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The Problem-Solving Cycle: A model of mathematics professional development

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ABSTRACT:
There is a growing consensus that mathematics teachers need
to significantly expand their content and pedagogical content
knowledge in order to make instructional improvements and
provide increased opportunities for student learning. Long-
term, sustainable professional development programs can
play an important role in this regard. Our research team has
spent the past several years developing a program called the
Problem-Solving Cycle (PSC). This professional development
model is grounded in a situative perspective on learning and
draws upon theoretical and empirical evidence regarding the
importance of professional learning communities and the
value of using artifacts of practice to situate teachers’ learn-
ing in their classroom experience. The model takes into
account the complexity of classroom teaching, the wide array
of knowledge teachers need to promote the mathematical
thinking of their students, and the long-term commitment
required to develop such knowledge. In this article, we pres-
ent the conceptual framework for the PSC, details of its
enactment, and initial findings regarding its impact on
teachers’ knowledge.
T
he Problem-Solving Cycle (PSC) model of mathe-
matics professional development is an iterative,
long-term approach to supporting teachers learn-
ing. One iteration of the PSC consists of three
interconnected workshops in which teachers share a com-
mon mathematical and pedagogical experience, organized
around a rich mathematical task. This common experi-
ence provides a structure within which the teachers can
build a supportive community that encourages reflection
on mathematical understandings, student thinking, and
instructional practices.
During the first workshop of the PSC, teachers collabora-
tively solve a rich mathematical task and develop plans for
teaching it to their own students. Workshops two and three
focus on teachers’ experiences implementing the task in
their classrooms (see Figure 1). The teachers consider more
about the mathematical concepts and skills entailed in the
task, their instructional strategies, and their students’
mathematical thinking. In all three workshops, there is an
emphasis on using artifacts of practice to situate teachers’
learning opportunities in the context of their work.
One iteration of the PSC roughly corresponds to an academ-
ic semester, so that teachers can participate in 2 iterations
(6 workshops) per school year. Each iteration focuses on a
unique mathematical task and highlights different aspects
of teachers instructional practices and students’ mathe-
matical thinking. Successive iterations of the PSC build on
one another and capitalize on teachers expanding knowl-
edge, interests, and sense of community. The PSC model is
designed to be implemented by a knowledgeable facilitator,
who carefully plans and conducts each workshop and
continually monitors the participating teachers needs and
interests. The facilitator might be a teacher leader, mathe-
matics coach, department chair, professional development
specialist, or other teacher educator.
The PSC model is flexible with respect to the domain of
mathematics that is selected as well as the specific learning
goals and instructional strategies that are addressed. In our
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The Problem-Solving Cycle:
A Model of Mathematics Professional Development
Jennifer Jacobs and Hilda Borko, University of Colorado at Boulder
Karen Koellner, University of Colorado at Denver and Health Sciences Center
Craig Schneider, Eric Eiteljorg, and Sarah A. Roberts, University of Colorado at Boulder
work, we have focused on algebra because of the growing
concern regarding students’ inadequate understanding and
preparation in this domain of K-12 mathematics (U.S.
Department of Education and National Center for
Educational Studies, 1998). Algebra operates as a gate-
keeper” to higher mathematics and future educational and
employment opportunities (Ladson-Billings, 1998; NRC,
1998). Students’ difficulties in learning formal algebra are
well documented (Kieran, 1992; Nathan & Koedinger,
2000), and our schools’ approaches to algebra instruction
are lacking. For example, first-year algebra courses have
been characterized as “an unmitigated disaster for most
students” (NRC, 1998, p. 1).
The enhancement of teachers’ professional knowledge
about algebra and the teaching of algebra is considered to
be a central component in the effort to support students’
algebraic reasoning (Blanton & Kaput, 2005; Lacampagne,
Blair, & Kaput, 1995; NCTM, 2000). The PSC model was
developed and implemented as part of the Supporting the
Transition from Arithmetic to Algebraic Reasoning
(STAAR) project
1
, which aimed to help teachers enhance
their professional knowledge for the teaching of algebra
and improve their instructional practices. We focused on
middle school because it is becoming more common for
school districts to require that algebra be taught during
the middle school years, yet many middle school teachers
have limited experience in teaching algebra. Furthermore,
their experiences as algebra students typically emphasized
learning procedures and manipulating symbols rather
than reasoning about algebraic ideas (Ball, Lubienski, &
Mewborn, 2001).
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1
The professional development program is one component of the STAAR Project, supported by NSF Proposal No. 0115609 through the
Interagency Educational Research Initiative (IERI). The views shared in this article are ours, and do not necessarily represent those of IERI.
FIGURE 1: The Problem-Solving Cycle model of professional development
Workshop 3:
Student Thinking
Videotaping
the Lesson:
Teachers’
Implementation
of Problem
Workshop 1:
Solve Problem and
Develop
Lesson Plans
Workshop 2:
The Teacher’s Role
Note: The arrow from Workshop 3 to Workshop 1 represents movement from one iteration of the PSC to the next.
CONCEPTUAL FRAMEWORK FOR THE
PROBLEM-SOLVING CYCLE
In this section, we present the conceptual and empirical
grounding for the goals and processes of the Problem-
Solving Cycle. We first explore the professional knowledge
that mathematics teachers need and then discuss critical
elements in designing professional development from a
situative perspective.
PSC Goals: Enhancing Teachers’ Professional
Knowledge
Researchers and policymakers have come to agree that
objectives for teacher learning should include becoming
more proficient in the content they teach, gaining a better
understanding of student thinking and learning, and
improving their skills in content-based instructional prac-
tices (Secretary’s Summit on Mathematics, 2003). These
learning objectives provide the foundation for the PSC
model of mathematics professional development.
In his seminal work in this area, Shulman (1986) identi-
fied subject-matter content knowledge and pedagogical
content knowledge as two central domains of teachers’
knowledge. Both domains are unique to the profession of
teaching and can be enhanced over time as teachers gain
expertise in their fields and participate in programs
designed to foster such knowledge development (Wilson,
Shulman, & Richert, 1987). Ball and her colleagues have
extended Shulmans work in the field of mathematics edu-
cation. Specifically, they have identified and elucidated
“knowledge of mathematics for teaching” the mathe-
matical knowledge that teachers must have in order to do
the mathematical work of teaching effectively (e.g., Ball &
Bass, 2000; Ball, Hill, & Bass, 2005; Ball, Thames, & Phelps,
2005; Hill & Ball, 2004). This conception of knowledge of
mathematics for teaching is multifaceted and incorporates
both content and pedagogical content knowledge.
Mathematics content knowledge. Ball and Bass (2000)
describe the mathematics content knowledge needed for
teaching as including common and “specialized” knowl-
edge of mathematics. Common content knowledge can be
defined as a basic understanding of mathematical skills,
procedures, and concepts acquired by any well-educated
adult. Specialized knowledge involves a deeper, more
nuanced understanding of mathematical skills, procedures,
and concepts. Specialized knowledge enables teachers to
evaluate the multiple, and novel, mathematical representa-
tions and solution strategies that students bring to the
classroom; to analyze (rather than just recognize) errors;
to give mathematical explanations; to use developmentally
appropriate mathematical representations; and to be explicit
about their mathematical language and practices (Ball &
Bass, 2003). It is what Ma (1999) characterizes as profound
understanding of fundamental mathematics” (p. 120).
Pedagogical content knowledge. Mathematics teachers
need a sophisticated understanding of instructional prac-
tices and student thinking related to specific mathematical
content. Ball and her colleagues consider these two types
of understanding as distinct components of pedagogical
content knowledge: knowledge of content and teaching,
and knowledge of content and students (Ball, Thames, &
Phelps, 2005). Knowledge of content and teaching
includes, for example, the ability to recognize instructional
affordances and constraints of different representations,
and to sequence content to facilitate student learning.
Teachers draw upon this knowledge when they plan for
the use of pedagogical strategies and instructional materi-
als in a lesson, when they modify a task or introduce a new
representation during instruction, and when they consider
how to improve their instructional practices the next time
they implement a lesson with related mathematical con-
tent. Knowledge of content and students includes the abili-
ty to predict how students will approach specific mathe-
matical tasks, and to anticipate student errors. Teachers
draw upon this knowledge when they create lesson plans
that take into account the thinking that a task is likely to
evoke in their students, when they interpret incomplete
student ideas during a lesson, and when they consider how
to respond to the various correct or incorrect pathways
that students explore.
Although these domains of knowledge of mathematics for
teaching can be separated for purposes of analysis, they are
inextricably intertwined in teachers’ instructional prac-
tices. Teachers routinely make decisions that draw upon all
aspects of their knowledge as they engage in the numerous
and complex activities of classroom instruction activi-
ties such as selecting, modifying, and using mathematical
tasks; selecting mathematical representations that are
appropriate for a specific learning goal and group of stu-
dents; understanding and building upon student concep-
tions; and establishing and maintaining a discourse com-
munity that enhances students’ mathematical understand-
ing and their capacity to reason mathematically.
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Whereas knowledge of mathematics for teaching includes
all strands of school mathematics, our research and profes-
sional development as part of the STAAR project focused
specifically on algebra; hence we use the term “knowledge
of algebra for teaching (KAT)
2
. Drawing upon the frame-
work developed by Ball and colleagues, we conceptualize
enhancing knowledge of algebra for teaching as enhancing
both specialized content knowledge related to algebraic
reasoning and pedagogical content knowledge related to
algebra instruction.
Designing Professional Development from a
Situative Perspective: Community and
Artifacts as Tools for Teacher Learning
Situative perspectives on cognition and learning provide
the conceptual framework that guided the design of the
PSC. In the field of professional development, a situative
perspective supports the value of creating opportunities
for teachers to work together on improving their practice,
and of locating these learning opportunities in the every-
day practice of teaching (Ball & Cohen, 1999; Putnam &
Borko, 1997; Wilson & Berne, 1999).
Professional learning communities. Situative theorists
draw our attention to the social nature of learning and the
central role that communities of practice can play in
enhancing teachers’ professional knowledge and improv-
ing their practice (Greeno, 2003; Lave & Wenger, 1991;
Little, 2002; Putnam & Borko, 2000). To create an environ-
ment in which teachers collectively explore ways of
improving their teaching and support one another as they
work to transform their practice, successful professional
development programs must establish trust, develop com-
munication norms that enable challenging yet supportive
discussions about teaching and learning, and maintain a
balance between respecting individual community mem-
bers and critically analyzing issues in their teaching
(Frykholm, 1998; Seago, 2004). Research also indicates that
the development of teacher communities is difficult and
time-consuming work. Although conversations in profes-
sional development settings are easily fostered, discussions
that support critical examination of teaching are relatively
rare (Grossman, Wineburg, & Woolworth, 2001; Stein,
Smith, & Silver, 1999).
Artifacts of practice. Another central tenet of situative
perspectives is that the contexts and activities in which
people learn become a fundamental part of what they
learn (Greeno, Collins, & Resnick, 1996). This tenet sug-
gests that teachers’ own classrooms are powerful contexts
for their learning (Ball & Cohen, 1999; Putnam & Borko,
2000). It does not imply, however, that professional devel-
opment activities should occur only in K-12 classrooms.
An alternative is to use artifacts of classroom practice—
such as instructional plans and assignments, videotapes of
lessons, and student work produced during a lesson—to
bring teachers’ classrooms into the professional develop-
ment setting (Kazemi & Franke, 2004; Little, Gearhart,
Curry, & Kafka, 2003; Nikula, Goldsmith, Blasi, & Seago,
2006; Sherin & Han, 2004). Such records of practice make
the work of teaching a central focus of professional learn-
ing experiences and anchor conversations in specific class-
room events.
Video records of classroom practice are becoming increas-
ingly popular as a tool for teacher professional develop-
ment. Short video clips can be selected to address particu-
lar professional development goals. They can be viewed
repeatedly and from different perspectives, enabling teach-
ers to closely examine one another’s instructional strate-
gies and student learning, and to discuss ideas for
improvement. Although any video of classroom instruc-
tion can situate professional development in a setting that
is likely to prove meaningful for teachers, there are con-
ceptual and empirical arguments for using video from
participants’ own classrooms. Video from teachers’ own
classrooms situates their exploration of teaching and
learning in a more familiar, and potentially more motivat-
ing, environment than does video from unknown teachers’
classrooms (LeFevre, 2004). In one comparative study,
teachers who watched video from their own classroom, in
a computer-based professional development environment,
found the experience to be more stimulating than did
teachers who watched video from someone else’s class-
room, and they believed that the professional development
program had greater potential for promoting instructional
change (Seidel et al., 2005). The “video club mathematics
professional development program by Sherin and col-
leagues (Sherin, in press; Sherin & Han, 2004; Sherin &
van Es, 2002) and the Video Case Studies in Scientific
Sense Making Project by Rosebery and colleagues
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2
The term KAT is also used by the Knowing Mathematics for Teaching Algebra project at Michigan State University (Ferrini-Mundy et al.,
2005). These two projects are unrelated, although our work draws upon their conceptualization of knowledge of algebra for teaching.
(Rosebery & Puttick, 1998; Rosebery & Warren, 1998)
informed our thinking about how to create an effective
professional development program that incorporates video
from participating teachers’ own classrooms.
Establishing community around video. . Establishing and
maintaining a strong community is particularly important
when teachers are asked not only to discuss teaching and
learning but also to share video clips from their own class-
rooms with colleagues. Because classroom video clearly
exposes actual teaching practices, sharing video is likely to
seem more threatening to teachers than sharing other arti-
facts such as student work and lesson plans. To be willing
to take such a risk, teachers must feel confident that show-
ing their videos will provide valuable learning opportuni-
ties for themselves and their colleagues, and that the
atmosphere in the professional development setting will be
one of productive discourse.
In an appropriate professional development setting, ana-
lyzing video from teachers’ own classrooms can help to
foster a tightly knit and supportive learning community.
As teachers share video records of their teaching with col-
leagues, they have the opportunity to create an atmosphere
of openness and bonding that is rare in professional learn-
ing environments (Sherin, 2004). Creating and maintain-
ing a productive learning community around video is an
integral component of our professional development
model (Borko, Jacobs, Eiteljorg, & Pittman, in press).
3
THE PROBLEM-SOLVING CYCLE
In this section, we describe the three workshops that make
up one iteration of the PSC, discuss decisions central to
planning each workshop, and identify some of the varia-
tions enacted by the STAAR team. In another paper, we
provide vignette descriptions of each workshop from one
iteration of the PSC, illustrating the opportunities teachers
had for learning about mathematics content, pedagogy
and student thinking (Koellner et al., in press). In addition,
our website (http://www.colorado.edu/education/staar/)
includes a Facilitator’s Guide to Planning and Conducting
the Problem-Solving Cycle. The guide is intended to help
professional development facilitators learn about the
Problem-Solving Cycle and prepare to implement it.
WORKSHOP 1: Doing for Planning
The major objective of Workshop 1 is to support the
development of teachers’ mathematics content knowledge.
Most of the workshop time is devoted to teachers collabo-
ratively working on the selected mathematical task and
debriefing their solution strategies. Additionally, teachers
spend a significant portion of Workshop 1 developing
unique lesson plans that will meet the needs of their stu-
dents. Specifically, they identify learning goals, predict stu-
dent solution strategies, and structure their lessons with
specific pedagogical moves. Teachers then implement their
lessons prior to Workshop 2. Thus, another aim of the
workshop is to enhance teachers’ pedagogical content
knowledge through discussions about designing a lesson
plan and considering different ways of teaching the select-
ed task. We call the framework for this workshop “Doing
for Planning” to highlight the dual focus on teachers’
problem solving and instructional planning.
Selecting the task. As described above, the PSC is built
around a rich mathematical task. Teachers work through
the task, design a lesson incorporating the task, teach that
lesson to their students, and discuss their classroom expe-
riences in two subsequent workshops. For the PSC to be
successful, facilitators must select a task that can foster a
productive learning environment for the teachers over the
course of three workshops. In our development and
implementation of the PSC model, we have found that
appropriate tasks meet the following criteria: (1) address
multiple mathematical concepts and skills, (2) are accessi-
ble to learners with different levels of mathematical
knowledge, (3) have multiple entry and exit points, (4)
have an imaginable context, (5) provide a foundation for
productive mathematical communication, and (6) are
both challenging for teachers and appropriate for students.
Given our focus on algebraic reasoning, for each iteration
of the PSC conducted by the STAAR team, the facilitators
sought problems that contained mathematical ideas cen-
tral to the middle school algebra curriculum. Facilitators
selected problems that focused specifically on the algebraic
concepts of patterns and functions; enabled teachers and
students to utilize different representations of functions
such as graphs, tables, and equations; and had connections
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3
A discussion of the STAAR professional development programs approach to building professional community is beyond the scope of this
article. For more information about the program, including our approach to building community, please see Authors 2005; and Authors,
in press a.
to other areas of the mathematics curriculum such as
number and operations, and geometry.
Conducting the workshop. The “Doing for Planning”
framework guides the structure of Workshop 1. Teachers
first read the selected problem and share ideas about the
mathematical concepts and skills that are likely to be
embedded in the solution strategies. They then work on
the problem in small groups. During this time, the facilita-
tor encourages the teachers to think about how they would
create a lesson for their students incorporating the problem.
At various points in the workshop, the teachers come
together as a whole group to share their solution strategies
and their ideas for using the problem in their teaching. As
teachers create lesson plans tailored to their own students,
they talk with colleagues and the facilitator about such issues
as their mathematical goals for students, prior knowledge
students will need for the lesson, and how they will adapt
tasks to make them more accessible for their students. By
the end of the workshop, teachers have explored the mathe-
matical opportunities presented by the task, considered
how their students might attempt to solve it, and devel-
oped a lesson plan for using the task in their classrooms..
Implementing and Videotaping the Lesson
Between Workshops 1 and 2, each participant teaches the
problem in one of his or her mathematics classes, and the
lesson is videotaped. In the STAAR program, we used two
cameras to film each lesson. One camera followed the
teacher throughout the lesson, and a second camera
captured one group of students as they worked during
small group activities. One of the most important compo-
nents in Workshops 2 and 3 is the analysis of teachers
pedagogical moves and students’ mathematical reasoning
using video clips of the PSC lessons. Therefore, after the
videotaping occurs, the facilitator selects short clips to
serve as anchors for discussions about teaching and learn-
ing during Workshops 2 and 3.
WORKSHOP 2: Considering the Teacher’s Role
The central purpose of the second workshop is to foster
teachers pedagogical content knowledge by guiding them
to think deeply about the role they played in teaching the
selected problem to their students. The majority of time in
Workshop 2 is spent watching and discussing short video
clips from one or more of the teachers’ lessons, and
exploring aspects of the teacher’s role such as how they
introduced the problem or orchestrated the classroom dis-
course. The workshop provides teachers the opportunity
to critically reflect on their own instructional practices,
along with those of their colleagues, as they analyze video
clips and participate in guided discussions. The rich task
and accompanying video situate the workshop in particu-
lar mathematical content and classroom practices, and this
interaction between content and pedagogy is highlighted
throughout the workshop
Planning the workshop. In planning for Workshop 2, the
facilitator identifies one or more aspects of the teacher’s
role to explore. This decision depends on the particular
needs and interests of the group of teachers as well as
overall goals of the professional development program.
Another key set of decisions for the facilitator involves
selecting video clips to show and developing guiding ques-
tions for discussions during the workshop. We have found
that video clips that work well in the PSC model have the
following characteristics: (1) are relevant to the teachers,
(2) are valuable, challenging, and accessible to the teach-
ers, (3) cover a relatively short time period, and (4) pro-
vide an anchor for considering new instructional strate-
gies. In addition, we have learned that it is important to
prepare questions to help frame teachers’ viewing of and
conversations about each video clip.
During our three iterations of the PSC, the STAAR facilita-
tors focused on topics related to the teacher’s role such as
introducing the task; posing questions to elicit, challenge,
and extend students’ thinking; deciding when to provide
explanations, ask leading questions, and let students follow
their own line of reasoning; and wrapping up the lesson.
Conducting the workshop. Workshop 2 typically begins
with teachers reflecting on and sharing their experiences
teaching the problem. Subsequent activities are designed
around the selected pedagogical topic and associated video
clips. Teachers view the clips in both small group and
whole group contexts, and the facilitator guides conversa-
tions about the instructional episodes they capture. Often,
a video clip is viewed multiple times, as the conversation
suggests another perspective to take or another interpreta-
tion to explore. Teachers are also given time to reflect criti-
cally and to consider ways of improving their instruction
that they can take back to their classrooms.
WORKSHOP 3: Considering Student Thinking
The central objectives of Workshop 3 are to deepen
teachers understanding of students’ thinking about the
mathematics in the selected PSC task, and to extend their
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ideas about how to foster and support students’ mathe-
matical reasoning. To situate teachers’ explorations in their
classroom practice this workshop relies heavily on clips
from the videotaped lessons as well as additional artifacts
that represent student thinking, such as students’ written
work and reflections. Throughout the workshop, teachers
have opportunities to gain further insight into the com-
plexities of both the mathematical concepts entailed in the
problem and students’ learning of those concepts.
Planning the workshop. A major task in planning
Workshop 3 is selecting artifacts of practice that will pro-
vide opportunities for teachers to explore the various
forms of mathematical reasoning their students applied to
the problem and the different solution strategies they
used. To select video clips, the facilitator considers the
same characteristics as in planning for Workshop 2; how-
ever, rather than choosing clips to provide an anchor for
examining instructional strategies, the facilitator selects
clips to provide an anchor for considering student think-
ing. In a similar manner, facilitators select rich examples
of student work such as individual student work on the
task, posters created by groups of students, and written
reflections. As in Workshop 2, the facilitator prepares guid-
ing questions to help frame teachers’ conversations about
each video clip and example of student work, encouraging
them to focus on the mathematical concepts and reason-
ing evident (or lacking) in the artifact.
The STAAR facilitators often chose video clips and student
work that featured novel ways of solving the mathematical
problem—in particular, solution strategies that none of
the teachers noted during Workshop 1. We also addressed
topics such as how students explained their solution
strategies, and misconceptions or naïve conceptions.
Conducting the workshop. . In Workshop 3, teachers
spend the majority of the time watching and discussing
video clips and students’ written work. Close analysis of
the mathematical content in the clips and other artifacts
often leads the teachers to rework the problem, and to
engage in mathematically sophisticated conversations. For
example, they may be prompted to discuss the affordances
and constraints of various solution methods, the progres-
sion from naïve to more formal understandings of the
content, and mathematical ideas embedded in the problem
that they had not previously considered. Workshop 3 also
includes time for teachers to reflect on what they have
learned, in this workshop and over the course of one itera-
tion of the PSC. As they reflect, individually (in writing)
and collaboratively (in small or whole group discussions),
the teachers not only consider how they might improve
their instructional practices based on knowledge gained
thus far but also provide valuable input that the facilitator
can use to shape successive iterations of the PSC.
RESEARCH METHODS
The STAAR professional development program began in
2003 and continued through spring 2005. During that
time, we worked with a group of middle school mathe-
matics teachers to develop and refine the PSC model. In
fall 2003, we conducted three professional development
workshops that focused on pedagogical practices associat-
ed with algebra. A central goal of these workshops was to
develop norms for viewing and analyzing classroom video
before conducting the first iteration of the PSC. We con-
ducted the first PSC in spring 2004 and two more itera-
tions during the 2004–2005 academic year. The three itera-
tions used different mathematics problems and focused on
different aspects of the teacher’s role and students’ mathe-
matical reasoning. During the three iterations of the PSC,
we utilized a design experiment approach (Cobb et al.,
2003; Design-Based Research Collective, 2003) to study
and refine the model
Participants
Eight teachers participated in the STAAR professional
development workshops during the 2003–2004 academic
year. All eight were middle school mathematics teachers,
with classroom experience ranging from 1 to 27 years.
They represented six different schools in three school dis-
tricts within the state. In 2004–2005, seven teachers con-
tinued working with us and three additional teachers
joined the project, as we further refined the PSC. Each new
teacher was a colleague of one of the current participants.
Data Collection and Analysis
Throughout the professional development program we
collected and analyzed a large amount of data on processes
involved in developing and enacting the PSC model (see
also Borko et al., in press and Koellner et al., in press). We
also collected data on the teachers experiences and learning
outcomes over the course of the two years that they partic-
ipated in the STAAR program. At the end of the second year
we conducted both a written survey and individual face-to-
face interviews asking the teachers to consider the impact
of the professional development program on their learning
of algebra, beliefs about learning and teaching algebra, and
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instructional practices. In addition, we conducted a follow-
up interview with each teacher during the school year
following the conclusion of the professional development
workshops, in order to assess their perception of the con-
tinuing impact of the professional development program.
To examine teachers’ perceptions of the impact of the
program, two coders analyzed three sets of self-report
data: the written surveys completed during the final PD
workshop, the post-program interviews conducted shortly
after the final PD workshop, and the follow-up interviews
conducted the next academic year. All of the interviews
were transcribed. The coders independently marked all
instances where the teachers wrote about or discussed the
following categories:
• Impact on content knowledge,
• Impact on pedagogical content knowledge related to the
teacher’s role,
• Impact on pedagogical content knowledge related to
student thinking, and
• Impact of watching video (including video of themselves
and of their colleagues).
The coders then met to discuss and reconcile their coding
decisions. In our analyses we report on the number of
teachers who brought up these categories in at least one of
the three data sources.
IMPACT OF THE STAAR PROFESSIONAL
DEVELOPMENT PROGRAM
In this section we present initial results regarding the impact
of the STAAR professional development program on the
participating teachers professional knowledge from their
perspectives. In particular, we illustrate the perceived impact
of the program on teachers mathematics content knowledge
and pedagogical content knowledge — specifically the
teachers role in promoting discourse and student thinking.
We also explore the teachers perspectives regarding a central
component of the PSC model: watching video of themselves
and their colleagues.
Impact on Content Knowledge
Analyses of the three self-report data sources suggest that
the teachers believed their content knowledge was fostered
through participation in the professional development
program. Specifically, we examined three categories of
coded data related to content knowledge: a) learning
mathematics content (generally), b) learning by working
on the mathematics tasks that were part of the PD, and c)
learning from using multiple approaches to solve the
mathematics tasks. Looking across the three data sources
for the eleven teachers
4
who participated in the program,
six teachers mentioned learning mathematics content, all
eleven mentioned learning by working on the mathematics
tasks, and ten mentioned learning from using multiple
approaches (see Table 1).
Impact on Pedagogical Content Knowledge
We considered the impact of the professional development
program on two aspects of teachers’ pedagogical content
knowledge that are emphasized heavily in the PSC model:
the teacher’s role and student thinking.
Knowledge about the teacher’s role in promoting
discourse.
Based on the participants’ stated interests and instruction-
al goals, the STAAR project focused on the teacher’s role in
improving classroom discourse. Therefore, in our analyses,
we coded the three sources of self-report data for teachers
perceptions of the impact of the PD on their role in pro-
moting discourse. All eleven teachers reported that the
program helped to increase their knowledge about pro-
moting classroom discourse, including learning about the
importance of meaningful discussions and techniques for
fostering discussions (see Table 2). Most of the teachers
talked about the program as having an impact on specific
aspects of their knowledge about classroom discourse. For
example, ten teachers noted that they learned something
about conducting groupwork, such as how important it is
to provide time for groupwork or how to group their stu-
dents more effectively. Nine teachers said that they learned
how to foster better conversations in their mathematics
classrooms, either within small groups or during whole
class discussions. Eight teachers mentioned that they
learned something about asking questions, including what
types of questions are most effective and strategies for ask-
ing questions to elicit student thinking.
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4
The data discussed in this section are from eleven teachers: seven participated in both years of the professional development program
and 4 participated in one year.
Knowledge about student thinking.
Teachers’ comments on all three self-report data sources
suggest that participation in the PSC strongly impacted
their knowledge related to student thinking. All eleven
teachers commented that they gained a general awareness
of students’ mathematical thinking, including learning
about how to listen to and promote their students’ think-
ing (see Table 3). In addition, all of the teachers said that
they learned about the importance of giving students
more authority, for example by making their classrooms
more student-centered or by decreasing their own role as
the mathematical authority. Eight teachers said they
became more knowledgeable about how to use or build on
their students’ mathematical thinking. Seven teachers
reported learning about how to use mathematical tasks to
promote student thinking, such as using rich problems
that emphasize exploring processes rather than generating
answers, or using fewer problems and exploring them for a
longer period of time.
Impact of Watching Video
Because watching video is such a prominent feature of the
PSC — and new to most teachers — we wanted to exam-
ine participants’ perspectives on the value of this compo-
nent of the professional development program. We coded
and analyzed the teachers’ self-report data to explore how
they felt about watching video from their own lessons and
from their colleagues’ lessons.
Watching video of themselves. Ten teachers told us that
being videotaped, although sometimes nerve-racking, was
one of the most valuable aspects of the professional devel-
opment. Many of these teachers pointed out that watching
their own lessons on videotape enabled them to see what
they were doing well and to identify areas for improve-
ment. A number of teachers commented that by watching
video they gained insight into what their students were
thinking and what assistance they needed..
I was filled with anxiety when I thought someone was
going to come in and videotape everything I was doing
during classes with kids. But it turned out to be a power-
ful learning experience for me. (Pam, post interview)
Watching the video clips was great to see me in action
and actually get to see what the students see. It allowed
me to see the parts of my lessons that need improvement
and what is good. (Laura, written reflection)
I think the most helpful [thing] was the videotaping, to
watch myself on videotape, sometimes painfully so.
Wanting to say, “Shut up, shut up. Why do you keep
going on with that?” But it’s so helpful to see how you
come across to kids and how they are or are not respond-
ing … and to think about what I might have changed in
that lesson … or how I could have connected with kids
better. (Celia, final interview)
Watching video of other teachers. . Eight teachers men-
tioned that they learned something by watching videos of
their colleagues, such as new pedagogical strategies or how
other students solve mathematical problems. Several teachers
mentioned that it was informative as well as reassuring to
watch their colleagues struggle with familiar issues.
We never get to see our colleagues doing what we’re
doing. We just assume they’re doing the same things that
we are, and that’s not necessarily so. Its a great window
into how other kids look and it’s comforting when you see
things that are the same. (Penny, final interview)
When I watched other teachers’ videos, it wasn’t critiquing,
it was seeing what they do in their classroom and realizing
[that] a lot of what’s going on in their classroom is what’s
happening in mine. Or this person really does a great
job at opening a lesson. Maybe I could try something
they’re doing. (Linda, final interview)
CONCLUSION
As this sample of findings from research on the STAAR
professional development program illustrates, the
Problem-Solving Cycle appears to be a promising model
for enhancing teachers’ content knowledge and pedagogi-
cal content knowledge. Our data suggest that as teachers
engage in the PSC, they are prompted to think deeply
about mathematics content and instruction as part of a
collaborative and supportive learning community. In par-
ticular, teachers who participated in the STAAR program
report a strong impact on specific areas of their profes-
sional knowledge that were targeted during the three itera-
tions of the PSC: mathematics content (including the
importance of working on tasks and generating multiple
solution strategies), methods for improving classroom dis-
course (including how to conduct groupwork, foster con-
versations about mathematics and mathematical thinking,
and ask effective questions), and ways of fostering and
exploring student thinking (including giving students
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N C SM J o u r n a l S P R I N G 2 0 0 7
authority, building on students’ thinking, and using tasks
that promote student thinking).
The PSC model provides a structure for the participating
teachers to work together as professionals, to establish trust
and develop communication skills that enable constructive
yet respectful discussions about teaching and learning, and
to share and expand their knowledge base. Drawing on a
situative framework, the model emphasizes the use of
classroom artifacts within a supportive professional com-
munity. Any professional development effort that fore-
grounds the analysis of video from teachers own class-
rooms is entering into relatively uncharted, and murky,
territory. However, our experience suggests that when the
necessary structure is in place, the impact on teachers can
be extremely powerful and fundamentally positive.
We are particularly encouraged by the fact that teachers at
four of the six schools represented in the STAAR project
are spearheading new professional development efforts
within their schools that contain some or all of the PSC
elements. Teachers at several schools have decided to observe
and videotape one another and then meet to discuss these
videotapes. At one school, the mathematics instructors
plan to all work on and then teach a selected problem, and
get together to share their experiences. When asked about
their reasons for initiating these professional development
activities, the teachers explained that they felt empowered
by their experiences in the STAAR program and wanted
to share what they had learned with their colleagues. The
following remarks, from three of the teachers’ final written
reflections, are illustrative of these ideas:
I proposed this sort of “community” to my principal and
next year we will meet once a week as grade-level math
departments. The problem is that teachers have a men-
tality of “shut the door and let me teach. I hope my
school’s math people can get the same sense of communi-
ty as we have here. (Peter, written reflection)
I have learned to become a leader in my professional
community. I have been able to share my classroom with
other teachers so they can take ideas about teaching and
learning from me. (Nancy, written reflection)
I want to get my entire department involved with this
process. As we put students in groups to work together, we
as teachers need to do the same. We need to be doing
math together. (Laura, written reflection)
Although our implementation of the PSC has been
restricted to middle school teachers and focused on algebra
content, the model is intentionally designed to be flexibly
implemented and responsive to the needs of facilitators,
teachers, and school district personnel. We anticipate that it
can be adapted for use with teachers at elementary and high
school levels and with different strands of the school math-
ematics curriculum. While our research and development
work on the PSC model will continue, we encourage others
in the mathematics education community to adapt, extend,
and refine this approach and further explore its effectiveness.
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TABLE 1. Perceived impact of the professional development program on teachers’ content knowledge
Number of
Teachers Representative Quotes
Learning mathematics 6 One of the things I was really weak at was tr ying to develop equations from
content (generally) patterns. I just could not do that for the life of me before [the STAAR
program]…. I actually forced myself to use those strategies… and it’s really
beginning to open my eyes.
(Nancy, final interview)
I used to think in algorithm mode. Now I try to see or picture patterns.
(Deborah, written reflection)
Learning from tasks 11 [I learned] just how much insight you can get from working problems with other
adults… When you do it on your own, you’ve got a much narrower view on it to
start with. Whereas if you solve it with other adults before teaching, it broadens
your view. And then the kids broaden it even more.
(Kristen, final interview)
Before the STAAR workshops, I have to admit honestly that I did not try ever y
single new rich problem, or non-rich problem, myself all the time. I’d look at
the parameters of the problem, but not necessarily sit down and work them.
You know what the STAAR project taught me? Feel their pain. Look at the prob-
lem, and work it either yourself or with someone else.
(Pam, post interview)
Learning from 10 I have learned that there’s more ways than I could have imagined to solve
multiple approaches problems. Without this program I would not have realized all the ways to
solve a problem and the importance of looking at student work and thinking.
(Linda, written reflection)
This group has enhanced my algebraic knowledge by listening to others’ ideas
to the same problem. Learning that multiple solutions do exist and [that it’s
important] to study them purposefully with kids.
(Penny, written reflection)
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TABLE 2. Perceived impact of the professional development program on teachers’ knowledge
about their role in promoting discourse
Number of
Teachers Representative Quotes
Teacher’s role in 11 I no longer think that math class is about me. It’s about them and their
discourse (general) learning. And it’s about my facilitating… I can say, ‘OK. Let’s look at this and
talk about it.’
(Celia, final interview)
I think learning to struggle is as important as anything else in math. [STAAR]
helped me to know that because you put me through it! Now when kids say to
me in class, ‘Well I can’t do it. Give me a hint,’ I say, ‘Maybe you better go
talk to your group.’ I step back and I step back for a good long time until we
bring the large group back together again.
(Pam, final interview)
Conducting groupwork 10 Working with other teachers on the math problems was really beneficial. And
that led me to understand why it’s so important for students to work in groups
in the classroom.
(Linda, final interview)
I used to just kind of let the kids pair up and I didn’t have much thinking
behind it. Now I structure it and have a purpose between who’s with whom.
(Peter, post interview)
Fostering conversations 9 I realized the importance of talking about our thinking, and giving kids the
opportunity to share their ideas. (Ken, written reflection)
I now tr y to say to them, ‘Please share that with the rest of your group….
Explain that to ever yone.’…. I want them to pursue that and ask those
questions of each other.
(Celia, final interview)
Asking questions 8 I feel like I went from a lot of lecture a very broad kind of questioning [style]
to asking deeper level or higher level thinking questions, provoking
more of their thinking, rather than just “Is this right?” kinds of questions.
(Ken, final interview)
I try to get them to explain more about what they did. And so my questions
ask for more than the answer. ‘How did you get there?’ ‘Why did you do it
that way?’ ‘Does anyone else want to tell us how they did it?’ Those kinds of
things. I think it’s all about digging in deeper.
(Kimber, post interview)
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TABLE 3. Perceived impact of the professional development program on
teachers’ knowledge about student thinking
Number of
Teachers Representative Quotes
Awareness of student 11 I thought so much about looking at kids’ work and trying to figure out what
thinking (general) they were thinking with the STAAR program.
(Kimber, post interview)
Watching other teachers allow their students to think and discover and
digest a problem makes me realize that is a change I must make.
(Kristen, written reflection)
Giving students 11 I learned how to not just tell students how to do things, but have them
authority participate… and share their information. Instead of me just standing up
there and blabbing the hour and a half.
(Linda, final interview)
I don’t want to keep pushing them to get my answer and to follow my path.
I want them to find their own path.
(Kristen, final interview)
Building on 8 Previously, I wouldn’t allow my students to continue their thought process.
student thinking I would stop them and have them go my way. Versus now, when I’m not quite
understanding what they’re doing, I will continue to ask questions.
(Laura, post interview)
Before STAAR I would have said immediately, ‘Oh yeah, that’s right. Move on.’
Now we explore it deeper than that. And they know, too, that I’m going to say
to them, ‘How can you prove it?’
(Celia, post interview)
Using tasks to 7 Student thinking takes time. [This knowledge has] helped me determine
promote student what part of the curriculum is more impor tant, so I can do away with ‘less
thinking important’ problems.
(Nancy, written reflection)
STAAR showed me that there are problems out there that have so many things
to offer kids. So many things that they can talk about and experience and try
to strategize.
(Pam, final interview)
... This type of professional work intentionally orients teachers' interpretations of signiicant classroom interactions around student thinking, which is a critical step toward teaching for understanding [4][5][6]. Video clubs ofer an efective collaborative structure for supporting, sustaining, and assessing the growth of developing professional teacher communities in relationship to student learning [7,8]. Smith described engagement in such activities as "practicebased professional development," where teachers develop the capacity to see speciic events that occur in the practice of teaching as instances of a larger class of phenomena. ...
... Moreover, although the video club model has strong potential for advancing collaborative professional learning and making space for developing teacher learning communities in ways likely to transform classroom practices over time, van Es cautions that "simply bringing teachers together does not ensure community development" ( [10], p182). Watching and discussing video footage of a colleague's classroom is inherently vulnerable work, especially for spotlight teachers whose classroom video excerpts are viewed and discussed [7]. Even in the best of circumstances it is hard not to be guarded when one's teaching is the subject of discussion [21,22]. ...
... It is important to note that James is not alone in his discomfort, as is it normal and typical for teachers to ind the experience of peer video review and discussion nerve-racking, especially in the beginning [7]. The study shared in this chapter captured James' perspective at a moment in time when he was relatively new to teaching and brand new to classroom videotaping. ...
... The Problem-Solving Cycle (PSC) is an iterative, long-term approach to mathematics PD Jacobs et al., 2007;Koellner et al., 2007;Koellner, Schneider, Roberts, Jacobs, & Borko, 2008). The key characteristics of the PSC are derived from the research on the nature of high quality, effective PD (Borko, 2004;Borko et al., 2010;Desimone, 2009): ...
... Implementing the PSC with integrity entails using a rich mathematics problem as a shared experience; facilitating productive discussions about the mathematical content, student thinking and instructional practices; focusing attention on multiple representations and solution strategies; and using video from the teachers' own classrooms. Based on prior research in which the developers of the PSC model also served as the PD facilitators, we have documented preliminary evidence of the effectiveness of the PSC as well as an emerging understanding of the characteristics of successful facilitation Borko, Jacobs, Seago, & Mangram, in press;Clark et al., 2005;Jacobs et al., 2007;Jacobs, Koellner, John, & King, in press;Koellner et al., 2008). This research identified basic supports that novice facilitators of the PSC would be likely to need in order to (1) create a professional learning community, (2) facilitate mathematics discussions with teachers, and (3) facilitate video-based discussions to help teachers examine student thinking and classroom instruction. ...
... And videoclips of classroom incidents, used in ways similar to those Bishop described, can provide the context and substance for their collective inquiry. Two such examples of collaborative communities are described by Kazemi and Franke (2004) – who worked with teacher workgroups in an elementary school that were examining student thinking and classroom practice, and the STAAR Project (Jacobs et al., 2007) – where teachers focused explicitly on student thinking in the third and final workshop of the Problem Solving Cycle. Building on early Cognitively-Guided Instruction work that laid out an " organized set of frameworks that delineated the key problems in the domain of mathematics and the strategies children would use to solve them " (Franke & Kazemi, 2001, p. 43), Kazemi and Franke (2004) organized and facilitated workgroups as places for teachers to share the mathematical work that was occurring in their classrooms. ...
... This exploration helped the teachers to develop a better understanding of their students' thinking, benchmarks in students' learning trajectories, and instructional trajectories to support students (Kazemi & Franke, 2004). Similarly, the third workshop in the Problem-Solving Cycle focused primarily on student thinking, addressing topics such as how students explained their solution strategies, and their misconceptions or na¨ıvena¨ıve conceptions (Jacobs et al., 2007). To foster these discussions, the facilitator selected videoclips and student work, often centering on a student's or students' novel approach to solving a problem. ...
Article
Full-text available
Complex cognitive processes underlie the thoughts and decision making of teachers engaged in planning and carrying out instruction. Research on teacher planning is summarized with respect to reasons for planning, how teachers plan, influential factors, and differences between experienced and inexperienced teachers' planning. Research on teachers' interactive decision making is summarized with respect to the conceptions, components, antecedents, and decision making of experienced and inexperienced teachers. Research investigating the relation between teachers' decisions and student outcomes is presented. A cognitive psychological framework for analyzing teacher thinking is presented and applied to existing research on teacher planning and decision making. Finally, new directions for research on teacher thinking are explored.
... This case study (Yin, 2009) received institutional review board approval and was part of a larger research project: Implementing the Problem Solving Cycle (iPSC), which was centered around an adaptive model of professional development with three interconnected workshops: The Problem-Solving Cycle (PSC). In this model, teachers experienced and debriefed solving a rich mathematical problem with multiple entry and exit points (Workshop 1); teachers discussed videotape of their enactment of this same lesson within their own classes, focusing on the teacher's role in executing the problem with their class (Workshop 2); and teachers discussed the role of student reasoning from that same lesson (Workshop 3) ( Jacobs et al., 2007). The goals of the second two workshops built on and elicited student thinking (Jacobs, Koellner, John, & King, 2014). ...
Article
This paper examines turn-by-turn interactions [Wagner, D., & Herbel-Eisenmann, B. (2014). Mathematics teachers’ representations of authority. Journal of Mathematics Teacher Education, 17(3), 201–225. doi:10.1007/s10857-013-9252-5] in mathematics professional development, looking specifically at facilitators’ positioning of supporting English language learners (ELLs), using ethnographic microanalysis [ Erickson, F. (1992). Ethnographic microanalysis of interaction. In M. D. LeCompte, W. L. Millroy, & J. Preissle (Eds.), The handbook of qualitative research in education (pp. 201–226). San Diego: Academic Press, Inc.]. This study took place in a weeklong professional development for seven teacher leaders preparing to facilitate mathematics professional development at their schools. Two facilitators led the professional development. The findings demonstrate how facilitators positioned supporting ELLs as important for all and important for some and how their participating teacher leaders responded to this positioning, either accepting or rejecting it. This research highlights the link between positioning authority, which attends to the relationships between comfort, knowledge, and authority, as well as how individuals might develop such authority. This paper begins a conversation about the role of positioning in professional development, specifically linked to facilitators’ positioning and the importance of authority in creating professional learning experiences focused on supporting ELLs.
... The Problem-Solving Cycle (PSC) is a long-term approach to mathematics PD designed to increase teachers' MKT, improve their instructional practices, and foster student achievement gains (Borko et al., 2005;Jacobs et al., 2007;Borko et al., 2008). In a number of previous articles we have articulated the theoretical and conceptual underpinnings of the PSC at length (see Borko et al., 2005. ...
... We report on the PSC, a school-based model of PD that was implemented in a number of middle schools in one large, urban school district over a period of several years. In other articles, we have highlighted the design of the PSC and its implementation in various contexts (Borko, Koellner, & Jacobs, 2014;Jacobs et al., 2007;Jacobs, Koellner, John, & King, 2014;Koellner et al., 2007), but here our goal is to bring together a variety of quantitative impact measures to consider what this body of evidence suggests about the promise of the model. Despite a number of important limitations of the study-such as the lack of an experimental design and fluctuations in the sample-it demonstrates the potential for conducting an empirical, longitudinal evaluation of an adaptive model of mathematics PD on multiple outcomes of interest. ...
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We posit that professional development (PD) models fall on a continuum from highly adaptive to highly specified, and that these constructs provide a productive way to characterize and distinguish among models. The study reported here examines the impact of an adaptive mathematics PD model on teachers' knowledge and instructional practices as well as on students' achievement over time. Results indicate at least modest impacts in each of these areas. Our findings demonstrate that adaptive models of PD can be subjected to investigations of impact based on quantitative research methodologies; moreover, we argue that utilizing a wider variety of methodologies to study adaptive models is increasingly needed as these models gain in popularity and usage.
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Video has become increasingly popular in professional development (PD) to help teachers both learn subject matter for teaching and systematically analyze instructional practice. Like other records of practice, video brings the central activities of teaching into the PD setting, providing an opportunity for teachers to collaboratively study their practice without being physically present in the classroom. In this chapter, we explore how video representations of teaching can be used to guide teachers’ inquiries into teaching and learning in an intentional and focused way. We draw primarily from our experiences developing and field-testing two video-based mathematics PD programs, Learning and Teaching Geometry (LTG) and the Problem-Solving Cycle (PSC), and preparing PD facilitators using those programs to lead video-based discussions. On the basis of evidence from these projects, we argue that PD leaders can guide teachers to examine critical aspects of learning and instruction through the purposeful selection and use of video footage. Furthermore, we use data from the LTG and PSC projects to build a chain of evidence demonstrating that video-based PD can support improvements in teachers’ mathematical knowledge for teaching, their instructional practices, and, ultimately, student learning.
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Breaking traditional instructional patterns is a notoriously challenging endeavor, particularly on a broad scale. However, a number of professional development (PD) efforts in mathematics have produced promising results, even within a relatively short time frame. In this chapter, we focus on the impact of one such effort and report on teachers’ instructional changes after they participated in the Problem-Solving Cycle (PSC) model of PD. We discuss quantitative patterns from a dataset of 51 videotaped lessons obtained from 13 participants, highlighting changes in their PSC and typical lessons over a 1.5-year period. We also present a case study analysis to illustrate the specific nature of the classroom improvements made by one participant. Overall, teachers experienced the most change in their ability to work effectively with students’ productions around meaningful mathematics. These findings add to the literature that demonstrates instructional growth potential among teachers who take part in PD for less than 2 years.
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In this article, Mary Kay Stein, Margaret Schwan Smith, and Edward A. Silver identify and describe the challenges that practicing teacher educators and professional developers are likely to encounter as they design and implement new programs to help teachers learn new paradigms of teaching and learning amidst current educational reforms. The authors call attention to the fact that, just as teachers will need to relearn their teaching practice, so will experienced professional developers need to relearn their craft, which traditionally has been defined as providing courses, workshops, and seminars. This article focuses on two professional developers who engaged in long-term efforts to work with teachers in new ways, identifying the tensions that each actually faced. The cases illustrate the challenges that professional developers may encounter in supporting the transformation of teachers, including learning how to work with groups of teachers in school settings, expanding their repertoires beyond workshops and courses and balancing interpersonal sensitivity with the need to challenge prevailing practices and beliefs. The final section of the article looks across the two cases and begins to map out common features of the terrain through which practicing professional developers can expect to travel.
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Current educational reform efforts in the United States are setting forth ambitious goals for schools, teachers, and students (e.g., National Council of Teachers of Mathematics, 1989; National Education Goals Panel, 1991; National Research Council, 1993). Schools and teachers are to help students develop rich understandings of important content, think critically, construct and solve problems, synthesize information, invent, create, express themselves proficiently, and leave school prepared to be responsible citizens and lifelong learners. Reformers hold forth visions of teaching and learning in which teachers and student engage in rich discourse about important ideas and participate in problem solving activities grounded in meaningful contexts (e.g., American Association for the Advancement of Science, 1989; National Council of Teachers of Mathematics, 1989, 1991). These visions of teaching and learning depart significantly from much of the educational practice that currently typifies American classrooms — practice that is based on views of teaching as presenting and explaining content and learning as the rehearsal and retention of presented information and skills.
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We present here results of a case study examining the classroom practice of one third-grade teacher as she participated in a long-term professional development project led by the authors. Our goal was to explore in what ways and to what extent the teacher was able to build a classroom that supported the development of students' algebraic reasoning skills. We analyzed 1 year of her classroom instruction to determine the robustness with which she integrated algebraic reasoning into the regular course of daily instruction and its subsequent impact on students' ability to reason algebraically. We took the diversity of types of algebraic reasoning, their frequency and form of integration, and techniques of instructional practice that supported students' algebraic reasoning as a measure of the robustness of her capacity to build algebraic reasoning. Results indicate that the teacher was able to integrate algebraic reasoning into instruction in planned and spontaneous ways that led to positive shifts in students' algebraic reasoning skills.
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