Mathematics teaching as a deliberate practice an investigation of elementary pre service teachers reflective thinking during student teaching

AMY ROTH MCDUFFIE

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE: AN
INVESTIGATION OF ELEMENTARY PRE-SERVICE TEACHERS’
REFLECTIVE THINKING DURING STUDENT TEACHING

ABSTRACT. In this case study I examine the reflective practices of two elementary preservice teachers during their student teaching internship. I extend current views of reflective
practice to create a framework for a ‘deliberate practitioner’. With this framework, I investigate the pre-service teachers’ thinking with regard to reflective processes and how they
use their pedagogical content knowledge in their practices. My findings indicate that the
pre-service teachers use their pedagogical content knowledge in anticipating problematic
events, and in reflecting on problematic events in instruction. However, limits in pedagogical content knowledge and lack of confidence impede the pre-service teachers’ reflection
while in the act of teaching. They were more likely to reflect on their practices outside
of the act of teaching. Implications for teacher educators and pre-service teachers are
discussed.
KEY WORDS: mathematics education, pedagogical content knowledge, reflective
practice, student teaching, teacher education

With the emergence of recent reforms in education in the United States
(e.g., National Council of Teachers of Mathematics [NCTM], 1989, 1991,
2000), researchers and educators have re-examined teaching by moving
away from a technical model of teaching by prescribed methods to one
that regards it as a, complex, demanding practice. Two separate but
compatible perspectives have made substantial contributions as to how we
view teaching, and correspondingly, to how we approach teacher education. First, viewing teachers as reflective practitioners has underscored
the problem solving nature of teaching (McIntyre, Byrd & Foxx, 1996;
Russell & Munby, 1991; Schön, 1983, 1987; Valli, 1992; Zeichner, 1993).
Consequently, the focus of many teacher education programs is on the
development of reflective practitioners (Christensen, 1996). This focus
is consistent with a constructivist perspective for teaching and learning
that is the basis of many teacher education programs (e.g., McIntyre,
Byrd & Foxx, 1996). Second, the conceptualizing of pedagogical content

knowledge (Grossman, 1990; Shulman, 1986, 1987) as a unique type
of knowledge for teaching has helped researchers, teachers, and teacher
educators gain an understanding of the knowledge base that teachers need
for successful practice.
Journal of Mathematics Teacher Education 7: 33–61, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.


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In studying the knowledge of mathematics teachers, Ball, Lubienski
and Mewborn (2001) call for research on “how teachers are able to use
mathematical knowledge in the course of their work” (p. 450) and “what
[teachers] are able to mobilize mathematically in the course of teaching”
(p. 451). One approach to understanding the use of knowledge is to investigate how teachers think about their practice. The purpose of my study was
to intersect the constructs of reflective practice and pedagogical content
knowledge (PCK) in order to examine how pre-service teachers use their
mathematical PCK in thinking about their practice, both in planning
and classroom teaching. I investigated the reflective practices in mathematics of two pre-service elementary teachers during their student teaching
internship.

THEORETICAL FRAMEWORK
I shall provide a summary of reflective thinking and pedagogical content
knowledge as I applied and merged these two concepts in my work with
the pre-service elementary teachers.
Reflective Thinking
While a range of interpretations exists for what is considered to be
reflective practice, I referred primarily to the ideas of Dewey (1910) and

Schön (1983, 1987). Reflection is a practice that has gained considerable
attention in the past two decades, yet Dewey began this discussion early
in the last century. Dewey argued that reflective thinking begins when
teachers experience a difficulty or troubling event (i.e., a problem). A key
aspect of the reflective process is that teachers act on their reflections:
Reflection involves not simply a sequence of ideas, but a consequence – a consecutive
ordering in such a way that each determines the next as its proper outcome, while each
outcome in turn leans back on, or refers to, its predecessors . . . Each phase is a step from
something to something . . . There are in any reflective thought definite units that are linked
together so that there is a sustained movement to a common end. (pp. 2–3)

Thus, acting on reflections distinguishes reflective practice from just
thinking back and may be an important aspect in the development of
teaching.
Schön (1983, 1987) developed these ideas further and separated
reflection into two forms, reflection-in-action and reflection-on-action.
Reflection-on-action is a deliberate process of looking back at problematic
events and actions, analyzing them, and making decisions. Russell and
Munby (1991) explained that reflection-on-action “refers to the ordered,


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35

deliberate, and systematic application of logic to a problem in order
to resolve it; the process is very much within our control” (p. 165).
Reflection-in-action is a more immediate consideration and resolution of
an identified problem in the act of teaching and learning (Schön, 1987).
Both types of reflection and the triggering (problematic) events were

considered in this study.
Mewborn (1999) studied the reflective practices of pre-service teachers
during field experience as part of their mathematics methods course.
Mewborn examined the elements of mathematics teaching and learning
that were problematic for the teachers and the thinking they engaged in
to resolve those problems. She found that pre-service teachers did engage
in reflective thinking. However, this reflective thinking was not evident
until they internalized their own authority to generate, reason about, and
test hypotheses in order to examine children’s mathematical thinking.
Mewborn argued that to facilitate reflection pre-service teachers need a
non-evaluative atmosphere and relationship with cooperating teachers and
university faculty so that they are encouraged to generate hypotheses and
to arrive at resolutions to problematic events without fear of judgment.
Inherent to all discussions of reflection is a problematic or puzzling
event triggering reflection and, thus, I sought to investigate reflection that
arose from problematic situations. While it could be argued that reflection might occur during or after a successful lesson (a non-problematic
situation), I chose to adopt the constructs for reflective thinking of
Dewey (1910) and Schön (1987) by focusing on pre-service teachers’
thinking that was inspired by identified problems. I synthesized the various
views and definitions of reflection considered for this study as a cycle
in teaching practice. In this cycle, the problematic event initiates the
process of reflection. Following this problematizing, either during instruction (for reflection-in-action) or after instruction (for reflection-on-action),
the teacher analyzes the problem and the options and/or approaches for
resolving the problem. Next the teacher decides on a resolution or plan
for action. Finally, the plan is implemented in practice and the resolution
is tested. At this point the process either ends for this event or results in
a subsequent problematic event (an unresolved issue), and the reflective
cycle continues. As Dewey (1910) contended, the cycle does not necessarily move directly from one phase to the next. One could, for example,
re-define the problem while in the process of planning an action.
Using Pedagogical Content Knowledge to Guide Reflective Thinking

Along with a focus on teachers developing reflective thinking practices,
researchers have been concerned about teachers developing a sufficient


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knowledge base to guide their thinking about children’s ways of understanding mathematical concepts and process (e.g., Ball, Lubienski &
Mewborn, 2001; Borko & Putnam, 1996). Indeed, pedagogical content
knowledge is an important resource for teachers to use as they reflect in
practice. Shulman (1986, 1987) first defined pedagogical content knowledge as,
The blending of content and pedagogy into an understanding of how particular topics,
problems, or issues are organized, represented, and adapted to the diverse interests and
abilities of learners, and presented for instruction. Pedagogical content knowledge is the
category most likely to distinguish the understanding of the content specialist from that of
the pedagogue. (1987, p. 8)

Building on Shulman’s work, Grossman (1990) delineated four central
components of pedagogical content knowledge. These components are:
conceptions of purposes for teaching subject matter (i.e., forming goals);
knowledge of students’ understanding, conceptions, and misconceptions
of particular topics in a subject matter; curricular knowledge; and knowledge of instructional strategies and representations for teaching particular
topics. Grossman acknowledges that “these components are less distinct
in practice than in theory” (p. 9), but this general framework was useful
in thinking about the ways in which the pre-service teachers in this study
used their pedagogical content knowledge.
Researchers have found that novice teachers tend to have inadequate or
underdeveloped mathematical pedagogical content knowledge for use in
practice (e.g., Borko & Putnam, 1996; Borko, Eisenhart, Brown, Underhill, Jones & Agard, 1992). In elementary mathematics education, several

projects have as a goal the development of teachers’ understanding about
children’s learning such as Cognitively Guided Instruction (e.g., Carpenter,
Fennema, Franke, Levi & Empson, 1999) and SummerMath (e.g., Schifter
& Fosnot, 1993; Schifter, Bastable & Russell, 1999). These projects have
produced materials (e.g., case studies for exploring teaching and learning) that have the potential to build pedagogical content knowledge for
pre-service teachers before entering the classroom. However, research
is needed to investigate how these teachers use this knowledge in their
practice.
My study investigated the role of reflection in practice by examining
the reflections and experiences of pre-service teachers during student
teaching. Although Mewborn (1999) focused on pre-service teachers, her
research was conducted earlier in the teachers’ educational preparation.
In my study, I focused on the form of reflective thinking pre-service
teachers’ exhibited and how they employed pedagogical content knowledge in mathematics in their thinking. The specific research questions


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for pre-service teachers’ mathematics instruction were: (a) When the preservice teachers demonstrate reflective thinking, what forms does it take
(i.e., reflection-in- or reflection-on-action)? (b) During reflective thinking,
how do these teachers use their pedagogical content knowledge?

METHODOLOGY
I studied two pre-service elementary teachers, pseudonymous Gerri and
Denise, during their semester-long student teaching internship. I conducted
this study from a perspective that combined ideas of interactionism and
constructivism in viewing the process of becoming a teacher. According to
the perspective of interactionism, people construct and sustain meanings

through interactions and patterns of conduct (Alasuutari, 1995; Blumer,
1969). This position is in accordance with the constructivist perspective of
learning in that individuals develop understandings based on their experiences and knowledge as it is socially constructed (Cobb & Bauersfeld,
1995). This framework supported this study in that the pre-service teachers
reflected and constructed meanings based on their participation in, and
observation of interactions and patterns of conduct with their students and
colleagues. Because I was interested in describing and interpreting the
thinking and experiences of pre-service teachers during student teaching, I
selected a qualitative case study as the most promising mode of inquiry
(LeCompte, Millroy & Preissle, 1992; Stake, 1995). The cases were
bounded by the semester-long student teaching experience and focused
on reflection in regard to their mathematics instruction using incidents of
reflecting-in and reflecting-on practice as units of analyses.
Data Collection
Context and Participants
Teaching program. Gerri and Denise were enrolled in a Master in Teaching
program at a state university. This two-year master degree program
served pre-service teachers who already held a baccalaureate degree in
a field other than education but desired to become teachers. Two primary
objectives of this program were:
1. To educate teachers to become effective practitioners who are
informed scholars with the leadership and problem solving skills to
help schools and communities meet the needs of the 21st century and
to enlighten thought and practice by bringing the inquiry method of a
research university to bear on the entire educational process.


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2. To empower teachers as reflective practitioners by helping them
develop the multiple and critical decision making skills essential for
today’s classrooms. (University program description document)
This research-based approach to developing reflective practitioners was
evident in the design of the student teaching internship. Requirements
of the internship included: 12 weeks in a K–8 school placement; writing
weekly in reflective journals; setting goals and reflecting on meeting these
goals weekly; writing lesson and unit plans; observing and reporting
on other teachers’ instruction; and completing a classroom-based action
research project on their own teaching. Gerri and Denise completed student
teaching during the spring semester.
For the action research project, the pre-service teachers designed their
studies during the previous semester as part of a course titled “Classroom
Focused Research”. Using two texts (Hubbard & Power, 1993; McNiff,
Lomax & Whitehead, 1996) as a framework for study, the pre-service
teachers studied methods of designing and conducting action research, and
planned original classroom-based research projects. The action research
project focused on a specific teaching strategy or approach. Each teacher
worked with a faculty committee (a supervisor, with expertise in the
selected area for research, and two additional faculty members). The preservice teachers wrote literature reviews in their areas of study as part of
a full study proposal. These proposals were submitted to their supervisors
for feedback and review before submitting a final version at the end of the
semester. Then they implemented their studies during student teaching. In
the month following their student teaching internship, Gerri and Denise
analyzed their data, wrote, and presented reports of their studies to a
faculty committee.
Given that pedagogical content knowledge is a focus of the study, I shall
describe the mathematics methods class that served as Gerri’s and Denise’s
primary source of this knowledge prior to student teaching. In the first

semester of the program, Gerri and Denise completed Elementary Mathematics Methods for which I was the instructor. The primary goals of this
course aligned with Grossman’s (1990) four components of pedagogical
content knowledge. This course emphasized developing understandings
for: reform-based visions of teaching and learning mathematics (goals
in teaching and learning); how children think and learn about mathematics, including common misconceptions in elementary mathematics
(knowledge of students); the range of resources and curriculum materials
available for mathematics instruction (curricular knowledge); and developmentally appropriate strategies for teaching and learning (knowledge of
instructional strategies).


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Van de Walle’s (1998) methods text was used along with supplemental research-based readings. Instructional approaches included in-class
explorations with textbook activities and extensive use of theoretical
frameworks and case studies from Cognitively Guided Instruction (e.g.,
Carpenter, Fennema, Franke, Levi & Empson, 1999) and Developing
Mathematical Ideas (e.g., Schifter, Bastable & Russell, 1999). Course
assignments included: interviewing children and analyzing their thinking,
understandings, and dispositions for mathematics; writing lesson plans that
included an analysis of possible children’s approaches and misconceptions; and collecting and critiquing problem-centered tasks from reformbased publications (e.g., Teaching Children Mathematics and Mathematics
in the Teaching in the Middle School).
Gerri. Gerri held a Bachelor’s degree in engineering and consequently
had substantial college-level coursework in mathematics. She entered the
master’s degree program to begin a career in teaching after staying home
with her children for several years. While Gerri had volunteered extensively in her children’s schools, she did not have any formal teaching
experience prior to entering the program. Gerri was regarded by University faculty and her field specialist as having strong content background,
especially in mathematics and science.
Denise. Denise was a recent graduate and held a Bachelor’s degree in

French with mathematics coursework through first-semester calculus. She
entered the master’s degree program two years after completing her undergraduate program. During those two years, she had worked as an educational assistant at an elementary school. During this time, she was exposed
to hands-on, student-centered approaches to teaching. Denise explained
that, as an educational assistant, she had recognized the value and benefits to student learning in using these approaches. However, Denise felt
that she lacked the theoretical foundations and framework needed to plan
effectively and to manage student-centered instruction because she did not
have an academic background in education prior to entering the program.
Researcher. I served as a participant observer in that I researched the
pre-service teachers’ practice while acting as their university supervisor.
I also had an established relationship with Gerri and Denise prior to the
study as their mathematics methods course instructor. Additionally, as their
Master’s degree Committee Chair, I advised Gerri and Denise on their
action research projects during all phases of their research.
Prior to the study and heeding the advice of Mewborn (1999) regarding
the creation of a non-evaluative environment to promote reflection, I


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discussed with Gerri and Denise my role as a researcher and as their
supervisor. I explained that the purpose of my study was to understand
how they reflected on their practice. I also explained that, as their supervisor, I perceived my primary role to be a resource to them and to provide
support during their internship. During my observations and conferences
with Gerri and Denise, I followed constructivist frameworks for student
teaching supervision that emphasized the developing of self-directed,
reflective practitioners (cf., Sullivan & Glanz, 2000). Correspondingly, my
focus was on facilitating the pre-service teachers’ skills in analyzing their
practice. For example, every conference began with my asking the preservice teachers for their perceptions of: “What went well?” and “What

would you like to change?” I encouraged them to identify their strengths
and problem areas; helped them to clarify their thoughts and plans; and
asked about their progress with their goals in follow-up meetings.
As is the case with most pre-service teachers in this program, Gerri
and Denise were confident they would pass student teaching (only pass
or fail grades were assigned). Therefore, they were more concerned about
their professional growth than official evaluations, and Gerri and Denise
stated that they also viewed me primarily as a resource. Indeed, neither
Gerri nor Denise was ever at risk of failing student teaching. Throughout
the semester, both Gerri and Denise commented that they viewed me as
a support and resource rather than an evaluator. However, ultimately, as
the university supervisor, I was responsible for evaluating their internship,
and thus it would be naïve to think that we had an entirely non-evaluative
relationship.
Data Sources
The primary data sources were: audio-taped interviews and conferences
with the participants; my observations of classroom teaching; reflective
journal entries and weekly goal statements; lesson and unit plans; and
participants’ data collected as part of their action research projects and
their final research reports. Nine observations and semi-structured interviews and/or conferences, occurring approximately every week to two
weeks, were conducted with each participant throughout the semester. I
recorded field notes for the observations, and each observation lasted about
one hour. Each interview lasted about 30 minutes and was transcribed. The
participants wrote at least one journal entry weekly, and wrote lesson plans
daily.
Data Analysis
To address the first question, I analyzed the data by analytic induction: I
searched for patterns of similarities and differences for when reflection did

MATHEMATICS TEACHING AS A DELIBERATE PRACTICE

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and did not occur and also for the forms of reflection (Bogdan & Biklen,
1992; LeCompte, Millroy & Preissle, 1992). I classified events as reflective
thinking if evidence existed that the pre-service teachers completed all
components of the reflective cycle. To find evidence, I first observed for
problematic events that initiated reflective thinking, and/or I read the preservice teachers’ journals for their reporting of surprising or puzzling
events. During observations, the pre-service teachers usually demonstrated
that something puzzling occurred through facial gestures, pauses in their
speech, talking aloud, and/or a lesson that departed from the previously
written lesson plan. As part of journal writing, the pre-service teachers
focused on problematic events and wrote about them directly.
Next, I investigated whether the pre-service teachers analyzed and
developed a possible resolution to the problem. I obtained this evidence
from questioning and from their journal writing about considerations
in making an instructional decision. In analyzing the journals, to help
sort out whether their reflections were in-action or on-action, the preservice teachers placed an asterisk next to any thoughts or ideas that
occurred to them after teaching rather than just writing about their thoughts
during the lesson. Finally, I examined the data for a resulting action or
implementation of a plan in instruction.
It should be noted that the findings were often supported, at least in
part, from the participants’ self-reporting of reflective experiences. While I
endeavored to validate these reports through triangulation of data sources,
the nature of this research did rely on participants’ reports of reflection.
Consequently, I made efforts to ensure that the participants had operationalized reflective experiences in a manner consistent with my understandings of reflection-in-action and reflection-on-action as presented in this
manuscript. We discussed these notions thoroughly in the initial interview
(with examples and non-examples of reflection), and then summarized
these ideas again throughout the semester.

For the initial coding of data, I used codes of reflection-in-action
and reflection-on-action to identify and track the forms of reflection.
Through the data analysis process, I found a need to distinguish further
forms of reflection into subcategories (immediate reflection-in-action,
delayed reflection-in-action, short-term reflection-on-action and long-term
reflection-on-action), and a new category emerged: deliberate practice.
Each of these forms of reflection is represented in Figure 1 and described
briefly below as they were used in coding and analyzing data. They are
exemplified further in the presentations of the case findings.
Immediate Reflection-In-Action (referred to as immediate reflection
hereafter) represents the thinking when pre-service teachers made immediate decisions while completely in the act of teaching. This form of


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Figure 1. Processes of deliberate practice.

thinking was difficult to identify, and thus I relied heavily on focused
observations and follow-up interview questions to determine if the
pre-service teachers identified a problem during instruction, and then
completed the reflective cycle. Moreover, I asked the participants directly
about occurrences of and their ability to analyze and make decision
about problems identified during teaching. Delayed Reflection-in-Action
(referred to as delayed reflection hereafter) represents the thinking the
pre-service teachers exhibited when a pause or break occurred in the
act of teaching (e.g., students completing individual work or recess).
Similar to immediate reflection, delayed reflection resulted in analysis, a
decision, and an action for the lesson in progress or for the plans for the

day. However, a break in activity distinguished delayed reflection from
immediate reflection. Immediate reflection and delayed reflection both
correspond to Schön’s (1987) description of reflection-in-action, but they
differ in the level of instructional activity and demands occurring during
reflection. In Figure 1, the double arrow between Reflection-In-Action and
the Teaching and Learning Episode represents how instruction triggers
reflection and consequently, reflection-in-action influence teaching while
instruction is taking place.
With regard to forms of Reflection-On-Action, Short-Term ReflectionOn-Action (referred to as short-term reflection hereafter) was exhibited


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when the pre-service teachers thought back over a short period of time after
a lesson or day was over (e.g., reflecting on a lesson as they drive home
or on the week’s instruction over the weekend). Short-term reflection is
different from delayed reflection in that the pre-service teachers were not
under pressure to reflect, resolve, and implement the action to address a
problem immediately during the problematic lesson or day. This reflection was often about the success of a lesson in contributing to learning
goals in order to guide planning for the next lesson or unit. Long-Term
Reflection-on-Action (referred to as long-term reflection hereafter) was
exhibited when the teachers systematically analyzed and examined their
practice over an extended period of time for the purposes of understanding
and improving practice more globally. They looked for emerging patterns
and developed personal theories about teaching and learning. Most often,
the reflective cycle for long-term reflection took place over several months.
Deliberate Planning was exhibited when pre-service teachers purposefully used existing knowledge, theories, and reasoning about teaching and
learning to design plans for particular students’ learning. While reflections

on past experiences may be part of an existing knowledge or theory base,
this form is different from reflection. Reflective thinking was initiated by
a problematic event. Conversely, deliberate planning involved analysis of
instructional options prior to teaching, often preparing to avoid anticipated
problematic events. All five forms of thinking compose a framework for
the deliberate practitioner.
After establishing this framework for deliberate practice, I used codes
of immediate reflection, delayed reflection, short-term reflection, longterm reflection, and deliberate planning to classify the forms of thinking,
and correspondingly, revised my research questions to consider thinking in
these five forms (versus Schön’s [1987] two forms). Additionally, for more
precise coding, I used qualitative data analysis software with a combination of open coding (Strauss & Corbin, 1990) and indexing of text (Miles
& Huberman, 1994). Once I identified and classified reflective and/or
deliberate events (the first research question), I analyzed these events for
how pedagogical content knowledge was used (the second research question). That is, I examined how these teachers applied pedagogical content
knowledge (as described in Grossman’s [1990] four component model) in
planning for or reflective thinking about a problematic event. Throughout
this process, the multiple data sources were compared to generate a
complete picture of each event and to confirm or refute emerging patterns.


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FINDINGS
For each teacher case study, I describe an instructional episode. Following
the description of each episode, I present my analysis and interpretations
by discussing the forms of deliberate practice and how pedagogical content
knowledge was used in reflection and deliberate planning. I chose these
episodes because they represent Gerri’s and Denise’s thinking and practices, and they provide examples of how the various processes of deliberate

practice connect to each other and to PCK.
Gerri
Gerri’s Philosophy and Field Placement
Gerri’s philosophy of teaching seemed to be focused on ensuring that
each student was learning and was forming a conceptual understanding
of the subject areas. In an interview prior to beginning student teaching,
Gerri made it clear that her foremost interest was in student learning. She
was distressed that current educational environments present barriers to
focusing on students’ understanding of concepts. In referring to a field
experience prior to the student teaching semester, Gerri stated,
I have seen that there is a real expectation . . . to get a lot of information to the kids . . . .
In the practicum that I just completed, I saw the kids pushing forward in the curriculum
without really having a good basis of what we were trying to teach . . . . Kids don’t really
have the concepts (Interview, December 30).

Gerri’s concern for students gaining strong conceptual understanding
was evident in her choice for her classroom-based action research project.
She investigated the efficacy of her teaching practices in developing her
students’ conceptual understanding of multiplication and division.
Gerri completed her internship in a third-grade classroom. While
her field specialist, Mrs. Baker, was well respected in the school and
the community, she was not perceived as a reform-based teacher. Mrs.
Baker had a predominately teacher-centered style and focused primarily
on skill mastery. Both Gerri and I observed this more traditional style.
Gerri described Mrs. Baker’s teaching as, “Based on my observations,
[Mrs. Baker has] a pretty traditional classroom. They [the students] are
encouraged not to visit with their neighbors, to stay on task” (Interview,
December 30). With regard to teaching mathematics, and just before taking
over the class, Gerri said, “The students are learning their multiplication
facts . . . . [Mrs. Baker] is a traditional teacher, relying on the text1 to teach

math. Her students have not really worked in small groups or with manipulatives, so I can expect some problems when I first start teaching math”
(Journal, January 3).


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Consequently, Gerri understood that Mrs. Baker was not likely to
be able to support her in her efforts to implement more reform-based
approaches to teaching mathematics. Additionally, Gerri’s journals and my
observations indicated that Mrs. Baker tended to be a reactive mentor,
offering suggestions to solve existing problems or dilemmas, but not
consulting with Gerri prior to teaching a lesson.
Making Connections: Skip Counting Patterns and Multiplication
The instructional episode. Early in Gerri’s student teaching experience,
she taught a lesson on multiples. While the students had not necessarily
acquired all of their multiplication facts, Mrs. Baker had provided instruction on all of the basic facts for multiplication. Gerri’s goals for this lesson
were: “Students will find the multiples of one digit numbers using their
knowledge of 0–9 multiplication facts. Students will understand what is
meant by the term multiple in mathematics” (Lesson plan, January 21).
The lesson began with Gerri writing, “0 2 4 6 8 10 12” on the board,
and asked, “What is the pattern for these numbers?” A student responded,
“Multiples of 2”. After another example with fives, Gerri wrote, “0 3
6 9 12 15” on the board, and again asked, “What’s the pattern here?”
Another student responded, “Goes up by three”. After showing a few
more examples with similar questioning, Gerri handed out a worksheet
with similar exercises on it (e.g., “Find the first 5 multiples of 6”). Up
to this point, I observed that Gerri had followed her lesson plan exactly.
Gerri then went back to her desk, where four students promptly lined up

for individual help (a practice established by Mrs. Baker during individual
seatwork time). The students did not know where to start on the questions
when a pattern had not been presented (as the example listed above). As
Gerri worked with individual students, she asked the students how they
found the next number in a pattern. When the responses relied on adding
to the last number, she realized that while the students could complete the
pattern, they did not relate this pattern to multiplication.
Indeed, from my observation many of the students seemed to bring only
a recursive interpretation to the pattern, observing that the next number
increased by a fixed amount from the last number. When one student
said, “Multiples of 2”, it was not at all clear that the student linked the
word “multiples” to multiplication. Perhaps she just used “multiples” as
a word that preceded the quantity of increase in the pattern. Gerri did not
emphasize this connection, and accepted responses such as “goes up by
three”, without asking for further explanation or exploration. Later during
seatwork, Gerri recognized this lack of connection in the concepts, and
identified a problem in meeting her stated goal of developing an under-


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standing for the term multiple. She explained in an interview immediately
following the lesson,
We started with the patterns, and they caught right on. They were skip counting. But then,
I didn’t make the connection very strongly that, besides skip counting, you could also do
the multiplication facts. I felt like I kind of missed that. And when they came back [to my
desk] and I was working one-on-one [with students], I realized that I felt like I could have
made that [connection] stronger. (Interview, January 21)

After realizing the problem, Gerri stopped the students during their
individual seatwork and began a whole class discussion at the board using
problems on the worksheet to build the connection she recognized was
missing.
Forms of thinking. This episode illustrates aspects of deliberate planning, delayed reflection during the lesson, and long-term reflection. While
presenting the patterns, Gerri deliberately planned for the students to
understand the term multiple through patterns. However, she did not
consider that simply being able to provide a correct response to the question as posed (i.e., “What’s the pattern?”) was not sufficient to determine
if her goal of understanding the term multiple had been met. Once she
worked with students individually, she began to probe their thinking further
by asking, “How did you find the pattern?” and through this deeper
questioning, she was able to assess that they, at best, had a limited understanding of the term multiple. In not planning to question the students on
how they arrived at their solutions (showing limited deliberate planning),
an approach emphasized in reform-based approaches to teaching (e.g.,
NCTM, 2000), she missed an opportunity of on-going assessment and a
chance to encourage the students to build their understandings. However,
once a break in the lesson occurred, and Gerri worked with students individually (without the demands of facilitating a whole class discussion),
she had time to consider and assess individual students’ approaches and
identify a problem in instruction: the students were not understanding
“multiple” as referring to multiplication of numbers. She analyzed the
situation and implemented a different approach in instruction, emphasizing
the connection between the pattern of products and each product’s corresponding multiplication facts. Given that this all transpired during a lesson,
it represented delayed reflection. Because the problem identification and
reflective thinking did not occur until Gerri had a break in action, I did not
classify it as immediate reflection.
Long-term reflection also was evidenced in Gerri’s conclusions about
questioning and seeking explanations from students. This lesson represented one of a series of efforts for Gerri to examine the efficacy of

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teaching multiplication and division for understanding as part of her
action research project. Based on earlier field experience prior to student
teaching, Gerri identified a problem of students having only procedural or
fact-based knowledge of these operations. Reflective thinking was exhibited as Gerri planned her study on a global level, and then planned each
lesson as part of implementing her study on a daily level. As Gerri implemented her plans to facilitate her students’ conceptual development of
multiplication and division, she completed the action of the reflective
thinking cycle.
Use of pedagogical content knowledge. In planning the lesson Gerri knew
that it is important for students to understand the meaning of multiple
and to look at patterns in different ways (multiples as a pattern and as
products), consistent with reform goals (e.g., NCTM, 2000) and representing Grossman’s (1990) notion of PCK for goals in teaching and
learning. Indeed, once Gerri realized that her goal of understanding had not
been met, she deemed that this idea was important enough that she should
adjust the lesson, and return to a whole class discussion to emphasize the
connections. Although use of PCK was evident in planning the lesson,
Gerri had not yet developed the capacity to apply this knowledge when
probing the students’ thinking during a whole class discussion. Yet, during
individual instruction (while other students were working at their seats),
Gerri demonstrated that she did have PCK for questioning strategies and
for understanding students’ thinking (Grossman, 1990) by recognizing the
problem (students’ limited understanding of the word multiple). These
finding are consistent with Corwin (1996) that emphasized the complexities involved in listening and questioning students to support mathematics
learning in whole class instruction. In sum, Gerri more effectively applied
her PCK when she was not orchestrating the lesson: during planning,
during a break in teaching, and after the lesson.
Denise

Denise’s Philosophy and Field Placement
Like Gerri, Denise’s fundamental philosophy for teaching and learning
was to focus on students’ understanding in learning. This focus was evident
in that her planning and self-analysis consistently relied on students’
conceptual learning as a referent over concerns for covering textbook or
external curricular guidelines, as will be illustrated in the episode below.
Moreover, Denise’s action research project was developed around her
concern for students’ attitudes and anxiety as factors in learning math.
Denise theorized that by using approaches from multiple intelligences in


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teaching math, her students would learn to utilize their individual strengths
and feel more successful in learning math. Her plan was to implement
a multiple intelligences approach (e.g., Gardner, 1993) in mathematics
instruction in order to facilitate her students’ learning. Denise was aware
that Gardner’s work has been questioned for its scientific merit in the
research community. In choosing her approach to action research, she
acknowledged limitations of the theoretical basis, but still wanted to see
the effects of these approaches in practice.
Denise was placed in a fourth- and fifth-grade combination class
that was team-taught. Both team members, Mrs. Knight and Mrs. Earl,
served as mentors to Denise, although Mrs. Knight was her official field
specialist. Mrs. Knight and Mrs. Earl used predominately student-centered,
reform-based approaches and materials. They focused on concept development and active, hands-on learning, and they frequently used cooperative learning strategies. Mrs. Knight was a teacher leader in the region.
Mrs. Knight tended to be both a proactive and reactive mentor. Mrs.
Knight discussed, on a daily basis, possible options and considerations

with Denise prior to Denise’s teaching lessons and reviewed events and
decisions after lessons.
Developing Meanings for Decimals: Representing Decimals on Grids
The instructional episode. The lesson was part of a unit that served as an
introduction to decimals. Prior to this lesson, the students had discussed
everyday uses of decimals, decimal equivalents for common fractions (e.g.,
1/2 and 1/4), approximate values of decimals (e.g., 0.49 is almost equal to
0.5 which is the same as 1/2), and why we need both decimal and fractional
forms of numbers (Lesson plans for March 10 and March 13). Denise
developed these lessons in collaboration with Mrs. Knight. Denise relied
primarily on Mrs. Knight’s advice and outside resources (e.g., materials
from Investigations [TERC, 1998]) and, indeed, I did not ever see Denise
using or referring to the district textbook, Addison-Wesley Mathematics
(Eicholz, 1991).
For this lesson, Denise’s goal was for students to gain an understanding
of “representing decimals on grids; reading, writing, and ordering decimals
[using grids]; and adding decimals on a grid” (Lesson plan, March 14).
Immediately prior to teaching the lesson, Denise and I had a pre-lesson
conference. Denise explained her goals to me and said that she thought the
decimal grids (see Figure 2) were an effective representation for students
to gain an understanding of the value of various decimal numbers. Denise
had been exposed to decimal grids both in her mathematics methods course
and as an educational assistant.


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49

Figure 2. Decimal grids Denise used in March 14th lesson.

Denise discussed her lesson plan with me and said that she planned
to have a whole-group discussion on representing various decimals on
the grids. She planned to ask them to color one tenth (written as “0.1”
on the board) on each of the four grids and then to choose the best grid
to show one tenth. Next, she wanted students to express the decimal of
one tenth in fractional forms that correspond to each grid (e.g., 10/100 for
the hundredths grid, 100/1000 for the thousandths grid, etc.). Denise then
planned to “Write on the board 0.1 = 0.1000. Ask the students to figure
out how they know these decimals are equal. Have the students share their
answers with their partners” (Lesson plan, March 14). Denise planned to


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end the lesson with students showing various other numbers on the grids
(e.g., 0.75, 0.125) and discuss fractional equivalencies for each.
After Denise shared her plans, I asked what challenges students might
have using the grids. She replied: “They have to show one tenth, so it’s easy
on this one [showing one column on the tenths grid]. But I’m afraid they’re
not going to know it’s the same shape on here [the hundredths grid]. I’m
wondering if they’ll color in one little box [representing one hundredth]”
(Pre-lesson conference, March 14). At this point Denise revealed that this
lesson was the first time her students had seen or used decimal grids. Upon
hearing this information, I suggested that she may be incorporating too
many ideas in a first experience with grids, and that perhaps she should
have the students just work with the tenths grid and become comfortable
representing various numbers on this grid before moving on to comparing

the grids and the multiple representations for decimals. As we ended the
conversation, Denise decided that she was not comfortable changing the
lesson at this point, but she would consider slowing down the discussion if
the students seemed to be having difficulties.
Denise began the lesson as she had planned it and asked a student to
come to the overhead and color in one tenth on the tenths grid. The student
correctly colored one column. Next, another student (Maria) colored in one
tenth on the hundredths grid by coloring a row of ten of the one hundredths
squares, and then labeled it as “0.01”. Denise then explained that this
second representation was “still one tenth because it was ten hundredths”,
but she did not address Maria’s incorrect labeling of one tenth as “0.01”.
Pointing at the thousandths grid, Denise said, “100 times 100 is a thousand okay?” Indeed, Denise was describing the ten-thousandths grid while
pointing to the thousandths grid. She paused, looked back at her notes, and
said, “Well this is still one tenth”, as she pointed to a column representing
one tenth on the thousandths grid. She paused again, and corrected herself,
“Wait, 100 times 100 is 10,000, right? But this is still one tenth”, again
pointing to one column on the thousandths grid. Somewhat rushed in her
speech, Denise said, “So all of these are one tenth right?” as she pointed
to each of the tenths, hundredths, and thousandths grids. Uncharacteristic
of her typical approach to teaching, Denise did not call on students and
the students were silent throughout this part of the lesson. Denise, seeming
flustered, omitted what she had planned for developing ideas of which grid
is most appropriate for which numbers and fractional equivalents of the
decimal numbers (as described in her plans above) and quickly began the
last part of her lesson plan.
Denise asked a student to represent 0.75 on a grid. The student colored
seven columns on the tenths grid. Denise questioned further and asked


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51

the student, “Now what are you going to do with the rest?” The student
looked again and shaded half of another column, and he explained, “five
[pointing to the five in the hundredths place] is half of ten”. Next, Denise
had another student show 0.75 on the hundredths grid. This student colored
the same shape as the previous student but on the hundredths grid, coloring
in 75 squares. Denise asked the class, “Do you agree?” Without much
response, Denise asked the student to explain her work. Without shading
anything on the thousandths grid, Denise asked, “What do you think it will
look like on the thousandths grid?” Another student responded, “The same
thing”. Denise asked this student to show this, and asked again if everyone
agreed (again little response from the class). Next, she turned to the tenthousandths grid, and asked a student (Carl) to show 0.75 on this grid. Carl
shaded seven and a half hundredths squares across the top row (showing
0.075). Again, Denise asked who agreed. With no response, Denise asked
another student (Mark) to show how he thought this should be represented.
Mark correctly colored the same pattern that had been shaded on the other
grids. Denise stated that they were both correct (but did not explain how
these two representations could be considered correct), and began a final
discussion reminding students how to pronounce the numbers correctly
(e.g., seventy-five hundredths).
From my observations, it was not at all clear that most students understood how to use the various grids. While a few students were able to
provide correct responses at the overhead, based on facial expressions
and lack of response from other students (not typical for this class), the
students working at the overhead seemed to be the only students who
were confident in their understandings. Moreover, all of the students who
contributed volunteered to do so. Maria’s and Carl’s work indicated that
not all students understood these representations and the values of the
decimal numbers. For this second part of the lesson (starting with representing 0.75), Denise’s teaching style seemed to be more representative of

what I had observed of her teaching in that she asked students to show
and explain their work, rather than Denise doing most of the talking (as
in the first part of the lesson). Yet, unlike her usual practice, Denise did
not delve into deeper explanations for their thinking and solutions, and
skimmed over the idea that both Carl’s and Mark’s representations “were
correct”.
Immediately following the lesson, Denise and I discussed the lesson.
As soon as we sat down Denise said, “I wasn’t sure how to use that ten
thousandths grid! You can tell! . . . I still don’t know if I was completely
correct”. I asked her to explain what she was unsure about, and she said,
“The representation of the examples”. She went on to say that she became


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confused when the two boys represented 0.75 on the ten-thousandths grid.
I reminded her, “You said that they were both correct”. She replied, “They
weren’t!” I went on to explain how they could both be considered correct
if the first boy defined the unit one to be a row (what had been one tenth in
other representations), but that if we were to assume that the unit consisted
of the large square (as was the case, but was never directly stated in the
lesson), he was not correct. We continued to discuss this idea of deciding
and declaring the unit one as a first step in using grids, and Denise indicated
that she had remembered this kind of discussion in the methods class.
Indeed, it seemed as if Denise’s vague recollection of the methods class
discussion led her to the idea that they both could be correct, but her
memory and understanding was not sufficient to recall how both students
could be correct.

Next we discussed her confusion with her statement that 100 times 100
is 1,000. She immediately said, “It’s not one thousand; it’s ten thousand!
It gets confusing!” After more discussion Denise said, “This makes sense
to me now . . . . But you think this is [only] decimals, why should it be
so hard! . . . There’s a lot of stuff here, and I didn’t know that until I got
up there I guess”. Denise continued to think about these issues, and in a
subsequent lesson on March 15 and 16, revisited the topics in this lesson,
incorporating the ideas from our discussion in facilitating her students’
understanding of and representations of decimals.
Forms of thinking. This episode illustrated deliberate planning, the difficulty of immediate reflection, and short-term reflection. In planning the
lesson, Denise had clear goals for students to understand the value of
decimals (number sense with decimals) and for using grids as a representation to facilitate this learning. She was aware of the possible difficulties
that students might encounter in using the grids (e.g., seeing that a number
would cover the same area on any of the grids), and had specific questions planned to draw out connections among the decimal number, the
grid representation, and the fractional equivalent. These plans and ideas
were consistent with recommendations from NCTM (2000) and from her
methods course and text (Van de Walle, 1998).
While teaching the lesson, Denise encountered problems in her own
understanding of what the grids represented (i.e., her confusion with the
thousandths grid representing 100 times 100) and how to use the grids
to represent decimal values (i.e., her confusion with Carl’s and Mark’s
different representations for 0.75). While she identified a problem, she
did not resolve her confusion until after the lesson was completed. She
admitted in the post-lesson conference that indeed she did not realize the


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53

limits of her understanding until she was teaching in front of the class.
In her confusion, she began a short teacher-led explanation of how to
represent one-tenth on each grid, without following her plans for developing these ideas with the class. In doing so, Denise essentially moved
away from her goals and reform-based approaches of facilitating understanding, towards more traditional approaches of presenting material.
Later in the lesson, Denise’s inadequate understanding of the representation again appeared in trying to sort through Carl’s and Mark’s two ways to
represent 0.75. Because Denise had not developed the idea of identifying
the unit (either in her mind or in her students’ minds), while she had some
sense that both representations could be considered valid (from a vague
recollection from the methods course), she was unable to make sense of
this in the act of teaching. So, she avoided a conceptual discussion and
moved on to other parts of the lesson.
These moves were consistent with the findings of Borko et al.’s (1992)
case of Ms. Daniels. When Ms. Daniels was confronted with her own
confusion over understanding division with fractions while teaching, she
attempted to use a representation she had learned in her math methods
course. Misapplying this representation, she realized that she had made
an error, but while teaching, Ms. Daniels did not fully understand the
problem. With no resolution to the problem, Ms. Daniels resorted to a
procedural, rule-based explanation. Laying my framework for deliberate
practice over Borko et al.’s findings, I would interpret this episode as
that, when Ms. Daniels was faced with a gap or weakness in PCK, her
immediate reflection on the problem led her to follow the pull of tradition.
Similarly, Denise, when challenged to analyze and make sense of
her confusions regarding the grid while teaching, evinced inexperienced teachers’ difficulties in accessing instructional options to resolve a
dilemma. Without a repertoire of options to consider during immediate
reflection, Denise opted for more familiar traditional methods. Denise
resorted to telling her students, “that they all show one tenth”, and indeed
omitted a significant portion of her plans to develop and unpack the ideas
behind how they each show one tenth. That is, when confronting an instructional dilemma, Denise demonstrated immediate reflection and resolved
the problem by moving toward the tradition of telling and abandoning

plans to implement more innovative approaches.
To resolve further this dilemma, Denise later exhibited short-term
reflection. Starting in our post-lesson conference and continuing in her
planning for the two lessons, Denise made sense of the representations in
her own mind, planned ways to facilitate her students’ understanding, and
implemented these plans. Thus, given time to analyze the problem, Denise


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reflected further and returned to her goals for developing the students’
understandings.
Use of pedagogical content knowledge. Denise demonstrated PCK in
each of Grossman’s (1990) four areas; however, her underdeveloped PCK
caused the problems identified above and her difficulties with immediate
reflection in resolving those problems while teaching. First, in establishing her goals for the lesson, Denise demonstrated that she understood
purposes and strategies for learning decimals, consistent with reformbased goals (e.g., NCTM, 2000). In planning for the lesson, Denise
was aware of and chose materials effectively (i.e., drawing from Investigations and using the decimal grids) for facilitating understanding of
decimal values, thus demonstrating curricular knowledge and knowledge
of instructional strategies. Moreover, in our pre-lesson conference, Denise
indicated her understanding of students when she explained her concern
for how they might misuse the grids and the importance of connecting
decimals, fractions, and the grids in learning.
Denise’s PCK needed further development in understanding fully what
each grid represented (e.g., her error in confusing the thousandths and the
ten-thousandths grids), and in understanding the role of identifying the unit
as part of understanding the multiple representations. Her limited PCK was
not apparent to Denise until she began to teach, despite her careful planning. Moreover, these limitations impeded Denise in her understanding of

the complexity of teaching and learning decimals (c.f., Ball et al., 2001),
and correspondingly resulted in a lesson plan that had too many concepts
packed into an introductory lesson, without sufficient time to explore and
develop each of these issues. In sum, these limitations in Denise’s PCK led
to some confusion and a dilemma during instruction, and upon immediate
reflection, she decided to depart from her plans for developing students’
understandings of how to use the grid representation in favor of more
traditional approaches of telling what the grid represented.

DISCUSSION AND IMPLICATIONS
The cases of Gerri and Denise illustrated forms of thinking in deliberate
practice and how they used each of Grossman’s (1990) four components of
PCK during deliberate practice. Below, I discuss and suggest implications
for the various forms of reflective thinking as they were exhibited, and how
the pre-service teachers’ use of their PCK played a role in this thinking.


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55

Supporting Pre-service Teachers through Collaborative Deliberate
Planning
Using their PCK Gerri and Denise deliberately planned for many of
the possible problems of teaching and learning. It is important to note
that the examples presented here in which Gerri and Denise demonstrated thoughtful, research-based practice were just a sample of the many
instances I found. It seemed that the use of research-based resources
(e.g., Carpenter et al., 1999; Van de Walle, 1998) in their teacher education program prepared Gerri and Denise to approach instruction as a
problem solving endeavor, focusing on facilitating students’ understanding
and anticipating problems in teaching and learning. However, they still

encountered problems in teaching that most probably would have been
avoided by more experienced teachers. For example, in Denise’s case,
while her plans indicated strong PCK for decimals, it was not until she
actually taught the lesson that gaps were revealed in her PCK. Despite
the PCK Gerri and Denise had gained through their professional coursework (as evidenced in their pre- and post-conferencing and planning), their
understandings were limited without having used it in practice. This is
consistent with Borko and Putnam’s (1996) argument for inadequate PCK
in pre-service teachers. In considering implications from this study, the
question then became, “Do we just accept these gaps as part of novice
teaching? Or, can teacher educators and field specialists better utilize and
develop pre-service teachers’ existing PCK and, correspondingly, enhance
their skills in deliberate planning?”
As I considered Denise’s episode in particular, I realized that if I had
had the pre-lesson conference on decimals with Denise prior to the day of
the lesson, I could have better supported her in understanding the complexities and issues in teaching this lesson. As it was, while I suggested some
ideas for her to consider, she was not in the position to act on them just 15
minutes ahead of teaching. Mrs. Knight, as an expert mathematics teacher,
did serve as this kind of resource at times. Indeed, Denise’s decision to
use the decimal grids was motivated by a planning discussion with Mrs.
Knight and access to Mrs. Knight’s extensive reform-based resources.
However, even this discussion did not include in-depth, lesson-specific
PCK that Denise might need to bring to this lesson. Moreover, it is not
clear that it is reasonable for mentor teachers to serve in this role. Mentor
teachers’ primary focus needs to be on supporting pre-service teachers’ in
their learning to do the daily work of a teacher, and this may mean that
developing pedagogical content knowledge is left in the background.
Conversely, university supervisors could, more aptly, support preservice teachers’ development of pedagogical content knowledge by using
the pre-service teachers’ experiences as the context for learning. To

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support effectively our pre-service teachers’ knowledge development and
correspondingly their skills in deliberately planning for instruction, university supervisors need opportunities to discuss in depth the teaching and
learning issues for specific lessons to pose various problematic scenarios,
both after teaching episodes and well in advance of lessons. Through
these discussions we might be able to provide opportunities for pre-service
teachers to consider possible mathematical teaching and learning problems that might not occur to pre-service teachers due to their lack of
experience. The model used for supervision in this study that emphasized reflective practice in general (Sullivan & Glanz, 2000), using a
pre- and post- conference on the day of the lesson, did not allow sufficiently for subject-area specific support prior to a lesson to build on
the pre-service teachers’ deliberate planning. The pre-service teachers’
experiences did serve as rich contexts for reflection and making sense of
mathematics for teaching and learning after the lessons were over, and
these discussions were an important part of the pre-service teachers’ development. However, it seems that opportunities for teachers’ learning could
be expanded by conducting in-depth conferences to plan lessons well in
advance of a lesson. The purpose of these conferences could be to examine
and explore the mathematics and appropriate pedagogical approaches
related to a lesson. Correspondingly, in order to enact this recommendation, pre-service teachers need subject-area specific supervisors and/or
field specialists that could structure and prompt these conversations.
Using Pedagogical Content Knowledge during Immediate
Reflection-in-Action is Difficult
In the instructional episodes, I found substantial evidence for the difficulty
of immediate reflection. Both Gerri and Denise stated that they wanted to
teach lessons as they had planned them, and did not have the confidence or
flexibility to change these plans in the middle of the lesson. Correspondingly, they found immediate reflection to be difficult and it rarely occurred.
Indeed, I only found one instance of immediate reflection that resulted in
a continued effort to develop students understanding (i.e., continue with
plans to implement reform-based goals). Instead, for all other observations,

when immediate reflection occurred this reflection resulted in a decision
to abandon innovative approaches in favor of more familiar traditional
approaches (as illustrated in Denise’s case). Delayed reflection as a specific
form of reflection-in-action recognized the pre-service teachers’ need for
a brief break in activity to provide the opportunity for reflection during
instruction, and short-term reflection allowed the pre-service teachers to
examine their practice outside of the act of teaching. Gerri’s perspective on
her reflective practice illustrated how it was easier for her to find opportuni-


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ties for delayed reflection or short-term reflection than it was for immediate
reflection:
You almost need a quiet time [to reflect] . . . A big time for me was a half an hour before
lunch, we’d have prep time. That was a big time to think about, “How did the morning go,
and what am I going to do in the afternoon? And is there something in the morning that’s
going to affect what I was planning to do in the afternoon? . . . Then, maybe I had to make
a quick shift there. It’s hard to do on the fly because that crunch of time. You don’t get
much of a chance to say, “Did that work okay?” . . . You do have to have that quiet time.
(Interview, April 26)

Gerri’s and Denise’s reflective practices were consistent with Borko
et al.’s (1992) report of the experience of Ms. Daniels and with Ball et
al.’s (2001) notions regarding the complexities of analyzing a problem and
its representations during instruction and the need for confidence in one’s
own mathematical understanding enough to adjust instruction “on the fly”
(p. 438).

It is important for teacher educators, pre-service teachers, and their field
specialists to consider how pre-service teachers find it difficult to reflect in
the form of immediate reflection but have the capacity to reflect through
delayed reflection and short-term reflection. While this study examined
only two cases, it seems that reflecting while in the act of teaching might
be difficult for many novice teachers, especially given the limits of PCK
for novice teachers (c.f. Borko & Putnam, 1996). Teacher educators, preservice teachers, and field specialists might benefit from recognizing that
this form of thinking could be difficult for novice teachers by: developing strategies for fostering this form of thinking through pre-teaching
conferences (as described above with regard to deliberate planning); and/or
recognizing that immediate reflection might be beyond the skills of some
novice teachers who are wrestling with heavy cognitive demands during
initial teaching experiences. In those cases where we recognize that immediate reflection might not be possible, we might instead focus on delayed
reflection, and help pre-service teachers to structure their instructional
plans so that slight breaks in activity are included to allow time for
reflection and adjustments to teaching and learning. Additionally, we can
facilitate short-term reflection through post-lesson conferences and journal
writing, as illustrated in Gerri’s and Denise’s cases.
Encouraging Long-term Reflection for Sustaining Growth
The long-term reflection exhibited by the pre-service teachers seemed
to be an important part of their reflective practice for future teaching.
Through long-term reflection, the pre-service teachers put together the
pieces of individual reflective episodes to realize a pattern that informed
their practices. These patterns over time may be even more important