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Professional Development 1

Running Head: PROFESSIONAL DEVELOPMENT

Relating professional development to the classroom

Elham Kazemi

Anita Lenges

University of Washington

Correspondence to:Elham KazemiAssistant Professor, Mathematics EducationUniversity of Washington122 MillerBox 353600Seattle, WA 98195-3600Office: (206)221-4793Fax: (206)[email protected]

Anita LengesDoctoral Student, Mathematics EducationUniversity of [email protected]

mailto:[email protected]

mailto:[email protected]


Abstract

Much professional education with elementary teachers is designed to support them to recognize

the mathematical ideas inherent in students’ work and use the diversity of mathematical thinking

in the classroom to advance students’ understanding and reasoning. In this study, we present

classroom episodes that characterize the diversity of instruction we observed in the classrooms of

six teachers who had participated in a substantial amount of professional learning focused on

understanding children’s mathematical thinking. We explain the diversity in classroom practice

theoretically by viewing the professional development seminars and the classroom as forming

distinct communities with their own norms, practices, and tools. Teachers may become full

participants in a professional development community in which they have a high degree of

support to puzzle over and deliberate students’ mathematical thinking. Yet the classroom,

responding to both external and internal goals, with its own practices and tools, may offer

different resources for learning and being. What teachers do in any given day in their classroom

is a result of heeding many—sometimes contrasting or competing—images of what it means to

teach mathematics. The episodes we present in this paper raise a number of questions about

characterizing the coherence of instruction and understanding the complex relationship between

professional development experiences and classroom practice.

Key Words: professional development, classroom instruction, using children’s thinking in the

classroom, instructional coherence


Relating Professional Development in Mathematics to the Classroom

Much of mathematics professional education with elementary school teachers focuses on

examining children’s mathematical reasoning (e.g., Carpenter et al., 1996; Saxe et al., 2001;

Schifter et al., 1999; Stein et al., 2000). This emphasis is designed to help teachers recognize the

mathematical ideas inherent in students’ work and use the diversity of mathematical thinking in

the classroom to advance students’ understanding and reasoning (Driscoll, 1999; Fennema et al.,

1998; Lampert & Ball, 1998; Stein et al., 2000). A focus on children’s thinking can also deepen

teachers’ understanding of mathematics (Ball & Cohen, 1999; Schifter, 1998). Even when

teachers find their experiences in such professional development to be transformative, they may

not consistently engage in practices in their own classrooms that elicit and build children’s

reasoning. To explore the questions raised by this issue, we describe the variations in instruction

we saw in the classrooms of six teachers who had participated in 120 hours1 of professional

education focused on understanding children’s thinking. The diversity in lessons suggests that

students and teachers have varied experiences with what it means to learn and know

mathematics. These findings underscore the need to understand the complex relationship

between professional development experiences and classroom practice.

Literature on Professional Development

Current curricular goals for teaching mathematics are commonly viewed as ambitious—

enacting them in classrooms is anything but simple. In the research literature there are at least

three different kinds of studies that have examined the connection between professional

development and classroom practice. Each of these bodies of literature rests on the assumption

that professional development experiences enable teachers to develop new skills, knowledge, and

dispositions, which can lead to instructional change. We review these literatures briefly in order


to frame our approach to understanding the relationship between professional development on

classroom practice.

One set of studies focuses on describing the change individual teachers undergo over

time as they learn to understand children’s reasoning processes (Fennema et al., 1993; Fennema

et al., 1996; Heaton & Lampert, 1993; Lampert, 1984; Schifter, 1998; Wood et al., 1991).

Changes in teacher beliefs and knowledge accompanied by changes in teaching practice are

central to these studies. This body of work has shown that teachers’ beliefs about the role of a

teacher shifted from one who demonstrates procedures to one who actively supports children to

build mathematical knowledge through engagement in mathematical argumentation and problem

solving (Ball & Bass, 2000). More recently, studies of individual teacher change have addressed

the situated nature of teachers’ learning trajectories by emphasizing how the development of new

knowledge and skills is necessarily linked to the nature of teachers’ participation in professional

development and to their evolving intellectual and professional identities (e.g. Franke et al.,

2001; Franke & Kazemi, 2001; Hammer & Schifter, 2001; Rosebery & Puttick, 1998).

Importantly, these studies have primarily focused on the nature of individual change and have

not addressed the institutional or sociopolitical forces that impact teachers’ instruction.

A second set of studies, in contrast, has been concerned with the relationship between

policy and practice, documenting how policy environments influence teacher practice. Such

policy initiatives are often accompanied with professional training linked to new curriculum

adoptions. Such research has emphasized how teachers’ own conceptions and interpretations of

the goals embodied in new policies or standards documents impact their classroom practice

(Ball, 1990; Cohen, 1990; Grant et al., 1996; Heaton, 1993; Spillane, 2000; Spillane & Zeuli,

1999). This work has helped us understand that the connection between new policies and


classroom instruction is not seamless. Teachers filter and adapt their professional development

experiences through their own experiences. The tenor of much of this work has been that

teachers have different visions of what the discourse about mathematics reforms mean in their

classrooms, and these visions may not coincide with the intents of policymakers and standards

documents. Researchers have argued that this misalignment is not surprising given that teachers

are being asked to create forms of instruction that they themselves did not experience as students

(Little, 1989; Little, 1993). McLaughlin (1990) has described this as a process of mutual

adaptation—policy may change teachers’ practices, but teachers, through the ways they enact

policy, in fact change policy.

A third set of studies has focused on evaluating the merits of particular forms of

professional development in mathematics on teacher learning from intensive institutes focused

on content to Lesson Study and study groups (e.g., Crockett, 2002; Lewis, 2000; Saxe et al.,

2001; Simon & Schifter, 1991; Simon & Schifter, 1993). From these studies, we have learned

that professional development efforts that have a clear focus, are ongoing and more closely tied

to teachers’ own classrooms have a stronger impact on classroom practices than either one-shot

workshops or collegial meetings where teachers share new ideas but do not necessarily work

towards achieving a particular goal (Cohen & Hill, 1998; Garet et al., 2001). Recent studies

have also begun to explore what teachers gain from professional development that is centered on

the study of artifacts of practices, such as written or video cases and student work. (e.g., Barnett,

1998; Franke & Kazemi, 2001; Sherin, 2002; Smith et al., 2001)

Our study contributes to these bodies of work because it too is concerned with the

relationship between teachers’ professional development experiences and their classroom

practices. We aim to contribute to understanding the relationship between professional


development and classroom life as issues of understanding connections between two distinct

communities of practices (cf., Wenger, 1998). Viewed in this way, the problem of assessing the

impact of professional development on classroom practice means not only attending to teachers’

knowledge, skills, and beliefs but also to the varying values, tools, practices, and sociopolitical

goals of each community.

This Study

Developing Mathematical Ideas (DMI, Schifter et al., 1999) is an example of professional

development curricula that heeds many of the current calls to engage teachers in long-term

experiences that develop their knowledge of children’s mathematics by situating discussions in

real episodes of classroom instruction. In each of five published seminars, teachers discuss

written and video cases, engage in doing mathematics together, and have opportunities to explore

student reasoning in their own classrooms. Each seminar focuses on a different mathematical

domain: place value, operations, measurement, geometry, and statistics.

Participation in DMI encourages teachers to discuss many vivid examples of classroom

discourse that productively elicit student thinking. The seminar materials are meant to achieve

several goals with teachers, including (a) developing teachers’ mathematical knowledge, (b)

supporting teachers to make sense of children’s thinking and connect those understandings to

instructional goals, and (c) encouraging teachers to engage their own students in discussions so

that they can analyze and support their mathematical ideas. In this study, we asked how teachers

who had experience with DMI and summer content institutes enacted principles of eliciting and

building on children’s thinking in their own classrooms. We collected classroom data from six

teacher leaders who spoke positively and confidently about their participation in DMI to see how

they interacted with students and facilitated mathematics lessons.


We characterize all of the study participants as skilled teachers who have learned how to

help children articulate, deliberate, and extend their mathematical understandings. We have

evidence of each teacher’s ability to interact with children in those ways. And we also found that

the teachers did not always facilitate lessons in which they actively attempted to elicit and build

on children’s thinking. The mathematics instruction tended to vary within classrooms. We will

present five vignettes to represent these varied enactments in the classrooms that we visited and

explain from the teachers’ perspective why they were engaging in those particular forms of

instruction. The lessons varied in the cognitive demand of the mathematical tasks (Stein,

Grover, & Henningsen, 1996) and the nature of the mathematical discourse.

We interpret the findings of our study in light of the settings in which these teachers

work. We have evidence that the teachers in this study have learned about children’s reasoning

because of their experiences in DMI. In fact, as mathematics leaders in their district, the teachers

in this study committed much of their time to learn how to help other teachers learn about

students’ reasoning. We view DMI as a tool that has helped teachers learn about student

thinking and raise questions about classroom practice. We interpret teachers’ choices about how

to teach and what to teach, during the lessons we observed, not just a matter of their own

personal preferences or a direct result of their experiences in professional development. Put

simply, we do not view the variation within classrooms as a classic issue of “transfer.” Rather,

we recognize that these teachers work in a particular historical moment in a particular place.

State and district policies, curricular resources, and the particular school and district cultures in

which they work interact with their own personal commitments to create varied forms of

teaching mathematics.


Theoretically, we understand these experiences by viewing the contexts in which teachers

work as forming distinct communities of practice with their own norms, practices, and tools.

Teachers may become full participants in a professional development community in which they

have a high degree of support to puzzle over and deliberate students’ mathematical thinking (cf.,

Wenger, 1998). But what does it mean for teachers to draw on that participation in their

classrooms in which the resources for learning and being, in terms of practices and tools, are not

always the same? We argue that what teachers do in their classrooms is a result of heeding many

—sometimes competing—voices of what it means to teach mathematics. The vignettes we

present in this paper raise a number of questions for us about characterizing the coherence of

instruction as well as both students’ and teachers’ experiences during mathematics lessons.

Further, we suggest implications for the way we design continued professional development if

our goal is to produce coherent mathematical experiences in classrooms where teachers work to

advance children’s thinking.

Method

Participants and Data Collection

The participants in this study included six teachers who were among the first tier of

“volunteers” in a multi-district five-year project aimed at enhancing teachers’ professional

development in mathematics and developing leadership capacity within each district. The

teachers in this study taught in different schools in the same district that had adopted Everyday

Mathematics (EM; Bell et al., 1999) several years prior to the study. As a central part of the

leadership project, teachers first participated in DMI seminars and were later given opportunities

to develop the knowledge and skills needed to facilitate seminars for other teachers. At the time

of the study, the six teachers had participated in two number sense modules (Building a System


of Tens, Making Meaning for Operations) and were participating in their third module, Statistics:

Working with Data. They had also attended two week-long summer institutes designed to

extend their experiences with the content of the number sense modules and their skills in

facilitating them. All of the teachers in this study reported their experiences in DMI to be both

positive and powerful, characterizing DMI as some of the best professional development they

had experienced in their careers. At the time of the study, their teaching experience ranged from

5 to 27 years. One teacher taught first grade, the other teachers taught third or fourth grade.

During the 2000-2001 academic year, four classrooms were observed by a member of the

research team at least three times, but scheduling problems allowed only two visits to the two

remaining classrooms. The researcher stayed for the entire duration of the mathematics lesson,

which typically ranged from 60 to 90 minutes. During each visit, the researcher took detailed

fieldnotes of classroom instruction and collected artifacts from the lesson. After each

observation, the researcher reviewed the fieldnotes, filling in any additional details not captured

in the moment of observation. During whole group discussions, the researcher scripted the talk

as closely as possible, reproducing any representations drawn on the board or overhead. During

small group or independent work time, the researcher noted the teacher’s movement and

interactions with students around the room. When students worked independently, the researcher

also talked to individual students about their problem solving efforts on the assigned task. At the

end of the year, each teacher was interviewed about her experience in the professional

development and leadership project (see interview protocol in appendix). The authors of this

article also interacted regularly with the participating teachers during DMI seminars and other

activities related to the larger leadership project.


Data Analysis


Fieldnotes and transcribed interviews were entered into a qualitative data analysis

software package for easy retrieval and coding. We reviewed the fieldnotes for each observation

several times. We discussed the nature of classroom lessons for each teacher. For each lesson,

we noted the nature of the task, the role of the teacher in eliciting student thinking, the

mathematical goal for the lesson, and the nature of discourse during the lesson. When teachers

used the EM curriculum, we compared the way the lesson was enacted to the directions in the

teacher’s manual for the lesson, noting deviations from the instructional guidelines. In the

process of our analyses of each lesson, we found that we could characterize lessons into three

macro categories: (a) Lessons that focused on the teacher presenting students with a particular

approach to solving a problem that could then be practiced by all; (b) Lessons that focused on

eliciting student reasoning and facilitating discussions so that students could compare their

approaches; and (c) Lessons that involved an indirect method (i.e., computer programs) for

students to practice particular mathematics skills. We noticed that while we made few visits to

the classrooms, we saw evidence of these various types of lessons across the teachers. We made

a matrix to look at which of these kinds of lessons we observed for which teachers. We further

noted that the type of lesson was linked to the use of particular materials. Lessons that focused

on presenting students with a particular approach were drawn rather faithfully from a set of

problem-solving steps (see Figure 1) used widely in the district or from the EM curriculum.

Lessons that elicited and focused on student reasoning either stemmed from tasks teachers were

posing as “homework” for a DMI seminar or modifications to the EM curriculum or another

curricular resource. The third kind of lesson was linked to the use of a computer program called

Accelerated Math ematics in which students practice skill and fact-based multiple choice

problems (see Table 1). In the findings below, we will present vignettes of each kind of lesson,


showing variations we saw across teachers and provide our analytic commentary to describe

differences in the nature of the task and the classroom discourse that each kind of lesson

generated.

Findings

Based on our analysis of the classroom data, characterizing each teacher’s instruction

globally became a difficult task. We found instead that mathematics instruction differed

according to how the teachers used particular materials to frame the task. We draw on our

fieldnotes and interviews with teachers to characterize these different lessons. The findings are

organized around five main vignettes; together they represent the range of kinds of lessons we

observed in teachers’ classrooms (refer to Table 2 to see in which teachers’ classrooms each type

of lesson was observed).

All six teachers described DMI as helping them learn more about student understanding.

They recognized the work they had done to explore and deepen their own mathematical

knowledge and to examine students’ mathematical reasoning. The goals of DMI are well

reflected in their own words (see Table 2). Each teacher emphasized that DMI helps them know

what “kids are thinking” and to use that knowledge to make decisions about how to “move kids

toward what my goals are for them.” We share the teachers words to provide some evidence of

each teachers’ enthusiasm for and commitment to building students’ understanding of

mathematics.

Vignette 1: Using Problem-Solving Steps

Following the adoption of EM in 1997, the district (in which the participants taught)

created grade-level mathematics specialists. These specialists supported classroom teachers to

implement EM and to prepare students for the state mathematics assessment. They felt that EM


needed to be supplemented with word problems, and so they gathered, developed, and distributed

them to all the teachers in the district. From her position as a classroom teacher, one of the six

participants in this study advocated for district-wide use of a set of problem-solving steps and

strategies (see Figure 1). She felt that their use would help focus students on important features

of word problems and help them to succeed on the state mathematics assessment. These

resources, then, were developed before any of the study teachers began their participation in

DMI seminars. The steps (e.g., underline the question, circle the data) lead students through a

procedure to use while the strategies (e.g., draw a picture, use objects) direct students to consider

general approaches one might use to solve a problem.

During classroom visits, we observed three teachers conduct six lessons during which

they guided students to use the Problem-Solving Steps. One other teacher, Ms. Carlson, told us

in interviews that she uses the problem-solving steps, but we did not observe her using them. In

all six lessons that we observed teachers using problem-solving steps, the lesson began with a

discussion in which students identified and enacted the first three steps (i.e., underline the

question, circle the data). We begin with one episode that typifies what we observed as teachers

worked with their students to prepare to solve their word problems. Then we describe how the

three teachers proceeded with the lesson. Finally we analyze the images of problem solving that

are created through teachers’ varied uses of the Problem-Solving Steps.

The following episode from Ms. Foster’s third-grade class shows how students and

teachers negotiated the first three problem-solving steps in the six word problem lessons we

observed.

One March day, Ms. Foster posted an overhead transparency of the Problem of the Day:

The students in Mr. Fischer’s class wanted to roller skate in the gym. They had to change the wheels on their skates to rubber ones. They had 36 rubber wheels. How


many skates could they change? How many students could skate at a time? (Fieldnotes, 3/12/01)

The class began with the following talk:

Ms. Foster: Circle the data. You should be pros at that since you just did the ITBS test. Write the date, March 12th. This is like the ITBS. What information do we circle?

Mark: 36 rubber wheels ((suggesting this is important data in the problem))Ms. Foster: ((Moving along since she agreed with the student’s answer)) What are you

supposed to do?Mark: Change the wheelsMs. Foster: You need to change to rubber wheels for the gym. Bring the data down

[which is] “Rubber wheels” ((writing it on the board)) Other questions? ((Encouraging students to determine what other parts of the problem-solving steps they need to address prior to solving the problem))

Jen: Questions. ((They need to underline the questions that the problem is asking them to solve.))

Ms. Foster solicited the questions that were posed in the written problem from students,

paraphrased them and wrote them on the overhead below the data that she had “pulled down” so

that the overhead now read:

Rubber wheels36? How many skates change? How many students could skate at a time

The discussion continued with another student offering that they should agree on how many

wheels skates have.

Tess: [We] need to know skates have 4 wheels.Ss: some have 3

The class negotiated whether skates have 3 or 4 wheels each.

Ron: Can we use 3?Ms. Foster: Yes, support your data ((allowing students to use either three or four wheels

to solve the problem as long as they were clear about their choice))Ms. Foster: ((ended the discussion by encouraging students to)) Show words, numbers

and pictures. (Fieldnotes, 3/12/01)


All three teachers in the six problem-solving lessons we observed, led their students to

underline the question, circle the data and write the data below the word problem on their own

papers while the same steps were modeled on the overhead. After this preparation, each teacher

engaged students differently in the process of solving the word problems.

Ms. Foster maintained dialogue throughout the process of solving problems. In the

lessons we observed, Ms. Foster taught math for one and half hours. The first 45 minutes were

devoted to solving word problems, the second 45 minutes were for EM. Ms. Foster asked

students to take out their journals and glue the day’s problem to the top of the page. The class

proceeded with following the problem-solving steps. After the class circled the data, underlined

the question and so forth, the preparation blended into a class effort to represent and solve the

problem. Over the three lessons using the problem-solving steps in Ms. Foster’s classroom,

between 8 and 12 students moved in and out of class discussion while solving the problem. In

all three lessons observed, Ms. Foster’s class had ongoing dialogue through the 45 minutes of

working on the problem, often procedural in nature, and sometimes gave students a minute or

two to work on the problem quietly or in groups. Several student-volunteers were invited to

share their ideas, representations, strategies and solutions with the class, either drawing on the

overhead or acting out a situation with their peers. Ms. Foster requested that students write their

explanations in their journals, and on one observation reminded students to enumerate their work

so it is clear what they did first, second and so on. Finally when the solution was written on the

overhead, Ms. Foster drew a box around the answer. Allowing the children to share strategies

reflected what she said she gained from participating in DMI. In the end of year interview, Ms.

Foster said, “I think the communication piece is really, really strong with DMI. You know, the

opportunity for dialogue, diverse thinking, and honoring every student’s thought process….”


Problem solving, in Ms. Foster’s class, was accomplished at a whole group level. Students who

were not part of the discussion, waited for the answers that were developed by the other students,

solved the problem alone, or did not solve it at all.

Ms. Bryant reported that her students solved word problems one day per week. Her

reasoning was that on Monday through Thursday she worked with students from a split-grade-

level class next door. She could not advance work from EM on Fridays without the students

from the other class. On Fridays, students solved problems independently so she could evaluate

how well students matched a scoring guide for effective problem solving. She used this as a way

to learn how her students were solving problems. The problem-solving steps gave students a

framework for focusing on and writing about the steps they took to solve problems.

Contrasting with Ms. Foster’s approach, Ms. Bryant used the problem-solving steps as

students solved problems independently. We observed Ms. Bryant lead two problem-solving

lessons. On each problem-solving day that we observed, she put the problem on the overhead

and asked students to take out their laminated copies of the problem-solving steps. She then

gave students about 45 minutes to solve the problem. Students, familiar with the routine, worked

quietly and independently with three-way folders framing their desks to prevent collaboration.

Occasionally two students peeked around their folders to whisper to each other. When asked

why she had students work alone during problem solving Ms. Bryant explained, “I want to start

hearing what they are thinking. So I just want their thoughts in that. And I think so often with

that kind of thing if [students] work together, one person does it and the other people write it

down. And so, the other person isn’t learning.” Ms. Bryant moved among the students

responding to those who had their hands raised. She helped students with suggestions of

strategies to help them when they reached an impasse, and guided them to check the Problem-


Solving Steps for completion. Ms. Bryant identified for the researcher which students had

unique approaches to solving the problem or students who had particular difficulties with

mathematics, reading, or self-confidence. It was clear in these informal conversations with the

researcher that Ms. Bryant sought to understand how students are thinking about and making

sense of the mathematics.

Ms. Aster, like the other two teachers, used the problem-solving steps with her first grade

class. Ms. Aster used a combination of individual work time, as Ms. Bryant had, and group

sharing time, as in Ms. Foster’s class. During one of our observations, Ms. Aster explained that

she modified an EM problem to match the context of the Halloween season.

8 bats were flying over the haunted house. When a ghost came out, 3 bats flew away. How many bats were left flying? (Fieldnotes, 10/18/00)

Ms. Aster asked students to take out their math journals. She, a parent volunteer and two

student teachers glued the problem into students’ journals as students wrote the date on top. She

took students through the problem-solving steps and added one more step. She included a “units

box” in which students would write the units associated with the numerical answer and draw a

box around it. Students worked independently for 15 minutes to solve the problem and then the

class moved to the floor in the back of the room where Ms. Aster invited students to publicly

share their thinking and solutions in a one-on-one dialogue with the teacher.

Ms. Aster: Did anyone build a model?Kim: I built one with a block.Ms. Aster: How many?Kim: 8 blocks and took 3 away. I also used my fingers. I took 8 blocks. One

stands for every one. ((with some questions and prompting from Ms. Aster.))Ms. Aster: One what? ((Pushing students to remember to use units.))Kim: One bat.Ms. Aster: You started with 8. Tell us about your fingers.((Kim showed her 8 fingers, closing 3 of them.)) Ms. Aster: ((Ms. Aster praised Kim for ignoring the ghost which was extraneous

information)). Did you draw a model?


((Kim acknowledged that she had. She drew it on butcher paper so her fellow first-graders could see her work. She explained it.))

Ms. Aster: So you crossed out 3. ((In conversation back and forth/)) What is your answer?

Kim: 5Ms. Aster: 5 zucchini? ((again, pushing for inclusion of units))Kim: 5 bats.Ms. Aster: What did you do first? Kim: I drawed a modelMs. Aster: ((Ms. Aster writes and says out loud, using student’s natural language)) First,

I drawed a modelKim: then I built a model with blocks. Then I counted them up.Ms. Aster: I’m confused. How many were in your model? Then I counted them and got

5. How could there be 5? Kim: I took 8 away.Ms. Aster: how many?Kim: 3Ms. Aster: ((writing and asking Kim about units)) I got 5 bats. (Fieldnotes, 10/18/00)

Ms. Aster went on to have three more students share their solutions in front of the class, asking

each time if someone had done it a different way. In Ms. Aster’s classroom, students were

encouraged to work independently for a period of time to solve the problems, accessing desired

manipulatives, and then various students shared their strategies where Ms. Aster pushed students

to clarify their thinking. Ms. Aster helped students make connections between physical models,

diagrams and numeric representations. Students in the lessons we observed experienced problem

solving as both individual work and collaborative sharing and discussing. They experienced the

importance of clear and correct representations, understanding other students’ solutions

strategies, and mathematical notation. Ms. Aster created word problems related to what students

were learning. She described “a blank space” in the EM materials designed to allow teachers to

insert what they deemed important. For Ms. Aster that meant inserting word problems.

1 3


Across the three classrooms we observed using Problem-Solving Steps, the teachers

reported valuing problem solving enough to incorporate it into their regular mathematics

instruction. All three teachers paid attention to student thinking and sense-making. Written

communication was important in each of the classrooms—students were expected to show their

work, draw diagrams, explain their thinking and write numerical expressions to represent their

problem-solving processes. As our research team examined these lessons, we asked ourselves

about the image of mathematics conveyed by the problem-solving framework. While each

teacher adapted its use in their classroom, and in their adaptation there is clear evidence of their

interest in students’ ideas, the framework itself portrays a rather narrow view of problem solving

as a series of linear steps. The steps were advocated for, originally, to help prepare students for

the yearly state assessment. How might the framework serve that purpose and how might it

shape their ideas about problem solving? As students use the steps, they are asked to select a

strategy from a list and make sure all components of the problem are addressed and all questions

answered. Does “circling” and “bringing down the data” shortcut students’ understandings of

what it means to make sense of a problem situation? How will students make use of the

framework when they solve routine word problems as opposed to more complex, ambiguous

ones that may take several attempts at entering the problem and revising strategies?

Vignette 2: Using Everyday Mathematics

Two teachers in this study conveyed to us that EM did not provide them with much

guidance about how to elicit students’ reasoning during a lesson. They felt it suggested that they

demonstrate procedures for students to use. We observed Ms. Denis lead a lesson that generally

followed the guidelines in the EM teacher’s guide. This lesson shows what the teachers in our

study characterized as typical EM implementation—students are taught a particular strategy for


solving a type of problem, and the students practice that strategy. EM differs from traditional

texts in that strategies are often non-standard. The curricular guidelines stand in contrast to

practices described in DMI cases where children generate multiple algorithms in any one

episode. In DMI seminars, teachers see students as capable of generating their own algorithms.

In the episode below, students learned a particular strategy for solving subtraction

problems without borrowing. Before getting to that work, the class went over their homework

from the day before where they used a “same change” rule for subtraction—the same quantity is

added to or subtracted from both the minuend and the subtrahend, which maintains the difference

between numbers. The quantity that is added to both numbers is chosen so that one of the

numbers has a zero in the one’s place. So if the problem is 82 – 47, 3 can be added to both

numbers so that the problem becomes 85 – 50. Next they reviewed the ‘partial sums’ rule for

adding three digit numbers to prepare to learn the ‘partial differences’ rule. For the partial sums

rule, 473 + 589, one student showed how the hundreds, tens, and ones could be combined

separately:

Ally: 400 + 500 = 900 70 + 80 = 1503 + 9 = 13900 + 150 + 13 = 1063

Ms. Denis: Today we will do partial differences. In the additions, what were we looking at?

((After several guesses a girl said, “100s, 10s and 1s.”))Ms. Denis: Today we will look at just the 100s, 10s and 1s…Why can’t you take 8 from

4?Jeff: It will be a minusMs. Denis: With four pens, why can’t I take 8 away?...Jenna: It will be a negative number because it would go past zero.((Ms. Denis pointed to a number line high on the wall that includes both negative and positive numbers. She started at the 4, and moved left 8 spaces.))


Ms. Denis wrote the problem94

- 47

Ms. Denis: We are going to look at 10s and 1s separately. What is 9 minus 4? Ss: 5, 50 (both answers were shouted)Ms. Denis: 50. What’s next? 4 minus 7, Ss: 3, -3 (students gave both answers)Ms. Denis wrote on the board:

94- 4750

- 347

((The class discussed how they could check the solution to see if it is correct by adding 47 + 47. They tried another problem.))

Ms. Denis wrote another subtraction problem, this time using three-digit numbers for students to

practice the strategy individually on their small white boards. Then she asked students for their

answers to that problem, listing five different answers, only one of which was correct. She went

over the procedure, showing the correct answer. This cycle was repeated one more time, and

then students were given a sheet of problems to practice quietly at their desks. At the end of the

lesson Ms. Denis explained to the researcher that the next day they would build subtraction with

base 10 blocks, and then the following day she would teach them the standard algorithm, and

relate that to the base 10 blocks, showing the borrowing and trading.

This lesson, as described in the EM materials is intended to provide students practice

with a particular algorithm for subtraction. Ms. Denis led students through the procedure,

verifying each step as she went along. It appears that the goal was to help students use this

particular strategy accurately. When the students reviewed their work, they provided their final

answer to the problem. In order to clarify the procedure and verify the correct answer, Ms. Denis

reviewed the steps again. When we observed teachers lead a lesson like this, following the


guidelines in the EM materials, what is notably absent is discussion about why this algorithm

works or when students might want to use a partial difference strategy instead of a same

difference strategy. This type of lesson, analyzed in terms of cognitive demand, can be

characterized as doing procedures without connections—students review the steps to an

algorithm without necessarily exploring the underlying conceptual issues (Stein, Grover, &

Henningsen, 1996). This kind of lesson demonstrates what teachers in our study meant when

they said that EM does not provide many opportunities to solve word problems. This lesson

does not press students to consider problem situations to which subtraction may be applied. The

lesson shows the teacher demonstrating non-standard algorithms for students to practice rather

than having students construct strategies that allow them to solve problems.

Vignette 3: Modifying Everyday Mathematics

Teachers reported to us that they sometimes made adjustments to the EM curriculum.

Next, we provide an episode showing one such adjustment we observed in Ms. East’s classroom.

Ms. East, a third grade teacher, felt that EM was restrictive because it did not support the work

she was doing in DMI. She described that, “Everyday Mathematics is so teacher directed that to

make time for student thinking is to diverge from EM.” When deciding what to teach, Ms. East

reported that she looked at the title of the new chapter to identify key ideas and concepts,

developed those ideas using manipulatives, and then used lessons from EM that related to those

skills and concepts. She described her lessons as a combination of EM, Math their Way (another

curriculum she had experience with), and “East-math.”

The following episode illustrates how Ms. East made modifications to an EM lesson. She

said that the goal of the day’s lesson was to help students build and make sense of base 10 in the

100-numbers chart. The EM worksheet asked students about patterns in the number chart with


questions like, “Which digit is used the greatest number of times on the 100-number chart?” Ms.

East divided the lesson into three parts. In the first segment she had students explore base ten

blocks looking at rods and small cubes, where the small cubes represented one unit. In the

second part of the lesson, students worked with a large 100 number chart with pockets for

number cards. Finally she had students work on the journal page from the EM curriculum. The

lesson begins with students getting bags of base ten manipulatives.

Ms. East: Tell me about these((Students described that there were rectangles, cubes, a long thin one. Using the manipulatives on the overhead, Ms. East directed students to make observations about the units cubes and rods. Students described them geometrically. Then Carlos spoke up.))Carlos: You can use them to count by tens. ((He showed how to count up to 10, and

then modeled counting by tens at the overhead.))Ms. East: How did you know counting by tens?Carlos: I counted themMs. East: If I lined up 10 little ones, it would be as long as one rectangle

((summarizing the boy’s observation)). ((To the class)) Count your pieces, using tens for the long, and 1s for the cubes ((Asking students to count the pieces that were in their bags. Each student had different amounts. Students counted aloud, checked, compared with each other, and were proud when they had more than those around them.))

Ms. East: It is okay that they are different numbers of cubes. Now separate them by tens and ones.

After students separated the blocks into piles, Ms. East gives the following instructions:

Ms. East: Now I want you to build seven((One boy used two rods to create the numeral 7.))Ms. East: Some [students] made the ‘numeral 7.’ Anyone do anything different?Lisa: I used seven cubesMs. East: Anyone else do that?((Lots of hands go up. Ms. East clarified that she wanted the value, not the numeral. In the second part of the lesson Ms. East handed out number squares, so each student had four or five cards.))Ms. East: please [arrange] your numbers in order((Ms. East went over to an empty 100s grid, which had slots for 10 rows and 10 columns. She pointed those out to students. She asked what should be put in the top row.)) Jess: Zero to nine((Ms. East explained that there was no zero, so what would they start with?))Ss: One


((Students were invited to come and put their numbers in the first row. So students with the numbers 1-9 put their numbers in the row in correct order. But there was one space remaining. She asked what would go there?))Ss: 10((So the student with the 10 put his number at the end of the row.))Ms. East: Are all the numbers two-digit?Ss: NoMs. East: If you have a number that goes in this column ((pointing above the five)),

bring it up. ((Students brought up most of the correct cards that had a five in the ones place.))Ms. East: What is the same with the numbers in this column?Ss: They all end in fiveMs. East: If we built with cubes, what would be there?Ss: Five cubesMs. East: What is different [about the numbers]?Ben: They grow by 10Ms. East: Now, I want you to bring the numbers that fit in this row ((pointing to the

41-50 row))((Many students came up. Ms. East continued to lead a discussion with students, making observations about the numbers and patterns in the hundreds chart. The students noticed that all the numbers in the fifth row would be in the forties except the last number, which was 50. They noticed that the numbers in the last column ended in zero and increased by ten each time. Then she shifted into the EM lesson, handing out a worksheet. Ms. East asked students to answer problems 1-4. They were to work independently, but if they wanted help, they should come up to the floor and work with her.)) (Fieldnotes, 10/16/00)

Ms. East’s modifications to the EM lesson allowed children time to make numbers using base ten

materials and create a class hundreds chart noting patterns the students themselves observed.

Then, she gave them the EM page which directed the children to notice particular patterns in the

hundreds chart, namely the number of times particular digits were used. She invited students to

think about ideas and share their thinking through the use of manipulatives and an enlarged chart.

She ended the lesson by using an EM journal page, according to the guidelines of the text. Her

modifications to this lesson reflected her ideas that children need to have experiences with

concrete materials, something she told us in an interview is not emphasized in the third grade

EM materials, so she built in more experiences with manipulatives in her lessons.


Vignette 4: Exploring Student’s Reasoning

The teachers in this study occasionally facilitated a lesson in which they explored what

students thought about a particular idea. These lessons became “cases” that teachers brought to

the DMI seminars for discussion. We observed two of the teachers leading such a lesson

(although all of the teachers participating in this study completed such homework tasks as part of

their participation in DMI seminars). This particular homework assignment asked Ms. Aster to

“do a short data activity with your students.” They were asked to predict and take note of how

students discussed the data. Ms. Aster facilitated a discussion in which she asked her first-grade

students about the kinds of potatoes they liked. What follows is an extended excerpt drawn from

fieldnotes that follows the course of the lesson. As our research team examined these kinds of

lessons, we noted that they were typically imbued more with students’ ideas than any of the other

kinds of lessons described in this article. This exchange raises many of the complexities of

working with data and more consistent with what Stein et al., (1996) characterize as cognitive

activity that reflects “doing mathematics.” Let us examine the flow of the discussion from the

beginning. The first thing that Ms. Aster did was ask her students to draw a picture of their

favorite potato.

Ms. Aster: What kinds of potatoes do you like?((Students enthusiastically shared various kinds of potatoes: French fries. Sweet, hash browns, baked, mashed, tater tots, potato pancakes, buttered, meatloaf with mashed potatoes on top, Joey potatoes (aka Jo Jos,) potato boats, hot dog potatoes; hot dog bun and potato shaped like a hot dog, scalloped, slices with milk and cheese, potato with butter and cheese, potato man, potato boats but not sliced in half.))Ms. Aster: Think of your favorite potato. Go to your desk, write the name and draw the

kind you like best. You have 5 minutes to draw.

Students drew potatoes, and then brought their drawings back to the circle again. Ms. Aster had

a big sheet of butcher paper in the center of the circle. She asked a student to make a prediction


about what kind of potato would be the most popular (the modal category). This motivated the

need for students to think about how to organize the data.

Ms. Aster: Sophie, make a prediction for a kind of potato that will be most popular.Sophie: Hashbrowns.Ms. Aster: Is that the most popular in the class? We need to figure out a way to figure

out which is most common.Joshua: Each type could go to a different location, count them up, and then we would

know.Ms. Aster: With the paper or people moving?Joshua: With names (pointing to the butcher paper)David: We could call a type of potato and put it in a row and see which line is the

longest.

Instead of providing a particular organizational scheme, Ms. Aster invited students to generate a

way they could organize the data. Two ways were suggested, grouping the same kinds of

potatoes together or stacking the same kinds of potatoes on top of each other (like a bar graph).

The class decided to use the latter idea. Ms. Aster returned to the prediction made earlier that

“hashbrowns” would be the most popular kind asking Sophie if she wanted to revise her

prediction. The lesson then continued with students counting and stacking potatoes by category.

As students watched each other, one person noted “Mashed potatoes and gravy are winning!”

while another students notes the general shape of the bars, “Huge, little, teeney, little,” making

comparisons across categories, and yet another says, “That one isn’t as much as that one by

two.” These comments opened up opportunities for Ms. Aster later in the lesson to ask whether

the shape of the data (with categorical data) tells you anything important since the order of the

categories (unlike numerical data) holds no mathematical or descriptive meaning. Once the data

was collected, students further commented on what they found. In what follows, Ms. Aster steps

into the conversation, picking up on students who were making comparisons between the heights

of various bars. As more students shared their favorite types of potatoes, the class was faced


with figuring how to classify certain kinds of potatoes such as angel potatoes, latkas, and plain

mashed potatoes.

Matthew: It lost by two. That one isn’t as much as that one by two. ((Comparing the tallest stack to the potato boats, which was Matthew’s choice.))

Alex: Potatoes with cheese and butter. ((two cards were contributed.))Ms. Aster: Make a sentence.Allyson: It is less than four. It is less than six.Ms. Aster: Name the food your group is less than.Allyson: The potato boats.Ms. Aster: What is yours? ((Asking students who still had a card in their hands.))Angela: Angel potatoes. Andrew: LatkasRick: ((The students had to make decisions about how to fit the rest of the cards.

They were running out of room in the width of the paper. They considered combining some groups. Rick had lots of ideas and scooted the papers around.))

Matthew: Mashed potatoes.Rick: It could be with gravy group. ((The mashed potatoes and gravy.))Max: What if there is another category. ((Implying that there won’t be enough

room.))Kenji: You could put it close to mashed potatoes.Rick: I could put it here. ((He put it next to the top of the bar for mashed potatoes

with gravy.))Max: You could put it half and half. ((Where the mashed potatoes card is next to

the top of the mashed potatoes and gravy bar, but offset so that half of the card is above the bar and half is next to the top card.))

Ms. Aster: Matthew likes it where it is.Reed: Buttered potatoes ((1 card was placed in its own column.))Yoon: Walking potatoes. Ms. Aster: I had baked potatoes. ((Putting her card down))

The students noted they were running out of room but also tried to think about whether plain

mashed potatoes could be justifiably grouped with mashed potatoes and gravy, prompting one

student to suggest that “You could put it half and half,” overlapping only half of the card over

the mashed potatoes and gravy column. Voicing these considerations is an important aspect of

making public the various choices that students need to make as they try to organize their data.

Once the data was organized, Ms. Aster asked an open-ended question, “What can we figure out

from the graph?”


Ms. Aster: What can we figure out from the graph?David: Mashed potatoes and gravy won. It had the most.Ms. Aster: What else?David: Mashed potatoes and gravy got most. Hash browns had two. The rest were

“oners.”Ms. Aster: The rest of the categories were “oners?”Sasha: Potatoes with butter and cheese and hash browns were stuck with the same

amount.Rick: Hashbrowns and potatoes with butter and cheese were tied because they have

2s.Rick: The rest were with 7, 2, 3, 4. One and up, lots of ones.Ms. Aster: Is this the best way to get the most information?Sasha: I have information. There would be ten in a row.Ms. Aster: If I understand, if everyone had one, … there would be ten categories.Sasha: Another way, you could have gone across ((horizontal – she compared the

dimensions of the paper, and showed how it would shift. She wanted them in order from most to least frequent.))

((Ms. Aster continued to urge kids to make mathematical observations about the graph.))Ms. Aster: Tell me one thing that has not been said before. ((Hands go up.)) Leaping

Lizards ((one of the groups in the class)), give a statement about the graph.Sam: If you take out these, it would look like a “T.” This looks like a “1.”Rick: Sometimes, if you look at them you could have them to look like an E.Ms. Aster: I want to know about the sizes of the groups.Sophie: It would look like a giant group and that group won.Joshua: It looks like an M.Ms. Aster: Notice which are most and least.Zoe: 4 kids chose potato boats.Ms. Aster: [See if you can] use “more than” or “less than” [when you make an

observation.]Yoon: 2 people made hashbrowns.Ms. Aster: If 2 people said 4 boats, and 2 hashbrowns, who can say something about

both?David: 4 + 2 = 6Ms. Aster: So 6 kids liked either potato boats or hash browns.Matthew: If you take off gravy from mashed potatoes and gravy, the mashed potato

group has the most.Ms. Aster: Give me a numberMatthew: 7 and one more, 8.Ms. Aster: How do you now that’s the most?Matthew: Cause it’s the one that has the most potato. It is the biggest category.Ms. Aster: You didn’t count all of the others. I can tell because it is longer.Matthew: I can tell the same way. It is longer than the others, all the others. (Fieldnotes, 3/14/01)


One student noted which category “won” or had the most. Others noted how many were in each

category. Ms. Aster prompted them to notice other things about the graph, and students

comment on the shape the bars make, alternatively suggesting that the whole graph looks like a

“T,” “M,” or “E.” Looking at the shape of the distribution is important when looking at

numerical but not categorical data (an idea explored in the DMI seminar on data) so Ms. Aster

redirected students to look instead at the size of the groups, to “notice which is most and least”

and further suggested students use words such as “more than” and “less than” to compare the

frequencies in each category. The exchange ends with a student restating an idea Ms. Aster had

just suggested, that the length of the bars can be compared to tell which is bigger.

What is notable in this lesson we observed is that students appeared to feel comfortable

and flexible moving things around and taking charge of their work. They talked about ideas and

discussed relevant mathematical ideas, such as why they might want to shift the graph around or

what could be included in a category. When asked about how her experiences with DMI had

influenced the way she taught this lesson, Ms. Aster discussed specifically how the data module

had affected her approach. In her own words, she describes how she was trying to draw out

students’ ideas so that they were the ones thinking about how to organize the data.

And I think about the number of times I have asked students to graph things, and it has been in the hundreds. And I think about how many times I have robbed them of the opportunity to think about organizing the data, because I didn't know I was supposed to let them do it. I've made this cute little chart (in a tone of self- mockery). I've got this nice little hand-lettered type. Oh, I might have even outlined the letters in a fancy way and laminated it, you know. And I've got the columns, and I've got the axes and those kids are plunking in the data, and on a good day I might have even asked them to do some language experience with the chart. Right, so tell me what you're… that's what we call, in elementary school when kids start making some observations about the data they have collected. So, you know, I'm noticing that January had more rainy days than September. And I'm noticing that October and March were exactly the same for temperature. So we might take some statements down like that. But heaven forbid that I ever let them think about whether it should be organized in a horizontal way or a vertical way, and just that whole idea of numerical data being different. (Interviewer: the


categorical vs. numerical?) Yes, yes. I mean, like, hello! (sarcastically about herself) That was a totally brand new idea. I have no recollection of ever having thought about that idea before. And that was significant in the way that I did graphing from that moment on in that class. (Interview, 7/10/01)

Comparing Ms. Aster’s own perceptions with the lesson we observed, we note that she

purposefully drew out students’ ideas both in terms of organizing the data and interpreting it. But

she is not merely stepping back and letting students “discover” ideas; she presses them to make

comparisons among categories, noting which has most, least and which are more than or less

than others and steering them away from focusing on the shape of the data.

Vignette 5: Accelerated Math

In response to the perceived need in some schools to bolster students’ basic skills in

mathematical topics, some schools in the district in which the teachers in this study worked

began using a computer program called Accelerated Mathematics as a supplement to their daily

curriculum. We observed one of our study teachers using this program with the students, which

we describe next. Ms. Carlson’s building had recently adopted the use of the supplemental

program, and each teacher had been asked to make regular use of it. Ms. Carlson herself was

uncertain whether this program was helpful for students and had consulted a member of our

project’s staff to talk about it further.

When Ms. Carlson mentioned to the class that I had never seen Accelerated Mathematics, some of the students said, “How did you pass the 4th grade without Accelerated Math?” The kids showed me how it worked. Students all got out their individual folders with pages of problems. Ms. Carlson fired up the computer, and they worked like machines. It was suddenly very quiet, and kids were not distracted by each other. Students sat at their desk marking answers on a scantron sheet. Then they would line up next to Ms. Carlson’s desk where Ms. Carlson would feed the scantron through the computer, and the printer would print another set of problems. There was no conversation about mathematics except the number of correct and incorrect.

Ms. Carlson explained that they would take a diagnostic test. They mark answers on a scantron form, and they feed it into the computer. The computer reads the answers, determines areas of strength and weakness. Then the computer prints several sheets of problems for kids to work on from their weak areas. Kids work independently. The problems had multiple-choice solutions. There is no instruction or assistance that


accompanies the problems. After they finish another set of problems they feed those answers in, get feedback on whether answers are right or wrong. The computer prints another page of practice.

During the accelerated math time I sat with one girl as she worked on the problems. She was marking answers. I asked her to talk through problems and reasoning with me. She described the problems and her reasoning and chose some correct and some incorrect answers. I let a few go, and then decided to help her a bit. They were factual things, like she was misunderstanding what square, rhombus, parallelogram and rectangle were. Those incorrect understandings made her miss questions relating to them. The problems that I saw were pretty fact driven. Could the student read and interpret a graph? Could she make statements about whether a square was a rhombus or a rhombus was a square, etc.? The program gave feedback about whether answers were right or wrong. It did not evaluate the reasoning students used. If a problem was incorrect, the program did not explain why they were wrong. It didn’t seem to instruct, but kept track of skill level.

The computer kept a log of what kids were getting and not getting. Ms. Carlson could easily keep track of where kids were, how they were progressing and what kinds of skills students showed proficiency in one time or multiple times. (Fieldnotes, 3/12/01)

No other teachers in this study were using Accelerated Mathematics, which was adopted

and paid for with building, rather than district funds. Since we collected this data, however,

another building with two study teachers had adopted its use.

This final vignette illustrates yet another kind of mathematics that students in these

classrooms could participate in. Unlike other forms of lessons, the Accelerated Mathematics

lessons are centrally focused on students’ independent work in what are viewed as “essential

mathematical skills.” While some might argue that essential mathematical skills should also

include learning to engage in mathematical argument and deliberation (Ball & Bass, 2000),

Accelerated Mathematics communicates that mathematics is about developing computational

and factual skills. In terms of cognitive demand, Accelerated Mathematics falls either into levels

of “memorization” or “procedures without connections.”

Discussion

The vignettes presented in this study show that teachers and their students participated in

a range of mathematical experiences; the nature of the task and the nature of the discourse varied


across lessons. The differences across the lessons we observed raise several key issues for us: 1)

Why might teachers use a range of approaches to teaching mathematics lessons? 2) How does

instructional diversity shape student learning and conceptions of mathematics? 3) What more is

there for us, as teachers, professional developers, and mathematics educators, to consider about

our visions for mathematics education and classroom practice?

We puzzle over the diversity of instructional practices apparent in teachers’ classrooms

even though all teachers have commitments to a conceptual understanding of mathematics. Our

argument is different from that found in classic cases of teacher learning such as that of Mrs. O

(Cohen, 1990). We do not claim, as Cohen did about Mrs. O, that these teachers did not learn

enough or that they filtered their professional development experiences through their own

beliefs, understandings, and routines. What we aim to emphasize through our discussion below

is that we must attend both to our visions of mathematics instruction and to the very different

communities of practice that constitute teachers’ professional lives in order to continue our

efforts in professional education.

Why do we see diversity in instructional approaches?

We did not find that every lesson was built centrally around eliciting students’ reasoning

about mathematics. This observation, we argue, should be considered from a number of

different perspectives. The professional development context has its own set of practices norms,

tools and ways of being. The classroom offers yet another set of practices, norms, tools, and

ways of being. To understand the relationship between professional development and teachers’

practice, we must attend to the degree of continuity or coherence across these settings and

communities. When teachers attend DMI sessions, the tasks are designed to center discussion on

student thinking. Students’ computational fluency integrates understanding with accurate,


efficient and flexible use of algorithms (Russell, 2000). The primary tools—written and video

cases—highlight how students solve mathematical problems. Math activities embedded in each

seminar are meant to advance teachers’ understanding of mathematics. Thus children’s

reasoning dominates discourse in DMI seminars.

In the classroom, other goals are present—visible in our data by the use of Accelerated

Mathematics and even the Problem-Solving Steps. Proficiency with skills is a clear dimension

of national and state standards in mathematics. Individual teachers and whole schools feel

considerable pressure, however, to have their students master these skills quickly in order to

perform well on state assessments. Despite efforts to balance mathematics reform talk to reflect

both understanding and skill development, debates continue to pit one against the other.

Accelerated Mathematics, built to have students recall factual information and practice skills,

does not by design promote discussion of mathematical thinking. It focuses on diagnosing which

kinds of problems students answer correctly or incorrectly. Similarly, the Problem-Solving Steps

were generated to respond to children’s test performance. On state-mandated tests, word

problems are prominent. Students must show both their solution and explain their strategies.

Test results are shared with the public broken down by district and by school. This degree of

pressure and publicity was Ms. Foster’s rationale for sharing the Problem-Solving Steps across

the district, and the reason why a bank of word problems were developed for each grade level.

And while teachers are pressed to support their students in performing on this explanation-rich

state-mandated test, they continue to have the more skill-driven multiple-choice exams. The

teachers in our study describe those tests as advocating different kinds of knowledge that require

different kinds of instruction. Thus, the difference in goals between a DMI seminar and the

classroom may account for some of the diversity of instruction we observed.


While practices in a professional development community and the classroom community

are driven by different goals, our data suggest that the very nature of the tools also provide

varying supports for teachers as they shape their lessons. The tools readily available in

classrooms may or may not emphasize student reasoning. Curriculum materials available in

mathematics, even ones that are meant to be more aligned with current standards in mathematics,

may shape teachers’ behavior to demonstrate procedures for students to practice rather than elicit

and then build on children’s reasoning. Some of the teachers in our study felt that Everyday

Mathematics, if followed faithfully, results in many lessons in which teachers demonstrate

procedures for all students to practice at once. It may not explicitly provide room for teachers to

elicit a variety of strategies in any one lesson for comparison with one another. Teachers in our

study felt they needed to adapt EM or create their own lessons in order to make the discussion of

student thinking a central part of the lesson.

We suggest that teachers’ own advocacy and positioning in schools and districts

influences what kinds of practices they incorporate into their teaching. The sociopolitical

context matters. The teachers in this study are all leaders in their district. They may believe that

they have less choice about diverging from curricular programs advocated in the district because

of their identifiable political positions. Their experience with DMI has continued to expand their

understanding of children’s reasoning and the mathematical ideas they are working on in school,

yet they were also among the people who advocated for certain kinds of tools, such as the

Problem-Solving Steps and Everyday Mathematics. Some may feel tied to and accountable for

using the materials their district is advocating. And there is little time, given the demands of

teaching and the various initiatives present at any one time in a district or school, for teachers to

work together to investigate how they are making use of curricular resources. Researchers


working on curriculum development have recently made arguments that curriculum guides

should be designed in ways that are more supportive of teacher learning by including explicit

features that can support teachers to enact lessons in ways that foster productive learning for

students (Ball & Cohen, 1996; Remillard, 2000; Schneider & Krajcik, 2002). As one of the early

reform curricula, EM does not reflect such explicit design intentions, yet new guides could be

developed to assist teachers in enacting lessons so that children’s reasoning and argumentation

abilities are more central.

What are students’ experiences?

A second major issue raised within this study relates to students’ experiences. When

mathematics instruction includes an array of ways in which teachers and students engage in

discourse, how then do we understand students’ experience of mathematics?

We know that the way students’ experience mathematics influence what they think it

means to do mathematics (e.g., Boaler, 1998; Franke & Carey, 1997; Lampert, 1990). These

studies have compared students who squarely have experienced one kind of instruction (open-

ended, problem solving) versus another (didactic, procedure-based). Such comparisons have

revealed that students develop very different conceptions of what it means to do mathematics and

what it means to be successful in mathematics, influencing students’ interest to pursue

mathematics as well. For example, Boaler (2001) describes the kinds of agency students wield

when they experience mathematics as problem solving versus procedural. Students with

problem-solving experiences were able to better evaluate a mathematical situation using a variety

of strategies, and employ appropriate mathematical strategies to solve problems. On the other

hand, students who primarily spent their time practicing procedures that were demonstrated by

their teacher relied on cues from the text and teacher to point them toward the appropriate


strategy to use. When students were asked to solve more complex problems that did not provide

the same kinds of structural or verbal cues, they had more difficulty in making sense of the

problem. Moreover, procedure-focused students believed that school mathematics looked quite

different from the mathematics they practiced in their daily lives, whereas problem-focused

students felt school math and math in life were quite closely tied.

In current discussions of mathematics education, we aim not to dichotomize problem

solving versus procedures. So, although we have evidence that children benefit from programs

that emphasize conceptual understanding, we still need to understand what the relationship is

between problem solving and skill development. Teachers, too, face the important question of

understanding how particular practices affect student learning. Our data, we argue, raise just this

question for educators and researchers to consider together. How do students’ experiences

impact their conceptions of what it means to be successful in mathematics, their motivation to be

engaged in mathematical work, their ability to solve complex problems, and their understandings

of mathematical ideas?

What role can professional development play?

Finally, we argue that this study has implications for the kind of work that is necessary in

professional development. Participating in DMI has allowed the teachers we observed to

engage in close study of students’ reasoning. What DMI begins to do is to challenge teachers to

interrogate how they shape instruction in their own classrooms and how they make sense of

students’ mathematical work. At the same time, attending and participating in a DMI seminar is

protected space. The very real debates and concerns with student test performance, with use of

particular curricular resources, for example, can be suspended. When they return to their

classrooms, teachers must respond to such demands. With their colleagues, in a DMI seminar,


the teachers in this study have become full participants in a community that values and probes

deeply into making sense of students’ ideas. The teachers engage actively in the work of the

seminar, ask questions about student reasoning, bring tasks from their classrooms in which they

have worked with their students. The DMI seminars are organized to allow for such

participation to dominate.

Viewing the classroom and the professional development setting as two distinct

communities of practice has raised some significant questions for us to consider as we continue

our work in professional development. For example, our approach to facilitation, as we have

enacted it in our particular leadership grant, has been to limit discussions about what is

happening in teachers’ classrooms—mainly to avoid discussions that may spiral into a show-and-

tell, facilitators steer conversations towards focusing on children’s reasoning. The seminar itself

serves to introduce teachers’ to the central mathematical ideas that students encounter in a

particular domain and the kinds of issues they have to work through as they make sense of those

ideas. This study suggests that teachers need to have continued conversations about what they

are doing in their classroom—to compare the kinds of mathematical experiences they are

providing for the students and the conflicts inherent in dealing with a politicized environment.

Teachers experience clear tensions in meeting standards and goals in mathematics. Judith

Warren Little’s (in press) observation about professional development is relevant here. In a

recent paper, she situates professional development “in relation to two central impulses in

teaching and teacher development in the United States: an impulse to locate and support teacher

learning more fully in and through practice; and a countervailing impulse to direct and control

teacher practice more firmly through instruments of external accountability” (p. 3). We argue

that our data can open up important conversations among teachers about mathematical goals,


student understanding, and teaching practice. What is the relationship between skill

development and conceptual understanding? How can discussions and tasks be facilitated so that

they raise important mathematical issues and help develop skills? What are the outcomes of

adopting multiple ways of engaging in mathematics for student achievement, attitudes and

interests in mathematics? Teachers’ participation in DMI helps develop a shared understanding

of what is important in student reasoning. It provides a strong foundation for teachers’ continued

work. Based on our observations in classrooms, we see the pointed need for teachers and

professional educators to consider together the curricular resources used in classrooms and the

way lessons are enacted. What coherence is there in lessons from day to day, from month to

month? Is coherence important?


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Appendix: End-of-the-year Interview Protocol

1. Tell me what you think DMI is all about? (probes: What do you think DMI’s goals or purposes are? How was experience impacted your teaching? How has your perception of DMI changed from your initial experiences to now?)

2. How do you think Everyday Mathematics links to your work in DMI?(probes: How do you use your curriculum? How do you decide what to teach everyday?How do you decide what to emphasize? What to cut? What to keep? When you go off the lesson, why have you done that? Probe for specific example)

3. How do you learn about student thinking in your classroom? How do you know when a student understands an idea?

4. Describe what happens when you teach something that’s related to a DMI module that you’ve taken? (e.g., decision making, children’s thinking, mathematics)

(probes: What happens when you’re teaching something that you haven’t had a DMI seminar on? What do you feel most confident/comfortable with in terms of teaching? What do you think you still need to learn about (or get better at) teaching?)

5. Do you think your knowledge of mathematics has changed? How has it changed? What mathematical ideas do you think you’ve grown in understanding? What remains challenging for you (mathematically)?

6. What have been a few key mathematical goals that you’ve had for your children in mathematics this year?


Author Note

This research was supported in part by the National Science Foundation (Award No. 9819438).

The opinions expressed in this article do not necessarily reflect the views of NSF. We thank

Philip Bell, Megan Franke, Leslie Herrenkohl, Carolyn Jackson, Deborah Schifter, and Virginia

Stimpson and the members of the Urban Teacher Education Network for their comments. We

also thank the teachers who participated in this study for their willingness to share their

successes and concerns with us.


Footnote

1The teachers had completed two modules of Developing Mathematical Ideas on Number Sense

and were in the middle of participating in a third module on working with data. Each module

consists of 24 hours of professional development. In addition, at the time of this study, the

teachers had participated in two week-long summer institutes designed to extend their work with

DMI with further experiences with mathematics, children’s thinking, and facilitation of DMI

seminars. Each summer institute consisted of 30 hours of professional development.


Table 1: Distribution of type of lesson per teacher

Type of Lesson Ms. Aster

Ms. Bryant

Ms. Carlson

Ms. Denis

Ms. East

Ms. Foster

Problem-Solving Steps O O R OEveryday Mathematics O O O O O OModifying Everyday Mathematics O OExploring Student Reasoning O R R O R RAccelerated Mathematics ONote: O = observed, R = not observed, but teacher reported using in interviews


Table 2: Teachers’ conceptions of Developing Mathematical Ideas (DMI)

Teacher Response to interview question, “What do you think DMI is all about?”Ms. Aster It has, I think, less to do with kids as learners than as me as a learner. And it has to do with

being able to think or reflect on what I am doing and why I'm doing it so that it is meaningful to move kids toward what my goals are for them. I think the DMI has to do with me and the professional development. And the way that I become more confident in math and be able to use the understanding of how I'm learning to help lead other people to think about where they are going and how they want to get there. And the idea of using the model of not being an answer but being a question.

Ms. Bryant I feel it is a way to help you start understanding how your students are thinking and what they are doing. And then when you start understanding them, then you are better able to instruct who they are based on what is going on. … That is basically what I see it as being is helping us focus and listen to the kids and to best help them. And then too, the thing it has done for us is to make me more math aware and deepen my thinking about math. Especially last year with the fractions.

Ms. Carlson Well, what I think it's doing is trying to get teachers to look at student work, look at what students are thinking about, and really try to figure out where they are and trying to move them along, and to use kind of their developmental peak to kind of go forward. And so part of that is recognizing that kids are in lots of different places and they solve problems in different ways. And that they can learn from each other by giving their ideas and presenting them. Um, that some of them are not ready to move on… as a teacher I need to stop and not think that these kids don't get it, as much as, what do they understand and how can I get them to move to the next step?… And the other thing I think, actually about DMI, is that it really is there to give teachers better math skills and look at how they learned… I just felt like, like maybe I wasn't as strong as I should be in my mathematical skills. And so I think that what DMI does is that it pushes teachers to think as well.

Ms. Denis To me DMI is, it's not a curriculum. …Well it's a philosophy, of questioning the kids, and see what kids are thinking. And being able to you know, to bring them to a better, a deeper level, or a higher level of understanding. Or you're not, just yeah you have to present some material, but in that presentation, you're questioning to see what are they getting, what are they not getting, rather than here it is, let's just do it and then you move on.

Ms. East I think there is probably two that would be the main focus. One is to develop teachers' understanding of math and to help them understand that there are instructional techniques to help student understanding of math. [DMI] helps you with looking at where the kids are, what they are doing, and what does that tell you about what they understand and then challenging them from there.

Ms. Foster I think mathematics is one of those areas that has to do with thinking. Thinking and creativity. Being able to think outside of the box, not memorization. Not, here’s something, memorize that. I think it’s about thinking. I think it’s about understanding. So you look at something that’s presented, you don’t understand what it is at first. You play with it, you think about it, you get to express your thoughts about it, so that your thinking increases, and you play with your thinking, with other people. It’s like tossing your thinking back and forth and playing ball with it. …All those things are part of DMI. I think the communication piece is really, really strong with DMI. You know the opportunity for dialogue, diverse thinking, and honoring every student’s thought process. … So I saw DMI as helping teachers understand concepts.


Figure 1: Problem-Solving Steps Used in Instruction

Problem-Solving Steps Problem-Solving Strategies

1. UNDERLINE the question2. CIRCLE the data3. PULL down the needed data4. CHOOSE a strategy and SOLVE

the problem5. CHECK :

- Did you answer the question being asked?- Did you label your work?- Is your answer reasonable?

6. Explain in SENTENCES your THINKING and STRATEGIES for solving the problem

Guess, check, and reviseDraw a pictureAct out the problemUse objectsSolve a simpler or similar problemMake a table or chartLook for a patternMake an organized listWrite a number sentenceUse logical reasoningWork backward

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