What Teachers Take From Professional Development

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JEFF D. FARMER, HELEN GERRETSON and MARSHALL LASSAK

WHAT TEACHERS TAKE FROM PROFESSIONAL DEVELOPMENT:CASES AND IMPLICATIONS

ABSTRACT. In this article, we report on an 18-month long mathematics professionaldevelopment project with elementary school teachers. Using a model we developed, threeparticipant case studies were analyzed with respect to not only the professional develop-ment milieu, but also how these teachers interacted with the professional developmentexperience. In particular we found that having teachers reect on new, authentic reform-oriented mathematics learning experiences leads some teachers to take an inquiry stanceconcerning their own teaching, resulting in self-sustaining changes in their mathematicsinstructional practices. This implies that professional development for elementary mathe-

matics teachers should include challenging mathematics learning experiences completewith opportunities to reect on personal and professional implications.

INTRODUCTION

Entering the third decade after the publication of An Agenda for Action(National Council of Teachers of Mathematics [NCTM], 1980) and thenational conversation that it spawned, visions of reform have been articu-lated by many organizations (American Association for the Advancementof Science [AAAS], 1993; NCTM, 1991; NCTM, 2000). With the recentpublication of Principles and Standards for School Mathematics {PSSM}(NCTM, 2000), the question of how to support practicing teachers inimplementing the reforms it envisions looms ever larger. One of the twocore premises from the Glenn Report (US Dept. of Education, 2000)is that better teaching is the lever for change and effective professionaldevelopment is the indispensable foundation for high-quality teaching.

There are many kinds of mathematics professional developmentprojects possible (Loucks-Horsley et al., 1998; Sparks & Loucks-Horsley,1989). They vary in scale, purpose, audience, length of intervention,content, and structure. The question of which are most effective inimproving instruction is not trivial. Ball (1995) encourages mathematics

professional developers (and mathematics education researchers) to takean inquiry stance toward this question, experimenting to discover what canwork. Moreover, there is the question of what it means for professionaldevelopment to work: What do teachers actually take from it?

Journal of Mathematics Teacher Education 6: 331360, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.


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332 JEFF D. FARMER ET AL.

Teachers may participate in professional development to gather specicactivities for classroom use or in order to learn how to implement particularcurricula, improve their instructional practices, obtain college credit, and

for other reasons. It may not be that teachers embark upon a professionaldevelopment activity in order to change their attitudes or beliefs, althoughthis is a common hoped-for outcome on the part of providers (Shifter& Simon, 1992). If we take Balls inquiry stance, we must uncover therelationships among professional development design, individual teachercharacteristics and actual outcomes.

To deal with these issues, we describe some of the general complexi-ties of designing professional development, and relate how our designemerged from negotiations in our design team. To discuss the resultingresearch project, we rst describe our theoretical perspective and the emer-gence of our own model and research questions from our interactions withteachers and early formative feedback. Second, we present descriptions of our methodology and three specic case studies that we chose to examinein greater depth, using the model. We conclude with implications for thedesign and implementation of mathematics professional development.

TENSIONS IN MATHEMATICS PROFESSIONALDEVELOPMENT

There are many things that can be meant by mathematics education reform.In addition, many reform documents articulate a rather coherent vision,but fall far short of explicating clear paths of implementation (Ball, 1995).The importance of problem-solving, attending to students thinking andencouraging sense-making were obvious to the authors (as mathematicseducation researchers), while the daily choices of problems, tasks, ques-tions, and processes are constrained by multitudinous factors which differamong grade levels, schools and individual teachers. This tension betweenthe vision of reform and the practicalities of teaching was the rst tensionconfronted.

Another important tension revolves around ownership ; namely, thatfor teachers actually to implement changes in instruction, they must beinvolved in creating and redesigning it. Yet, the very essence of changeis that something novel must happen; without the clear articulation of adifferent vision, what is designed is destined to resemble closely what has

come before. This conundrum is difcult to address and often overlooked(Ball, 1995).

There is also a conict between time and content. We must recognizethat many current elementary teachers mathematical understanding is far


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WHAT TEACHERS TAKE FROM PROFESSIONAL DEVELOPMENT 333

from ideal (Ma, 1999). Certainly there is not enough time in a one-week workshop to support teachers in the learning of mathematics to the depththat is considered normative in the PSSM (NCTM, 2000). Moreover, a

single workshop, without periods of gestation and sustained support, doesnot afford sufcient time for teachers to develop a deep understanding of the mathematics they teach; such understandings develop, if at all, overlonger periods (Ma, 1999).

An additional dimension to the time vs. content dilemma, madeapparent by PSSM (NCTM, 2000), concerns teachers knowledge of cogni-tion and research on how children learn mathematics. Given the numberand kinds of topics in most elementary curricula, familiarizing teacherswith even a representative portion of this knowledge base seems daunting.Yet we know, from experience with such projects as Cognitively GuidedInstruction [CGI] (Carpenter, Fennema & Franke, 1996) that such knowl-edge can be protably and effectively used by teachers to improve theirinstruction.

ASSUMPTIONS AND PERSPECTIVES

Given these tensions and complexities, designing effective mathematicsprofessional development is no simple task. Among the designers of ourprogram (the rst two authors of the article, a master elementary mathe-matics teacher and two district curriculum specialists), two fundamentalareas of agreement emerged which guided us.

First, while a professional development project provides raw materialfor change, teachers themselves ultimately determine what the impact willbe, and they bring a wide variety of experiences and needs to any project.We viewed mathematics content and effective reform-oriented pedago-gical practices as important needs, while teachers might view themselvesas having other needs (professional support, specic classroom activitiesthey could use, methods for teaching particular skills, etc.). Our designintended to take into account teachers expressed needs, and to supportchanges they might decide to make in their instructional practices, ratherthan prescribing our own paradigm.

Second, teachers mathematics learning must have a central positionin the project; such learning can serve as an entry point for addressing

other goals. For example, by modeling constructivist pedagogies in ourmathematical activities, we entered into discussions of their usefulnessin elementary mathematics instruction. Furthermore, by using activitiesthat were adaptable to the elementary level and sufciently challenging to


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the teachers, we could address the desire of some teachers to add to theirrepertoire while still meeting a need to learn more mathematics.

We selected mathematical topics for their maximum relevance to our

teachers perceptions of their current needs, given our states mandatoryhigh-stakes testing environment. We also were forced to cull ruthlessly thepedagogical topics and focus on nurturing oral and written communication,understanding constructivist principles and using mathematical problem-solving, with some attention to a few additional principles from the PSSM (NCTM, 2000), such as equity and assessment.

Next, and perhaps more importantly, we realized that the only way tohave sustained impact would be to address explicitly teachers fundamentaldispositions and beliefs about the teaching and learning of mathematics.We focused on supporting teachers to become life-long mathematicslearners and inquiring, reective practitioners, given the projects timeframe. This development in our thinking parallels that of Rhine (1998):

I propose that our human, bounded rationality dictates that the value of educationalresearch to the teaching community is not the acquiring of research-based knowledgeof student understanding, but the process of teachers engaging with that knowledge andconsidering implications for their instruction. (p. 27)

Instead of focusing mostly on research-based knowledge of learning,we decided to engage our teachers in authentic mathematics learningexperiences. We use the word authentic here to mean being relevantto teachers professional needs and intellectually challenging, giventheir current mathematical understanding. By focusing the workshops onsupporting teachers both in learning mathematics and in reecting on how

they learned it, we had the potential of inuencing teachers (a) mathema-tical knowledge, (b) view of mathematics learning and teaching, (c) atti-tudes toward mathematics and mathematics learning, and (d) beliefsabout the nature of mathematics, mathematics learning and mathematicsteaching. These are important components of instructional change (Ernest,1989).

THE EMES PROJECT DESIGN

The objectives of the resulting Enhancing Mathematics in the ElementarySchool (EMES) project were to:

1. Increase participants knowledge of mathematical content relevant toelementary instruction in ways that model appropriate standards-basedinstructional practices.


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2. Increase participants familiarity with national and state teaching andlearning standards for mathematics, the state mathematics assess-ment plan and increase their ability to implement standards-based

instruction and assessment.3. Increase participants awareness on issues of diversity and equityrelated to mathematics education and enable participants to changetheir instructional practices to be more supportive of all studentsmathematics learning.

4. Enhance participants skills in problem-solving, critical thinking andmathematical communication in ways that can be directly applied toinstructional practices.

5. Support participants in integrating and applying knowledge gainedfrom project activities into their curriculum and instructional practices.

6. Support participants in professional collaboration and networking.

In general, we agreed that the teachers needed positive, reform-orientedmathematics learning experiences (such as solving interesting problems insmall groups, discussing ambiguities and hidden assumptions in problems,considering multiple representations of mathematical ideas, engaging inmathematical writing, etc.), discussion of pedagogy that would challengetraditional views of mathematics teaching and learning, and opportunitiesfor ongoing support to put new ideas into classroom practice. All of thiswas presented in two one-week summer institutes with regular Saturdaysessions held during the academic year. The Probability and StatisticsInstitute was offered the rst and second summers, while the Arithmetic,Algebra, and Geometry Institute was also offered the second summer.

Although we were clear about what mathematical content to examineand how to facilitate learning of mathematics and reection on pedagogy,we were less clear about how teachers implementation of reform prin-ciples would (or should) actually look. This was compounded by thediversity of assignments: among the approximately eighty participantswere teachers of kindergarten through grade six. We decided to encourageand support implementation, but to leave its actual direction and focusto the teachers. To assure their work was well-directed and focused, wemonitored their choices and engaged in some suggestion and negotia-tion. Our hope was that the changes they chose to make would be moreauthentic, effective and permanent than anything we could create andimpose. In short, we applied the principles of constructivist epistemology

to think about what and how our participants would learn (Loucks-Horsleyet al., 1998).

Virtually no one involved in the project (neither facilitators nor parti-cipants) experienced the kind of elementary school mathematics instruc-


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tion envisioned in reform documents: rich in problem-solving, orientedtowards conceptual understanding and sense-making, supportive of allstudents mathematical thinking and focused on students activity and

cognition rather than on textbook material. Thus, we designed everyproject session to include each of the following:

Mathematical communication Discussion of mathematical concepts and principles Analysis of the learning process Reection on pedagogical principles (particularly those which chal-

lenge traditional views of teaching and learning) Discussion of implications for classroom instruction and/or planning/

debrieng implementation activities

We describe one illustrative activity that took place in the proba-bility and statistics institute both summers. It began with the entire group

reviewing the denitions of the three measures of central tendency ( mean ,median and mode ) and range . This was accomplished by questioning parti-cipants and discussing the denitions they provided; the group settledfairly quickly on the usual denitions.

Participants, working in small groups, were then given six problems.Each described attributes of a data set; participants were to construct adata set with the given properties. For example: The mean of the dataset is 8, the median 10, the range 16; construct a possible data set.Several problems were easy but some required more thought and onewas impossible (or barely possible, depending on rounding conventions).Groups that became stuck while working were encouraged to clarify theirthinking and explain their difculties. After all the groups solved at leasta majority of the problems, solutions and problem solving methods werediscussed in the whole group.

Each group shared their data set for each problem. The participantslooked for similarities in these data sets, which provided clues to the rela-tionships among the measures of central tendency. When the impossibleproblem was discussed, other issues arose: What kinds of numbers arepermissible in a data set, can rounding be used, exactly how do thestated constraints combine to make the problem impossible, etc. Discus-sions often led into interesting mathematical byways. Participants reportedgaining a deeper understanding of the measures of central tendency.

This activity models the kind of question that Sullivan and Clarke

(1991) call a good question in their book, Communication in theclassroom: The importance of good questioning . This book was providedto participants in the project. Some properties of such questions are thatthey:


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have more than one correct mathematical answer; require more than recall of a fact or reproduction of a skill; are designed so that all students can make a start;

assist students to learn in the process of solving the problem; support teachers in learning about students understanding of mathe-

matics from observing/reading solutions.

Answering these questions provides opportunities for teachers to exper-ience rsthand deeper levels of conceptual knowledge construction, andshows them how small groups can work together to solve such complextasks.

As part of this activity, we discussed explicitly with participants howwe used Sullivan and Clarkes (1991) procedure to create this task. Inanother session we asked teachers to modify traditional recall tasks fromtheir own curriculum to create similar good questions. The mathematical

and pedagogical processing of this activity takes over two hours. At theend of an activity (but before the full discussion of pedagogy), teachersmay write in their mathematics journal about the content learned. After thepedagogical discussion, teachers may be asked to reect either in writingor orally in pairs or groups.

An important aspect of our facilitation involves the stance we takein leading the discussions and responding to questions. While bringingto light mathematical connections that were relevant to material beingdiscussed, we also made clear that we learned from the discussions. Wepointed out proposed solution methods that were novel to us, and tried tobe explicit about the expectation that we would all learn from the prob-lems, solutions, and discussions. The teacher-leader who co-facilitated theworkshops with us was also explicit about how she gains new insights inthe process of teaching mathematics.

In addition to small and large group discussions, participants engagedin a structured activity called a dyad (Weissglass, 1996) in which twopeople take turns talking, without interruption, comment or evaluation, forabout two minutes concerning the question at hand. Dyads were sometimesused during the discussion of mathematics problems, to help participantsfocus on prior knowledge at the beginning of an activity or to facilitatereection and planning. Each institute day included signicant time forteachers to plan implementation of their projects based on their reectionand learning. Participants wrote in mathematical journals, gave written

feedback at the end of each session and wrote expository papers describingtheir instructional innovations; those participating in one institute wereasked to email the facilitators periodically with reports on their classroomimplementation.


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THEORETICAL PERSPECTIVE AND EMERGENCE OF THERESEARCH PROJECT

The research project emerged for two reasons: a desire to know what wasbeing accomplished in EMES, and what our participants were getting fromit. We were hoping to see teachers change their disposition toward mathe-matics (viewing mathematics as being constituted by numerical and logicalrelations rather than in terms of skills), toward themselves (as mathe-matics learners) and toward mathematics teaching (viewing mathematicsteaching as supporting learning by facilitating conceptual understanding).A preliminary answer was provided by an external evaluator who usedpre- and post-tests, written comments from teachers and observations of workshop sessions to show that the project was operating consistentlywith the original proposal and that the project goals (stated earlier) were

substantively met (Shaw, 2000). The research team (authors of this paper),however, became interested in looking deeper into how individual teachersinteracted with the project, and particularly in describing and analyzing itsimpact on them.

Our theoretical perspective can be described as holistic constructivism(Noddings, 1990). We accept that knowledge is constructed individu-ally by an organism through interaction with its environment, whileacknowledging that for humans, our environment is largely social. Weexplicitly reject the tendency to view either the psychological or the socialperspective of mathematical learning as primary, but instead maintain thatboth have a vital role to play in a rich description of the reality that emergesin a mathematics classroom.

This position is not taken for naively pragmatic reasons, but withthe principled assumption that reality is multilayered and complex, bothidiosyncratic and socially mediated. To locate cognition solely eitherwithin the individual brain or within the social milieu is fundamentallyto give up on understanding some important aspect of it. We locateknowledge construction (meaning-making) in the interactions of an indi-vidual (complete with prior knowledge schemas, experiences and socialidentities) with her or his (primarily social) environment.

This position is reected in the design of the workshops through ouremphasis on individual reection and conceptual understanding along withcreating group norms, social interactions and a supportive professional

learning community. It is also reected in our stance as researchers; weare engaged in an inquiry process, constructing for ourselves and ourcommunity new understandings of the mathematics professional develop-ment processes. We worked, through cycles of design, implementation,


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data collection, analysis, and description, to develop our individual under-standings of what we were studying and to negotiate them with each other,with our participants and nally with the broader mathematics education

community.As the project progressed, it became clear that various participantshad rather different ways of interacting with it, and hence, seemed toexperience different effects from their participation. Some appeared to bemostly interested in obtaining specic activities for use in their classroom,or in receiving credit for their participation. Others were interested inenhancing their professional skills, and their understanding of the subjectmaterial. Still others seemed to be turned on to a different way of thinking about and doing mathematics, and eager to uncover implicationsfor their students and classrooms. We began to create a model describinghow teachers were interacting with the project, and tried to learn in greaterdepth what they were taking from it.

A REFLECTIVE MODEL OF MATHEMATICS PROFESSIONALDEVELOPMENT

In order to capture the different ways we observed teachers interacting withthe EMES project, we developed a reective model of our professionaldevelopment milieu and its relationship to the participants classrooms.This model, though designed to capture some of the specic aspectsof the EMES project, is adaptable to other settings. In particular, itallows for analysis of connections between project activities and teachersprofessional practice when professional development is designed to modelreform-oriented mathematics instruction.

A model with a similar purpose can be found in Fennema et al.(1996). That model is also an attempt to characterize teachers interactionswith a professional development project. One difference is that it focusesentirely on beliefs and actions regarding CGI, while our model attempts tocharacterize a broader collection of interactions.

A fundamental agreement in the design process was that we wouldmodel for participants, in ways that were appropriate to their learning asadult professionals, activities consistent with the PSSM (NCTM, 2000).To describe the effects of this on teachers, we developed a model of levels

of appropriation that represents elements of both the professional develop-ment milieu and the classroom milieu.. The model is displayed in Figure 1.The elements of the professional development milieu parallel those of theclassroom milieu, displayed on the right.


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Professional Development Milieu Elementary Classroom Milieu

P1: Teacher as (mathematics)

learner

C1: Student as (mathematics)

learnerP2: Project mathematics

activities and contentC2: Classroom mathematics

contentP3: Professional developer as

facilitator/instructorC3: Teacher as teacher

P4: Facilitators knowledge(of participants, theirknowledge, mathematics,mathematics learningin adults and children,facilitation skills, etc.)

C4: Teachers knowledge(of students, theirmathematical knowledge,how children learnmathematics, mathematics,teaching techniques)

P5: Professional developer asinquirer into professionaldevelopment

C5: Teacher as inquirer/scholarwithin the classroom and itsprocesses

Figure 1. Model of the interaction between the professional development milieu and theelementary classroom milieu.

The rst three entries of each milieu can be thought of as the tradi-tional part of the model. These describe the participants and their rolesin a mathematics problem-solving activity. These elements appear in thetraditional instructional triangle joining teacher, student and mathematicscontent. There are two additional elements in each milieu. One is thefacilitators knowledge of professional development content and pedagogyparalleling the teachers knowledge. The second represents the facilitatorin the inquiry role, as in Ball (1995), of learning what works in the profes-sional development milieu; the parallel element on the right is the teacheras learner in the process of teaching. These two elements are sometimesconsidered in a larger instructional triangle (Mumme et al., 2003). Weinclude them because we consider mathematics, not instruction, to be ourprimary content, although mathematical activity serves as an entry point toprofessional conversations.

Levels of Appropriation within the Model

In order to characterize how mathematics professional development inter-acts with the teachers professional selves, we identied three levels of appropriation:


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Level one: Concrete activity and content. In level one, teachers parti-cipating in a mathematics professional development activity appropriatecontent such as specic mathematical skills or concepts that they will

actually teach or specic pedagogical techniques to implement in theirclassrooms. Furthermore, they look for specic mathematical problems,tasks or games to use with their students.

Participants who appropriate at this level are focusing on the specicparallel between the mathematics activity in the seminar and mathematicsactivities in their own classrooms. They are likely to report that they didmany similar activities with their students that we did in the professionaldevelopment. The teacher may report that it worked well, their studentsliked it or difculties were experienced, but without much analysis. Inthe model above, these appropriations could be represented by parallelhorizontal lines from P1 to C1, P2 to C2 and P3 to C3.

Level one is the most basic level of appropriation. Russell et al. (1995)point out that some teachers characterize mathematics as an ad hocaccumulation of facts, denitions and computational routines (p. 18).Likewise, some teachers may characterize teaching as an accumulation of various kinds of skills and knowledge of practical routines, uninformedby general principles. Such teachers might only be able to make level oneappropriations.

In our experience, teachers often evaluate single professional develop-ment sessions based on whether they were able to appropriate material atthis level. The expectation of being able to make concrete appropriationsmay create dissonance when teachers participate in sessions not primarilydesigned for this purpose. In the EMES project, teachers were generally

happy from the start, nding that they were almost always able to appro-priate a concrete activity or some directly relevant mathematics content, asmany of our activities were easily adaptable.

Level two: Professional principles and understandings; attitudes and beliefs. In these kinds of appropriations, participants view themselves asprofessionals who are gaining additional knowledge from the session.In content, they look for and construct mathematical ideas that willallow them to integrate, connect, and explain the mathematical conceptsthat they will teach. In pedagogy, they may attempt to gain an under-standing of strategies that can be useful in mathematics instruction, suchas cooperative learning, journal writing for mathematical understanding,constructing challenging mathematical tasks. Level two appropriationsarise when participants view themselves simultaneously as learners (P1)and as teaching professionals (C3), and appropriate skills and knowledgeto improve as professionals. They may build their knowledge (C4) from


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reecting on any of the elements in P1 through P4, either individually orwith other participants.

Level two appropriations can look quite similar to those of level one

in some cases. The difference lies in whether teachers are taking indi-vidual activities or bits of content which they can use in practice, orwhether they are organizing them into general categories and developingprinciples that they can exibly use as professionals. Repeated appropria-tion of concrete elements can lead to changes in attitude or belief, or to theconstruction of general principles. These level two appropriations can beinferred when teachers create new classroom activities that are consistentwith reform principles, in addition to adapting and extending activitiesgained from a session. Teachers who appropriate mathematical under-standings at level two are interested not only in knowing the mathematicsthat they teach, but also in understanding connections among variousmathematical ideas, and in understanding more background and depth incontent, to better teach their students. They are interested in principles of mathematics learning and teaching.

Examples of level two appropriations have been described in othermodels by placing the traditional instructional triangle (joining C1, C2 andC3) as the content inside a larger instructional triangle involving teachersas participant-learners and facilitators as teacher-instructors (Mummeet al., 2003). Such a model assumes that teachers in a professionaldevelopment setting are learning mostly about teaching. Our model ismore comprehensive in that it allows description of teachers mathe-matics learning and their learning about learning (through self reectionand group discussions with peers about current mathematics learning

experiences) as well as their learning about pedagogy. Level three: Teaching as inquiry . Teachers who are constructing knowl-edge at this level have a different perspective. In addition to being ableto use and adapt concrete elements, learning mathematical ideas, andapplying general principles for mathematics teaching, these teachers alsosee themselves as learning from (or, perhaps more appropriately, in)the process of teaching. They view themselves as mathematical learnersalongside their students, acknowledging that they can never know enoughmathematics to support and teach each student perfectly as they strugglethrough a school year. They also view themselves as learners about theirstudents cognition, striving to understand how their students are thinkingand why, and how to pose interesting worthwhile tasks.

Teachers at this level appropriate the same elements that are found atlevels one and two; the main difference is in how the elements are viewed.Here they become tools of inquiry for the teacher/learner who is facilitating


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mathematical learning in the classroom. Teachers who are constructingknowledge in this way view themselves as learners all the time (not justin the professional development session), and particularly as persons who

learn from their students.This stance probably requires a signicant amount of professional andpersonal maturity. It certainly requires both the opportunity and the will-ingness to reect critically on ones own (and others) practice. Because of this, we say teachers who appropriate elements from professional develop-ment largely at level one are taking a practitioners stance toward theproject, those who also appropriate and integrate knowledge (level two)are taking a professional stance and those who, in addition, begin to takeon the role of a learner in their own teaching process (level three) are takingan inquiry stance.

In constructing our model, we wanted to capture not only the reectivenature of the project, but also the ways teachers were interacting withit. Relating the professional development milieu to the classroom milieuallows us to describe not only how teachers view themselves as learnersand professionals in the professional development setting, but also theknowledge they construct and its relationship to practice.

There is an interesting parallel in this model with one developed byRussell et al. (1995) to describe beliefs about learning and teaching mathe-matics that guide teachers in making instructional decisions. The nature of the levels in the two models is strikingly similar: they describe a develop-ment ranging from seeing mathematics as an ad hoc collection of factsand procedures (their level one) to systematic inquiry organized aroundbig mathematical ideas (their level four). The difference is that in our

model, the levels revolve around teachers professional learning about bothmathematics and mathematics teaching, encompassing a larger range of interactions.

The two milieus of our model also roughly parallel the rst twofoci of the model developed by Tzur (2001) to describe the multi-layered reective processes involved in developing as a mentor of teachereducators. The elements of his model (like ours) allow for a rich descrip-tion of the reective processes involved in learning from the practice of teaching (or facilitating professional development).

DEVELOPMENT OF THE RESEARCH QUESTIONS

Our model represented an attempt to create a tool to aid in the descriptionand analysis of what we were observing. To study the interactions between


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professional development design and teacher change, we then developedthree questions:

1. What does an inquiry stance toward mathematics teaching look like?

2. How does this stance develop?3. What was the role of the EMES project in its development?

We focused on these questions because they have signicance for theoverall understanding of what makes for successful mathematics profes-sional development.

While it is clear that there may always be a need for professionaldevelopment activities that teachers can learn from as practitioners andprofessionals, the development of an inquiry stance represents a self-sustaining level of autonomy on the part of the teacher. This is not tosay that such teachers no longer need support; however, by learning withand from their own students, they become the directors of their own

professional development.If real reforms are to be sustained, this type of learning from profes-

sional practice must become a reality (that such continual learning fromreection on teaching is possible, see again, Ma [1999]). Thus the inquirystance embodies the target for effectiveness; analyzing cases may help uscreate opportunities for more teachers to develop this stance. In addition,because of the autonomy the teachers enjoyed in implementation, the casemethod seemed likely to reveal teachers varied experiences.

METHODOLOGY AND CASE STUDIES

We chose three teachers to study based on several factors. One, we wantedto look at teachers who appeared to be learning from the project. Weselected teachers who (a) had been involved from the beginning or nearthe beginning of the project, (b) appeared to be interested and enthusi-astic in some way, (c) appeared to be learning from the project, and (d)were willing to allow us to interview them and observe their classes. Thethree teachers chosen were at different places in their careers and teach atdifferent schools. They and their students vary in ethnicity. Our selectionof participants did not include any who embraced a practitioners stance,since analysis of such cases would be unlikely to uncover any new insightsrelated to teacher change and the development of an inquiry stance.

For each case, at least two interviews and two classroom observationswere completed. Emails teachers sent and nal implementation reportswere collected. We also had access to teachers daily reections on work they did during both the summer institutes and the Saturday seminars.


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The data were coded, codes were rened and each case was analyzed forimportant themes. These were then related to the model, and a descrip-tion of the kinds of appropriation was generated for each case. Each case

description was written by one person from the research team, with theother researchers providing feedback. Individual participants read theircase descriptions and were given the opportunity for comment regardingsuggestions for corrections, clarity of descriptions and their consistencywith that persons experience. The researchers are still in contact withall three teachers in the case studies, and have interacted with themoccasionally since the completion of the initial data collection.

Note: In the case studies, all names used are pseudonyms. Directexcerpts from data are indicated in the following way: (I) interviewexcerpt, (D) daily reections during a workshop, (E) e-mail report froma teacher participant, and (R) participants nal written reports on theirwork.

Case One: Donna McBride

Donna has been teaching elementary school for three years. During hersecond year of teaching, she attended EMES Saturday sessions. Shefollowed this by participating in the Arithmetic, Algebra, and GeometryEMES summer institute. For her pedagogy project, Donna focused onpromoting and improving written communication in her classroom. Thecontent project Donna chose focused on number sense. Specically, shewanted her students to gain a better understanding of decimals. In additionto her projects, Donna indicated that she used many ideas and specicactivities from the EMES workshops during her third year of teaching.

In an early interview, Donna said she had been trained in the use of manipulatives and cooperative grouping. However, she indicated that shedid not know how to implement the use of those techniques effectivelyin her own classroom. The EMES project gave Donna the knowledge shedesired:

And so its . . . been a great benet of the project because its heightened my renement of a lot of the methods I was taught in college. And so, its made me a much better teacher . . .as a result of that. (I)

However, Donna also expressed a desire to better understand the whysbehind mathematical concepts and believes the project provided this: Ifeel more grounded in the concepts. I mean, . . . Ive known the conceptsand I can . . . I can do math, but . . . (it) . . . has never really felt a part of meas much as it does now (I).

Donnas pedagogy project involved using more written communicationin her classroom. But during the course of her third year of teaching,


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Donna observed that written and oral communications are closely linkedand realized that communication promotes conceptual understanding:

I really wanted the kids to take time and reect why is this answer correct, what was your

thinking, how did you get that, where are you going with this. Which takes a long time andeffort. And it gets very painfully laborious but . . . but their thinking is just heightened andlight bulbs start going off when you slow down and say. Why? And they really have tothink through it and when . . . they dont have the Why right there, they slow down andstart thinking and start talking aloud and all of a sudden the concept I wanted them to getin the rst place nally is there. (E)

One area of communication Donna focused on was her studentstendency to describe mathematical processes themselves instead of theirrationale. This was illustrated by a writing assignment on division wherestudents explained how division works. Her students wrote descriptionsof the process one goes through in order to divide and not the reasoningneeded to understand the process. In this case, when her students perfor-

mance fell short, Donna did not seem to contemplate why; instead shefocused on improving their performance.

Donnas work on communication resulted from her experiences atEMES sessions where group work and dyads were used to discuss mathe-matical ideas, along with good questioning and reective writing. Donnatook the information acquired and adapted it for use in her own classroom.

Donnas mathematics content project revealed another area of change.To promote the learning of decimals, Donna used real world problemscenarios from the Summer Olympics integrating technology, good ques-tioning, manipulatives, communication, and grouping. Her students wroteabout how decimals are used in the Olympic contests; they watched someof the Olympic games and studied the statistics from them.

She gave her students activities to encourage exible thinking aboutplace value as well. One of these games involved taking three numbers andmaking as many decimal numbers as possible and developing strategiesto systematically check that the list was complete. Donna said that herstudents were more successful than when following the district-adoptedtextbooks approach to these topics.

Extensions and connections to decimals were made throughout the restof the semester and were not limited to mathematics lessons. One follow-up lesson dealt with money. Donna tried a lesson from the textbook aboutmoney and felt that it was entirely unsuccessful because her studentsfound it uninteresting. So, she had the students look at foreign currency,

research the countries in which the currency originated and then, comparethe currency values. Donna wrote about how:In science and reading, we had been learning about earthquakes. We studied the Richterscale and learned what range of magnitudes the scale represents. We constructed a model of


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the earth with shoeboxes and they replicated an earthquake. If they chose a magnitude of,say, 7.0, I then asked them to show me what a 7.8 might look like to see if they remembered. . . decimals. They remembered, all right! They loved creating massive destruction withtheir models! (E)

As her methods of instruction changed, Donna felt she began to growas a teacher. She realized she had previously been bound by the text-book stating, Textbooks simply do not offer this type of teaching thatthe seminar does (I). Her students were empowered by what they werelearning and were now able to connect their new knowledge to othersubject areas.

During the academic year that followed the EMES summer institute(her third year of teaching), Donna began to move away from the text-book and use explorations with other standards-based teaching techniques.She observed how her students beneted from inquiry-driven experiences.

Furthermore, despite anxiety about the annual state-mandated assessment,she knew this was better than teaching to the test. Donna found that, WhenI let go of my dependency on the text and began moving towards theseother approaches, I nally felt comfortable and excited to teach math (I).Now she is able to integrate the exam material into her daily teaching. Butshe still must resist going back and teaching the old way because of heranxiety.

In the Reective Model, Donna appears to be taking a professionalstance toward the professional development activity. Given her back-ground, Donna was already having experiences and taking actions thatappear consistent with this stance. Moreover, Donna did not come to theEMES sessions with the sole purpose of gathering new teaching tech-niques: she wanted to learn not only effective methods of teaching, butalso the whys behind mathematical concepts.

Donna did begin her interactions with the project by appropriatingmany specic elements (level one). This emphasis is consistent withwhat is known about beginning teachers, namely that one of their maindevelopmental tasks is to develop standard procedural routines that inte-grate classroom management and instruction (Kagan, 1992, p. 129). Butshe also developed her views about what constitutes good mathematicsteaching, adapted activities and processes to her own situation and creatednew activities that were consistent with reform principles (level two). Instating that I hope I am never where I think I want to be because then

my effectiveness will begin to diminish (I), she seems to embrace profes-sional learning as a lifelong process. This indicates her commitment to herown professional development. This interest in continued learning signalsthat she may develop an inquiry stance as she matures professionally.


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Case Two: Eva Pantoja

Eva Pantoja has been teaching elementary school for 13 years, 10 of themas a fourth and fth grade teacher. She started the project the rst summer,completing the institute on Probability and Statistics and returning to herfth grade classroom to implement her project. During the second summershe participated in the Arithmetic, Algebra and Geometry institute, andcame to two of the three Saturday seminars held in the fall.

Eva made signicant changes in her classroom practice as a result of herparticipation in the project. During the rst year, the changes were ratherclosely tied to either the summer institute or the Saturday seminars, andrequired some courage to attempt:

I tried a lot of things last year that were completely new to me because I had learned themin the class. And as I was going to the Saturday seminars and I kept nding out moreand trying more things and sometimes I felt like I was just stepping off of a cliff and notknowing what I was doing and not knowing where I was going. I just knew that it wasbetter than what I was doing before. (I)

Two events helped Eva solidify her commitment to change the rst year.One involved students in her fth grade class who had been in her fourthgrade class the previous year:. . . the year before last I had some kids who, when I was teaching the traditional way, werevery low, but last year when I started trying new things, those were the kids who werehaving the light bulbs go on left and right. (I)

The other signicant event involved a change in her students. The fthgrade teachers, at the end of the year, ability-grouped their classes and putone group with each teacher. Eva was assigned the high-ability group:

. . . I had the high group but I had to get through so much material in a short amount of time . . . and so I went back to my traditional teaching thinking these are the high kids andtheyll catch on fast and well just go right through it . . . they learned very little . . . theywere bored, they didnt like it, the whole atmosphere changed. (I)

Eva came to the institute the second summer highly committed toimproving her instructional practices. Eva decided on place value asher content emphasis, and on mathematical journaling as her pedagogyemphasis.

For her place value unit, Eva created her own instructional design. Shetold her students she had been abducted and taken to Pluto, and whilethere, learned about their strange monetary system (base three). After the

students worked on the Plutonian system for a while, she modied it tobase 10. Eventually she moved her students into working with base-10blocks and creating strategies for addition and subtraction of multi-digitnumbers with chip-trading. All of this took quite a bit of instructional time,


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and her students ended up working on place value and associated conceptsand operations for a number of weeks.

There are many possible questions about this instructional design. Was

it effective? How were students assessed? Was the trajectory appropriate?Efcient? Did all students nd it within their grasp? Was the motivationalstory engaging? From the point of view of our research questions, theseissues do not interest us directly. What is signicant, however, is that thesequestions were of great importance to Eva. While some participants tendedto emphasize positive aspects of their implementation, Eva reports mostlyabout what she learned and what she would change:

In my action research questions, one question I did not pose was, What will I learn aboutplace value when I teach this unit? I thought I understood our place value system becauseI knew that there was an exponential relationship between each place. I had hoped to guidemy students to some understanding about how to move around in our place value system.I had hoped that they would become more mentally exible when using our number system

(to add, subtract, solve problems, etc.) if they could understand how our system works. Ithink I achieved this goal to some extent, but what I learned is that our place value system is just a system of symbols designed to simplify the representation of numbers. Other systemsexist which do the same. I did not make this clear to my students and I realize now that Ishould have. (R)

Evas report is replete with things that she found surprising which ledher to revise the unit. In one case, she realized that students nd it difcultto deal with large numbers in base ten, even after developing a concreteunderstanding of the system by using base-10 blocks. She shared this withother teachers at a Saturday seminar and found not only that some hadsimilar experiences, but that one teacher had an activity designed to dealwith the situation.

This report shows that Eva is open to learning new mathematical ideasas well as new methods of instruction. As she says:

One issue I was constantly reminded of while tackling this unit is that I myself amconstantly learning. I am not the best teacher of place value, but because I was braveenough to try this unit and reect on it, I know I am better now than I was at the beginningof this school year, and next year I will be even more effective. (R)

In choosing to use mathematics journals with her fourth grade students,Eva joined a large portion (about two-fths) of the teachers in the insti-tute who chose to implement some aspect of mathematical writing. Thismay have been because writing was an important (if small) part of theprocessing of almost every mathematics activity during the summer insti-

tute; teachers kept journal entries describing mathematical ideas they hadlearned and, on occasion, these were collected and read by the facilita-tors. On the other hand, the state-mandated mathematics assessments havenumerous open-response items that require students to write solutions;


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implementation of these assessments occurred at the same time as theEMES project activities. Discussions with teachers indicated that both of these factors likely contributed to the choice of so many to implement

writing.During mathematics time, Eva gave the students a problem, had themwork on it and discuss it, either as a whole class or (less often) in pairs orgroups, and at the end of the period, write about it. She initially character-ized their writing as very disorganized, but she says it has improved. Evasought out information by reading Writing in Math Class (by elementaryeducator Marilyn Burns) but still created her own methods. She has modi-ed the prompts she gives (making them more specic to the lesson). Atrst she picked up every students notebook, but has learned to collect afew each day. She uses them as an informal assessment tool, and some-times uses a student question from a journal as a starting point for classdiscussion.

Other changes Eva identies are tied to the idea of mathematicalcommunication. She has worked consistently to implement the idea of creating good questions, and spends considerable time in class askingstudents to explain their responses. She and her students call this back-wards math. She listens carefully to student explanations, at times askingthem to clarify if she or other students do not understand. Eva believesthat it is important for students to have time to work out solutions withoutinterruption from the teacher, and to see multiple solutions. She now givessome explicit attention to problem-solving strategies. This contrasts withher previous emphasis on having students practice saying numbers and onthe basic operations of arithmetic.

During the rst year of the project, Eva took a professional stancetoward the professional development experience, appropriating bothconcrete activities and content and principles and ideas. She used specicactivities that were modeled or developed in either the summer instituteor the Saturday seminars, modifying them as appropriate for the level of her students. She also signicantly altered her approach towards classroomcommunication, spending much more time on problem-solving and payingattention to student explanations. A key factor in creating many of thesechanges involved a shift in her understanding of mathematics learning:

And I realized that there is a whole different way to approach things kind of like the whole-language versus skills-based reading. I didnt realize that math was constructive, you know,I thought it was sequential and so I guess that was a big catalyst. (I)

Later, she claried these ideas:

Giving the students experiences giving them opportunities . . . to construct their under-standing of math. You know that is the main thing that I learned in this program is that. . . I know that kids construct literacy. But I had no idea that they construct their math


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understanding too. (Laughter) You know, I thought I just fed it to them. (Laughter). So Ithink . . . that good questioning is very important. (I)

By the middle of the second year, after two summer institutes and half adozen Saturday seminars, Eva developed a strong inquiry stance towardsher mathematics teaching. Eva took the suggestion (made in the secondsummer institute) of keeping a journal of her mathematics teaching exper-iences quite seriously. She uses the journal to reect on her own practice,which she says helps me think of what to do next, it is a great opportunityfor me to problem-solve and trouble-shoot my own math teaching (R).Her nal report on the progress of her plan is deeply reective, containingsignicant analysis of the practices in which she engaged, as well as alengthy discussion of what lies ahead: what she will try next and why,and what outcomes she is looking for. Her stance is also indicated byher careful attention to students thinking, and her orientation towards

difculties; if things do not go well, she says, at least Im learning (I).It is clear that, at least with regard to mathematics teaching, there was

no real sense of inquiry on Evas part before participating in the project.She entered the project with a signicantly felt need for improving hermathematics teaching, combined with a willingness to try new things, bothof which may have contributed to the development of her inquiry stance.In addition, it is worth noting that the engagement and reactions of herstudents to the way that she teaches mathematics is her touchstone forassessing and modifying her practices. This is clear from both classroomobservations and the way Eva talks about what has been effective for herstudents. Modications can occur in as short a time frame as a single lessonor in the overall planning for her mathematics curriculum, and are based onstudent engagement with tasks and communication about what they havelearned. This nding is consistent with Fennema et al. (1993) who foundthat increased learning on the part of students was a highly signicantmotivating factor in one teachers change process.

Another factor in Evas development appears to be the reectionthat participation in the project engendered. While not every teacherwho is encouraged to reect on practice engages in the same level of analysis, Evas reections are thoughtful, self-critical and highly focusedon improving instruction for her students. Some of the action researchquestions included in her plan were: Will I ask questions that lead studentsalong at a good pace? Will base ten seem harder than base three? What

questions will my students have and what problems will hang them up?What discoveries will they make? (R). The importance of a teachers own(written) reection on her instruction as a catalyst for change was alsonoted by Edward and Hensien (1999).


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teaches, and gaining new understanding and insight. She was very ener-gized by these ahas and by the support she received in thinking creativelyin a mathematical context. She appreciated being encouraged to think of

herself as a mathematician/mathematics learner, and readily took on thisidentity for herself.Vera is a professional writer as well as a teacher; empowering her

students in literacy has been a strength for her because of this identi-cation:

Three years ago, four years ago, ve years ago in my classroom . . . I would have told mychildren . . . you are a writer, just like Im a writer. I never did that with mathematicsbecause I was on the outside looking in. I wasnt a mathematician. And so I never said,Oh, mathematics! Im a mathematical thinker. (I)

From the beginning, Vera was interested in giving her students similarexperiences to those that she had in the EMES class. The similarity,however, did not relate to the specic nature of the activities, nor did itonly encompass the kinds of interactions that occur between teachers andstudents; mostly, for Vera, the point was for the students to have similarinternal or personal experiences (the aha experiences that she had) asthey created their own understanding of mathematical content.

Early on in the project she stated, I want to excite, challenge andincrease the math competence of my students, to teach them to persevereand to teach them to be mathematically condent (E). Veras desire toempower her students and her constructivist teaching philosophy led to herinterest in the concept of nding ways to ask good questions of students.She latched on to this concept and constantly expressed the desire to applyit to her entire mathematics curriculum.

For her pedagogy emphasis, Vera chose to work on creatingchallenging, non-standard problems to send home with students. Shecreated problems (sometimes in the form of games that pupils wereinstructed to play with family members) that were closely related toconcepts being covered in the project, often using the results of thehomework in her instruction.

Observations of Veras Teaching Clarify the Picture

At the beginning of one period, Vera asked students to write in their note-books the rules of the function game, which they had learned the previousday, with Vera choosing the rule and the students guessing. After a few

minutes, Vera asked for volunteers to read their description to the studentswho had not been present the day before.

To play the game today, students were asked to volunteer a rule andrun the game. After listening to the rst volunteer explain his rule, Veradecided he did not yet understand the game and asked him to sit down and


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TABLE I

Student rule data from Veras class

7 569 90

8 725 30

1 20 03 12

watch for one turn. The next student had a rule much more complicatedthan the ones previously used. So it took a long time for other students toguess it. Table I shows how the chalkboard read.

The students were a little frustrated but eagerly engaged. At one point,Vera asked the students to discuss at their tables what might be going on(students sit in groups of 4 or 5), giving them a hint by suggesting that therule might be doing more than one thing. After some further discussion,students at one table were able to characterize the rule as multiplying bythe same number and then adding it on.

Fairly early in the process, one student who felt he had the rule begansaying 7 times 8, 9 times 10 and so on, specifying different rules foreach set of numbers. Vera pointed out that it had to be the same rule eachtime. After someone came up with the rule that worked for every case, theclass checked it collaboratively (again working at tables) to see if it reallyworked and if they understood it. Vera showed the students the algebraicnotation for the rule [(n n) + n] on the board, explaining it as involvinga variable (a term she had discussed with them earlier), and they checkedthat this representation worked in each case. Then all students took about10 minutes to write in their notebooks what the table was and what the rulewas.

This episode illustrates how Vera has worked to create activities toempower her students as mathematicians. Her instruction contains richmathematical discourse, both among students and between student andteacher. Students explanations of their thinking are listened to and valued,and this atmosphere is reected in the interactions of the students.

The description also illustrates how Vera changed her homework

assignments. Now, instead of practice problems, homework is either aproblem to solve or some sort of game. These assignments integrate withher classroom instructional activities.

In terms of the reective model, what did Vera appropriate from theproject? She did take some activities directly to use with her students


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and also improved her professional knowledge in a variety of ways,using processes (such as the idea of math notebooks) or principles (usingmultiple representations of a mathematical concept) that were used in the

project. Mostly, however she took a sense of empowerment, inquiry, andnew modes of interacting with students around mathematics:

I can truly say now, If youre good at language you can be good at math. Math is alanguage . . . its more natural than I ever thought it was. I thought it was, I truly believedthat mathematics was for a limited number of people who had mathematical minds. I reallydid believe that. And Id read many places that this simply isnt true. But I didnt believeit, in my heart. And I believe it in my heart now thats changed in three years. (I)

When asked what it was in the project that helped her get to this point, shemade clear that it was not just that we were saying the same things that shehad read before, rather:

It was your doing with me . . . what I need to do with children. Modeling with me . . .

letting me live through the experience . . . of being . . . a constructor of a mathematicalworld. Which Id never been allowed to be before, I just had to nd the right answer andspit it out enough times in the right context. (Laughter) And . . . pray a lot, you know. Butnow (laughter) . . . I get to . . . I get to play . (I)

It is this sense of play that connects Veras interactions in the project asa mathematics learner and interactions with her students as a teacher. Sheclearly receives great enjoyment from creating opportunities to investigatemathematics with her students and from watching them reectively as theylearn. It is clear that Vera has adopted an inquiry stance in her mathematicsinstruction.

DISCUSSION

These cases show three different teachers, at different stages in theircareers, working to improve mathematics instruction through professionaldevelopment. Vera, the most experienced teacher, came into the projectwell-prepared (almost primed) to act as an inquirer, explicitly uninterestedin level one appropriations and looking for something more, which shefound in her own learning experiences. Her reactions to, and reections on,her own mathematical learning were immediately and explicitly applied toher thinking about her students and her instruction. Her experiences increating and reectively improving student-centered pedagogy in literacy

allowed her to create such pedagogy in her mathematics instruction, onceshe reected on her personal experiences.

Eva, in the middle of her career, moved more slowly into the inquirystance, beginning with level one and two appropriations and graduallycoming to see herself as a mathematics teacher-learner, driven at least


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partly by her intense interest in her students reactions to her pedagogyand their mathematical thinking. Donna, a new teacher, began the projectas a practitioner, interested primarily in learning how to do the things that

she had been taught that she should do while in her teacher educationprogram. As the EMES project progressed, she became more interestedin the principles behind the activities and worked to adapt and create newinstructional elements with them in mind, moving her into a professionalstance.

Just as mathematics learning can look very different from one studentto another, so these cases show signicant variation among teachers forwhom this particular project worked. For both Eva and Vera, theirown mathematical learning and their attention to their students thinkingbecame important inuences for instructional change. This dynamic hasbeen seen among pre-service teachers as well (e.g., Ball, 1988) and, there-fore, is not entirely attributable to the teachers classroom and careerexperience.

Both Vera and Eva remarked that the project allowed them to bringto their mathematics instruction student-centered elements that had onlybeen previously present in their literacy instruction. A common elementfor them was the catalytic nature of the mathematical learning experiencesin the project. In attempting to explain the success of professional develop-ment programs (such as CGI) which focus on providing teachers access toresearch-based knowledge of learning, Rhine (1998) speculates:

. . . the power of these projects may not be primarily due to developing teachers knowledgebase of specic strategies students employ. Perhaps the major impact of these projects onadvancing educational reform is, instead, due to teachers engagement with research that

serves as a catalyst for their new orientation toward inquiry into students thinking andvaluing students knowledge and thinking processes. With this focus, teachers then inte-grate their assessment of students understanding into their instructional decision-makingprocess. (p. 28)

In our project, a different element (teachers own mathematics learningexperiences) served a similar purpose. Level two and three appropriationsin content and pedagogy are the goal of many professional developmentprojects. Schifter (1998) explores the need for examination of disciplinarycontent and examination of student thinking and, in addition, discoversthat powerful synergy can arise between the two. We found that teacherswho take an inquiry stance toward professional development utilized this

synergy for their own learning.It is interesting to speculate about the role of literacy instruction in the

two cases of Eva and Vera. How important for these teachers was the factthat they already had a well-developed constructivist theory of learningrelated to literacy? Did the fact that they had this schema allow them to


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develop more quickly such a theory for their mathematics instruction andact on it?

For Donna, the new teacher, the dynamic was quite different, though

also inuenced by her experiences. She appeared not to have any conictwith the pedagogical aspects of the project; these were consistent withher preservice teacher education program and she had not built up anyrepertoire of teaching experience that might have conicted with it.

Contrary to this, as a new teacher she viewed herself to be in need of specic ideas for implementing student-centered pedagogy (Kagan, 1992).As she continued making these kinds of appropriations during the courseof the project, her classroom experiences led her to construct professionalknowledge for this purpose. It seems that for Donna, this met her currentneeds; we may speculate that as she gains condence as a practitioner andprofessional, she may develop an inquiry stance toward her own teachingand future professional development opportunities.

IMPLICATIONS FOR PROFESSIONAL DEVELOPMENTDESIGN

We return to Balls (1995) suggestion and examine the implications of thisstudy for the design of mathematics professional development. In the caseswe studied here, the following elements were effective:

Inclusion of authentic and readily adaptable student-centered mathe-matics learning activities

Rich opportunities for discussion and reection An open, learner-centered implementation component An inquiry stance taken by the facilitators

Including authentic mathematics learning experiences that are alsoeasily adaptable to the classroom milieu allowed teachers to adapt theactivities directly and also to reect on mathematical concepts or pedago-gical principles (either in the workshop or as a result implementation). Themodeling of student-centered instruction also allowed teachers to adoptinstructional techniques directly, integrating them into instructional prac-tices and activities that were obtained from other sources. Using suchmathematics as the entry point into professional discussions allowsteachers to interact with content in ways that are specically useful for

teaching (Shulman, 1986).Furthermore, opportunities for discussion, journaling and reective

writing, centered on mathematical ideas and issues of pedagogy, allowedteachers to construct mathematical and professional meanings for them-selves from the project activities. This is true, even though the construc-


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tions clearly varied considerably among our research cases and based onour impressions even more among other project participants.

In addition, the implementation component of the project was individu-

ally designed by each teacher to meet her or his own perceived needs asa classroom teacher. Our input usually involved suggestions or questionsdesigned to clarify issues. This not only created a sense of ownership onthe part of the teachers for the instructional changes, it also supportedteachers as professionals in making a wide variety of appropriations of project elements, encompassing all three levels identied in the reectivemodel.

Finally, all three of the above components were supported by theinquiry stance of the EMES project facilitators. We entered both theprofessional development project and later the research project with adeep awareness of how little we knew. We know that the teaching weare trying to help teachers learn is not completely dened. It is embeddedin a vision of reform, rather than a collection of procedures. Likewise,our understanding of professional development that can support teacherslearning is a mix of myth, belief, and conjecture (Ball, 1995). We made noattempts to hide from our participants the fact that we were learning: facili-tators worked with teachers as co-learners of mathematics and as inquirersinto their classroom milieus. Facilitators read participant feedback aftereach session and not only used it to modify the next days activities, butoften shared this feedback and our modications explicitly with the group.Questions from teachers about how they should implement what they werelearning were usually met with comments such as Thats up to you andWe dont know how this can be adapted to your grade level thats

something we expect to learn from you. Such explicit attempt by thefacilitators to model an inquiry stance in discussions with participants maybe an important factor in stimulating teachers thinking about themselvesas inquirers (through the teaching process). Those who design professionaldevelopment and wish to support teachers in their adoption of an emergingmodel of the profession of teaching may reap signicant benets fromtransparency regarding their own inquiry processes.

ADDENDUM

We thank the editors for pointing out to us the similarity of our work

with that of Cochran-Smith and Lytle (1999). Although there are someinteresting parallels that could be drawn between our levels of appropria-tion and the three views of knowledge that they identify, the parallelsare not exact. There is a close match, however, between what we foundand called the inquiry stance and their view of inquiry as stance;


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some differences are that (a) they seem to explicitly locate this stancein communities of teachers, whereas we identied it in several individualcases (who were however indeed part of a learning community) and (b)

they include a critical stance toward the current educational system as apart of their description whereas our cases (while quite possibly possessingsuch critical views) were not analyzed from that perspective.

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Shifter, D. & Simon, M. (1992). Assessing teachers development of a constructivist viewof mathematics learning. Teaching and Teacher Education , 8(2), 187197.

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JEFF D . FARMER & HELEN GERRETSONUniversity of Northern Colorado Mathematical SciencesGreeley, CO 80639 E-mail: [email protected]@unco.edu

MARSHALL LASSAK

Eastern Illinois University Dept. of Mathematics and Computer ScienceCharleston, IL 61920 E-mail: [email protected]

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What Teachers Take From Professional Development