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Preservice Secondary Teacher ‘Moves”

ABSTRACT

This paper discusses the experience of one preservice secondary mathematics teacher as

she and her group prepared a pre-calculus lesson in the context of lesson study. The study

looks at how this preservice teacher used different compensation ‘moves’ to direct the

conversation away from her mathematical knowledge in order to protect her identity as a

knower of mathematics. Although the preservice teacher was a strong undergraduate

mathematics student, she feared being labeled as ‘dumb’ and sought to deflect attention

away from her weak conceptual understanding of secondary mathematics. This paper

seeks to investigate the culture in which preservice secondary mathematics teachers

develop their beliefs and how those beliefs influence prospective teachers’ behavior

during mathematical conversations. The study acknowledges that belief systems are

formed from years of experience within the mathematics school culture and provide clues

as to the motivation for using ‘moves’ to keep from revealing a weakness in mathematics.

DRAFT—Please do not quote without authors’ permission.

Preservice Secondary Teacher ‘Moves’

A PRESERVICE SECONDARY TEACHER’S MOVES TO PROTECT HER VIEW OF

HERSELF AS A MATHEMATICS EXPERT

Julie Stafford-Plummer

Blake E. Peterson

INTRODUCTION

In 1990, Ball reported on a study in which she investigated three beliefs that were

commonly held among mathematics education majors. The beliefs were that high school

mathematics was not difficult, the mathematical knowledge needed to teach secondary

mathematics was primarily gained prior to college, and any additional mathematical

knowledge that was needed was gained as part of a bachelor’s degree in mathematics.

She found, however, that many preservice teachers who held these beliefs lacked a

conceptual knowledge of high school mathematics and tended to explain concepts in

terms of procedures. The purpose of this study was to investigate a form of Japanese

lesson study as a possible vehicle to dislodge these commonly held beliefs described by

Ball (1990).

LITERATURE REVIEW

To lay the groundwork for this study, we will discuss beliefs in general as well

more specific beliefs related to intelligence, mathematics, and mathematics teaching.

Beliefs

One focus of the reports of the Third International Mathematics and Science

Study was the culture of mathematics teaching. Stigler and Hiebert (1999) defined


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teaching as a cultural activity which possesses cultural scripts that are “learned implicitly,

through observation and participation and not by deliberate study” (p. 86). They also

stated that the culture of mathematics teaching was unique to a country and greatly

influenced the way in which mathematics teachers viewed their profession (Stigler and

Hiebert, 1999). This meant that the culture largely impacted what the teachers believed

about their subject matter, their philosophies about student learning and their role as

mathematics instructors (Stigler and Hiebert, 1999). These beliefs then contributed to

cultural norms and actually “generat[ed] and maintain[ed] cultural scripts for teaching”

(Stigler and Hiebert, 1999, p. 89). This set up a cycle where beliefs about teaching were

influenced by the culture, and the culture in turn preserved these commonly held beliefs.

Members of the culture took on the cultural beliefs, and then over time, as one acted on

those beliefs, he or she helped to perpetuate the cultural norms.

Cultural beliefs about intelligence and aptitude for learning

Cross-cultural research has highlighted the different views that societies have

about an individual’s aptitude for learning and capacity for retaining information. One of

the most striking differences between the U.S. and Asian countries, like Japan, is the

relationship between a student’s capacity for learning and the student’s innate ability.

Stevenson et al. (1990) found that when Japanese students and their mothers heard the

quote, “The best students in the class always work harder than the other students,” the

majority agreed with the statement. They disagreed, however, with “The tests you take

can show how much or little natural ability you have.” These responses from the

Japanese highlight the different beliefs about schooling held by U.S. students, who

offered the opposite response to both statements by disagreeing with the first quote and


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agreeing with the second. Stevenson et al. (1990) summarized this idea by saying that in

the United States “achievement is attributed to innate ability” (p. 23). Thus mathematics

students in the U.S., who do well, are thought to have natural mathematical ability.

Williams and Montgomery (1995) looked at how gifted high school students’

“academic self-concepts are determined in relation to both internal and external

comparisons. An external comparison refers to the belief that one’s mathematics

performance is better than other students in the mathematics class: subsequently, this

student reports high mathematics self-concept” (p. 401). They found that there is a high

correlation between how students perform in mathematics and how they perceive

themselves in the community (Williams and Montgomery, 1995). Students from the

United States learn, subconsciously, how their ability to perform in school mathematics is

closely associated with how the community views them. This U.S. focus places an

interesting pressure on prospective mathematics teachers. Because they are majoring in

mathematics, many look at them as having a “natural ability” for mathematics. Since

their mathematical success is not measured by effort as much as it is by ability, any

stumble in mathematical achievement strikes at how these prospective teachers see

themselves.

Cultural beliefs about the mathematics teacher’s role

The role of the teacher in a traditional U.S. classroom could be described as a

supplier of knowledge. The teacher’s responsibility then, is to oversee students’ access to

mathematical information. This leads to teachers who impart knowledge in “pieces that

are manageable for most students” (Stigler and Hiebert, 1999, p. 92) through direct

modeling (Hiebert et al., 1997). This style of instruction is common in the U.S. traditional


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mathematics classroom and strengthens the image that the teacher knows more than

students. It also leads to the belief that the teacher is the all-knowing mathematical

authority in the classroom (Stigler et al., 1996). Students then learn that what the teacher

says is correct, and rarely develop enough confidence to depend on their own knowledge.

Cultural view of mathematics majors and their views of themselves

Through years of being a participant in the mathematics classroom, students

acquire beliefs about what it means to teach and be a teacher (Clark, 1988; Cooney et al.,

1998). As a result, prospective teachers bring preconceived notions about the role of the

mathematics teacher with them to teacher education, such as the teacher being the

mathematical authority in the classroom (Stigler et al., 1996). Over time teacher

candidates assimilate into their belief systems that the two, teacher and all-knowing

authority, are synonymous. Teacher candidates grow accustomed to the direct modeling

method of learning mathematics. Researchers (Cooney and Shealy, 1997; Cooney et al.,

1998) have elaborated on this idea of learning but in relation to learning how to teach.

Many times “preservice teachers press [teacher educators] to tell them the right way to

teach” (Cooney et al., 1998, p. 311). Not only do preservice teachers expect to learn

mathematics in a delivery and receiving mode, but they also expect to learn how to teach

in this manner as well.

This particular cultural norm seems to have the most significant impact on the

way society and teacher candidates view mathematics education majors. Cooney (1994)

commented that this “orientation of authority can provide a way of conceptualizing how

[preservice] teachers view mathematics and their roles as teachers of mathematics and

thereby provide a basis for impacting their beliefs” (p. 628). When teacher candidates


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feel that a teacher is the mathematical authority, it seems natural that during teacher

education they will slowly take on this identity. The culture and even the preservice

teachers, for that matter, expect that once teacher candidates have completed a teacher

education program, they will become the mathematical authority in the classroom (Stigler

and Hiebert, 1999).

Ball (1990) studied three commonly held cultural beliefs surrounding teacher

education. Ball first reported that often the culture of mathematics teaching views high

school mathematics as lacking in difficulty. There seems to be an assumption that high

school mathematics consists of procedures and prescribed operations. Therefore, when

the culture comments that high school mathematics lacks difficulty, this is in reference to

secondary procedures and not the mathematical principles underlying the procedures. The

second cultural belief Ball reported was that mathematics courses prior to college had

prepared prospective teachers sufficiently to teach high school mathematics. The last

belief was that subject matter knowledge could be secured through the process of

pursuing a mathematics degree.

Other researchers have come to similar conclusions about the culture’s view of

mathematics majors. Stigler and Hiebert (1999) pointed out how “in the United States,

teachers are assumed to be competent once they have completed their teacher-training

programs" (p. 110). This includes the belief that preservice mathematics teachers know

secondary mathematics as well as how to teach it. Once the preservice teacher graduates

from a mathematics teacher program, the culture assumes that a beginning teacher is

prepared to take on the sole responsibility of teaching students mathematics. Peterson and

Williams (2001) studied conversations between student and cooperating teachers. They


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found that the topic of conversation rarely focused on mathematical content but rather on

administrative or managerial issues (Peterson and Williams, 2001). Studying the same

group of subjects, Durrant (2001) found that the cooperating teachers felt “intimidated by

the mathematical knowledge student teachers supposedly obtain[ed] during their

preparation for teaching” (p. 952). As a result, both student and cooperating teacher

assumed discussing mathematics was not important (Durrant, 2001). What these student

and cooperating teachers did not realize was that the teacher candidates’ experiences with

secondary mathematics were usually limited to their initial exposure in the secondary

grades (Cooney, 1994). The student teachers actually needed discussions of mathematical

content to become better teachers.

Comparing this cultural practice to a Japanese norm, one finds the opposite to be

true. In Japan “beginning teachers…are considered to be novices who need the support of

their experienced colleagues” (Shimizu, 1999, p. 111). This particular belief is just a part

of the overall beliefs that the Japanese culture holds about teachers. In Japan teaching is

viewed “more as a craft, as a skill that can be perfected through practice and that can

benefit from shared lore or tricks of the trade” (Stigler et al., 1996, p. 216). Thus it makes

sense that novice teachers would not be left alone. However, in the U.S. there is a

prevalent belief that “teaching is an innate skill, something you are born with” (Stigler

and Hiebert, 1999, p. 86). Therefore, beginning teachers are expected to be capable of

teaching mathematics from day one. This belief influences not only how beginning

teachers are viewed, but also the cultural attitude toward preservice teachers. Not only are

they viewed as possessing a natural ability to teach, but also after four years of college

they should know how to teach secondary mathematics. Therefore, this cultural belief


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traps prospective teachers in a system that prevents them from being exceptionally

knowledgeable about secondary mathematics (Stigler and Hiebert, 1999).

Summary

It has been established that the teaching culture places certain expectations on

preservice teachers that they are or should be the mathematical authority. These

expectations often force these new teachers to accept the cultural belief that they should

have no weaknesses in mathematics even though their understanding of the underlying

meanings of high school mathematics is fragmented and fragile. To further investigate

this conflict between personal and cultural beliefs about new teachers being mathematical

authorities and the reality that their mathematical understandings are actually weak, we

chose the environment of Japanese lesson study to study these cultural beliefs.

RESEARCH DESIGN AND METHODOLOGY

Setting

Stigler and Hiebert (1999) describe the practice of lesson study as a commonly

used method of professional development in Japan. They state “In lesson study, groups of

teachers meet regularly over long periods of time (ranging from several months to a year)

to work on the design, implementation, testing, and improvement of one or several

‘research lessons’ (kenkyuu jugyou)” (p. 110). A research lesson is the actual lesson that

results from the collaborative efforts of a lesson study (Lewis, 2000). Although

collaborative work between teachers can quickly become a case of “show and tell,”

where each teacher describes what they do when they teach a certain topic, Stigler and

Hiebert (1999) emphasize the critical component of focusing on student thinking in the


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lesson study process when they say, “The lesson-study process has an unrelenting focus

on student learning. All efforts to improve lessons are evaluated with respect to clearly

specified learning goals, and revisions are always justified with respect to student

thinking and learning” (p. 121).

The basic format of lesson study below is drawn from Stigler and Hiebert (1999),

Lewis (2000, 2002), and Fernandez et al. (2001).

1. Goal Setting: Select a learning goal for the research lesson (the lesson that

results from the lesson study process). Identify goals for student learning

and long-term development.

2. Planning: Plan, as a group of teachers, the learning activities and the

sequence of these activities that will aid students in achieving the learning

goal. This planning often begins by referring to a variety of resources

including textbooks and individual teacher experience.

3. Research Lesson: One member of the lesson study group teaches the

research lesson to a group of students while the other members of the

lesson study group carefully observe and take note of how the students are

responding to the learning activities and making sense of the mathematics.

4. Revise: Based on the data gathered regarding student responses and how

those responses fit with the learning goals, revise and refine the research

lesson to better meet the desired learning goals.

5. Teach Research Lesson a 2nd time: After the research lesson has been

adequately revised, have another member of the lesson study group teach

the lesson again to a different class. Again the other members of the lesson


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study group as well as other teachers in the school or community observe

the student reactions and responses to the learning activities.

6. Report: Write a report describing the lesson study group’s reasoning and

rationale behind the purpose and sequencing of the components of the

research lesson and summarize the student responses.

Although most Japanese lesson study is done by practicing elementary teachers,

this study adapted the above principles to preservice secondary mathematics teachers in

the United States. This lesson study format was chosen to look at preservice mathematics

teachers’ beliefs because Lewis (2000) found that “competing views of teaching bump

against each other” (p. 17). Lewis (2000) stated “it would be interesting to look at

research lessons as a potential influence on teachers’ content knowledge development”

(p. 17), which would indicate that lesson study could also be fertile ground for investing

teachers’ mathematical understandings.

Class Structure

The study participants were members of a methods of teaching secondary

mathematics class that was taught at a large private university in the western United

States during fall semester of 2001. One of the authors was the instructor of the course,

while the other author was a graduate research assistant (GRA) who observed but did not

participate in the course.

As a large part of the course, the students were placed into lesson study groups

midway through the semester in which they planned a lesson that they would teach in the

public schools. Each group was paired with a practicing teacher in the public schools

with whom they met to select a goal for their research lesson. Once the lesson goal was


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selected, each group met for approximately 2 hours a week to refine their lesson. After 2

weeks of meeting, one member of the group taught the lesson to their peers. The

remainder of the group observed the lesson and watched how their peers reacted and

responded to the questions and tasks that were posed. The lesson continued to be refined

for another 6 weeks until another group member taught the lesson in a public school

classroom. Each group then wrote a report on the development of their lesson and how

the public school students responded to the questions posed.

The textbook for this methods class was a compilation of chapters from

commonly used public school textbooks. The chapters were arranged so that the

preservice teachers would be exposed to the general progression of the mathematics

curriculum from seventh grade mathematics through geometry and a second year of

algebra. Each student in the class had the opportunity to teach two lessons to their peers

from this textbook. The peers were given the task of role-playing the appropriate grade

level for the lesson that was being taught. Because of this ongoing experience, the

research lesson that was taught to the peers was a natural part of the course.

The preservice teacher who was selected for this case study belonged to a group

which was given the topic of introducing the concept of logarithms. The public school

teacher to whom they were assigned taught pre-calculus at a public high school. The

lesson study group met with this teacher toward the end of the first half of the semester.

They jointly selected the topic of logarithms as the focus of the lesson study.

Participants

In order to select the participants in the study, each student in the methods of

teaching secondary mathematics course was given a Likert-scale survey and asked to


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respond in writing to the following two questions: “What is mathematics?” and “How do

you learn mathematics best?” The responses were analyzed focusing on two issues: 1) the

level of confidence the student reported on the Likert-scale statements, in particular: “I

would consider myself an expert when it comes to secondary mathematics” and 2)

whether the student viewed and learned mathematics conceptually or procedurally. The

term conceptual referred to the idea that concepts underlie mathematics problems, and

these problems can be solved by understanding and wrestling with these underlying

concepts. On the other hand, procedural was taken to mean that mathematics problems

are solved through the practice of following prescribed procedures. Of course strict

classification without overlap was impossible. However, the most dominate views of self

and mathematics that the students exhibited were documented.

Each preservice teacher was classified as C+, C-, P+ and C-. The plus (+) and

minus (-) symbols represented confident or not confident in secondary mathematics,

respectively and the letters C and P represented conceptual and procedural views of

mathematics, respectively. Students were also observed in the class setting to identify

those who readily vocalized their thoughts. Lesson study groups were created so that a

variety of perspectives were represented in each group. The more vocal students were

placed in the groups that would be part of the study so that the thoughts and ideas could

be more easily documented.

From the five lesson study groups that were created in the class, two were

selected to be part of the study. From these eight students, Janica was selected as the

subject of the case study because of her background and her experience in the lesson

study environment.


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Data Collection

Several instruments were utilized in the data collection process. These included a

dialogue journal which was submitted almost weekly over the course of the semester,

field notes taken during the teaching of the methods class as well as during the lesson

study group meetings, audio taped interviews (initial and exit), audio taped conversations

during the lesson study group meetings, videotaped lessons taught during the methods

class by each of the subjects in the study and a final written individual and group report.

Of the data sources that were required parts of the course, only the dialogue journals were

evaluated by the GRA. Of the remainder that were formally assessed, all were done so by

the instructor. The lesson study group meetings were not observed by the instructor but

were observed and audio taped by the GRA. Similarly, all interviews were conducted by

the GRA who had limited evaluation responsibilities over the students.

Data Analysis

A general grounded theory approach was used to analyze the data and to develop

Janica’s case study. In addition to the survey questions, an initial interview was

conducted to get a more holistic picture of Janica and to provide a baseline perspective

before she engaged in lesson study. This initial interview was used along with the survey

questions to form an initial description of Janica’s beliefs about mathematics and her

confidence level.

The journal responses provided evidence of Janica’s inner thoughts about the

lesson study experience. These data were used to further refine the description of Janica’s

beliefs and were also used to monitor whether she reported any changes in her thinking

about herself or mathematics.


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Data collected during the lesson study conversations showed how Janica

interacted with the other members of her group, and how the members of the group

interacted with one another. These conversations were also monitored to see whether or

not ideas were received and supported by various members of the group and whether

those ideas were conceptual or procedural. The ongoing journal responses were analyzed

to gain insight into the nature of the conversations and comments that occurred during the

lesson study meetings. These conversations and journal responses were analyzed to

further refine the description of Janica’s beliefs and confidence and most importantly

identify any movement in her view of herself as an expert.

The final interview provided an opportunity to question Janica about comments

made in the initial interview as well as follow up on ideas that emerged during the lesson

study conversations and lessons. Evidence of agreement with Janica’s earlier ideas as

well as evidence of changes from those original descriptions was sought. Of course,

descriptions pertaining to mathematical ability and the perception of mathematics were of

particular interest. An analysis of this interview supported further refinement of the

description of Janica’s beliefs and confidence.

In an effort to find coherence between Janica’s statements and her actions, a

follow-up interview was conducted in which Janica was asked to address some of this

apparent incoherence. The data from this follow-up interview was then used as a lens to

examine all of the previously gathered data and as a member-check to resolve questions

and issues.


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RESULTS

The following part of the paper provides an introduction to the focus of this case

study, Janica. To better understand Janica the next section discusses her background with

mathematics and mathematics education prior to the study.

Janica’s Background

Janica was a senior mathematics student possessing a strong mathematics

background and knowledge of current mathematics education research. Her course load

consisted of several upper level mathematics courses which typical mathematics

education majors did not take. She was a conscientious student and received good grades.

Her pursuit of college mathematics could easily be called a success. Outside of the

methods course, Janica worked as a research assistant for two mathematics education

professors.

During the initial interviews, Janica shared an experience from her youth which

had a large influence on her later success in mathematics. Janica remembered a routine

evaluation she received in an elementary mathematics class. As she recalled, the mark in

mathematics was a B+, which greatly disappointed her. Janica remembered thinking that

she was not a B+ student and that she was determined never to get anything less than an

A again. This powerful event caused Janica to strive to do better not only in this

mathematics class but also in mathematics classes thereafter.

Janica’s drive to excel continued on into her high school years. During high

school Janica enrolled in the top mathematics classes and received excellent grades. In

Janica’s high school the mathematics classes were uniquely combined with the advanced


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physics classes. Janica remarked that the physics and mathematics teachers often

conversed and shared ideas about lesson plans. Janica appreciated this connection

between the two subjects because it illustrated how mathematics could be applied in the

world of physics.

Janica’s success in the physics/mathematics combination classes in high school

influenced Janica into college. Early on in her college career Janica planned on majoring

in physics. She remembered heading down the physics path because she saw the

applicability of the mathematics when she studied physics. However, she eventually

decided that she would prefer studying mathematics and abandoned her pursuit of

physics.

About half-way through her college years, Janica took an 18 month break from

college. This break eventually led to a change in her career path. Janica spent 18 months

away from home engaged in missionary work. As a missionary, Janica developed a

strong affinity toward teaching. Once she returned from her missionary work, she

enrolled in a private religious university and added the mathematics education major to

her mathematics major requirements. At the new school, she began work with

mathematics education professors. Her job responsibilities included coding data, reading

articles about educational systems of other countries, transcribing mathematical

conversations, and working with preservice mathematics teachers. Janica’s work

experience had also briefly introduced her to lesson study. Because of her familiarity with

lesson study, Janica approached the project with enthusiasm, and was excited about

engaging in a lesson study experiment.


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Janica’s view: What is mathematics?

Toward the start of the semester, the instructor asked the teacher candidates to

answer the question: What is mathematics? Janica reported:

Mathematics is the study of numbers and the manipulation of numbers. Not only

that, but it is an explorative science that builds from basic concepts to more

complex ideas to abstract ideas. The basic building blocks of mathematics are

numbers and manipulatives with numbers or symbols representing numbers or

functions. Mathematics is used to solve day-to-day problems and more universal

problems. All sciences use mathematics as a way to represent and manipulate data

from their respective fields.

Janica’s reply showed that she valued the concepts of mathematics and applying

the procedures to real-world problems. Janica’s written response focused on the use of

variables, the manipulation of symbols and the utilization of mathematical tools to solve

problems. Although Janica’s response indicated a belief in mathematics as problem

solving, she really saw mathematics as a vehicle of manipulation of numbers and symbols

to solve problems in the sciences. This alluded to her high school experiences and

referred to the idea that mathematics should be couched in terms of real-world contexts.

It is important to note the type of problems Janica was referring to, though, were physics

problems that could be easily solved with algorithms. This perception, that mathematics

procedures were used to solve real-world problems, became a common theme for Janica.

During the initial interview, the interviewer asked Janica how she would describe

mathematics to a friend who knew nothing about the subject. Janica depicted

mathematics as “a way to represent everyday situation[s]… maybe apply the variable…to


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something and use it, or manipulate it into something else.” Another part of the interview

required Janica to describe what would be important for students to know or understand

about fractional division. She mentioned that it would be important for students to know

how to “just invert the fraction and multiply.” When asked to create a problem she shared

a practical example of “3 cups of flour” and wanting “to half the recipe.” After a while

she realized that this was actually multiplication of fractions, not division of fractions.

She came to this conclusion because if she took 3 divided by ½, the quantity of flour

would be larger, namely 6 cups, not the smaller 1 ½ cups of flour that she was looking

for. Later on she was asked how she might go about drawing a division of fractions

problem. In response to this question, Janica shared an example of cutting a 5-inch string

into half inch pieces. This model highlighted the underlying concept behind the problem,

however, as the interview progressed Janica indicated that she had recently found the idea

in a book.

In the early part of the course, Janica completed an assignment to prepare and

teach a mathematics lesson. The professor stressed that a critical part of writing lessons

was identifying and stating the “big mathematical idea” (BMI) of the lesson. He also

emphasized this BMI was to be the underlying concept that was to be taught and not just

the mechanics of a procedure. The BMI Janica submitted about integer division was,

“Quotient of 2 integers with the same signs…are positive and quotient of 2 integers with

different signs are negative. This is just applying rules of multiplication.” The

mathematical idea she considered to be most significant centered on the multiplication

and division rules. Janica’s actual teaching of the lesson followed her BMI closely, and

focused on the procedural process of integer division. Partway into the lesson, students


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began to question why the procedures worked. This resulted in Janica struggling to finish

the lesson and her coming to realize that she was not prepared or flexible enough to

handle the concerns the students had raised. Reflecting later on the presentation, Janica

wrote that the “lesson was a flop” and she “assumed that the class knew more” than they

did.

Janica’s view: Mathematical expert

Janica spoke about her feelings and confidence in relation to mathematics. She

said in the initial interview that during high school “it always seemed to come easier to

[her]” and she “got it,” meaning secondary mathematics. She remembered that she

learned mathematics quickly and felt comfortable with it. She even recalled helping other

students with the mathematics lessons after she had mastered them. As she looked back

to those high school years, Janica’s memory did not reveal any dramatic struggles when

learning mathematics.

Janica’s confidence and self-assurance during high school carried over into her

current beliefs about her secondary mathematics knowledge. Janica’s answer to a

particular interview question revealed her relative confidence in college and secondary

mathematics. The two-part question put to Janica was: On a scale from 1 to 10, how

would you rate your knowledge of 1) college mathematics and 2) secondary mathematics

when compared to other preservice mathematics teachers. When it came to her

knowledge of college mathematics, she said that she would give it “probably like a 7, and

then high school mathematics, probably an 8 or a 9.” Later she even admitted that she

would change the 7 to a “6… on the college” mathematics, showing some lack of

confidence with university mathematics.


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Summary

Janica’s early written statements about mathematics, views that she expressed in

class and her teaching of mathematics topics indicated a conflict between what Janica

said and what she actually did. Janica’s verbal comments in class consistently referred to

conceptual elements in teaching mathematics. However, her written responses to the

early journal questions and her lesson on fractional division were procedural in nature.

Based on her academic performance, Janica was an excellent student of

mathematics. She felt very confident in her understanding of secondary mathematics and

less confident with her college mathematics. Her interview responses, however, indicated

some weaknesses in her understanding of secondary mathematical topics. In addition to

these contradictions, Janica often said conceptual mathematics was important and yet

behaved as if mathematics were all about procedures. These contradictions made Janica

particularly interesting as a case study subject.

Lesson study

Janica began lesson study with three other preservice secondary teachers. During

the first meeting Janica and her group spoke to a teacher in the field to decide on a goal

that the teacher had for her students, and the mathematics topic that the preservice

teachers could teach in an effort to reach that goal. The preservice and in-field teachers

decided to focus on justification while teaching logarithms. Janica spoke often during this

meeting, asking questions about how the students comprehended mathematical ideas,

asking about the characteristics of the students and the logistics of the eventual teaching

of the mathematics lesson. Janica later referred to a book about lesson study, The

Teaching Gap (Stigler and Hiebert, 1999), and shared her impressions of the development


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of the lesson study process. She said “knowing a little bit about lesson study…they

actually chose like a math topic…talked about it, researched about it, and came up with a

lesson to more fully explain it and help the students understand it.” Janica then

questioned the researcher, “Is that what we’re trying to do…or, are we trying to work on

maybe just justification?” After discussing the ideal process of lesson study, Janica

narrowed the group’s purpose to the idea that “the lesson isn’t necessarily to justify what

we do, but to help [the students] have an understanding so that they can justify.”

During the first lesson study collaboration meeting Cindy, another participant,

started writing down the group’s ideas in the lesson plan format. Cindy kept track of the

ideas and suggestions she and the group generated. Half-way through the meeting Janica

commented that the group needed to write down their ideas. Even into the second

collaboration meeting, Cindy continued to ask the group specific questions about their

ideas and included this in her written lesson plan. During this meeting Janica stood at the

board and began to write the group’s ideas on the board, commenting that she enjoyed

writing on the board. This drew the group’s attention to Janica’s writing and made Janica

the official scribe even though Cindy had been taking notes for the group for the first

meeting and a half.

During collaboration, the preservice teachers worked together to brainstorm and

share ideas. These meetings centered on the underlying principles of logarithms and

exponents. Janica did not speak up much during the first half of the first planning

meeting. She spent much of her time looking in the textbook for information. At first she

was particularly interested in the logarithm tables and the symbolic representation of

logarithms. While working to prove the logarithm property: log (ab) = log a + log b,


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some of the group questioned whether they would be able to prove the property. Cindy

said “Maybe we don’t know how to do this.” And another member McKenna wondered,

“So is this ever gunna work?” Janica reached for the book and queried, “Did they develop

it anywhere?” Later, she told the group that even after taking the book home to study the

section on logarithms that she “didn’t read through the chapters very well.”

The group continued to work through the proof of: log (ab) = log a + log b. Cindy

questioned the group about a particular concept of logarithms and exponent asking:

Cindy: “Well, can’t ‘a’ be written as 10 or 5 or whatever to some power? That’s…what a

logarithm is, it’s writing a number as…the base to a power. And if we stress that big

enough, maybe [the students will] see that they need to write their number as a base to a

power.”

Janica: “The only problem is that it’s an arbitrary number, it could be anything. The

only, the only reason we choose it, is so we can prove it. We choose, we specifically

choose an ‘a’ and a ‘b’ so that we could show that it works. Otherwise if we’d…end up

with all these decimals if we choose numbers and stuff. But this was a clear way to show

that it was true.”

Cindy: “What do you mean?”

Janica: “‘a’ and ‘b’ can be anything.”

Cindy: “Yeah, I know. I just mean, like we need to stress that, that any number can equal

a base to a power. So that they can see that ‘a’ can equal the base to a power, see what

I’m saying? I think if we stress that enough they should be able to make the connection.

Don’t you guys think? Do you think?”

Robin & McKenna: “Uh-hu.”


21


Janica: “Well what did you mean when you said any number can be written as a base to a

power?”

Cindy: “Like any number, like 200 can, or 100 can be written as base 10 to the power of

2.”

Janica: “Cuz, there’s like 2.”

Cindy: “2 can be written as base 10 to some decimal power, see what I’m saying? Well

see, what do you mean? 2 where as in, 2 instead of a 100?”

Janica: “Or pi.”

Cindy: “Pi?”

Janica: “Or any rational number?”

Cindy: “Can’t they?”

McKenna: “Hmm?”

Janica: “Nuh-na.” (negative intonation)

Cindy: “Not all of them?”

Janica: “I don’t think so. Can they?” (looks at GRA in room)

It is important to note here that Cindy was considered to have the least knowledge

about logarithms. She proclaimed at the first meeting that in high school she had moved

right before logarithms and thus never learned about them. Even after hearing Cindy’s

statements, Janica continued to try and make sense of the idea but since she had never

heard this mathematical idea before, she struggled to determine the validity of the

statement. The discussion finally ended with Janica’s comment, “I’ve never thought of it

as actually a base to a power and that’s why, since I’ve never heard it, that’s why I don’t

know if it would be true or if it is true.”


22


Overall the lesson study collaboration session consisted of the group discussing

the rich mathematical principles underlying not only logarithms but exponents as well.

These meetings led to some rich discussions and helped the participants not only plan

their lesson but examine their own individual understandings of the mathematics topic.

DISCUSSION & CONNECTION TO RESEARCH

Janica’s belief system

Current research confirms the suspicion that Janica’s beliefs about mathematics

were comparable to beliefs held by preservice mathematics teachers. Janica viewed

mathematics as procedures that could be used to solve real-world problems. For her, this

provided the impetus for learning mathematics. Cooney (1999) similarly found that

prospective secondary mathematics teachers often displayed “a strong computational

orientation” (p. 165) and Ball (1990) noted that it was natural for secondary majors to

perceive mathematics as “a collection of arbitrary rules to be memorized” (p. 460).

Janica’s notion that mathematics consisted of procedures to solve real-world

problems spilled over onto her outlook of her knowledge of secondary school

mathematics. Her strength in procedural knowledge sustained Janica’s view of herself as

one who was good at mathematics. Meredith (1993) saw this perception of being

proficient at procedural methods when subjects “reported their mathematical knowledge

as being the one aspect of teaching about which they felt most confident” (p. 331). Like

the subjects in Ball’s (1990) research, Janica was a good student who received good

grades in college. Because these secondary mathematics majors were considered above

average in high school mathematics and then later had majored in the subject, the


23


preservice teachers believed that they knew the subject matter (Ball, 1990). Cooney

(1999) agreed that this was a common assumption but gave a convincing argument why

this assumption was not necessarily true. In his research Cooney (1999) determined that

“often preservice teachers have a poor understanding of school mathematics – having last

studied it as teenagers with all the immaturity that implies” (p. 165). So even though

teacher candidates, like Janica, had experienced success in college mathematics and felt

that their knowledge of secondary mathematics was “fine,” this did not signify a thorough

understanding of secondary mathematics.

Janica’s beliefs about mathematics and her mathematics knowledge lacked

significant change over the course of lesson study. Collier (1972) also found that one

teacher education class did not dramatically alter the views of preservice teachers and if

there was change it was minimal. One can not expect teacher education to alter preservice

teachers’ belief systems (Thompson, 1992). Therefore, the fact that Janica’s beliefs were

not modified should not be surprising.

Janica’s compensation moves

During the study Janica assimilated her experience and used several different

actions or moves to keep from having to reveal any weaknesses in her mathematical

understandings. While studying preservice teachers, Siebert et al. (1998) found that one

preservice teacher, Antonio, utilized compensation moves in an effort to avoid a

reassessment of his view of mathematics and to keep from having to deal with his

fragmented mathematics knowledge. Antonio’s first move came from a discussion about

why the invert and multiply rule worked. The researchers saw that Antonio’s

understanding of the conceptual meaning of the rule came during the interview, and that


24


previous to this interview Antonio only understood the procedure. Even though Antonio

was introduced to the rich meaning imbedded in the concept, Antonio countered with the

idea that “meaning can be found in procedures” (Siebert et al., 1998, p. 622). Antonio

ignored the significance of the concept and instead claimed that there was meaning in

procedural mathematics. In another interview, Antonio’s responses to fractional division

highlighted his second compensation move. Siebert et al. (1998) found that “Antonio

came to view what [they] saw as the meaning of division of fractions as just another

procedure for obtaining answers to division of fractions problems” (p. 623). This meant

that Antonio used the drawings simply as a means for finding answers and did not value

the meaning represented in the diagrams. The fact that this occurred was in line with the

work of Schram and Wilcox (1988) which showed that in situations of conflict,

preservice teachers seem to “adapt conceptual ideas to fit the framework of beliefs [they]

brought to the course” (p. 354).

Like Antonio, Janica used moves to avoid conflict; however, the motivation for

Janica’s moves was very different when compared to Antonio’s motivation. In this study,

Janica used the moves to keep from having to reveal any possible weaknesses in her

understanding of secondary mathematics. Janica felt that the cultural expectation for

teacher candidates was that they should completely know mathematics when they are

done with college. When Janica was not able to live up to this expectation, she took

certain actions to preserve her perceived identity in the educational culture. Janica’s

moves during lesson study collaboration included acting as the scribe. A second move

she used during both lesson study and the interviews came in response to a new idea. She

would look in a book, turn to the researcher for help or comment that it was a good


25


review, or she forgot the topic. A move exhibited during the interviews was that of saying

that the class structure was the culprit when it came to her inability to identify

mathematical ideas. The last compensation move was emphasizing that the reason she

was having trouble was her lack of pedagogical knowledge.

Acting as the scribe.

In her capacity as scribe Janica mediated the group’s concerns and the order in

which the mathematics was placed in the lesson. Often Janica would ask, what was the

question again? Or how did you want to word that question? While it is natural for a

scribe to ask clarifying questions, Janica used this role to protect herself. In one episode,

Janica started writing the group’s ideas down in the prescribed lesson plan format. As

Janica wrote, the group continued to discuss a particular detail. As the discussion ensued,

many of the individual comments and ideas were lost as Janica tried to write down all the

comments and make sense of the lesson plan on her own. At several points if Janica did

not agree with an idea she would not write down the comment. She then thought of how

she would ask the question and would ask for a group consensus.

The role of scribe was a safe role because she did not have to publicly express her

understanding of the topic. Frequently other members of the group would willingly admit

that they did not understand an idea and seek help in understanding it. Janica, on the other

hand, rarely, if ever, made such an admission. This allowed her to speak as the scribe to

ask questions of clarification and not speak as Janica trying to clarify her own

understanding. As a result, Janica was not only able to direct the conversations, but also

how much information the group’s members had about her mathematics knowledge.


26


I’ll need to review.

One of the first indicators of Janica diverting attention came during the

interviews. Janica’s comment that this was a good review or that she forgot the little

details of a topic focused the attention away from her knowledge of secondary

mathematics. In the initial interview, Janica attempted several times to draw a picture of 3

divided by 2/3rds but was only able to describe a model with different numbers that she

had seen in a book. At first Janica drew an incorrect model. However, after a moment of

thought she utilized an example she had seen in a textbook. The model consisted of

taking a string and dividing it into sections. When the interviewer pressed Janica to

elaborate, she struggled to describe her drawing, giving evidence that she did not

understand the conceptual nature of the model. Finally Janica turned and said, “Thanks,

this is a good review.” For her, the reason she had trouble communicating her ideas

effectively was related to her lack of recent experience with the topic.

While engaged in lesson study Janica often turned to the book for help reviewing

topics. During the log (ab) = log a + log b episode, Janica depended on the book to show

her how to step through the logarithm proof. Showing that Janica’s understanding of the

topic was weak and limited. About this same time, Janica also turned to the researcher

observer for clarification during this part of the meeting. The fact that Janica turned to the

researcher for the answer showed how she did not trust her own knowledge and the

knowledge of her group members. These two experiences during lesson study highlight

how Janica non-verbally communicated that she was reviewing the mathematics topic,

rather than initially learning it.


27


In the final interview, the interviewer questioned Janica about her confidence with

secondary mathematics. The question that was directed to her was “Does it bother you to

know that you’re…maybe not as complete in your knowledge…?” Janica’s response

revealed once again the tendency to point at forgetting the mathematics and the need for

review. She commented, “I’ve forgotten a lot of mathematics, especially the mathematics

that I learned in like high school, the stuff that I that I will be teaching.” This idea of

forgetting the mathematics needed for teaching seemed to her like a typical reaction for

most teachers. She also stated, “I’ve forgotten so much that I would have to just, and, and

every teacher does I’m sure when they’re starting out to teach a new subject just having

to review the material and really understand what’s going on.” Later in her final

interview, she commented, “If I’ve forgotten…that worries me…but I think I’m a little

more confident…that even though…I realize I don’t know a lot of the mathematics, and

I’ve forgotten a lot, it will come back and I’ll be able to teach it.” So for Janica it was

more an issue of forgetfulness, and that through the process of reviewing the material it

would all come back to her. For her, it was not as if she did not know the material, it was

simply that she forgot it.

Class structure to blame.

The second way Janica was able to deflect the focus away from the mathematics

was blaming the structure of the class for her lack of ability to identify the BMI. One of

the first indicators of this defense was when she blamed the professor for not helping her

recognize the main idea in the mathematics. “Early on [she] thought everybody was kind

of confused with the big mathematical idea. What exactly [the professor] wanted, what he

thought was the big mathematical idea.” She thought she had to play the game of trying


28


to figure out what the teacher wanted her to write rather than realizing that she struggled

to pinpoint the main idea in the mathematics lessons.

Janica also blamed the materials for her inability to summarize the big

mathematical idea. She went on to say that while writing her lesson plans she would pin

down the main idea but then would run into a problem. “That’s one of the problems with

the way the class was set up” she said, “a little bit with the text that we got, that, when I

was like writing some of my lessons I didn’t know what the big mathematical idea should

have been.” She felt that since they did not have the extra information in the teacher’s

edition or previous chapters that it was hard to isolate the main idea. In this particular

instance, she blamed the books for her struggle to come up with the big mathematical

idea. In a small way, she held the teacher and the materials responsible for her lack of

ability to spot mathematical ideas. In this instance, Janica’s willingness to be critical of

the course is an indication that she viewed the GRA who conducted the interviews as

someone who was not part of the course staff. Thus her “moves” to hide weaknesses were

not done out of fear of external evaluation but out of an internal concern for how she was

viewed by the community.

That’s a pedagogical question.

Another way that Janica kept from revealing a lack of knowledge was explaining

that she might not have an answer because the question was about mathematics

education. Janica often made this comment because she felt that ideas about how to teach

mathematics were new to her. She believed that she was learning a lot about how to teach

mathematics because of her recent change to the mathematics education major. Since the

idea of teaching mathematics was so new to her, Janica depended upon the book to help


29


her make pedagogical decisions. One particular experience during the initial interview

highlights this point. The interviewer asked Janica to write down everything that she

would want her students to know or understand about division of fractions. Her response

indicated that she would probably have to consult a book. She said “it’s hard just off the

top of my head; it’s easier if I have like a book so that I can see, you know, what students

are going to need to know and apply it.” This dependence on the book was couched in

terms of making curricular decisions and focused the attention on her personal knowledge

of the pedagogy. It also suggests that she believed that learning to teach was a procedural

activity as well.

After being asked about how she would teach slope, Janica responded in a similar

manner. She said, “Asking a question like that is sort of, I mean, not…its sort of dumb

because I don’t have a background in teaching up to that point and I don’t know exactly

what’s gunna be after.” For her the issue was that she did not have the teaching

experience to make such claims, rather than draw on her knowledge of secondary

mathematics. She even commented that the lesson study group members lacked

mathematical pedagogical knowledge since they were not teachers yet. Janica said that

this was part of the reason the group struggled to create a lesson plan. She said that they

were “inexperienced preservice teachers” and that “there’s only so much we know,

there’s only so much we know about pedagogy…about how to present the mathematics

to students.” Although much of the discussion during the lesson study conversations

centered on understanding the mathematics, Janica still attributed their difficulty to a lack

of pedagogical knowledge. Janica felt that if they would have had someone more

experienced then the process of creating a lesson may have gone quicker but then again


30


she admitted that it “might be good because that, that gives us sort of a fresh approach on

how we can teach the lesson…” In her eyes there was good and bad to the inexperience.

However, Janica’s reference to inexperience means inexperience in teaching

mathematics, not lack of mathematical knowledge.

Even with the sense of confidence that surrounded mathematical knowledge,

Janica freely admitted that she was not as confident when it came to teaching

mathematics. Janica saw mathematical and pedagogical knowledge as separate and

distinct knowledge bases. Janica felt that her knowledge of secondary mathematics was

“fine” but that she would not be able to describe how to teach it because of lack of

teaching experience. Even though she said that it was the “pedagogical” knowledge in

which they were weak, a large portion of the lesson study conversations centered on the

lesson study group members struggling to clarify their own understanding of logarithms.

Summary

During lesson study Janica took on the role of Acting as the scribe to keep from

sharing too much information about her mathematical knowledge. During lesson study

and the interviews, Janica utilized both non-verbal and verbal clues signifying I’ll need to

review to explain why she was not able to answer questions about mathematics content.

In Ball’s (1990) research with secondary education majors she too found the same

characteristic verbal response. Like Janica the mathematics majors felt that “If they could

not figure something out, they assumed they were ‘rusty’” (Ball, 1990, p. 464). This idea

of needing to review was a common theme with Janica but with other preservice

secondary teachers as well. Janica also mentioned that the class structure was to blame if

she could not find the big mathematical idea behind a lesson. Reference was made to


31


either the class textbook or the teacher if she could not develop the mathematical idea

behind a lesson. She would also say that’s a pedagogical question if she thought the

question referred more to mathematics teaching. Cooney (1999) found a similar

perception that content knowledge and knowledge needed for teaching are not integrated.

His research findings also agreed with Janica’s statements about preservice teachers’ lack

of pedagogical knowledge. Often teacher candidates do not have the kind of knowledge

needed to be able to teach secondary mathematics. Janica believed that she lacked the

knowledge necessary to make good pedagogical decisions. In her mind, her knowledge of

secondary mathematics, even though she felt comfortable with it, did not influence the

kinds of decisions that she made when teaching the subject matter. Janica saw

mathematical and pedagogical knowledge as separate and distinct knowledge bases.

Because Janica saw these knowledge domains as separate, she could admit a weakness in

pedagogical knowledge without threatening the perception of her mathematical

understanding.

Coherence of Beliefs and Actions

One of the challenges of this study was identifying the coherence and consistency

between the subjects stated beliefs and observable actions. As a person participates in the

culture, they typically assume the beliefs of the culture and act upon those beliefs. After

research with teachers, Thompson (1984) concluded that “the teachers’ views, beliefs,

and preferences about mathematics [did] influence their instructional practices” and that

even general conceptions about issues unrelated to mathematics “appear[ed] to play a

significant role in affecting…behavior” (p. 125). Investigating a subject’s words and

actions can help researchers expose the beliefs that motivated the actions. However, this


32


is not an easy task considering that from an outside observer’s perspective a subject’s

comments may seem at odds with actions. When this occurs, it may be difficult to make a

statement about the subject’s beliefs.

The researchers found this to be true with Janica. Janica made comments during

interviews and in class that pointed to a rather conceptual orientation about mathematics.

However, after watching Janica teach a mathematics lesson the researchers questioned

how her beliefs, her verbal comments and her actions were consistent. Her lesson focused

exclusively on procedures and lacked reference to the mathematical concept behind the

topic. Throughout the study Janica stated her confidence in secondary mathematics and

yet her actions indicated some weaknesses and a tendency to avoid revealing these

weaknesses.

According to coherence theory however, a person’s comments or actions are

consistent within the individual’s belief system (Alcoff, 1996). With regard to Janica, her

actions were indicative of the belief that secondary mathematics was about procedures,

since she was proficient at procedures, her confidence in secondary mathematics was

high. This does not mean that Janica’s behavior was logical to another individual but it

made sense to Janica. That means that when actions seem in disagreement even with

reality, it is necessary to look deeper at the subject’s activities in order to uncover the

motivation for the behavior. “The ‘coherence’ at issue in the coherence theory is a matter

of a proposition’s relation to other propositions—not its ‘coherence’ with reality or with

the facts of the matter” (italics in original, Rescher, 1973, p. 32). Therefore, even when a

subject’s beliefs disagree with truth (Alcoff, 1996) or with another belief, the challenge is

to see how they fit within the subject’s interpretations of the world.


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Follow-up Interview

Janica’s role as the group’s scribe and her comments during interviews changed

the focus of the discussions and often diverted the attention away from mathematics. The

question that arose was “Did Janica intentionally or unintentionally divert the attention so

that she would not have to address her mathematical knowledge?” She described herself

as being good at mathematics, and that she felt comfortable with her secondary

mathematics knowledge, however, during lesson study her behavior indicated otherwise.

In more than one instance, Janica looked to outside sources to help her make sense of the

topic. For example, she would often search the textbook for answers or turn to the

researcher-observer when she was unsure about an idea. These actions painted the picture

that she was still grappling with her understanding and knowledge of logarithms. Even

though Janica did report a process of learning about the specific precalculus topic,

logarithms, during collaboration, she still held tightly to the idea that she was comfortable

with her secondary mathematics knowledge. This meant that Janica’s overall belief

system remained unaltered even after reflecting with her lesson study group. Janica did

not present any evidence to suggest that she acknowledged her lack of mathematical

knowledge. Through Janica’s comments and actions she quietly defended her position as

a knower of mathematics.

Janica’s response to mathematical interview questions also brought her

mathematical confidence into question. When Janica was asked a mathematical question

she would shy away from answering the question directly. Instead, Janica would provide

an explanation for why she could not complete the task successfully. Janica’s repeated

comments about her confidence in secondary mathematics and yet her frequent behavior


34


that indicated a weakness caused the researchers to question the coherence between her

beliefs and her actions. Because of this, it was necessary to probe Janica further about her

beliefs, and to assess the nature and intent of Janica’s responses.

A follow-up interview was conducted and it revealed how Janica’s words and

actions were consistent within her own belief system, and shed some light on why the

attention was often shifted away from mathematical conversation. The interviewer asked

Janica to respond to a similar set of questions as her other interviews. Her answers

seemed consistent to those shared previously. The follow-up interview also consisted of

presenting Janica with the interviewer’s description of Janica. generated to reflect how

the interviewer would describe Janica. Janica agreed wholly with three of the statements,

disagreed with five and had a mixed response with the rest. Janica agreed with the

statement, “People look up to me because I know mathematics,” disagreed with, “Lesson

study did not change the way I view my knowledge of mathematics,” and reported some

agreement or disagreement with, “I never learned the type of mathematics that I know I

should teach.” With respect to this last statement, Janica reported disagreeing with the

idea that she had the knowledge needed to teach, but agreed that she had probably learned

the mathematics at some point in her education. Overall, the follow-up interview

highlighted that Janica still held onto many of her previous ideas about her mathematics

knowledge but she was slowly assimilating some new ideas about mathematics

education.

Near the end of the follow-up interview, Janica opened up and shared some very

personal feelings she had about her mathematics ability. She confided that the first

interview task of drawing 3 divided by 2/3rd had made her feel uncomfortable. In the


35


initial interview Janica’s thought was one of relief because she had read the book

example a few days before and was able to share it during the interview. She even

thought to herself, “What would I have done if I did not have the example from the book

to use?” Upon further questioning, Janica admitted that she would have felt “dumb” in

front of the interviewer if she had appeared to struggle with her mathematical knowledge.

She had been worried about revealing her lack of conceptual understanding of fractional

division and was glad that she was able to save herself some embarrassment by using the

book example.

After Janica’s admission of her true feelings, Janica pointed out a passage to the

interviewer that represented the feelings she had during the first interview. The following

quote from the NCTM 2000 Principles and Standards seemed to sum up Janica’s first

interview experience.

Many students have developed the faulty belief that all mathematics problems

[can] be solved quickly and directly. If they do not immediately know how to

solve a problem, they will give up, which supports a view of themselves as

incompetent problem solvers. (p. 259)

Janica identified with this quote because it fit well with her initial interview experience.

During the interview, she felt she should be able to provide an answer speedily if she was

to be considered bright in mathematics.

For Janica, this idea of being categorized as not knowing mathematics was very

real and came through in a journal entry and a subsequent conversation between her and

the researchers. She wrote in her journal that she would “begin to feel inferior/superior

depending on [her] level of knowledge and experience with the particular subject.” Janica


36


was referring to feeling inferior or superior in relation to her peers depending on how she

perceived their knowledge compared to her knowledge of mathematics. Thus, part of

Janica’s self worth was associated with her mathematical understanding, or lack thereof.

Janica’s reactions during the follow-up interview highlighted the cultural pressure

Janica felt to be an expert in the field of secondary mathematics. She felt the cultural

expectation that preservice secondary mathematics teachers should know secondary

mathematics. Similarly, Cooney et al. (1998) reported the comments of one preservice

teacher, Nancy, who had felt stress related to her inability to provide mathematical

explanations. “Later she said that she was afraid of letting her instructors down by not

knowing the mathematics” (Cooney et al, 1998, p. 325). Similarly, Janica did not want to

reveal that she lacked the mathematical knowledge necessary to adequately discuss the

mathematics problems. Janica feared being perceived as unintelligent or “dumb” if she

could not solve mathematics problems quickly. These fears stemmed from her belief that

the culture expected her to be knowledgeable in mathematics. However, it was different

for Janica than Nancy because the interviewer was essentially a stranger to her.

CONCLUSION

Although Janica claimed, in most of her interviews, to be very confident in her

understanding of secondary mathematics, the data revealed that she used various actions

or ‘moves’ during lesson study and her research interviews to unknowingly keep from

revealing any weaknesses that she might have in secondary mathematics. Acting as the

scribe, Janica was able to control the flow and pace of the conversations and ask

clarifying questions as the scribe and not as Janica. When faced with a topic that she felt


37


she should know she would say I’ll need to review or turn to an outside source, indicating

that she thought she knew it at one time but had not practiced it lately. She felt the class

structure was to blame, for her inability to identify the underlying concepts on the weekly

lesson plans. When Janica was unable to describe the mathematical concept associated

with the secondary topic, it was excused by saying, that’s a pedagogical question.

Because of the culture’s perceived connection between mathematical performance

and natural ability (Stevenson et al., 1990), Janica had come to view herself in a way that

suggested any falter in her secondary mathematics reflected on her intelligence. She truly

was afraid of being seen as mathematically incompetent in front of her peers and the

researchers. In the follow-up interview, Janica explained that she feared that the

interviewer would think she was ‘dumb’ if she struggled with mathematical ideas. With

this fear, natural defense mechanisms were used in order to protect herself. Janica had

pushed the attention away from her mathematical knowledge and pointed to other aspects

of her knowledge that she could admit were weak without threatening her self-image.

As Janica’s moves are compared to those used by Antonio in Seibert’s study

(1998), it can be seen that the motivation for these moves is very different. Antonio used

moves to avoid having to reassess and possibly change his beliefs about mathematics.

Janica’s moves, on the other hand, were used to protect her self-image as an expert in

secondary mathematics.

While other researchers (Ball, 1990; Cooney, 1999) have described some similar

actions, this paper offers a more comprehensive list of these moves and the motivation

behind them. As the follow-up interview revealed, Janica was subconsciously taking a


38


variety of different steps to avoid having to admit that she did not understand high school

mathematics, in particular logarithms, as well as she felt that she should.

While it appeared as if there was a conflict between Janica’s claims about the

strength of her mathematical knowledge and her moves to protect herself, the follow-up

interview revealed that Janica was in fact insecure about her mathematical knowledge.

This admission allows us to see that Janica’s beliefs and actions were, in fact, consistent

and coherent. During the follow-up interview, Janica admitted that she was aware of her

lack of secondary mathematics knowledge. She commented that her knowledge of

logarithms had grown as she engaged in lesson study collaboration and that lesson study

forced her to rethink how she viewed logarithms and helped her redefine her fragmented

knowledge of the subject. As a result of her experiences, Janica was able to openly admit

that the topic of logarithms was an area of difficulty for her.

Although we only have data that will allow us to describe Janica’s feelings and

actions, because the teaching culture as well as the beliefs about teaching and learning in

the United States were a source of Janica’s beliefs, we feel that this sentiment is held by

many preservice teachers. Teacher candidates may feel hesitant to talk about mathematics

topics with teacher educators or peers because they run the risk of exposing a weakness

in secondary mathematics which is closely associated with their personal view of

themselves. As a result these preservice teachers may take steps to avoid such

conversations. Preventing this avoidance and strengthening these weaknesses is the

challenge for teacher educators. This may be done by helping preservice teachers realize

that such weaknesses are not uncommon and by creating opportunities for these teachers

to share their feelings about secondary mathematics. Prospective secondary mathematics


39


teachers also need the opportunity to study secondary mathematics in situations where

they can develop their knowledge of mathematics without feeling that their self-worth is

being challenged. If the feelings about secondary mathematics are initially discussed,

lesson study can be a good environment in which mathematics is learned and some of the

beliefs imposed by the culture of teaching can be addressed.

REPLACE STRIKETHROUGH WITH THE FOLLOWING.

Since lesson study has a focus on student thinking, the mathematical conversations

among preservice teachers in a lesson study environment may be more open because it

allows participants to talk about their own thinking in the context of a student’s thinking.


40


REFERENCES

Alcoff, L. (1996). Real knowing: new versions of the coherence theory. Ithaca, NY:

Cornell University Press.

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to

teacher education. The Elementary School Journal, 90(4), 449-466.

Clark, C. M. (1988). Asking the right questions about teacher preparation: contributions

of research on teacher thinking. Educational Researcher, 17(2), 5-12.

Collier, C. P. (1972). Prospective elementary teachers’ intensity and ambivalence of

beliefs about mathematics and mathematics instruction. Journal for Research in

Mathematics Education, 3(3), 155-163.

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