2004 Inquiry Into Children’s Mathematical Thinking

8/15/2019 2004 Inquiry Into Childrens Mathematical Thinking

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RUTH M. STEINBERG, SUSAN B. EMPSON and THOMAS P. CARPENTER

INQUIRY INTO CHILDRENS MATHEMATICAL THINKING

AS A MEANS TO TEACHER CHANGE

ABSTRACT. In the context of U.S. and world wide educational reforms that require

teachers to understand and respond to student thinking about mathematics in new ways,

ongoing learning from practice is a necessity. In this paper we report on this process for

one teacher in one especially productive year of learning. This case study documents how

Ms. Statzs engagement with childrens thinking changed dramatically in a period of only

a few months; observations and interviews several years later confirm she sustained this

change. Our analysis focuses on the mathematical discussions she had with her students,

and suggests this talk with children about their thinking in instruction served both as an

index of change, and, in combination with other factors, as a mechanism for change. Weidentified four phases in Ms. Statzs growth toward practical inquiry, distinguished by her

use of interactive talk with children. Motivating the evolution of phases were two sorts of

mechanisms: scaffolded examination of her students thinking; and asking and answering

questions about individual students thinking. Processes for generating and testing knowl-

edge about childrens thinking ultimately became integrated into Ms. Statzs instructional

practices as she created opportunities for herself, and then students, to hear and respond to

childrens thinking.

KEY WORDS: discourse community, elementary mathematics, practical inquiry, teacher

change, teacher learning, teacher reflection

Mathematics educators have articulated a vision for teaching mathematics

that includes engaging students in problem solving, mathematical argu-

mentation, and reflective communication (NCTM, 1991, 2001). Calls for

instructional reform in mathematics have been accompanied by demands,

in many countries, for radical changes in teaching practices. Many teachers

have learned to teach in ways consistent with calls for reform (Cobb, Wood

& Yackel, 1990; Cobb & McClain, 2001; Fennema et al., 1997; Hiebert,

Carpenter, Fennema et al., 1997; Hiebert & Wearne, 1993; Jaworski, Wood

& Dawson, 1999; Schifter & Fosnot, 1993; Sullivan & Mousley, 2001).

Without attention to how teachers learn, however, our understanding of

instructional reform is seriously incomplete (Franke, Carpenter, Levi &

Fennema, 2001; Hammer & Schifter, 2001; Richardson & Placier, 2001;Schn, 1983; Sherin, 2002).

A small but growing body of research has focused on teacher learning

as practical inquiry into the problems of teaching (Jaworski, 1998, 2001;

Journal of Mathematics Teacher Education 7: 237267, 2004.

2004 Kluwer Academic Publishers. Printed in the Netherlands.


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238 RUTH M. STEINBERG ET AL.

Lampert, 1985; Richardson, 1994; Tabachnick & Zeichner, 1991). This

research has found that teachers who engage in practical inquiry are able

to change their teaching in ways that are sustainable and self generative

(Franke, et al., 2001). There has been little research, however, on theprocess of change towards inquiry-oriented practice.

In the current study, we focus on one teachers use of practitioner

knowledge and research-based knowledge as she learned to integrate

practical inquiry into her teaching. We focus in particular on the mathe-

matical discussions she had with her students, and argue that this talk with

children about their thinking during instruction served both as an index of

change, and, in combination with other factors, as a mechanism for change.

We concentrate on this latter feature of teacher-student talk because, we

contend, it provides insight into the nature of generative change (Franke et

al., 2001) in teaching.

This teachers mature teaching can be characterized as an integra-

tion of inquiry and instruction, in which both she and students learned.Although the process of change we have documented does not necessarily

represent the path to practical inquiry that all teachers should take, it lends

useful insight into how a teacher can combine practitioner knowledge and

research-based knowledge to ask and answer questions profitably about

teaching and learning.

The teacher, Kathy Statz,1 taught mathematics using the precepts

of Cognitively Guided Instruction (CGI) (Carpenter & Fennema, 1992;

Carpenter, Fennema, Franke, Levi & Empson, 1999; Carpenter, Fennema,

Franke, Levi & Empson, 2002). CGI is a research and professional devel-

opment program founded on the fact that children enter school with a rich

store of informal knowledge that provides a basis for engaging in problemsolving. We draw on previous research that documents levels of teachers

engagement with childrens thinking in order to track Ms. Statzs learning

(Fennema et al., 1996; Franke et al., 2001; Simon & Schifter, 1991). We

go beyond documenting the fact of change to describe how she progressed

from one level to the next, initially by reflecting on instruction as questions

about her students thinking were posed for her and, later, by posing and

answering such questions herself. Changes in her practice accompanied

these changes in her stance towards teaching.

CONCEPTUAL FRAMEWORK

Research suggests that teachers learn a great deal from teaching, but the

content of that learning varies from teacher to teacher (Richardson, 1990;

Richardson & Placier, 2001). Conditions that appear to be most condu-


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INQUIRY INTO CHILDRENS THINKING 239

cive to learning include: 1) membership in a discourse community that

provides tools for framing and solving the problems of teaching (Ball,

1996; Cobb & McClain, 2001; Stein, Silver & Smith, 1998; Wenger,

1998); 2) processes for reflectively generating, debating and evaluatingnew knowledge and practices (Ball, 1996; Jaworski, 1988; Wilson &

Berne, 1999; Wood, 2001); and 3) ownership of change, so that the prob-

lems of teaching that change is meant to address are problems that teachers

want to solve and feel capable of solving (Loucks-Horsley & Steigelbauer,

1991; Simon & Schifter, 1991).

None of these conditions, alone or in combination, assures ongoing

teacher learning. Perhaps the most important is teachers own stance

towards practice as inquiry (Jaworski, 1994; Schifter & Fosnot, 1993;

Tom, 1985). This inquiry can take several forms. It can be exercised in

interaction with students and the curriculum (Sherin, 2002) or removed

from classroom interactions, in reflection on action (Mewborn, 1999;

Schn, 1983, 1987; Wood, 2001). Little (1999) noted that the systematic,sustained study of student work, coupled with individual and collective

efforts to figure out how that work results from the practices and choices

of teaching may be one of the most powerful sites for teacher inquiry

(p. 235). Student thinking is not the only focus possible, but it is one

that has proven productive for teachers and students (Carpenter, Fennema

& Franke, 1996; Carpenter, Fennema, Peterson, Chiang & Loef, 1989;

Schifter, 2001; Steinberg, Carpenter & Fennema, 1994; Tzur, 1999).

Teachers who change in ways that embrace new knowledge and beliefs

about childrens problem solving do not necessarily sustain that change

or continue to change. Franke et al. (2001) found that the most profound

change among a group of 22 CGI teachers occurred for those who engagedin practical inquiry into childrens thinking. Those 10 teachers, more than

the rest, thought of the research-based framework for childrens problem

solving as their own to create, adapt, and investigate (Franke et al.,

2001, p. 683). Franke et al. (2001) called this learning generative change

because teachers used what they knew to generate new knowledge through

practical inquiry, and saw this inquiry as part of their identity as profes-

sionals. In particular, 1) these teachers believed understanding childrens

thinking was central to their work, and 2) their knowledge of childrens

thinking went beyond the frameworks first presented to teachers in staff

development four to eight years earlier.

Not all teachers who use problem solving in teaching (e.g., NCTM,

2000) take a stance of inquiry toward their practice. There are many profi-cient teachers whose instruction is based on problem solving but who do

not engage in practical inquiry. However, as Franke and colleagues (2001)


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argued, practical inquiry is a powerful means to the continued improve-

ment of practice. The case we report here is an example of a teacher

who not only taught in reform-oriented ways, but also developed a stance

of inquiry towards her practice. We examine in closer detail the processof change towards what Franke and colleagues (2001) called generative

learning and the mechanisms that appeared to stimulate this change.

Levels of Teacher Change in Cognitively Guided Instruction

The teacher in this study taught mathematics using Cognitively Guided

Instruction (CGI). In this approach to professional development, teachers

are encouraged to use research-based knowledge about childrens mathe-

matical thinking to make instructional decisions (Carpenter et al., 1999). It

differs from most curricular interventions in that lessons are not prescribed

for the teachers. Instead, teachers plan for instruction using what theyknow about their own students and their general knowledge of childrens

problem solving. CGI consists of research information about the develop-

ment of childrens thinking, portrayed through problem-type frameworks

that emphasize semantic differences among problems and solution strategy

hierarchies. Teachers learn which semantic features of problems are easiest

for children to understand, and which features are more difficult. In whole-

number addition and subtraction, for example, problems involving actions

on sets (e.g., joining two sets together) are generally easier than problems

involving relationships between sets (e.g., comparing the sizes of two sets).

Similar analyses have been developed for multiplication and division, and

the development of base-ten concepts and multidigit strategies for addition

and subtraction. (See Carpenter et al. [1999] for more information.)

A longitudinal study of a sample of 21 first through third-grade teachers

involved in CGI professional development documented five distinct levels

of teachers use of childrens thinking (Fennema et al., 1996). Franke et

al. (2001) revised the levels to reflect the integration of teacher beliefs and

practices, and called the classification scheme engagement with childrens

mathematical thinking (Table I). The levels are useful for characterizing

teacher change, for they go beyond dichotomizing teachers practice into

reform-oriented or not. Fennema et al. (1996) found that teachers who

engaged with childrens thinking at levels 3, 4a, and 4b of the scale taught

in ways that were distinctly different from teachers at levels 1 and 2. The

key distinctions hinged on students opportunities to solve and discussproblems. Further, student outcomes were higher in the classrooms of

teachers at the top three levels (see also Carpenter et al., 1989). We review

these levels here and use them to describe Ms. Statzs growth.


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TABLE I

Levels of Engagement with Childrens Mathematical Thinking from Franke, et al., 2001,

p. 662 (Copyright 2001 by the American Educational Research Association; reproduced

with permission from the publisher)

Level 1: The teacher does not believe that the students in his or her classroom can solve

problems unless they have been taught how.

Does not provide opportunities for solving problems.

Does not ask the children how they solved problems. Does not use childrens

mathematical thinking in making instructional decisions.

Level 2: A shift occurs as the teacher begins to view children as bringing mathematical

knowledge to learning situations.

Believes that children can solve problems without being explicitly taught a

strategy.

Talks about the value of a variety of solutions and expands the types of problems

they use.Is inconsistent in beliefs and practices related to showing children how to solve

problems.

Issues other than childrens thinking drive the selection of problems and

activities.

Level 3: The teacher believes it is beneficial for children to solve problems in their

own ways because their own ways make more sense to them and the teacher wants the

children to understand what they are doing.

Provides a variety of different problems for children to solve.

Provides an opportunity for the children to discuss their solutions.

Listens to children talk about their thinking.

Level 4A: The teacher believes that childrens mathematical thinking should determine

the evolution of the curriculum and the ways in which the teacher individually interacts

with students.

Provides opportunities for children to solve problems and elicits their thinking.

Describes in detail individual childrens mathematical thinking.

Uses knowledge of thinking of children as a group to make instructional

decisions.

Level 4B: The teacher knows how what an individual child knows fits in with how

childrens mathematical understanding develops.

Creates opportunities to build on childrens mathematical thinking.

Describes in detail individual childrens mathematical thinking.

Uses what he or she learns about individual students mathematical thinking todrive instruction.


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At level 1, teachers mostly used direct instruction, and, as measured by

interviews and beliefs scales, did not believe children could invent their

own strategies to solve problems. There was little to no opportunity for

children to solve problems. At Level 2, teachers began to believe childrencould solve problems on their own, but were inconsistent in implementing

this belief. At Level 3, teachers believed children should solve problems

using their own strategies because it led to deeper understanding. Children

were presented with a variety of problems to solve and discuss. Teachers

listened to childrens thinking, but did not necessarily build on it. Although

they understood the problem-types and solution-strategy frameworks, they

were not aware of individual childrens thinking in detail. Nonetheless, this

level marked a departure from the teacher-centered instruction that charac-

terized levels 1 and 2. At Level 4A, teachers believed childrens thinking

should drive the curriculum. Childrens thinking was elicited and teachers

could describe that thinking in detail. However, decisions about how to

build on that thinking were made at a global level, for the whole class.At Level 4B, teachers believed the curriculum should be driven by what

individual children know. They knew what problems individual children

could solve, what strategies children used, and how childrens strategies fit

with understanding mathematics. Teachers used this knowledge to build on

individual childrens thinking in instruction. Fennema et al. (1996) argued

that instruction at Levels 3, 4A and 4B epitomize the process standards

of the reform movement (p. 429). Thus, teachers who reach levels 3 and

above teach in ways that are consistent with U.S. calls for mathematics

education reform (National Council of Teachers of Mathematics, 1991;

2000).

Fennema et al. (1996) found 19 out of 21 teachers teaching at Level 3or higher at the end of a four-year intervention; that is, 90% of the study

teachers taught using reform-oriented practices by the studys conclusion.

Nine of those teachers began the study at Level 1, and seven teachers at

Level 2. Twelve teachers instruction was classified as Level 3 by the end

of the study, suggesting that attaining Level 4A or 4B is not common, even

among teachers who change.

The case study we report here deepens our understanding of how

teachers may attain a level of teaching in which instruction is based

on teachers generative knowledge of individual childrens mathematics

(Level 4b in the scale).


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METHOD

This study was conducted in collaboration with one fourth grade teacher,

Ms. Statz, who has worked as a classroom teacher and mathematicsresource teacher for grades K-5. We collected data at three points in time.

At the first point, the first author acted as an observer participant for a

five-month period, in Ms. Statzs third full year of teaching. At the second

point, the following year, Ms. Statz was observed by the second author,

over a period of several months. These data are used to ascertain whether

Ms. Statz maintained the changes documented here, and are only briefly

reported in this paper. At the third point, several years later, Ms. Statz

was interviewed about her growth as a teacher, looking back on her career

beginning with her pre-service teacher education.

Data Collected in Ms. Statzs Third Year of Teaching

At the time of the observations, Ms. Statz had been teaching three years,

all of which were with fourth grade classes. She had implemented CGI

from the first year, after learning about it in her University certification

program. That years class consisted of 21 students from a racially, ethni-

cally, linguistically, and economically diverse population of families. A

third of the class was new to the school. All the childrens names reported

in this paper have been changed.

PROCEDURES

Classroom observations. Thirty-four complete mathematics lessons were

observed by the first author, over a five-month period. Lessons were

audiotaped and parts were transcribed. Field notes were taken on teacher-

student interactions, students solution processes, class organization, and

the teachers knowledge of, and efforts to build on, childrens thinking.

Nine children were randomly selected as target students and their solution

processes were documented regularly by observing them and asking them

how they solved the problems. The other children were observed on a

rotating basis.

Teachers meetings with researcher. The first author met with Ms. Statz 13

times during the first year of data collection, usually once a week for 3040minutes. All meetings were audiotaped and transcribed, and included

discussions about Ms. Statzs knowledge of her childrens thinking, her

decision making processes regarding content and classroom organization


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and their relation to childrens thinking, and the researchers observa-

tions of specific solution strategies in interviews with students or in class

observations.

Student assessments. Each child was interviewed at length, at the begin-

ning and at the end of the study, on solution strategies for word problems in

a variety of content areas. Students mathematics journals were examined

regularly by the researcher.

Data analysis. Observations and interview decisions were made in

response to the situations arising in the classroom and in the teacher-

researcher discussions. Themes consistent with teacher change and the

CGI framework for teachers engagement with childrens thinking were

marked, such as teachers knowledge and beliefs, building on thinking

in instruction, and teacher-identified dilemmas and resolutions (Denzin &

Lincoln, 2000).

Career Interview. Ms. Statz was interviewed by the second author several

years after the primary data were collected. The interview was designed to

elicit the story of her teaching career, in terms of formative events, such

as high points, low points and turning points. The interview was adapted

from interviews (Math Stories) used by Drake (under review)2 to elicit the

stories for teachers involved in reform.

RESULTS AND DISCUSSION

In her third year of teaching, Ms. Statz changed dramatically in her knowl-

edge of childrens thinking and the use of interactive talk to enhance this

knowledge and build on her students understanding. In this section, we

describe how Ms. Statz made the transition from good reform-oriented

teaching, corresponding to level 3 in Franke et al.s (2001) scale, to

outstanding teaching, corresponding to level 4b, that incorporated practical

inquiry as a way to continue to acquire knowledge and gain insights into

teaching. We documented four distinct phases of change. We frame the

changes Ms. Statz experienced in her third year of teaching by describing,

based on her own reports, Ms. Statzs earlier and later years of teaching

and teacher learning.


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Early Teaching Years

Ms. Statz reported that she left her teacher certification program with a

commitment to working with a belief in all childrens potential to succeed.

She began her teaching career in a fourth-grade classroom. She describedher first year teaching as one of two low points in her career because she

had no materials to teach mathematics other than a commercial textbook

series. She tried to use it, but felt lousy when she taught expository

lessons. She felt her students were not learning.

Ms. Statz decided to abandon the textbook in the middle of the year and

to begin using the framework for problem types and solution strategies,

the basis for CGI, to plan instruction. She had the support of her principal

and a mathematics resource teacher, Ms. J, who was herself experienced at

building on childrens thinking. During this time, Ms. Statz gladly accepted

help from Ms. J, who would say things like . . . Dont tell [the students]

that. Let them find it out. Ms. Statz reported that, with help from Ms. J, she changed her mathe-

matics instruction a great deal during that first year, but described the

second year and beginning of the third as more of a plateau. The

remaining stretch of the third year the time we report on here was

described by Ms. Statz as a big jump.

Third Year Teaching: A Year of Change

In her third year of teaching, Ms. Statz experienced intense growth in

understanding childrens thinking, knowledge of the content area, and

beliefs about her role as a teacher. Concurrent with this learning, Ms. Statz

developed a stance of inquiry into her teaching practice and its relationshipto childrens thinking. In this section, we document these changes, and

consider possible mechanisms of growth. These mechanisms include the

types of situations that prompted Ms. Statz to perceive a need for change,

and how her concerns and internal debates inhibited or contributed to

change.

PHASE 1: LEVEL 3 ENGAGEMENT WITH

CHILDRENS THINKING

In November of the school year, Ms. Statzs instruction was consistent with

calls for reformed classrooms (NCTM, 1989) and CGIs Level 3 of engage-ment with childrens thinking (Table I): students solved challenging word

problems using their own strategies; the teacher gave students opportuni-

ties to present and talk about their strategies; and children recorded how


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they solved problems in mathematics journals. No textbook was used. Ms.

Statzs goals for her students coincided with several of those of the reform

movement:

Being able to write about math. And being able to verbalize what theyre doing andthinking about math . . . Being able to feel comfortable enough about math to share what

theyre talking about. And to develop an appreciation for each other. That people solve

math problems differently and thats okay.

Ms. Statz often chose topics for word problems that related to a story the

children had read in class or to events in the childrens lives (e.g., selling

things in the school store, counting the number of names on a childs cast).

Talk with Children about their Thinking

Although students had opportunities to solve problems, Ms. Statz did

not build on or extend childrens solutions strategies in her interactions

with students. After children solved problems, there were short discus-sions of each problem, lasting 5 to 10 minutes, in which four or five

different strategies were presented. Although Ms. Statz encouraged the

children to talk about their strategies, she seldom challenged them to

justify them, think of alternative solutions, or relate their strategies to more

advanced strategies. She listened to what they said, and accepted it with

little questioning.

For example, one day late in the fall, Dan showed the class how he

had solved a word problem by calculating 68 + 37. He drew 37 tallies and

counted by ones from 68, using the tallies to keep track. Ms. Statz then

called the next child. She did not question Dans strategy or relate it to

more advanced strategies that used base-ten concepts (such as adding the

three tens and the seven ones) presented by other children.

Ms. Statzs belief in accepting and encouraging a variety of strategies

from children was so strong she perceived her role in helping children to

progress as passive. Later in the school year, Ms. Statz reflected on her

interactions with students at that phase: I would just accept what was put

on the board and that was all good, thats fine. Go have a seat. Next

person.

Further, Ms. Statzs knowledge of individual childrens thinking was

not well integrated with the research-based framework for childrens

thinking that was the basis for CGI. In December we asked Ms. Statz to

classify students in her class according to the strategies she predicted they

would use, in our clinical interviews with them, to solve a set of problems.She accurately predicted the strategies of students who direct modeled3

to solve problems, but did not predict the more sophisticated strategies five

of her children used. Although she knew who used more or less advanced


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strategies, when it came to strategies beyond direct modeling, Ms. Statz

did not describe or classify these strategies appropriately.

Not surprisingly, this state of knowledge influenced Ms. Statzs inter-

actions with students. For example, in October, Ms. Statz assisted a childto solve a problem in a way that acknowledged neither his understanding

nor the solution-strategies framework. David was trying to solve a Joining

problem with an Unknown Change (Raji has 17 dollars. He wants to buy

a pet snake that costs 33 dollars. How many more dollars does he need

to earn to buy the pet snake?) by counting up from 17 to 33. He was

keeping track of how many numbers were added by going back and forth

between each number sequence: first is 18, second is 19, third is 20 and

so forth until he got to ninth is 26. He lost track of his double count

here and stopped. His strategy was appropriate to the additive semantic

structure of the problem but the method of keeping track appeared to

make more demands on his working memory than he could handle. Ms.

Statzs first attempt to help David encouraged him to continue to thinkabout the problem additively, but did not address his specific difficulty

of keeping track. She suggested, instead, that he add 10 to 17, and then

asked if it would be enough. Before David could respond, another child,

Nick, said he had solved the problem by subtracting 17 from 33. Ms. Statz

suggested to David that he could use a strategy similar to Nicks and solve

the problem by separating 17 counters from 33. Although this strategy is

concrete it does not directly model the semantic structure of the problem,

which is additive. It re-represents an additive semantic structure in terms

of subtraction and so is a more difficult strategy (Carpenter et al., 1999).

When Ms. Statz asked him to solve the problem using subtraction, she

did not realize this strategy required knowledge of the inverse relationshipbetween addition and subtraction that David may not have had.

Beginning Inquiry. Discussion of episodes like this one began to help Ms.

Statz reflect on how she used knowledge of childrens thinking in instruc-

tion. In a conversation with the participant researcher a few days later,

Ms. Statz said she was often not sure how to respond to children, like

David, who were struggling with a specific strategy. She then recalled a

strategy she had recently seen in which a child used tallies to keep track of

a count. She realized it would have been an appropriate strategy to suggest,

because it would have addressed his specific difficulty of keeping track and

would have allowed him to build on the additive structure that he saw in the

problem. Thus, this conversation facilitated new connections for Ms. Statzamong different episodes involving interactions with childrens thinking.

In the next section, we document how, as the first author and Ms. Statz


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continued to have these kinds of conversations, Ms. Statz became increas-

ingly dissatisfied with her use of childrens thinking and began a transition

from Level 3 towards Level 4A engagement with childrens thinking.

PHASE 2: MOTIVATION FOR TRANSITION FROM LEVEL 3

TO LEVEL 4A ENGAGEMENT WITH CHILDRENS THINKING

In this phase of change, Ms. Statz became dissatisfied with what she knew

about childrens thinking. She realized that many of her students strategies

were basic or even wrong and did not necessarily show well developed

understanding. In response to this dissatisfaction, Ms. Statz formulated

questions for teacher inquiry that guided her in later phases. She thought,

in particular, about her knowledge of students strategies and her role

as a teacher in facilitating more sophisticated strategies and deepening

childrens understanding. Little action to answer her questions took placein this phase. Rather, this phase was distinguished by reflection-on action

(Schn, 1987), as Ms. Statz stepped back from her practice and began to

identify personal dilemmas.

The transition from satisfaction to dissatisfaction with Level-3 type

engagement with childrens thinking appeared to have been triggered by

discussions with the first author about the researchers problem-solving

interviews with the children. Ms. Statz sat in on some of the interviews and

was intrigued, often surprised, and sometimes troubled by how children

solved the problems.

For example, one child, Pang, wrote down every single number between

398 and 500 to solve a Join Change Unknown problem (Robin has 398dollars. How many more dollars does Robin have to save to have 500

dollars to buy a new bike?). Surprised by this strategy, Ms. Statz checked

Pangs journal and discovered she was using similar strategies there too:

The way she solved this is kind of strange . . . Can I go see what shes got in her journal?

. . . Yeah, shes doing similar things.

Ms. Statz became especially concerned about seven children who were

using the standard subtraction algorithm incorrectly in the interviews.

Ms. Statz had not introduced this algorithm in class; instead, she encour-

aged children to generate their own conceptually grounded strategies

for multidigit addition and subtraction. However, for problems involving

regrouping, these seven children always subtracted the smaller digit fromthe larger one, an example of a buggy algorithm when the smaller digit

is the minuend. When asked if she had seen the children using this kind

of strategy in class, Ms. Statz replied, Thats something that surprised


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me during these interviews. Because she believed children should always

solve problems with understanding, and not by rote, she was taken aback:

The borrowing with regrouping really disturbs me now. Thats all that Ive been thinking

about now that weve been interviewing the kids and we see that they dont have it. And

they dont; theyre not even coming up with a good way of explaining it. It doesnt make

sense to them.

Ms. Statzs growing knowledge of the basic character and inadequacy of

her students strategies, combined with a strong belief against grouping

children by ability or even by type of errors, presented a dilemma for her:

how to accommodate a wide range of childrens thinking without resorting

to ability grouping or remediation. Maintaining her belief in the centrality

of student-generated strategies to the development of understanding and

confidence, she started to think about how she could assist these children

to grow mathematically without directly telling them how to solve prob-

lems. She re-examined her belief that the teacher should accept whateverstrategies children chose to solve problems, and began to believe she

needed to be more active in helping children progress especially those

children who used incorrect strategies or still counted by ones to solve

problems involving quantities in the hundreds.

Ms. Statzs deliberations about childrens buggy subtraction algorithms

illustrate these nascent changes. She discussed with the first author the

fact that the children did not connect their paper-and-pencil algorithms to

their knowledge of working with base ten blocks. For the first time, she

debated whether to show children how paper-and-pencil algorithms could

be modeled using base-ten blocks to make sure children understood:

I was even struggling with the idea of getting the overhead projector and doing it for the

whole class. Here are the base ten blocks, here is my marker and this is what we are doing.

But maybe we should try it, let them construct it themselves first?

I am struggling with how to go about doing that. Give them lots of take away problems and

try to go around to each person individually? That is the hard part. Thats what Im trying

to figure out, how to do that.

In summary, in this phase, Ms. Statz became aware of the inadequacy of

her knowledge of childrens strategies and some of the consequences for

her students. As she learned more about their thinking by sitting in on the

participant researchers one-on-one problem-solving interviews with her

students, dilemmas arose regarding her teaching, because she saw evidenceof rote or underdeveloped understanding. This phase was an intense one

for Ms. Statz, characterized by uncertainty and unresolved questions; she

reported she thought about these issues a lot, even when I first wake up.


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The net result was that Ms. Statz developed a deeply felt motivation to

learn more about the childrens thinking.

PHASE 3: LEVEL 4A ENGAGEMENT WITH

CHILDRENS THINKING

This phase was characterized by the integration of practical inquiry into

Ms. Statzs practice. Motivated by a need to know more about her students

thinking, Ms. Statz began to assess systematically individual children in

depth and decide how to use the information in planning instruction.

Because Ms. Statz was interested in specific questions such as how David

used counting on to solve Join Change Unknown problems, and how to

help Pang go beyond counting by ones to find triple-digit differences, she

required more talk with individual students in the context of instruction.

Talking with Children about their Thinking. In January, Ms. Statz began to

spend much more time with individual children at their desks. Previously,

these sessions usually lasted no more than a minute. Now they often ran for

more than 10 minutes per child or a pair of children: What I am noticing

in these last couple of weeks is that I am spending less time with all kids

and more time with particular kids.

Ms. Statz was especially interested in children who were using inef-

ficient or incorrect strategies. She began to probe their thinking more, to

help them use base-ten blocks to represent quantities in the hundreds and

thousands, and questioned them in ways that built on their thinking. For

example, to solve 378 + ? = 600, Mark started to represent the 378 withbase-ten cubes. Ms. Statz asked him if he could do it in his head, and

he replied that he did not think so. So Ms. Statz scaffolded a strategy that

followed his use of base-ten materials, but focused on operating on number

relationships instead of base-ten blocks. She asked him how much was

needed to get from 378 to 380 and he immediately gave the answer 2. Then

she asked how much was needed to get from 380 to 400; he answered 20.

And to get from 400 to 600, he knew it would take 200. Mark then totaled

the addends on paper, to get 222.

This time spent talking with individual students was fruitful for Ms.

Statz. Her growing knowledge of their thinking helped her to adapt instruc-

tion to their needs and motivated her to consider childrens thinking in

specific content areas:

I definitely think I am noticing more than I noticed before. And not only just noticing it but

knowing where to take it and how to push further and how to question more. . . .


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This period was also stressful to Ms. Statz, because as she learned more

about her students thinking, her questions grew:

I think I am more frustrated now about teaching math. . . . I am more exhausted . . . because

I am then spending more time thinking about it. Although the things that I think are comingout of it are good, I am seeing what needs to be done. I am spending more time with kids

who need specific things. I think it is making a difference . . . Now there are all these other

questions that are on top of it.

After a few weeks of focusing on specific children and interacting exten-

sively with them, Ms. Statz felt more confident in her knowledge of

individual children, but she was frustrated she did not have enough time

to do this kind of in-depth work with everyone. A new dilemma arose:

But sometimes I feel like Im spending too much time and neglecting the rest of the

classroom. Ive felt like that, more frustrated almost, this year because Ive needed more

time with each kid.

Ms. Statz wanted the benefits of talking extensively with children about

their thinking to extend to all. Thus, she started thinking of new ways to

organize her class that would allow her to spend more time with individuals

but ensure that all children were working and progressing.

When the participant researcher suggested that Ms. Statz would be able

to talk to more children if they worked in pairs, Ms. Statz rejected the idea,

because she believed working in pairs would not be beneficial to students

who used significantly different strategies to solve problems. Two days

later, Ms. Statz devised a solution that accommodated her concern. She

asked the children to choose partners who solve problems in a similar

way to you. Each pair got a sheet with the problems and a space for

two strategies. Each child could use his or her own strategy or the pair

could generate two strategies together. The children were fairly accurate

in choosing partners who were solving problems in a similar way. Ms.

Statz had time to work with individuals while the other children worked

with their partners. Having ten pairs to work with instead of 20 individuals

made the class more manageable for Ms. Statz.

To help children move forward to using more sophisticated strategies,

Ms. Statz told them she would ask both children from a pair to explain

his or her partners strategy at discussion time. Most children were able to

explain their partners strategy. When a child had difficulties, the partner

explained the strategy. More children were called to the board in this way,

during the discussion time, and giving opportunities to more children toshare strategies helped to solve another issue that concerned Ms. Statz.

Ms. Statz started thinking of how to choose problems to help children

progress:


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I wouldnt say it is any easier (to come up with the problems for the children). In fact, it may

be more challenging to decide what type of problems to use . . . I think I am thinking more

of the problems . . . and the kids who are doing the problems that I am writing specifically

for them.

In summary, in this phase Ms. Statz learned about her students solution

strategies by talking with children individually about their thinking. The

new need to spend much time with individuals stimulated her to think of

how to change the class organization. She began to experiment, an activity

that continued in the fourth phase. She continued to generate new questions

and look for solutions. This phase corresponded to Franke et al.s (2001)

Level 4A classification of engagement with childrens thinking. Ms. Statz

interacted with students individually to learn more about their thinking.

She acquired detailed knowledge of what her students understood and

the kinds of strategies they could use. But decisions about what to teach

were still largely driven by the global notions of childrens thinking and

the curriculum. In the final phase of change, Ms. Statz began to build onindividual childrens knowledge in instruction.

PHASE 4: TRANSITION TO LEVEL 4B ENGAGEMENT WITH

CHILDRENS THINKING: BUILDING ON CHILDRENS

THINKING IN WHOLE-GROUP DISCUSSIONS

In this phase Ms. Statz considered how her teaching practices influenced

childrens thinking, and how what she learned about childrens thinking

influenced her teaching practices. She began to experiment with using

the knowledge she gained from working with the children individually to

guide whole-class discussions.

Ms. Statz wanted to increase the number of children who presented

strategies to the group, yet felt that the sharing time was not productive

for many children because they were not listening to or thinking about the

presenters strategies. Ms. Statz was especially concerned about children

who might not understand the more sophisticated strategies. To address

this concern, she started to get students more actively involved in the

discussions: she would typically stop the child who was presenting and

ask the class or specific children what they thought the childs next step

would be. Ms. Statz asked other questions, in addition, such as Can you

tell what she did? How is her strategy different than somebody elses?

How can we make this strategy easier? Clearer?The children became more involved during discussions and Ms. Statz

increased the discussion time considerably, from 510 minutes early in the

year up to two consecutive lessons of 45 minutes of discussion for one set


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of problems. The discussions in this phase lasted on average 21.8 minutes

(based on 18 observations), whereas discussions in phases 1, 2 and 3 lasted

an average of 7.9 minutes (based on 16 observations).

Ms. Statzs efforts to elicit childrens thinking in discussions andinvolve the audience in responding to this thinking led to a breakthrough

in her engagement with childrens thinking. She started to use the whole-

class discussions to help individual children progress. This practice marked

a change in instructional goals and orientation toward the use of interactive

talk:

I never used the sharing strategies as a time to move the kids along. That was just a time

for the kids to be able to talk about their strategies . . . And maybe thats why theyre being

more focused on it (now). Because Im including more of them in the discussion. I have

given the kids more time to discuss their strategies. And I have used the information that I

am getting from their strategies to move other kids. In the past I would just have [student]

show the class that problem and that would be all, but not use it as a teaching moment, to

teach the rest of the class about renaming fractions or whatever it was.

Ms. Statz remembered individual childrens strategies or difficulties that

were elicited while working one-on-one with children and addressed them

in whole-class discussions. The way in which Ms. Statz began to use

whole-class discussions to help individual children move from strategies

based on counting by ones to strategies that incorporated base-ten concepts

was especially striking.

The whole-group discussion of the following problem typifies these

kinds of discussions: Ellen has 287 books. How many more books would

she need to have 400 books? First Ms. Statz called on Anne, who usually

solved problems like this one by writing all the numbers between 287

and 400 and counting them by ones. This time, Anne began to solve the

problem, with help of her partner, by adding 200 to 287 to get 487. The

teacher stopped her and asked the class what problem Anne was facing at

that point. Some children said that Anne had 87 too many. Ms. Statz asked

a few children what they would do to continue and then asked Anne if

that was what she did. She helped Anne and the rest of the class to do the

calculation 200 87, needed for the next step, by counting down 80 by

10s to 120, and subtracting 7 from 120 by taking away 5 then 2 more.

When Ms. Statz called on Jared, he was hesitant to share his strategy

with the class, because, he said, It will take me years. He drew tally

marks and wrote next to each single number: 288, 289, 290, 291, . . . The

teacher stopped him when he got to 310 (in his notebook he had drawn

tallies all the way to 400) and asked him what he needed to get from 300to 400. Jared said 10 10s or 2 50s. Other children suggested one 100 and

Jared added the 100 to the 13 tallies he made to count from 287 to 300.

Jared concluded by saying the strategy was really easy.


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Another child presented the beginning of his solution, writing 287 +

100 vertically. Ms. Statz stopped him and asked Anne again what 287

plus 100 was. Then she asked a few of the children, who tended to use

ones instead of tens or hundreds to calculate multidigit sums, a seriesof problems in which 100 was added (387+100, 487+100 . . . 987+100).

The child who solved the problem showed the next step: 387 + 10. Ms.

Statz returned to Jared and asked him to solve the problem. When he had

difficulty, she asked another child from the group she was targeting that

day. The next steps, involving 397 plus 3, then 100 plus 10 plus 3, were

handled by Ms. Statz in a similar manner.

The fourth child who came to the board solved this problem using

the standard subtraction algorithm (400 287 written vertically, with

regrouping from right to left). He, as well as other children, explained the

conceptual underpinnings of each step. For example, Mary explained that

she saw 400 as 40 tens: if you take 1 ten, 39 tens are left, so you just write

39 tens and a 10 on top.In this example, Ms. Statz used whole-class discussion to help specific

children, such as Anne, to construct base-ten concepts and to use more

advanced strategies. Her agenda that day involved helping these children

by building on her knowledge of their strategies. Before this lesson, Ms.

Statz rarely, if ever, attempted to build on childrens thinking in whole-

class discussions. Afterwards, she regularly used whole-class discussions

like this one to assist individual children.

Whole-Group Discussion and Inquiry into Mathematics. As Ms. Statz

continued to grow in her use of group discourse to advance childrens

understanding, she broadened her inquiry into new domains, such as multi-digit multiplication and division, and fractions. With these investigations

came new uses of interactive talk to build content in ways that were greater

than the sum of the individual contributions to a discussion.

For instance, to elicit childrens fraction thinking, Ms. Statz began by

giving partitive and measurement division problems that have fractions

as answers, and asked children to solve them using their own strategies

(Baker, Carpenter, Fennema, & Franke, 1992). Although Ms. Statz did not

instruct the students in specific strategies, all students were successful in

solving the problems and depicting fractional quantities using drawings

and diagrams.

Near the end of March, Ms. Statz, with much excitement, told the first

author what had happened in class. The day before, the children wrote andsolved their own word problems. Kanisha, a student who routinely counted

by ones to solve multidigit problems, wrote a problem similar to the kind


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of problems they have been solving: There were 20 cakes and there were

7 kids. How much cake will each kid get? Kanisha solved the problem

using an unconventional partitioning strategy based on repeated halving.

To begin, she gave each child two whole cakes. She then partitioned theremaining six cakes in half, and wrote the numbers 1 through 7 on the first

seven halves to designate 1 half for each of seven children (Figure 1).

Figure 1. Kanishas repeated halving strategy for sharing 6 remaining cakes among 7

children.

Continuing, she halved each of the remaining halves, to create enoughfourths for seven people. She wrote the numbers 17 to indicate giving one

fourth to each of the seven children. A half and a fourth remained in the last

circle. Ignoring the fourth, Kanisha divided the half into seven pieces and

again wrote 17 on each. (Ms. Statz later helped her create an appropriate

representation of equal pieces). Thus, at this point, each sharer had one

half, one half of a half, and one seventh of a half. Ms. Statz reminded her

to share the remaining fourth among the seven children and Kanisha did

that. Therefore, each child also got an additional one seventh of a fourth.

Ms. Statz then helped Kanisha decide the sizes of the fractional pieces.

She helped her see that, since the half was divided into seven pieces, each

piece was 1/14: We said, if there are 7 slices in one-half, how many pieces

will be in a whole cake? Kanisha knew it was 14. In a similar manner, she

figured that each of the seven pieces in the fourth was 1/28. Thus Kanisha

gave as an answer for her problem: 2 + 1/2 + 1/4 + 1/14 + 1/28.

Although this kind of strategy is not common in the standard teaching

of fractions, it is common in classrooms where teachers encourage children

to generate their own strategies (Empson, 1999). As the following account

of the whole-group discussion suggests, strategies based on non-standard

partitions of sharing situations, such as this one, can be mathematically

rich (Streefland, 1991).

Ms. Statz accepted Kanishas strategy and was excited by her achieve-

ment: The fact that Kanisha did that was fabulous. I thought it was really

good. But then what came out of it (in the whole class discussion). . .

was really cool. After Kanisha had presented her strategy to the class,

Ms. Statz asked the children what the result would be if the fractions were

combined. She thus used Kanishas strategy as a basis for posing a new


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problem to the class. Ms. Statz prompted the children by asking them how

many twenty-eighths were in one-fourteenth. Then they figured how many

twenty-eighths were in one-fourth and one-half and found that Kanishas

fractions combined to make 2 and 24/28 cakes.Another girl then said she saw an easier way to solve the problem:

divide the six cakes that were left into seven pieces each (because there

were seven children sharing) and each child gets one seventh from each

cake for a total of 2 and 6/7 cakes for each child. Ms. Statz used the

two apparently different answers as an opportunity to explore the idea of

equivalent fractions, which was new to the children. She again posed a

new problem to the group by asking how they could decide whether the

two amounts were the same or not. Ms. Statz did not see immediately how

to help the children answer this question meaningfully and she resorted to a

symbolic technique based on reducing 24/28 to 6/7 by finding the greatest

common factor. However, the following year, Ms. Statz used opportunities

like this one to elicit childrens informal justifications about how equiv-alent fractional amounts were related (e.g., Empson, 2002, Figure 6; this

example of a students work came from Ms. Statzs classroom).

In this example, we see how Ms. Statz not only reacted to and built

on childrens strategies spontaneously in discussion, but also used the

problem and childrens different strategies as a basis for new, more

challenging problems. Through Ms. Statzs orchestration of the groups

discussion, the class explored mathematics topics that went beyond each

childs effort.

In summary, in this phase Ms. Statzs knowledge of childrens thinking

continued to grow. We classify her engagement with childrens thinking

as Level 4b on the CGI scale, because she used knowledge of specificchildrens thinking to inform classroom interactions. She found ways to

build on the childrens strategies in whole-class discussions and not just

individually.

Continued Inquiry

As Ms. Statz continued to grow in her use of group discourse to advance

childrens understanding, she broadened her inquiry into new mathematics

domains. With these investigations came new uses of interactive talk to

build content in ways that were greater than the sum of individual contri-

butions to discussion. The following year, the second author documented

Ms. Statzs continued inquiry into childrens thinking in the domains offractions and multidigit multiplication and division (e.g., Baek, 1998;

Empson, 2002). Later, Ms. Statz became involved in inquiry into childrens

algebraic thinking (Carpenter, Franke, & Levi, 2003).


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Retrospective Reflection

In a career interview several years after the events reported here, Ms.

Statz was asked about her growth as a teacher. She attributed her growth

in teaching to two main factors: a second pair of eyes in the classroomfocused on childrens thinking, and the freedom to experiment with

instruction based on childrens thinking.

A second pair of eyes focused on childrens thinking. When asked to

describe the most important turning points in her teaching, Ms. Statz

mentioned the times, such as in the year reported here, when she had a

second person in her classroom, who was knowledgeable about childrens

thinking and who could see and describe things to her that would have

otherwise gone unremarked. Interactions with people like the mathematics

resource teacher, or researchers such as us, provided some of the raw

material for Ms. Statz to reflect on her students knowledge. Her priorbeliefs in the value of childrens thinking provided some of the motivation

for this reflection.

Freedom to experiment. Ms. Statz credited the freedom she had to give

children problems to solve, to talk to her children about their thinking, and

to experiment with interactions with students for the growth she experi-

enced as a teacher. In her current role as a mathematics resource teacher,

she worried that an emphasis on teaching using printed curriculum mate-

rials even standards-based curriculum programs may prevent teachers

deep engagement with childrens thinking. Because of an increased

emphasis, in many districts, on following these programs, Ms. Statz said

that teachers do not have the same kinds of opportunities to experiment andfind out what their children know and can learn. They feel that they do not

have the freedom to have discussion sessions that last 45 minutes because

there is so much to cover. Ms. Statz believed that, unless teachers are able

to have lengthy discussions with children about their thinking, they will

not be able to learn from their teaching or at least not the same kinds of

things about childrens thinking that she had learned.

DISCUSSION

This case study documents how Ms. Statzs engagement with childrensthinking changed dramatically in a period of only a few months. In Phase

1, children talked about their strategies, and Ms. Statz listened, but rarely

challenged children to extend their thinking or referred to their strategies


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in later discussion. In Phase 2, as the participant researcher shared infor-

mation with Ms. Statz about how individual children were thinking, Ms.

Statz realized there was a discrepancy between what she knew about her

students problem solving and how her students actually solved problems.She realized, in particular, that some students used mistaken strategies and

others used very basic strategies, and was concerned by this information. In

Phase 3, Ms. Statz began to make time in her instructional routine to talk

to children in more depth about their thinking, for her own benefit. She

concentrated on students whose thinking she believed to be problematic

in some way, and struggled with how to support these students learning

by building on their thinking, rather than imposing her own. Her solu-

tion involved engaging students in talking to other students who solved

problems in a similar way. Finally, in Phase 4, Ms. Statz began to use

information gathered in one-on-one interactions to build on childrens

thinking in group discussions. Instruction was guided by knowledge of

individual childrens thinking. Ms. Statz continued to benefit from talkingwith children about their thinking, but now that talk was also designed to

help children advance.

The extent of the change over the course of the study is especially

striking, given the fact that, at the beginning, Ms. Statz already used

many reform-oriented ideas in her teaching and believed that children

should construct their own knowledge. Her engagement with childrens

thinking corresponded, at the beginning of the study, to Level 3 in Franke

et al.s (2001) scale. By the end of the her third year of teaching, Ms.

Statzs engagement with childrens thinking was characterized by gener-

ative learning, and corresponded to Level 4b engagement with childrens

thinking.We argue that the primary driving force behind the process of change

was Ms. Statzs need to know more about childrens mathematical

thinking, and her pursuit of this knowledge in interaction with students.

This need was founded on her beliefs about the importance of student-

generated strategies, first fostered in her pre-service teacher-education

courses, and on her realization of gaps in her knowledge of her students

thinking. It was nurtured by her participation in the discourse community

of CGI teachers and researchers. As she organized her interactions with

children to learn more about their thinking, new dilemmas arose about

how to increase childrens opportunities to express their thinking and to

learn by listening to other childrens thinking. These dilemmas, in turn,

led to solutions that allowed Ms. Statz to continue to learn about childrensthinking while children learned of each others thinking. Ms. Statz began

to use whole-group discussions not just as displays of thinking, but also


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as arenas for building on thinking. Each of these reflection cycles was

grounded by increasingly detailed knowledge of childrens thinking. Ulti-

mately, learning about childrens thinking was integrated with classroom

participation structures to elicit and build on childrens thinking. In thisway, Ms. Statzs learning about childrens thinking became generative.

CONDITIONS FOR TEACHER CHANGE

In outlining the conceptual framework for this study, we discussed three

conditions for teacher change based on the research literature. We revisit

these conditions here, and speculate about their contributions to Ms. Statzs

change as a teacher.

1) Membership in a discourse community. The CGI framework for

childrens thinking provided a basis for conversation and other kindsof interactions about a phenomenon central to Ms. Statzs work as a

teacher: childrens mathematics learning. By virtue of interactions with

old timers (Lave & Wenger, 1991; Wenger, 1998) such as Ms. J and

the participant researchers, Ms. Statz became an increasingly knowledge-

able member of this discourse community. Because the tools for thinking

about her teaching provided in this discourse community intersected with

the problems that were most pressing for her as a teacher, Ms. Statz was

motivated to use and adapt these tools for herself. Although there is no

single point at which Ms. Statz can be said to have joined this discourse

community, her opportunities for engaging in it were multiple and occurred

in a number of contexts.

2) Processes for reflectively generating, debating and evaluating new

knowledge and practices. The processes for producing new knowledge

and practices identified in this case study were, for the most part, infor-

mally organized. Other than during her pre-service teacher education, Ms.

Statz did not participate, in her early teaching years, in formally organized

learning opportunities, such as professional development workshops in

mathematics. However, these informal processes were powerful for her,

perhaps because, in partnership with old timers in the CGI discourse

community, she was able to formulate and address some of the most

pressing practice-based dilemmas.

We identified two sorts of processes in particular for generating newknowledge and practices at work in Ms. Statzs third year of teaching.

The first process involved the participant researcher making available new

information to Ms. Statz about her students thinking, through conver-


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sation, examination of students written work, and one-on-one problem

solving interviews observed by Ms. Statz. The process of examining

her students thinking in partnership with the participant researcher (a

second pair of eyes) served as a kind of scaffolding to Ms. Statzs owninquiry into childrens thinking by placing in the foreground aspects of

her students thinking she had previously not seen. The second process

involved Ms. Statzs independent inquiry into students thinking. As she

took ownership of questions about childrens thinking, as well as the

outcomes, she became engaged in practical inquiry. This inquiry included

questions about class organization such as how to assist struggling

children, how children learn from each other, and how to conduct mean-

ingful discussions. Thus, processes for generating and testing knowledge

about childrens thinking became integrated into Ms. Statzs teaching as

she created opportunities for herself, and then students, to hear childrens

thinking.

In contrast, at the beginning of the study Ms. Statz did not learn fromher students in the manner in which she did later, even though she gave

them opportunities to solve problems in their own ways and to talk about

their strategies. We suggest a change in Ms. Statzs perception of her role

as a teacher, from passive to active, provided a motivation to learn more

about what her students were doing in order to use that knowledge to help

them advance. She realized that, even as a teacher who valued childrens

informal thinking, she could have goals that called for childrens thinking

to progress.

We speculate this passive role is a common step in teachers develop-

ment. It seems there is a need at the beginning of teacher change to step

back, and not intervene in childrens problem solving very much (Jacobs& Ambrose, 2003). After becoming convinced children can generate their

own solution strategies, teachers become active again but in a different way

from before, by: helping students develop their own strategies; helping

students who do not understand the meaning of the problem; helping

them express their solutions in multiple forms; asking probing ques-

tions; and leading discussions that build on childrens ideas and stress the

mathematics content of those ideas.

3) Ownership of change. Ms. Statzs transition to practical inquiry is

evidence of ownership of the change she experienced. That Ms. Statz

made this transition may be due, in part, to the nature of the participant

researchers interactions with her. The participant researcher did not giveMs. Statz ready-made activities, but encouraged her to make her own


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decisions about instruction. Reflecting later on the process she experienced

in this collaboration, Ms. Statz commented:

You allowed me to voice my concerns. And you were somebody to listen to the things that

I had problems with. You gave suggestions. Yet you also said: Its up to you. Do it yourway. Try it your way. Its up to you with your class. I guess I learned to stop asking for

advice and I learned to start thinking on my own. Because I knew you would say, What

do you think? So then I was already doing some of the thinking and trying it out on you

more.

Because of the participant researchers insistence that Ms. Statz knew her

own students best, Ms. Statz developed a predisposition to ask and answer

her own questions, which led to a sense of professional autonomy. Further,

without the freedom to experiment with the curriculum, cited by Ms. Statz

as key to her development as a teacher, she may not have developed this

predisposition, no matter what the participant researchers stance was.

IMPLICATIONS FOR PRACTICE

What have we, as researchers and teacher educators, learned from

conducting this study? How has the collaboration between teacher and

researcher-teacher educator helped us educate other teachers? We present

some insights based on the first authors subsequent experience with

teachers from about 80 schools in Israel, many of whom made major

changes in their teaching and pre-service education. Some adjustments

were necessary in adapting teachers use of CGI to Israeli classrooms,

because of larger class sizes (3540 students), a different culture, and

the national curriculum. The findings of this study informed the profes-

sional development work in Israel in three ways. In her work with

teachers, the first author focused on 1) eliciting and interpreting childrens

thinking, 2) building on childrens thinking in one-on-one interactions,

and 3) building on childrens thinking in group discussions. The teacher

development program included study of childrens solution strategies, use

of challenging problems, encouraging a variety of solutions, discussion

of classroom organization, and examination of teachers beliefs about

the kinds of problems children can solve without direct instruction in

strategies. Special importance was attached to understanding and aiming

for the highest levels of teacher development in Table I. It is a difficult

task for teachers to obtain a clear picture of a students current level,to understand his/her difficulties and to help in a manner that builds on

his/her thinking. Because it was not feasible to have a second pair of

eyes in each classroom, the teacher development included analyses of


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written or videotaped examples of individual students and teachers inter-

acting with students from participating teachers classrooms. A second

important topic is how the teacher can stimulate class discussions that

build on childrens thinking and help them progress. The experience withMs. Statz was an important catalyst in bringing this topic to the forefront

of the teacher development. For example, one useful activity was to ask

the teachers to bring four examples of childrens strategies on a problem

and to think how they could build a discussion around them. This activity

helped the teachers understand what sorts of questions they could ask and

what mathematical ideas to emphasize.

CONCLUSION

An enduring problem in teacher change is the tension between inducting

teachers into new instructional practices and respecting teachers profes-sional autonomy. In this study, these tensions were represented, respec-

tively, by CGI and Ms. Statzs personal teaching dilemmas. Part of Ms.

Statzs learning concerned learning problem types and solutions strategies;

but the other, more important part had to do with learning how to use

this knowledge and how to generate this kind of knowledge by/for herself

in practice that is, to conduct practical inquiry into childrens thinking.

Ultimately, this practical inquiry was integrated into her interactions with

children and became generative. The result was a body of knowledge

for Ms. Statz that was richer and more complex than CGIs research-

based framework for childrens thinking because it was informed by

the concrete particulars (Lampert, 1985) of her own practice-based

dilemmas, and driven by her growing knowledge of her own students

thinking. Ms. Statz reported that she began teaching with a strong belief

in the value of childrens thinking. Many teachers have such a belief, but

without specific knowledge of childrens thinking, they may not be able

fully to implement it.

We conjecture that mechanisms that help teachers see their students

thinking in new ways, combined with the freedom to respond to, and

experiment with, this information about childrens thinking are key to

the development of practical inquiry in teachers. More specifically, a

turning point for Ms. Statz was the realization that talk with children about

their thinking was valuable, not only because it provided opportunities

for students to articulate their thinking, but also because it provided acontext for her to ask and answer questions about childrens thinking for

herself. As Ms. Statz learned how to use the information gathered in these

interactions, she began to influence the direction of these conversations


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through specific questions about cognitively, socially, and mathematically

appropriate extensions of individual childrens thinking.

Although Ms. Statz accomplished remarkable change during the course

of the study, the process was difficult. The experience of phases of uncer-tainty and conflict that had no obvious solutions was emotionally trying.

Yet Ms. Statz was open to seeing and responding to these dilemmas, even

though she was not sure what would result. We believe this openness is

attributable to a school atmosphere that was open to teachers experimenta-

tion, and the emphasis by the participant researcher on Ms. Statzs capacity

to ask and answer questions about her practice.

Finally, whatever form participation in a discourse community takes,

we believe it must emphasize the teachers professional autonomy. This

emphasis reinforces the capacity of teachers for practical inquiry, and

provides a means for the discourse community itself to adapt and remain

vital in response to new perspectives.

ACKNOWLEDGEMENTS

We would like to thank Kathy Statz, the teacher who collaborated in this

study, for her major contributions; Ellen Ansell, Linda Levi and Debra

L. Junk, for their feedback on earlier drafts of this manuscript; three

anonymous JMTE reviewers and editor Peter Sullivan for their suggestions

for improvement; and Lou Her for translating interviews from English to

Hmong for two students. The first and second authors thank the University

of Wisconsin-Madison for support as visiting scholars in the Wisconsin

Center for Education Research and the Department of Curriculum and

Instruction, respectively, during part of the time this study was conducted

and written. The research reported in this paper was supported in part by

the National Science Foundation under Grants No. MDR-8955346 and

MDR-8954629. The opinions expressed in this publication are those of

the authors and do not necessarily reflect the views of the National Science

Foundation.

NOTES

1 Her real name, used with her permission.2

Drake, C. (under review). Mathematics stories: The role of teacher narrative in theimplementation of mathematics education reform.3 Children initially direct model story problems, representing each object in a story

problem one for one in the strategy, and acting out the semantic structure of the story

with these objects. For example, to solve a Join Change Unknown problem, such as Lucy


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had 7 dollars. How many more dollars does she need to buy a puppy that costs 11 dollars?

by direct modeling, a child would represent the first set of dollars with 7 objects (e.g.,

counters, tallies), and join other objects to the set until there was a total of 11 objects.

The child would then count the set that was joined to the initial set for the answer to the

story problem. Strategies beyond direct modeling include counting, deriving facts, andimposing a different semantic structure on the problem (see Carpenter et al., 1999 for more

information).

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2004 Inquiry Into Children’s Mathematical Thinking

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