Correspondi

E-mail addre

twood@purdue

(J.D. Lehman).

Teaching and Teacher Education 21 (2005) 439–456

www.elsevier.com/locate/tate

Autonomy, beliefs and the learning of elementarymathematics teachers$

Janet Warfielda,�, Terry Woodb, James D. Lehmanb

aDepartment of Mathematics, Illinois State University, Campus Box 4520, Normal, IL 61790-4520, USAbDepartment of Curriculum and Instruction, Purdue University, West Lafayette, IN 47907-2067, USA

Abstract

Seven elementary teachers participated in a project designed to help them learn to teach mathematics according to

reform recommendations. Teachers were provided opportunities to learn through both private reflection and public

inquiry about their teaching and children’s learning. The teachers’ instruction, reflection, and beliefs were studied. All of

the teachers adopted some reform-based procedures including having children report problem-solving strategies.

However, only three of them developed more complex practice in which children were involved in inquiry into one

another’s strategies. The groups had different beliefs about the autonomy of children to construct mathematics and

their own autonomy to make instructional decisions.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Mathematics teacher education; Teacher beliefs; Teacher learning

1. Introduction

A goal of education during the last quartercentury has been to enable students to becomeindependent learners (Darling-Hammond, 2000;Feiman-Nemser, 2001). This is particularly truefor mathematics education where the goal is to

e front matter r 2005 Elsevier Ltd. All rights reserv

te.2005.01.011

ersion of this paper was presented at the Annual

American Educational Research Association,

cago, IL.

ng author. Tel.: +1309 664 0409.

sses: [email protected] (J. Warfield),

.edu (T. Wood), [email protected]

enable students to think creatively and flexiblyabout mathematical concepts and solve mathema-tical problems with understanding (NationalCouncil of Teachers of Mathematics, 1989,2000). To do this, teachers need to create situa-tions in which students are involved in exploringmathematical ideas, making conjectures aboutthose ideas, and justifying their mathematicalreasoning. This involves both reflecting on math-ematical situations oneself and communicatingabout one’s thinking with others (Hiebert et al.,1997). It is believed that students will developindependence or autonomy only through suchactivities (Kamii, 1982; National Council of

ed.

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J. Warfield et al. / Teaching and Teacher Education 21 (2005) 439–456440

Teachers of Mathematics, 1989, 2000). The wordautonomy is used here to mean that students arecapable of thinking about mathematical ideaswithout having the ideas ‘‘explained’’ to themand of solving mathematical problems withoutbeing shown a method by another person.

In order to realize the goal of enabling studentsto become autonomous learners, it is necessarythat teachers of mathematics also become auton-omous learners. Teaching, like doing mathematics,is essentially a problem-solving activity (Carpen-ter, 1989; Darling-Hammond, 2000; Grossman,Wineburg, & Woolworth, 2001; Lieberman &Miller, 1990). ‘‘Teaching is a complex practiceand hence not reducible to recipes or prescrip-tions’’ (National Council of Teachers of Mathe-matics, 1991, p. 22). Teachers need to recognizethe uncertainty inherent in teaching (Cooney &Shealy, 1997; Cooney, Shealy, & Arvold, 1998;Richardson & Roosevelt, 2004). They need torecognize that, just as children’s understandingof mathematics develops over time (Hiebertet al., 1997), so does their own understanding ofteaching mathematics and that their learningto teach should continue throughout their teachingcareers (Brown & Borko, 1992; Feiman-Nemser,2001).

Teachers need to become, as Franke and hercolleagues write, self-sustaining, generative lear-ners (Franke, Carpenter, Levi, & Fennema, 2001;Franke, Carpenter, Fennema, Ansell, & Behrend,1998; Franke & Kazemi, 2001). Teachers who areself-sustaining learners maintain changes in theirpractice adopted during a professional develop-ment project after the project ends. Teachers, whoare self-sustaining, generative learners both sustainchanges in their practice and continue learningafter the end of a professional developmentproject. These teachers make connections betweenwhat was learned in the project and their ownteaching and continue to reflect on and adapt whatwas learned as they teach (Franke et al., 2001;Franke & Kazemi, 2001). Teachers who are self-sustaining and generative see themselves as auton-omous decision-makers. That is, they see them-selves as capable of using what they know to makedecisions about teaching in ways that help childrenlearn rather than relying on others (the textbook,

the state tests, teachers of higher grades, etc.) tomake instructional decisions for them.

In this paper, we report the results of a projectthat had dual goals of investigating teacherlearning and designing an approach to profes-sional development that enabled teachers to learnto teach mathematics in ways consistent withcurrent recommendations and in which they hadopportunities to become self-sustaining, generativelearners. We investigated the teachers’ learning toteach, their learning to reflect on their teaching inways that would enable them to continue to learnafter the project ended, and whether their beliefssupported or constrained their learning. In thetwo-year professional development approach,learning opportunities were created to allow sevenbeginning teachers to learn by engaging in privatereflection on their teaching and the learning oftheir students. In addition, we provided teachersopportunities to learn through public discussionabout their teaching with us and with otherteachers. We focused these learning opportunitieson the teachers’ own experiences in their classeswith their students, thus situating the learning inthe context in which it would be used.

2. Background and theoretical orientation

According to Feiman-Nemser:

yif we want schools to produce more powerfullearning on the part of students, we have tooffer more powerful learning opportunities toteachersyUnless teachers have access to ser-ious and sustained learning opportunities atevery stage in their career, they are unlikely toteach in ways that meet demanding newstandards for student learning or to participatein the solution of educational problems (2001,pp. 1014–1015).

Feiman-Nemser suggested that the process oflearning to teach be seen as a continuum thatextends throughout a teacher’s career. She alsosuggested that the tasks of learning to teach differacross the continuum. During initial teachereducation, teachers learn subject matter content,develop understanding of learning and learners,

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and develop beginning strategies for teaching.During the first two or three years of teaching, aperiod that is often referred to as the ‘‘inductionphase,’’ teachers learn about the context in whichthey are teaching. They begin to develop ‘‘respon-sive curriculum and instruction’’

ynew teachers must bring together theirknowledge of content and their knowledge ofparticular students in making decisions aboutwhat and how to teach over time and then makeadjustments in response to what happens (2001,p. 1028).

The teachers involved in our project were all intheir first or second year of full-time classroomteaching, that is, they were in the ‘‘inductionphase’’ of their teaching careers. The professionaldevelopment focused on helping teachers learn todevelop ‘‘responsive curriculum and instruction.’’

In designing this project previous research onlearning was considered. This research shows thatlearning: is situated in the contexts in which itoccurs; is enhanced by reflection and communica-tion; and may be influenced by beliefs. In additionto considering literature on learning in general, wedrew on literature about learning to teach duringthe induction phase of a teacher’s career andresearch on learning to teach mathematics asrecommended in the mathematics education re-form movement (National Council of Teachers ofMathematics, 1991, 2000). A tenet underlying newrecommendations for teaching is that teachersshould base their instructional decisions on inquiryinto the mathematical understanding of theirstudents as well as into their own practice.Learning to do this is one of the tasks thatFeiman-Nemser says needs to occur during theinduction phase (2001). These recommendationsare founded in part on research on students’learning (e.g. Kamii, 1994) and are supported byresearch in classrooms (Cobb, Wood, & Yackel,1990; Fennema et al., 1996; Wood, 2001; Wood &Turner-Vorbeck, 2001).

2.1. Situated nature of learning

Studies of learning have shown that what islearned is often tied to the contexts in which it is

learned. Therefore, theories that learning consistsof acquiring knowledge that can be applied in avariety of situations have given way to theoriesthat learning is situated in communities of practice(Brown, Collins, & Duguid, 1989; Grossman et al.,2001; Lave, 1988, 1996; Resnick, 1987). Variousresearchers suggest that teacher educators considerthe situated nature of learning in designingprofessional development (Grossman et al., 2001;Loucks-Horsley, Love, Stiles, Mundry, & Hewson,2003; Putnam & Borko, 2000). Putnam and Borko(2000) say that there are several ways to do this: (1)professional development can take place in theteachers’ classrooms and be directly tied to what ishappening there; (2) teachers can bring artifactsand experiences from their classrooms to profes-sional development meetings away from the schoolsite; and (3) teachers can learn about content andpedagogy away from the school site and then besupported in using what they have learned in theirclassrooms. Our approach included elements of allthree approaches.

2.2. Roles of reflection and communication in

learning

Central to our approach was the understandingof learning as involving construction of knowledgethrough reflection and communication withothers. This understanding is drawn from Piaget(1985), in particular from his contention thatlearning involves assimilation and accommodationin thinking and occurs through reflection on theactivity of oneself and of others. In this view,learning occurs in three types of situations: (1)situations in which individuals are involved in aform of reflective thinking while comparing andcontrasting their ideas with those of others; (2)situations in which confusion, complexity, orambiguity occurs and individuals are engaged inthinking that involves questioning and reasoningto make sense of those situations (Entwistle, 1995);and (3) situations in which conflict arises andindividuals attempt to resolve the conflict bycritically examining and justifying their existingideas. Thus, according to this theory, in order forlearning to take place, it is necessary to reflect on

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both one’s own knowledge and actions and on theknowledge and action of others.

Researchers in mathematics education havefound that, when teachers focus their reflectionand communication on the mathematical thinkingof their students, they learn to teach differently(Cobb et al., 1990; Fennema et al., 1996; Wood,Cobb, & Yackel, 1991). The notion that teachersshould base their instructional decisions on inquiryinto the mathematical understanding of theirstudents and on their own practice implies theimportance of teachers reflecting on their students’learning and their teaching. Franke and Kazemiextend this notion by writing about teaching aslearning and about teachers engaged in ‘‘genera-tive growth.’’ They say, ‘‘Not only do teachersengaged in generative growth learn in the contextof their classrooms, but they also create commu-nities of learners for themselvesy’’ (2001, p. 47).Thus, both reflecting on one’s teaching andcommunicating with others about that teachingcontribute to learning to teach.

2.3. Role of beliefs in learning to teach

As indicated by the title of a recent book,Beliefs: A Hidden Variable in Mathematics Educa-

tion (Leder, Pehkonen, & Torner, 2002), it can beargued that more research on the beliefs ofmathematics teachers is needed. Research onteachers’ beliefs has generally focused on relation-ships among beliefs and practice and on therelationships among changes in beliefs and prac-tice. Although this research has not focuseddirectly on teachers’ beliefs and their learning, itis discussed here, inasmuch as there may be linksamong beliefs, practice, and learning. It hasfrequently been assumed that teachers’ beliefsabout content and about learning and teachingwould have a direct impact on their practice. Someresearch has shown a relationship between beliefsand practice (Cronin-Jones, 1991; Thompson,1984); however, the relationship between teachers’beliefs and their instruction is not as direct assometimes thought. Beliefs do not necessarily forma cohesive unit; it is not unusual for an individualto hold contradictory beliefs making it difficult to

determine how particular beliefs influence instruc-tion (Pajares, 1992; Pearson, 1985).

Although research on the relationships amongbeliefs and practice has not been conclusive, thereis evidence that reflection plays a mediating rolebetween beliefs and practice. Thompson (1984)contended that the different relationships betweenthe beliefs and practices of the teachers she studiedwere ‘‘related to differences in the teachers’reflectiveness—in their tendency to think abouttheir actions in relation to their beliefs, theirstudents, and the subject matter’’ (p. 123). Cobb etal. (1990) found that the beliefs of the teacher theystudied began to change only when, throughreflection, ‘‘she realized that her current instruc-tional practices were problematic’’ (p. 131). Theycame to believe ‘‘that beliefs and practice aredialectically related. Beliefs are expressed inpractice, and problems or surprises encounteredin practice give rise to opportunities to reorganizebeliefs’’ (p. 145). Fennema et al. (1996) had similarfindings. They concluded that teachers could makeprocedural changes in practice without changingtheir beliefs. However, for the practice of theteachers they studied to change such that theybased instructional decisions on the thinking oftheir children required that the teachers’ beliefschange first. They concluded that changing prac-tice in this way required teachers to reflect onevents in their classrooms. Such reflection createdopportunities for further changes in both beliefsand practice.

Cooney and his colleagues (Cooney & Shealy,1997; Cooney et al., 1998; Wilson & Cooney, 2002)recognized the importance of reflection in chan-ging teachers’ beliefs in ways that could have apositive impact on their practice. Their work withpreservice teachers focused on changing theteachers’ belief structures such that they came tosee themselves as authorities who were able to‘‘evaluate materials and practices in terms of theirown beliefs and practice, and be flexible inmodifying their beliefs when faced with discon-firming evidence’’ (Cooney & Shealy, 1997, p. 88).They contended that, through consideration of thestructure of belief systems, we can learn whysome teachers change their practice and others donot, ‘‘despite participation in the same teacher

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education program and rhetoric about reform thatmay be strikingly similar’’ (p. 97).

The importance of reflection in learning, as wellas the role of reflection in helping teachers connecttheir beliefs and practice, led to us to questionwhether there were relationships among teachers’beliefs, their reflection, and their learning. As aresult, we investigated the beliefs of the teachers inour project to determine whether such relation-ships existed.

3. Methodology

3.1. Participants

Seven beginning elementary teachers partici-pated in the project. At the beginning of theproject, one of the teachers was in her second yearand the others were in their first year of full-timeclassroom teaching. The teachers taught first,second, or third grade in three different schoolsin the same district, a district that encompassed theportion of a county surrounding a mid-sizedMidwestern city. In that district, some of theteachers used the materials developed in a previousproject (Cobb et al., 1990; Wood et al., 1991),rather than the commercial textbooks purchasedby the district. All but one of the teachers wasusing these materials. The other teacher taughtfirst-grade and had no prescribed curriculum; shedesigned her own curriculum but drew from theproject materials. In the second year of the project,some of the teachers changed grade levels. Duringthat year, one teacher taught fifth grade and used acommercial textbook. Also, one of the secondgrade teachers moved to a private Christian schoolwhere she alternated between using the requiredcommercial text and the project materials forinstruction.

3.2. Approach to teacher development

During the summer before the first year of theproject, the teachers attended a one-week work-shop led by two teachers who had used the projectmaterials and one mathematics educator. In thatworkshop, the teachers had opportunities to learn

about children’s mathematical thinking and theways in which they could use the project materialsto help them learn about the thinking of thechildren in their classes. At the end of the summerworkshop, all of the teachers agreed to be part ofan on-going project.

3.2.1. Private opportunities for learning: individual

reflection on teaching

Videotape was used to help the teachers examineand reflect on their own practice. In the first yearof the project, the teachers videotaped theirmathematics lessons once a month. They devel-oped a Personal Plan of Action (PPA) based on adilemma they had encountered in their teaching,worked on that dilemma throughout the month,and used a structured procedure to analyze andreflect on their teaching. This procedure consistedof: writing expectations related to the PPA prior toteaching the lesson; teaching and videotaping thelesson; watching the videotape and making de-tailed records of discourse that occurred in theclass discussion portion of the lesson; and compar-ing and contrasting their expectations with eventsthat actually occurred. The expectations, recordsof discourse, and comparisons of expectations andwhat they observed in their tape were written inreflective journals.

During the first year, the dilemmas identified bythe teachers differed with some focused on issuesof classroom organization (e.g., how to helpchildren work co-operatively) and others onchildren’s ways of thinking. During the secondyear of the project, the approach was altered in anattempt to shift those teachers who focused ontheir pedagogical dilemmas to center on themathematical thinking of their students. Teacherswere encouraged to write PPAs focused on theirchildren’s problem solving strategies and mathe-matical understanding such as, ‘‘I plan to learnhow my students think about subtracting two-digitnumbers.’’ That year they videotaped their classestwice a month and followed the same proceduresfor writing in their reflective journals. In February,the teachers decided that, instead of writing intheir journals, they would post their reflections toa listserv and that our responses should also beposted.

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3.2.2. Public opportunities for learning: collectively

communicating about teaching

In addition to teachers’ private learning in theclassroom, their participation in an on-goinggroup was an important part of the approach.As members of the group, the teachers wereexpected to publicly examine and critique theirown teaching and that of their colleagues. Theteachers participated in a series of workingsessions. In the first year of the project, they metmonthly and brought segments from their class-room videos to share with one another. Theteacher whose video was to be shown explainedhis or her PPA and then showed the video.Together the group discussed and asked questionsabout the segment, focusing on the dilemmapresented by the teacher.

In the second year of the project, teachers’attention was more explicitly focused on children’smathematical strategies and thinking during theworking sessions. Drawing on the approach ofCognitively Guided Instruction (CGI) (Fennemaet al., 1996) examples of children’s strategies forsolving word problems were used to draw teachers’attention to children’s mathematical thinking. Inthe working sessions for October and Novemberof the second year, teachers brought tapes ofinterviews with individual children in their classes.During those sessions we watched and discussedthe videotapes, focusing on how the children weremaking sense of mathematics. Another changethat was made in the second year was that theteachers held conversations about their teachingon a listserv as well as during working sessions.

3.3. Data sources

In order to investigate the relationship ofteachers’ beliefs to changes in their practice, wecollected two types of data for analysis. One set ofdata was used to analyze change in teachingpractice and consisted of the classroom videotapesthe teachers made of their lessons. We used thisdata source as evidence of ‘observed’ change inteaching and thus an indication of teacher learn-ing. Another set of data was used as evidence ofthe teachers’ views or ‘intended’ change inpractice. This data source consisted of teachers’

spoken comments given during the interviews andworking sessions and their written comments fromthe reflective journals and listserv messages.

3.3.1. Classroom videotapes

Each teacher taped a total of 12 mathematicslessons. These videotapes were collected and logscreated of the tapes. These logs were similar totranscripts, in that they consisted of teacher andstudent dialogue and interaction patterns.

3.3.2. Teachers’ reflective journals

The teachers’ reflective journals were collectedalong with each videotaped lesson. These journalsprimarily contained the teachers’ PPA for theirlesson and their reflections on the lesson afterviewing the videotape. The journals also includedresponses to specific questions asked of themduring working sessions (referred to as ‘‘journalresponses’’) and any notes they might have takenduring working sessions. The teachers’ reflectionson their lessons along with their notes takenduring working sessions were summarized. Theirjournal responses were copied verbatim.

3.3.3. Teachers’ listserv messages

The e-mail messages posted to the listserv werearchived, printed, and then grouped by teacher.Since the teachers usually used the reply commandto respond to others, these messages included themessage or messages to which they were replying.When necessary to ensure understanding of amessage, excerpts from other messages wereincluded.

3.3.4. Teacher interviews

Each teacher was interviewed twice, once at thebeginning of the second year of the project andagain at end of the project. The first interviewfocused on the teachers’ beliefs about learning andteaching mathematics; the second focused on theirbeliefs about mathematics and what it means to domathematics. The interviews were semi-structuredwith common questions asked of all teachers inorder to provide consistency across teachers andfollow up questions based on individual teachers’responses. The interviews were audio taped andtranscribed.

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3.3.5. Working session videotapes

Videotapes were made of the working sessionsand logs were made of substantive portions of thevideos—when mathematics and/or mathematicslearning and teaching were being discussed. Adocument was created for each teacher showinghis/her comments during the working sessionsalong with descriptions of the context of thosecomments and any statements of others necessaryto understanding those comments.

3.4. Data analysis

The data analysis adhered to procedures similarto those described by Strauss and Corbin (1990).The data were analyzed separately but simulta-neously and then compared in order to examinethe relationship between teacher’s beliefs andchanges in teaching practice. The data addressingteacher beliefs were first summarized as describedabove and then examined for evidence of teachers’beliefs about: mathematics; learning mathematics;and teaching mathematics. As we read, reread, anddiscussed the data, this initial categorizationscheme was modified. We found evidence ofstrongly held views of children that went beyondbeliefs about how children learn mathematics;thus, ‘‘beliefs about children’’ became a separatecategory in our scheme. Finally, we added acategory we called ‘‘constraints.’’ This beliefcategory included teachers’ statements aboutfactors that prevented them from teaching inways consistent with the reform movement.We developed descriptions of each teacher’sbeliefs in each of the categories and looked forsimilarities and differences in categories acrossteachers.

The transcribed logs of classroom videotapeswere used to look for changes in the patternsof teacher and student interaction during wholeclass discussion. Three researchers individuallycoded each statement made during the class-room discussion portion of each lesson, using asystem of codes we had developed in previousresearch. In the initial coding, the reliabilityranged from 69% to 90%. We discussed differ-ences until we arrived at agreement as to how tocode each statement.

3.5. Role of the researchers

We saw our role in this project as creatingsituations in which teachers could learn. Wecreated the structure that the teachers used toreflect on their teaching individually. We alsoplanned the working sessions and created thelistserv so that teachers had opportunities to talkabout their teaching with others. In addition, weresponded to the teachers’ individual reflectionsand to their comments in group discussions. In ourresponses, we attempted to challenge the teachers’thinking through questioning rather than by tell-ing them what to do. We recognize, however, thatwe did have a bias as to what and how we hopedthe teachers would learn by participating in theproject. For this reason, we used all of the data wecollected to allow for triangulation. In addition,we included graduate students who were notinvolved in the professional development in allstages of data analysis and interpretation.

4. Findings

4.1. Observed change in mathematics teaching

practice

4.1.1. Beginning practice

The teachers’ first videotape of their lessonsserved as baseline data of their classroom mathe-matics practice. The results of the analysis revealedthat at the onset their practice of teaching inmathematics was similar to that shown in the twoexamples from second grade given in Fig. 1. Theteachers’ practice still consisted of some traditionalforms of interaction and discourse but within a‘‘reform class discussion structure.’’ That is, allteachers created a strategy reporting (Wood et al.,2001) context in which children come to the frontof the room and ‘‘explain’’ their strategies forsolving a problem. The main focus in a strategy-reporting classroom is on children’s presentationof different strategies for the problems. Childrenpresenting their solutions may be asked to providemore information about how they solved theproblem by the teacher. The focus of teacherquestioning is on ‘‘how’’ and ‘‘what’’ to prompt

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Teacher A Teacher B

T: Okay, so we have 10 over here, another 10. Over here we have 9 and an empty box. Amy and Maddie, what did you guys get for the answer?

Maddie: 11. T: 11. Could you guys stand up there and

tell us how you got 11? [to the class] Raise your hand if you and your partner agree with 11. Okay girls how did you get 11?

Maddie: 10 plus 10 equals 20. T: She said 10 plus 10 equals 20. Why did

you want 20? Maddie: [inaudible]. T: She looked over on this side of the

balance [points to left side]. How much is over here?

Maddie: 20. T: How much was over here before we

found this [covers the 11 the girls had written in]?

Maddie: 9. T: 9. So she said, an easy way, she knew

that 10 plus 10 equaled 20, so 9 plus 11 would equal 20. Did anyone solve this a different way? . . . Dave, how did you solve that?

Dave: I knew that 10 plus 10 was 20 and 9 plus 9 equals 18. I knew that 10 and 10 would make 20 and then 9 and 11-- I knew that it would be 11, and so because it can’t be 9 and 10.

T: Okay. You knew that it can’t be 9 and 10, because 9 and 10 is what?

Dave: 19. T: 19. Good job. September 23, Year 1

T: . . . .The first box is a 12 and you can see that 5 is being taken away from the 12. We need to know what we need to put in this box to make both sides equal. Anybody have an idea what we need to put here in this side? Ellen?

Ellen: A 7. T: A 7. Okay, tell me how you got that

answer. Ellen: Because 5 plus 6 equals 11, so it

would be 1 more to equal 12. T: So you knew that 5 plus 6 equaled

11 and that 1 more would equal 12. Okay. Did someone think about this in a different way? Jenny?

Jenny: I just took the 12 and I counted back to 7. 11, 10, 9, 8, 7.

T: Very good. So Jenny counted back 5. She started with 12 and counted back 11, 10, 9, 8, 7. So, very good, so 12 minus 5 equals 7. Very good Jenny. Sally, how did you think of this problem?

Sally: I did 6 plus 6 was 12, so 7 plus 5 is 12.

T: Good. So you knew that 5 was 1 less than 6 and that 7 was 1 more than 6 and that was 12 because you knew the double of 6 plus 6. Very good. Nice.

October 5, Year 1

10 9 10 12 5

Fig. 1. First year’s practice.

J. Warfield et al. / Teaching and Teacher Education 21 (2005) 439–456446

students to describe what they did to solve theproblem. However, as these teachers were intransition from conventional mathematics teach-ing their ways of working still maintained somefeatures of conventional mathematics teaching. Itwas not uncommon after a child’s explanation forthe teacher to echo or repeat what the child said toensure all the students heard the details of thestrategy. Frequently, the teachers took over thestudent’s explanation to elaborate by adding detailor providing a rationale for what the child did. The

teacher then continued a reform orientation byasking if anyone used a different strategy. If so,that child described his strategy and the teacherrepeated or elaborated on it. Following this, theteacher typically moved on to a new problem.

4.1.2. Second year practice

The analysis of the lessons throughout the twoyears revealed distinct differences in the develop-ment of the teachers’ practice as the examples inFig. 2 illustrate. One group of teachers developed a

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Teacher A Teacher B

There are nine girls on the basketball team. Each girl needs a shirt and shorts for the games. A shirt costs $3.50, and a pair of shorts costs $5.00. What is the total cost of all of the outfits?

T: Vic and Katie, do our first balance. 35 on one side and 60 on the other. Why don’t you just tell us how you got it first and then we can write an answer?

Vic: We took 60 minus 35, no, 35 plus--I took the 3 from the 35 and the 5 from the 35 and I knew that 30 plus 20 is 50. 55 and added another 5 is 60.

T: Okay. So he said he started with 35 and added 20 to it which gave him 55, plus another 5 would have been 60. Katie, did you do it the same way?

Katie: Yeah, because I knew that 3 plus 2 is 5, and 5 plus 5 is 10, so that would have to be 60.

T: Was there a different strategy for solving this problem? Maggie, what did you do? Why don’t you go up and explain. Okay Maggie can you explain to us how you got your answer.

Mag.: 35 plus blank equals 60. 35, 45, 55, so that is 10 and 10 is 20, plus 5 is 25.

T: Okay. [to class] Maggie did it sort of the same way. She added on to 35, but she thought of adding tens.

March, Year 2

T: I want everyone to be good listeners. Brad and Keith real loud okay?

Brad: First we did 50 cents and listed it 9 times [Keith wrote a column of nine 50s].

T: Why did you list 50 cents first? Brad: Because there was 50 cents in the

$3.50, and we didn’t want to do the $3.50.

T: You added the change first? Brad: These two made a dollar [Keith

showing adding the 50 cents in pairs]. Then we had 4 dollars plus 50 cents. Then we picked the 3 dollars and added that 9 times.

T: Okay. So first you added the change, the 50 cents first, and now you are adding the 3 dollars 9 times?

Brad: Yes. T: Audience does everyone

understand this? Class: Yes. Adam: Nope. T: Adam, you don’t understand? Adam: No. T: Keith, why don’t you explain to

Adam and probably some other people here what it is you are doing?

Keith: Well, we kind of got confused with the dollars over there, so we just put the dollars and cents separate.

Adam: But why did you put the change and the dollars separate??

3560

. . . . Keith: We had four 6 dollars and then we

added the 3 dollars to one of the 6 dollars and got 9 dollars [on the board he had written a column of nine 3s. He showed putting the first eight 3s in pairs and indicated that each pair made 6 dollars. There was one 3 left. He added it to one of the 6s to make 9. Then he had 6 + 6 + 6 + 9]. And so we added that up and we got 27 dollars. So we put $4.50 plus $27.00.

T: So $4.50 that you added from the change, plus the money from the dollar bills…. Go ahead. Let’s see how you do that.

Brad: $31.50. March, Year 2

Fig. 2. Second year’s practice.

J. Warfield et al. / Teaching and Teacher Education 21 (2005) 439–456 447

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form of practice consistent with the reform agendawhile others maintained the same type of teaching.(It was interesting to us that one of the twoteachers who used commercial textbooks in thesecond year of the project was in each group.) Ascan be seen in the example, teacher A continued tomaintain the original strategy-reporting environ-ment with characteristics from conventional in-struction. The children’s explanations were neithercomplete nor clear, but the teacher did notquestion them in ways that enabled them tofurther explain. Instead she stated what sheinferred the child had done, again perhaps forthe benefit of the other children.

In other lessons taught by teachers in this group,the teachers curtailed their children’s thinking. Insome instances, the teachers introduced problemsin such a way as to indicate how the childrenshould solve the problem. For example, in oneclass the teacher posed a problem about snowmenwith three body parts and a hat. Each part was tobe colored a different color. The children were todetermine how many different snowmen therecould be if only four colors were used. The teacherprovided large sheets of paper and coloredmarkers and stated that a good way to solve theproblem would be to draw pictures. She was thensurprised when every child in the class drewpictures of snowmen to solve the problem. Inanother instance, a child who was sharing astrategy made an error and the teacher immedi-ately stepped in and corrected the mistake. Ratherthan let the child continue her explanation andperhaps discover her error or ask follow upquestions for clarification, the teacher instead tookover the explanation depriving the child of anopportunity to learn by recognizing and correctingher own error.

Teachers in the second group developed a morecomplex practice categorized as an inquiry class-room culture (Wood et al., 2001). This classcontext includes strategy-reporting characteristicsbut is further distinguished by the asking ofquestions for clarification by the listeners and thegiving of reasons by the explainer. Therefore,classrooms classified as inquiry are those in whichchildren offer different solution methods, as in thestrategy reporting classes, but they provide reasons

for their thinking in order to make sense to others.Student listeners and teachers in these classes mayask questions for further clarification and under-standing.

For example, teacher B’s class, which lookedmuch like that of teacher A in the fall of the firstyear of the project, looked different in March ofthe second year. In the illustrative example shownin Fig. 2, Brad and Keith were called on to sharetheir strategy. Brad talked while Keith wrote onthe board. The boys explained how they addednine 5 s to get the cost of the shorts. The discussioncontinued with other children asking questionsabout the strategy and Brad and Keith explainingtheir thinking. Teacher B added some commentsto their explanations and checked with the othersin the class, asking if they understood. At all timesBrad and Keith were responsible for explainingtheir strategy, clarifying their thinking and provid-ing reasons so that the others could understand. Inaddition, the other children tried to make sense ofthe explanations Brad and Keith gave, and theyasked questions to help them understand.

4.2. Teachers’ reflection on their practice

The teaching dilemmas and the ways in whichthe teachers intended to address those dilemmasdescribed in their PPA indicate how they inter-preted our suggestions about the structure ofmathematics lessons. They also provide insightinto their learning about their role in fosteringmathematical learning in their classrooms.

4.2.1. Group 1

The teachers in this group wrote PPAs that dealtprimarily with issues of classroom management.They interpreted our suggestions of having chil-dren solve problems in pairs and holding class-room discussions as procedures to be carried outin their classes and struggled throughout theproject with the implementation of those proce-dures. The teaching dilemmas that they presentedthroughout the two years were often related to themechanics of implementation, and the techniquesthey used to improve pair work and discussioninvolved adopting procedures to compel the

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children to work together and listen to theexplanations of others.

Many of the PPAs written by this group ofteachers dealt with ways of encouraging pairs ofchildren to work collaboratively to solve problems.One teacher wrote that she would give pairs ofstudents awards for working well together andlistening to one another. Another had her studentsadopt specific procedures during pair work. Eachpair would be given one worksheet that was to beplaced on the desk between them; they shouldshare their ideas aloud; they should listen to oneanother’s ideas; and they should encourage oneanother’s involvement in the problem-solvingprocess. Still another encouraged the children ina pair to take turns writing problem solutionsusing different colored pencils so that she couldtell they had done so.

Other PPAs dealt with the whole-class discus-sions. The teachers in this group saw the discus-sions as opportunities for children to describehow they solved problems and for those whowere listening to learn and adopt new strategies.They wrote about holding children accountableduring discussions. They did this by havingchildren show ‘‘thumbs up’’ or ‘‘thumbs down’’to indicate agreement or disagreement with ananswer, write the strategies of other children ontheir paper, and repeat the strategy that had beendescribed.

These PPAs indicate that the teachers in thisgroup saw their role as being to manage theinteractions in the class. These teachers frequentlydid not think they had solved the dilemmas in theirPPAs by the end of the month. They were to writenew PPAs each month, but instead they ofteneither repeated the PPA from the previous monthor referred back to earlier ones, writing statementssuch as: ‘‘My plan was to continue working on myprevious goal of letting class discussion runsmoothly,’’ or ‘‘I wish to continue to implementideas from earlier plans.’’ There was no mention ofinvolving children in questioning and challengingone another.

4.2.2. Group 2

The PPA written by the second group ofteachers were quite different. Although these

teachers also wrote about children’s working inpairs and the class discussion, they did not focuson management issues; rather, their PPAs centeredaround children’s understanding of mathematics.Also, their PPAs changed throughout the project.These teachers seemed to have visions of the kindsof mathematical interactions that were possibleamong children and to reflect deeply about theirrole in creating those interactions.

One teacher wrote about class discussions in herPPAs. The reasons she gave for children to listento others during discussions evolved over the firstyear of the project. In November, the childrenwere to listen, because they would be exposed tomany different strategies for solving problems.The following January, they were to listen,‘‘because they may be called on to share theirviews or answers to the problem being discussed.’’In February, they were to ‘‘think about thestrategies being discussed—share what they arethinking, listening and trying to understand whatthe others are trying to say and telling me if theydo not understand.’’ And, in April, they were to‘‘use active listening—share strategies as well asrespond or comment on anything they question orjust want to add.’’

The teachers in the group also included in theirPPAs expectations for themselves that weredifferent from those of the teachers in GroupOne. These teachers saw their role in the discus-sion more broadly than simply establishing rou-tines. The teacher described above wrote in Aprilabout what she expected children to do during thediscussion, but added: ‘‘I also expect to meet myPPA by (me) standing back—not being in thefront—and letting the kids take control. I wantthem to explain their thinking.’’ A month later, sheadded: ‘‘I want to be able to stand back and let thechildren take control. I want them to talk abouttheir thinking without me prompting them somuch.’’ Another of the teachers wrote the follow-ing PPA: ‘‘Don’t reason for my students. Let themthink through the problems and clarify unclearpoints of discussion.’’ These teachers learned thatchildren are capable of solving problems and ofexplaining and justifying their own thinking, andthey learned that their actions played a role inallowing that to happen.

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4.3. Teachers’ beliefs

As was evident in the above discussion the twogroups of teachers reflected differently on theirpractice and learned to teach differently. Weexamined the teachers’ beliefs as expressed in theinterviews and in response to specific questions inthe working sessions to determine if they wouldgive us insight into why the learning was different.Our initial look at the teachers’ beliefs led us toconclude that their beliefs were similar and thatthose beliefs were aligned with those underlyingthe current reform movement in mathematicseducation. In order to understand the teachers’beliefs and their impact on the teachers’ learning,we began by considering the teachers’ professedbeliefs about mathematics, learning mathematics,and teaching mathematics.

4.3.1. Beliefs about mathematics

The teachers’ beliefs about mathematics weresimilar. In the first interview conducted at thebeginning of the second year of the project, each ofthe teachers alleged that mathematics is ‘‘problem-solving;’’ however, the teachers’ views of mathe-matics were utilitarian. Six of the seven said thatmathematics was created by humans to fulfillspecific needs. As an example, one teachercommented, ‘‘Way, way back, somehow it had todo with living situationsymaybe as a way ofmoneyybartering and trading. It probably camefrom that.’’ (Interview 1). This utilitarian view ofmathematics was also apparent in their responsesto a question about why we teach mathematics.Each gave a response similar to: ‘‘Well, it’s animportant life skill. I mean you use math ineveryday life. Everything we do just about involvesmathy .’’ (Interview 1).

There was little indication that the teachers sawmathematical proof or reasoning as important indoing mathematics. Rather, they believed thatmathematical truth was verified by using concreteexamples. This was apparent when they talkedabout what they would do if a child made an errorin a subtraction problem. Four of the seven saidthey would ask the child to redo the problem usingbase-ten blocks; through using the blocks, thechild would ‘‘see’’ what he had done wrong. Two

others indicated they would demonstrate the‘‘correct way’’ to do the subtraction. Only onesaid she would allow the children to discuss theirstrategies for solving the problem and resolve theresultant conflict in answers among themselves.

4.3.2. Beliefs about learning mathematics

During a working session (October, Year 2), theteachers were asked to write in their journals abouthow they thought children learn mathematics.Their responses again were similar, expressing abelief that children learn mathematics throughexperience. One teacher wrote: ‘‘I believe childrenlearn mathematics through experiences with num-bers. They ‘learn mathematics’ by using mathe-matics and then make connections to moredifficult problems and continue to learn more.’’Another said: ‘‘Students learn mathematics bybeing involved in meaningful, real-life mathactivities.’’ The teachers also said that a childwas learning if the child was moving from a needfor concrete objects to solve problems to beingable to solve problems abstractly. ‘‘They first useconcrete ways to help them learn ideas and,hopefully, move on to more abstract ways ofsolving problems.’’

4.3.3. Beliefs about teaching mathematics

At the same working session (October, Year 2)the teachers were asked to write about how theyshould teach mathematics. The statements madeduring interviews were, for the most part, similarto what they wrote in the journal responses. Theyagain expressed similar views. According to one:‘‘Teaching is providing an environment of newsituations and new ideas that become meaningfulto the students.’’ Another said: ‘‘Teaching is manythings. It is about creating an atmosphere forchildren to learn in. It is about providing themwith materials and ideas to get them involved andthinking. Teaching is about supporting studentsand their ideas and values.’’ The teachers said theirrole was to encourage children to share theirthinking, to structure classes so that children coulddiscuss their ideas, and to guide children whenthey were stumped by asking questions to furthertheir thinking.

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The teachers’ written comments, both aboutlearning mathematics and about teaching mathe-matics, were couched in the ‘‘rhetoric aboutreform’’ (Cooney & Shealy, 1997, p. 97). However,other data, including their journal entries abouttheir videotapes and the discussions on e-mail andduring working sessions, indicated that, while theteachers might indeed believe what they wrote,they also held views that were not consistent withthis rhetoric. It was through looking at thecomments that the teachers made about thechildren in their classes and about the obstaclesthat they faced in teaching that we began tounderstand the differences in the teachers’ beliefsthat may have influenced their learning andultimately their change in practice. The teacherscould be separated into two groups based on thesecomments. These were the same groups describedearlier.

4.3.4. Beliefs about children

4.3.4.1. Group 1. This group of teachers oftenspoke about the ‘‘high kids’’ and the ‘‘low kids’’ intheir classes. Although they had written aboutchildren learning by being involved in meaningfulexperiences, building on those experiences, andmoving from reliance on concrete objects to moreabstract mathematical thinking, it became appar-ent that they believed that learning happened thatway only for the ‘‘high kids.’’ One expressed thisbelief during a working session:

I even think there are some kids that can’tthinky . I think that they are all trying to makesense, but I think there are some of them thatthere is something missing and they have a verydifficult time making sense of stuff, in that thereisn’t enough time in the day and in their lifetimeto let them finally make sense of some of it.

Another described one of her ‘‘low kids’’:

I have a little girl who is really low, and shewould just do better to have the standardalgorithm than to have to process learning anew way every timey . She is the kind ofstudent, if you give her a story problem, she justneeds to memorize right nowy . She just needsto be told what to do.

For these teachers, if a child did not learn, it waseither because the child was one of the ‘‘low kids’’incapable of understanding mathematics or be-cause the child did not want to learn. One of themdescribed a child in his class as a ‘‘passive mathstudent’’ who often relied on an algorithm to getan answer without thinking about whether theanswer made sense. Another wrote about hersecond-grade class in an e-mail message:

I have a wide range of abilities. A few high, fewlow, some clueless. I am very frustrated thisyear. The kids are bad kids. They just do notlisten. Their attitude seems to be ‘‘I don’t care ifI’m in school, but I don’t want to do anything.’’

We suggested that the teachers incorporate pairwork and class discussions into their teaching,because we believed that children learn by workingtogether and communicating about mathematics.This group of teachers did not understand why wesuggested pair work and class discussions andinterpreted our encouraging communicationamong children quite differently than we did.These teachers believed that the ‘‘high kids,’’ whocould come up with strategies for solving pro-blems, could tell the ‘‘low kids’’ their strategies,and thus the ‘‘low kids’’ would learn. One of themexpressed this view during a working session:

I think that is why it is so important to listen toother kids in the discussion. It helps them torefine their thinking or move or clarify. Some-times I think students do a better job in refiningother students’ thinking than we do.

Another of the teachers expected children toadopt the strategies used by other children and wasconcerned when that didn’t happen. She wrote inher journal: ‘‘y when they are in group discus-sions they don’t learn new strategies from othersy .Most of the students like to give differentstrategies, but it seems like many of the studentsdo not use new strategies given.’’

4.3.4.2. Group 2. The three teachers in Group 2rarely spoke about children as ‘‘high kids’’ or ‘‘lowkids.’’ They did, however, often speak about theiramazement at their children’s mathematical think-ing. For each of these teachers, there was a

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particular event or series of events that triggeredher fascination with the thinking of her children.

During the first year of the project, one of theteachers began including a problem to be solvedmentally in her morning calendar routine. Shetalked at the working sessions about the wonderfulmathematical discussions that resulted and thewide variety of strategies the children used to solvethe problems. She contrasted this with the discus-sions during her regular mathematics lesson; thenthe children were polite and listened to oneanother, but there tended to be few differentstrategies and no true discussion. The differencesin the discussions in calendar time and in hermathematics lessons, made clear to this teacherwhat was possible and the depth of children’smathematical thinking. In her journal she indi-cated her amazement at the strategies her childrenused to solve problems, strategies that she wouldnot have thought of.

The second teacher in this group becameparticularly fascinated by her children’s thinkingwhen they solved non-routine problems. In thesecond year of the project, this teacher was at aChristian school and using a commercial textbook.She continued to use non-routine problems fromthe project material, because:

We always had good discussions with them. Itwas very puzzling for the kids at first. It reallycreated a problem for them to solve, so theirpartner work was always a lot bettery . Iwould say [a good lesson was] when I would domy problem-solving problems and then discussthem, because there’s so many possible solu-tions to solving them. I would always have areally good discussion.

The third teacher taught first grade. Shefrequently shared her amazement about herchildren’s problem solving strategies. In the firstinterview she said:

My little first graders blew me away last year.They just picked up on so much that I neverthought would be possible y I would have anidea of where I would want to go, and it wouldgo in a completely different direction, but in a

good way. I just had to be really flexible withthem and take what they said and go with it.

She gave a specific example of this. During alesson on fractions, she was using fraction bars onthe overhead projector to show the children what 1

3

and 12

looked like, and

One of my students just said, ‘hey, when youadd 2

3and 1

3; you get 3

3and that makes a wholey .’

They just blew me away on that oney.and theywere amazed at any time there was a numberover itself, it was a whole. They were reallyinterested in that.

4.3.5. Beliefs about obstacles

4.3.5.1. Group 1. Cooney and his colleagues(Cooney & Shealy, 1997; Cooney et al., 1998)wrote about the importance of teachers beingaware of the complexity of teaching and seeingthemselves as authorities able to make pedagogicaldecisions in light of their own beliefs. The teachersin Group 1 did not see themselves as suchauthorities. These teachers realized that they werehaving difficulties teaching in the ways advocatedin the project, and they tried to justify theirdifficulties, partly by claiming that their childrencould not all learn in the ways suggested in theproject. However, these teachers also perceivedother obstacles to being able to teach as weadvocated. One wrote in his journal about a lessonin which his expectations were not met. ‘‘yI feelthe problem lies before in earlier grades, parentalinvolvement, and many students just being lazyand not trying to visualize what the words ask tobe done.’’ Another teacher wrote:

yI also feel we (teachers) receive many mixedsignals. On one hand we know what we think isbest for the child, but on the other hand we areforced to face reality. We have society telling usthat children must perform well on tests, beready for the work force, etc. Our principal tellsus she understands why we want to teachproblem solving math, but continually sends usmemos on testing and asks how we are going tobetter prepare our students, etcy . I try to giveall students an opportunity to make sense ofthings, but also fill in when I think it is essential.

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4.3.5.2. Group 2. The teachers all taught in thesame school district, so one would expect that theywould face the same obstacles in their teaching.For example, all of the children in the district tookstandardized tests required by the state. There wasimmense pressure for children to do well on thesetests. However, the teachers in Group 2 did not seethese tests as an obstacle to their teaching in waysthey thought best for their students. Only one ofthe teachers in Group 2 mentioned any obstaclesto teaching as advocated in the project. Thatteacher taught a multi-age class consisting of first,second, and third grade children. The teacherswho did this had decided to separate the childrenby grade level for mathematics lessons. Because ofthis, the teachers had to adhere to a strict timelimit for mathematics. This was considered by theteacher to be a problem, because it meant shesometimes had to stop meaningful discussions inorder to send children back to their own class-rooms. Otherwise, no obstacles similar to thosebrought up by the teachers in Group 1 werementioned by the teachers in Group 2.

4.4. Summary

We investigated the teachers’ learning to teachaccording to the reform recommendations, theirlearning to reflect on their teaching in ways thatwould enable them to continue to learn after theproject ended, and whether their beliefs supportedor constrained their learning. All of the teachersadopted practices advocated in the reform. How-ever, they responded to these opportunities forlearning that we created in different ways.

A group of four teachers did not learn to teachin ways that encouraged children to becomeautonomous learners. They often did not under-stand their children’s thinking and did notencourage the children to clearly explain andjustify their reasoning. These teachers also fre-quently interfered with their children’s thinking.They based instructional decisions on the expecta-tions of external voices rather than on theirchildren’s thinking. They did not reflect deeplyabout either their student’s mathematics or abouttheir own teaching (Warfield, 2003). Instead their

thinking about teaching focused on classroommanagement and procedures.

The remaining three teachers did begin to learnto teach in ways that enabled their children tobecome autonomous learners. They allowed chil-dren to solve problems in their own ways andexpected them to both explain and justify theirreasoning and to listen to and question thereasoning of other students. They also learned toreflect about their children’s mathematics andabout their own roles in developing children’sthinking (Wood et al., 2001).

5. Discussion

These findings offer explanations as to howteachers’ beliefs, learning, and practice might belinked and why the learning of teachers in the sameteacher development project differs. This explana-tion hinges on teachers’ beliefs about the auton-omy or authority of individuals.

5.1. Beliefs about autonomy

The teachers in Group 1 believed that indivi-duals do not have the autonomy to learn or tomake decisions for themselves. This belief appliedboth to the children in their classes and tothemselves as teachers. Previous research discussedthe importance of recognizing that children canbecome autonomous learners of mathematics(Kamii, 1982) and of seeing oneself as havingautonomy to make appropriate instructionaldecisions (Cooney & Shealy, 1997; Cooney et al.,1998). We found that these beliefs were linked inthe teachers we studied. The ramifications are far-reaching.

5.1.1. Children as autonomous

If teachers believe that there are ‘‘high kids’’ and‘‘low kids,’’ some children who can learn bymaking sense of mathematics for themselves andother children who are not able to learn that waybut who need to be shown how to solve problems,they will interpret the tenets underlying the reformmovement in different ways than those who do nothold such beliefs. When these teachers hear that

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they should not show children how to solveproblems, they may accept the admonition with-out understanding why it is given. Then when theyhear, as Hiebert et al. (1997) say, that reflectionand communication are necessary for learningmathematics with understanding, they are likely toconclude the reason for encouraging communica-tion is that it allows the ‘‘high kids’’ to show the‘‘low kids’’ how to do the problems, since they asthe teacher are not supposed to do so. It is notsurprising that teachers who interpret the role ofcommunication in this way struggle with theimplementation of reform recommendations.

In addition to helping children learn mathe-matics, communication allows teachers to gaininsight into their children’s understanding, insightthat can be used to plan subsequent instruction.However, teachers who do not believe thatchildren can be autonomous learners are not likelyto see a need for them to understand children’sstrategies or mathematical thinking. For suchteachers, the sole purpose of talking aboutmathematics is for children to see new problem-solving strategies that they can use in the future.

As was pointed out earlier, researchers inmathematics education have found that, whenteachers reflect on and communicate about themathematical thinking of their students, learningcan occur (Cobb et al., 1990; Wood et al., 1991;Franke et al., 1998, 2001). If teachers do notbelieve there is a reason for them to learn in detailabout their children’s thinking, if they do not thinkthat that thinking can be useful to them in makinginstructional decisions, then there is no reason forthem to learn to reflect on that thinking. Inaddition, if teachers believe that many childrenlearn only by being shown procedures, there is noneed for teachers to examine their teaching orreflect on instructional decisions. This may explainwhy some teachers do not learn to reflect on eithertheir children’s thinking or their own teaching anddo not learn to implement reform recommenda-tions.

5.1.2. Teachers as autonomous

When teachers do not believe that they have theauthority to make decisions about their teachingthey become frustrated and may eventually leave

teaching (Cooney & Shealy, 1997; Cooney et al.,1998). When these teachers encounter problems intheir teaching, they are unable, because of theirown perceived lack of autonomy, to find solutionsfor their problems. They do not see that it wouldbe useful to reflect on their teaching and theirchildren’s learning in order to find solutions.Instead, they blame others for the problems. Theymay blame their students for being bad or lazy,their students’ parents for not seeing that home-work is completed, the students’ previous teachersfor not adequately preparing them, the students’future teachers for expecting the children to knowalgorithms, the principal for emphasizing testscores, and even the tests themselves.

5.2. Implications for professional development

For the teachers we worked with, beliefs aboutindividual autonomy either supported or con-strained their learning. Although we cannotgeneralize from a sample of seven teachers, itmay be important for mathematics teacher educa-tors to attend carefully to these beliefs and tochallenge teachers who do not believe that theyand their children are autonomous individualscapable of learning and making decisions. If this isto be done, it may entail working closely andintensely with individual teachers to help thembecome aware of the creative thinking of all of thechildren in their classes, not just those they havecategorized as ‘‘high kids.’’ Cobb et al. (1990)found that the teacher they worked with changedher beliefs and practice only when she realized thather previous practice was problematic, only whenthey confronted her with evidence that herstudents had not learned. Similar confrontationrelated to children’s thinking and learning mighthelp alter teachers’ beliefs about children’s auton-omy and about the best ways of teaching. A biggerchallenge may be overcoming teachers’ beliefsabout their own autonomy.

5.3. Suggestions for further research

Further research is needed on teachers’ percep-tions of individual autonomy as discussed in thispaper. In particular, we need to know whether

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these beliefs are related to teacher change in otherteachers. Further, it has become apparent that notall teachers respond to professional developmentin the same way. Research that helps us learnabout why that is, how to identify differencesamong teachers, and how to adapt professionaldevelopment to reach different teachers wouldcontribute greatly to the field of mathematicsteacher education.

Acknowledgements

The research reported in this paper was sup-ported by a Grant from the National ScienceFoundation (RED 925-4939). The opinions ex-pressed in this paper are those of the authors.

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