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]
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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
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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
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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
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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
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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.
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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.
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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
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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.
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Data Analysis
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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,
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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
Professional Development 13
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
Professional Development 14
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)
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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….”
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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-
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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?
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((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
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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
Professional Development 20
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.))
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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
Professional Development 22
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
Professional Development 23
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
Professional Development 24
((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.
Professional Development 25
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
Professional Development 26
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
Professional Development 27
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?”
Professional Development 28
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)
Professional Development 29
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
Professional Development 30
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
Professional Development 31
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
Professional Development 32
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,
Professional Development 33
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.
Professional Development 34
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
Professional Development 35
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
Professional Development 36
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,
Professional Development 37
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,
Professional Development 38
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?
Professional Development 39
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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?
Professional Development 46
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.
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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.
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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
Professional Development 49
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.
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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