Academic journal article Educational Research Quarterly

Beyond Classroom Description: Methods of Understanding Reflection and Beliefs in Mathematics Teaching

Academic journal article Educational Research Quarterly

Beyond Classroom Description: Methods of Understanding Reflection and Beliefs in Mathematics Teaching

Article excerpt

In the present study I examine two methods of unpacking teacher's mathematics-content/pedagogy beliefs and their instructional decisionmaking The study took place over the course of a school year. For each the two participants, I observed & videotaped 10 classroom mathematics lessons, and held 6 follow-up dialogues (three of each type) of more than 60 minutes each. I then used data from the Video Reflection and Theory Reflection methods and qualitatively analyzed each for meaningful patterns and richness of response. Results indicated that both methods were useful for varied purposes in obtaining teacher belief data.

How have teachers incorporated recent reform efforts, particularly those in mathematics education (NCTM, 1989, 1991; NRC, 1989; Grouws, 1992)? Beliefs about mathematics influence how a teacher teaches mathematics. As teaching beliefs in general move from a behaviorist orientation to more constructivist and humane, are individual teacher's beliefs changing in this direction also? Are such belief changes carried into the classroom? Beliefs are not readily observable and are often tacit to the teacher, him/herself. Indirect methods of discovering and furthering teacher beliefs are needed. The study analyzes two reflective forms of inquiry that attempt to go beyond the observable level to understand two teachers` beliefs about teaching and learning mathematics.

Change and redefinition in teaching require reflection and a personal questioning of the underlying beliefs that drive practice (Schon, 1983). These actions, common in the assessment process of students may be new to the teacher as a form of professional self-study.

The study analyzed two different methods of reflection in accomplishing the following: (1) as a setting for deep reflection on beliefs and practices in mathematics, (2) as a context for teacher change in either belief or practice, and (3) as a personalized approach to professional development. The first method is referred to as The Video Reflection Method and the second, The Theory Reflection Method. Both will be described in detail later in the article.

Background

Reforms in mathematics education are dependent on the teacher in the classroom. However, many teachers' beliefs about mathematics and what learning mathematics entails are incompatible with the reform effort. Such beliefs not only hinder the progress of school reform but also result in an enacted curriculum "that is seriously damaging the mathematical health of our children." (Battista, 1994, p.462)

The need for reform and its success stemming from a change in traditional teacher beliefs suggests novel approaches to collecting, refocusing, and analyzing data. Duckworth (1986; 1987) used elicitation methods during actual science investigations in order to learn and change teachers beliefs about teaching science. Cobb, Yackel, & Wood (1992) worked intensively with one second grade teacher in the area of mathematics discussing classroom practice, challenging established beliefs. and providing her with actual constructivist lesson plans. Thompson's (1984) classic study of three teachers' conceptions of mathematics and how those conceptions effected their teaching practice employed the use of questionnaire, interview, and observation methods1.

Fenstermacher ( 1986, 1993) pursued the question of change in beliefs by drawing on Green's (1971, 1976) notions of the tendency of the mind to group ideas into "clusters" with internal but not necessarily external consistency. Through dialogue with an "Other" beliefs were brought out and examined in relation to each another, thus forcing the consistency issue. This process Fenstermacher calls a "practical argument." By applying these ideas to a contemporary analysis of teaching reading, Richardson i 1994) suggested a method to draw out and affect a teacher's beliefs through the use of video tape reflection and intense dialogue between the teacher and the staff developer.

Method

Participants. The participants included two elementary school teachers. chosen from a larger study, and myself as outside researcher. Both teachers volunteered for the study and worked in depth with me over the course of the 1991-92 school year. For each of the two participants, I observed & videotaped 10 classroom mathematics lessons, and held 6 follow-up dialogues (three of each type) of more than 60 minutes each. I then used data from the Video Reflection and Theory Reflection methods and qualitatively analyzed each for meaningful patterns and richness of response. Each session described in this article involved the pairing of the researcher with one teacher. From the classroom data collected. I selected the two fourth grade teachers to demonstrate the usefulness of and differences between the Video Reflection Method and the Theory Reflection Method. The teachers were representative in the sense that they both taught fourth grade. However, each teacher's approach differed in regard to the role of the student in the learning process. the role of the teacher, and teachers' classroom manner and method.

"Bob" was a fourth grade teacher with a background of college mathematics, preservice emphasis on "New Math," and a teaching specialization in science. Bob requested help in mathematics teaching in the form of "recipes" that work. Some teaching materials were provided. However. the main goal was to assist Bob in reflecting on his present beliefs and future goals, and how these related to classroom practice. He had a wonderfully warm and caring way with his students. In fact, the major portion of class time was spent with Bob moving around teaching each child individually. The students responded in kind which produced a family atmosphere where members were expected to do their part, but a certain comfort level was apparent as well. This was noticeable in the humorous exchanges, informal "rules" for turn-taking, freedom in seating postures, and the teacher's willingness for student criticism and comment.

The climate within Bob's classroom was one of informality. In contrast, he believed mathematics was a structured hierarchical syatem, and mathematics pedagogy traditional. Bob, having lived through the New Math era, was suspicious of mathematics reform in general. He was reluctant to subject his students to the new and different unless it could be justified to the point of assurance that it would "work." Bob's strongest motivation was that his students be successful in mathematics and in life. Everything that he did in the classroom filtered through that sieve. He wanted to change, but feared the consequences on the children of a trial and effort approach to his professional development. In summary, Bob requested recipes that work in teaching mathematics-- the stark opposite of the intentions of the study

"Pam " was a fairly new fourth grade teacher2. Her recent preservice education and her confidence in the subject area of mathematics earned her the name of fourth grade level mathematics teaching expert and mentor. Conversations with Pam centered around her expertise, and the strengths and weaknesses of her students. Her enthusiasm for the subject was apparent in her manner of teaching, the amount of time she allotted to the subject, and her desire to teach it "the right way." Pam incorporated immediately into her mathematics lessons suggestions she picked up from our discussions and others she had read about. Her level of energy and enthusiasm had the opposite effect on several of the students and an teacher-imitation effect on others.

In spite of her reputation and self declared expertise in the subject, her communication with me was difficult and at times defensive. She initially revealed that she thought that she didn't need professional development in the mathematics area, but would be willing to "just talk" with me. Pam's lessons were replete with the symbols of reform, such as manipulatives, students in groups, and writing in mathematics. Yet, her beliefs such as teacher as leader, knowledge source, and person to be imitated interfered with Pam's carrying out the new approaches in her classroom. The atmosphere in Pam's classroom during mathematics, though highly routinized, demonstrated her love for teaching the subject. Her energy level was high as she demonstrated, often using the overhead projector and manipulatives to introduce a concept or procedure. The content presented, however, was very traditional, as were her daily routines of 5-10 review computation problems for the students to solve while she took roll, and how she began each day's mathematics lesson with a sheet of 100 basic fact problems for the students to solve in 5 minutes or less. Her students appeared to enjoy the segment of the lesson where Pam taught in her dramatic and expressive way the content of the day.

Researcher. I came to the school as an outside researcher/mathematics teaching advisor/staff developer. My beliefs about mathematics and teaching evolved over the years from my experiences in the classroom as well as from my reading. These beliefs embraced a somewhat constructivist3 approach to learning with student concerns and understandings the goal of teaching. I believed in the reforms put out by the NCTM (1989, 1991) and the NRC (1989) to be moves in a positive direction. I also believe that there are multiple approaches to the teaching of mathematics and no one "right way." My role became that of listener, supportive colleagues resource and helpful critic. The dilemma of being there for the teachers while also allowing them the freedom to search out their own solutions caused us both to constantly adjust our thinking and self-awareness of our constantlyshifting roles. Having attended numerous staff development programs, the teachers expected me to present "all the new ways to teach mathematics." It took time, attention to, and reflection on our goals for a symbiotic relationship necessary for self-study and growth to develop. My background in the classroom helped the teachers to see me as colleague to some degree. Although in another sense I could never truly be colleague, since I was not a member of their faculty. Teaching is context sensitive, so that experience regardless of how much has its limitations. Teachers aware of this fact, have a tendency to doubt whomever they consider to be "outside" their beliefs or contextual experience. My initial goal was to cross over this outsider/insider line. The fact that my mission there was seen as "the mathematics teaching expert from the University here to do a study" did not ease my stepping over that line.

PROCEDURE

In order to understand each participant's beliefs, I had to devise ways of unpacking them in a caring and data-producing environment. Constant reflection on my relationship to the teachers as colleagues and peer professionals became critical to the study. I used data collected from observations in the teachers' classrooms as cultural context (FeimanNemser & Floden, 1986) and the starting point for our reflecting together. The observations also provided insight into the participants' tacitly held beliefs (Polanyi, 1966). Observation alone, however, could not produce the data necessary. In order to study the beliefs underlying the teachers' actions and witness possible changes in those beliefs, forms of interaction were necessary. These will be discussed at length in "The Comparison of the Two Methods" section. Both types of sessions were audiotaped and transcribed, becoming part of the corpus of data. Observational, interview, and other related data were analyzed for emerging patterns during and after the data collection period. The researcher read interview transcripts, viewed observation video tapes, discussed early findings with the participants and outside researchers, and interpreted the data. (See Erickson, 1986; Lofland & Lofland, 1984; and Eisenhart, 1988.) In this process. the data were examined recursively driving further research questions. Findings from the various data sources-- observations, discussions. and interviews, were given to the teachers to read. comment on. and which formed the basis for further indepth discussion. This method of triangulation allowed for input from both parties and collaboration on the finished product.

A Comparison of the Two Methods

METHOD I: VIDEO REFLECTION

Video Reflection sessions were private meetings with one teacher and myself. These feedback sessions were built around a video tape of the teacher's mathematics class in action. Participants were given copies of the video tape to take home and view in advance of the video reflection session. The preview afforded each participant the time to note where they wanted To stop to make comments, question, or explain. During the session, the tape was played through or scanned to find segments for discussion. This method was adapted from Richardson and Anders (1990) in their study of reading teachers' beliefs.

Collaborative viewing of the video tape was used to initiate discussion. Selecting an action or statement from the tape, the teacher's reasoning as explored. A typical example would be in the form of a series of "why" questions to unpack underlying reasons for actions. A process of active listening was used to verify connections between classroom actions and stated beliefs. As participant observer, I would often suggest an alternative premise or related journal article for the teacher to consider. This might take the form of a question. For example, the following is a piece of a reflective dialogue with teacher Pam after the viewing of a problem solving lesson.

Pam:

...That's very hard. We always go back to what-how much money did he spend, okay, so then we know t, so how do we work those numbers to make it that [answer]. I had one little girl draw 9 pencils with 52 cents under it....Some have a very difficult time with word problems.

Researcher:

I wonder if you put chips in the middle of the groups, like bingo chips, and said, "Use these if you care to." I wonder if that would be attractive enough to get them to do it.

Pam:

Like use chips for manipulating?

Researcher:

Right. Like for the pencils, 9 pencils, and they could use chips to represent them or for whatever they wanted. Maybe if they were there and they would be attracted to use them....

Pam:

Or each, what I could do is to have a jar in the back about 10 jars and say, "If you need chips, go and get them." I don't know if I could, you know.

Researcher:

Or instead of saying, "need them," that might be a put down. Maybe say, "If you want to they're here for you to use" or something like that...

Pam:

"If you'd like to." Where I'd get bingo chips who knows....I've never done that in the past but hmmmm.

The purpose of these meetings was to bring out teacher's underlying justifications for classroom actions. From there, beliefs acknowledged by the teacher earlier, or alternative beliefs were suggested. In the case above. Pam had stated a firm belief in and use of manipulatives in her classroom in her initial belief interview. Yet, here she is confronted with a video picture of her class struggling with problem solving where it appeared to me that the students would have been helped by their use. As she reflects on the discussion, Pam moves closer to aligning practice with belief

In a reflective dialogue with Bob, he provided a solution in the form of a question. Bob struggled with the discontinuity of his traditional beliefs of teaching-as-lecturing with strong concerns for student success.

Researcher:

...at the beginning of the year I noticed that the kids desks were in groups, now they're in rows. What is your thinking behind those arrangements?

Bob:

The main thinking was the kids talking out of context. I thought maybe putting them in rows would stop some of that and it did, and the other reason is when I do board work it's easier for them to see [when they're in rows] ...this class is exceptionally noisy--- talking. And it has helped. It didn't cut it all out, but it has helped with the talking out of context.

Researcher:

How do kids learn mathematics best? This might have to do with that question I just asked you.

Bob:

Different ways. Some kids work best by themselves, some kids work best with kids, you know, working with other kids. (pause) And I have agonized over changing my room set up. Really, I agonized over it. I talked to people.

Researcher:

It's a dilemma.

Bob:

Yeah, because, you know, shall I leave them in the grouping and then let them just talk? Or set them up, the only thing that finally swayed me was that when everybody is facing the same direction it is easier to view explanations on the board.... you have to make your groups very, very, very carefully.

Researcher:

Or change them often. Do you think adults in real world situations do more group-thinking or individualthinking when they're making decisions?

Bob:

Now, that is a tough question! (Pause) I don't know.

Researcher:

...if you have a real dilemma in your life, do you solve it just by yourself or do you--

Bob:

No, I get help, group help. For me, personally, I do more problem-solving in groups than by myself.

Researcher:

I do, too. I think that's one of the advantages I see to cooperative learning because it's a life skill.

Bob:

So do I hear-you suggesting that maybe I ought to reconsider my desk setup?

Researcher:

Well, it's just when I observe you doing the part after you do your lesson on the board, it seemed like you're pulled in so many directions that I think maybe the kids could do some of that without you.

pulled in so many directions that I think maybe the kids could do some of that without you.

Bob: What about a set up like this. What do you think of this? Instead of grouping them in 4 or 5 desks all facing toward the center of the little group so kids are facing in all directions in the room, leaving the rows like I have now, but with 2 desks [pushed together]?

Bob observed classroom actions, via video, while simultaneously reflecting on contingent beliefs. This is not a case of stimulated recall techniques to reproduce what he was thinking while in the act of teaching (Clark & Peterson, 1986). It was rather a case of discovering for himself inconsistencies and finding his own increments of change toward a solution that felt right for him now.

While interacting with the teachers and in examining the data, I attempted to distance myself from my knowledge of school culture to continually question classroom actions. My questioning of the reasoning behind normal" classroom procedures and content caused puzzlement initially. However, in time, the teachers began to expect my questions, often anticipating them by offering thick description (Geertz, 1973) without extensive prompts.

Method 2: Theory Reflection

Near the beginning of the second semester, the feedback sessions took on a different form. This happened partly as a result of the teachers distraction at viewing themselves on video tape, and partly as a way to isolate particular belief-related events from the myriad available in each lesson. This focus, though necessarily narrowing the breadth, permitted a depth of discussion. This method followed the spirit of Spradley's (1980) focused observations" in the form of focused narrative elaboration.

Teachers' classroom actions may suggest particular underlying values. or beliefs integrated over time or they may suggest a need for reframing beliefs or practice to achieve alignment. During or after observing a teacher's lesson, reflection on possible beliefs that could underlie the particular classroom action would come to mind. These were written as statements for the teachers to reflect on, determine their alignment with their true belief and discuss the practice/belief connection. Teachers were asked to clarify or verify my statements of their beliefs. Short answer responses were probed for further insight into underlying beliefs, values. and meanings. Contradictions between stated beliefs and classroom actions were discussed and clarified for mutual understanding. At times alternative premises were suggested that aligned more closely w-ith mathematics education reform efforts and constructivist pedagogy.

Transferring the statements to a 3 by 5 card deck that was placed in the center of the table, allowed the teacher to control the length of the discussion. The teacher would read the card, deny or support the statement, then elaborate to support their position. The method, its derivation. and participant roles were presented to the teacher before the session began. This was done primarilly to emphasize the tenuous nature and the origin of the "belief statements. Rich data emerged as the teachers began the discussion on the belief level and proceeded to issues of practice. The following example from a Theory Reflection session with Pam will illustrate the process.

MOST PROBLEMS HAVE ONLY ONE REAL SOLUTION AND THAT'S GIVEN ON THE KEY, IF STUDENTS COME UP WITH OTHERS, THEY'RE PROBABLY WRONG.4

Pam:

If it's a long division problem and if it's in the book, paper/pencil, yes.

Researcher:

What about the toothpick problems?'

Pam:

No, they could come up with many different [answers]. Although I don't know if they did. I don't think they could...A lot of kids came up with solutions. Well, they came up with two squares and then things sticking out. So [I said], "Well, you're right-- two squares there but what's this right here?" This [the toothpicks sticking out] has to be something to be a correct solution.

Researcher:

So you wouldn't accept those with the toothpicks sticking out even if they got the two squares? You wouldn't accept that as an answer?

Pam:

Well, it's an answer. It may not be the right one... but in reading the instructions, you have to take it for what it says.... What's good about this... [would be] to promote these kids if they believe this is true, sell it to me, tell me why you think that this should be right. In the lesson from which the theoretical belief above was taken, Pam held the position of gatekeeper to what was "right." Her position of authority and controller of the curriculum emerged early in the data-collection process. However, Pam's incremental change became evident with continued interviews using the Theory Reflection method. The change is salient in Pam's final interview.

Researcher:

...We also talked a lot about constructivism, what does that mean to you?

Pam:

... I think that [this] is kind of a new area I've hit toward the end of this year [exemplified] in the Math Menus where there's no set answer. In fact, some of the instructions are vague. When they come up with what they believe the instructions say and go about it, they have gone through a process that they have thought out themselves and they believe to be the right thing and it could be. No, I can't really say, "No, you're wrong."

The Theory Reflection process evolved during the course of the study as a direct means of teacher response to assumptions made by an outside observer. Both the Video- and Theory-Reflection methods were based on teacher communication of tacit beliefs. Their purpose was to make the teacher more aware of underlying beliefs and their relationship to mathematics pedagogy. Both methods of study allowed for suggestions and alternative premises to promote teacher reflection, logic building, and moves toward change. Bob's discussion of procedural and conceptual knowledge6 is another example of incremental change. Here again the Theory Reflection method was used.

MANIPULATIVES ARE ONLY NECESSARY WHEN KIDS ARE HAVING TROUBLE

Bob:

I think manipulatives are necessary when you're introducing a new concept plus [you must] go back to manipulatives when students have trouble. That's when they're necessary... That doesn't mean that you can't use them other times. too.

Researcher: For instance?

Bob:

For instance any time you are doing math if you could use manipulatives [for that topic]. My premise is that, that kids shouldn't learn to rely on manipulatives because they're not going to be able to carry around a pack of Cuisenaire rods with them out in the real world. But they're good for teaching concept and they're good for going to if they're having problems.... My goal, though, is for them to be able to do a problem or a procedure cause that's the real world.

Researcher:

So your goal is to get them to do the procedure. Do they have to understand the concept?

Bob:

I think they have to understand the concept first.

Researcher:

Before they can do the procedure. but the goal is really the procedure because that's real world?

Bob:

Well, yes, my goal is procedure because it's real world.

Researcher:

And concept isn't real world?

Bob:

Of course not. But you've got to have the concept before you can have the procedure...

Researcher:

Okay, but your goal is procedure.

Bob:

Right. Therefore concept comes first. They've got to have concept first.

Researcher:

But why isn't your goal concept?

Bob:

Well, it is, first.... See I was taught procedure. I wasn't taught concept.

Researcher:

How did you learn it?

Bob:

It came--dawning, oh oh yeah, Now I see! (Loudly and with expression.) It was an 'ah-ha'.

Researcher:

That's neat, do you think kids get those?

Bob:

Oh yeah and they get them with me. I have a couple of kids that we do the beans and we do the beans and we do the beans, and then we start on the procedure and then all of a sudden, 'ah-ha' !

Researcher:

'Ah-ha' it comes?

Bob

I don't know, now I don't have a lot of those, but I've had a couple...

Bob continued to insist that his own history of mathematics concept development was attained without manipulatives, but with an emphasis on algorithms. The data revealed a small change-- where Bob began to value a broader use of manipulatives by the students as means of connecting concept to procedure on their time schedule rather than his. This new belief he held simultaneously with his other unchanged belief that in the final analysis, the students' ability to perform the algorithm was more important than their ability to understand the concept. In Bob's interpretation. concept development with a wider use of manipulatives was a step toward the ultimate goal of algorithmic success. The Theory Reflection method respected Bob's increment of change or 'where he was' at that moment. An extended reflection period and subsequent discussion might encourage Bob to reframe his beliefs further. However, the fact that he did see concept development through an increased use of manipulatives by students on their own terms as rewarding, suggests some value to the method.

Although the Theory Reflection method of unpacking teacher beliefs emerged from the school setting as an adaptation of the Video Reflection method. both were useful for the collection of data. In fact. the Video Reflection sessions may have been a necessary first as a trust-building entry into the more straightforward nature of the Theory Reflection sessions. The focus went from wide-angle to zoom in an effort to discover deeper connecting theories and beliefs. The discussions that emerged from the Theory Reflection type quickly became challenging mentally in comparison to the Video Reflection method where the multidimensional classroom scene offered numerous distractions and possibilities for tangential conversation that interfered with the progressive flow of the dialogue at the belief level. The distractions in the Video Reflection Method provided an outlet for the intensity of the dialogue. Some teachers who are naturally deep thinkers may need the respite, while others who may take longer to get into such an intense thinking mode, may use the video distractions as substitutes for confronting the real issues.

Analysis/Discussion

Data from the two methods indicated that both offered a suitable setting for understanding teachers' tacit beliefs. Both provided insights into the change process. The Video- and Theory- Reflection methods produced rich data that revealed the relationship between beliefs about mathematics content/pedagogy and classroom practice. Both provided opportunities for personalized professional development. The Video Reflection discussion progressed from practice to belief to practice. The Theory Reflective discussion began with a theoretical belief, considered practice in light of that belief and ended with a movement toward a true belief or an aligned practice. Both methods were useful in understanding the relationship of belief to practice and for encouraging teacher change.

The findings indicated that both methods provided rich data in regard to teacher beliefs. The fact that the teachers were co-researchers in the study encouraged them to thinkabout their beliefs in relation to their practice. The rich data that was the result provided self-awareness of this relationship, forced them to question their practice. and set the stage for change. The study provided the setting and gave the teachers permission to expose and defend their true beliefs against the background of their classroom practice. The outside observer's inferences became questions posed for the purposes of discussion, communication, and redefinition of tacit beliefs.

In the case of the Theory Reflection Method, the teachers'beliefs were extracted from classroom observation and written down. Forms of language are context-sensitive in terms of time, place, audience, and purpose. Conversation and written text are not interchangeable modes of communication . As the teacher and researcher talked and clarified each other's positions. true meaning of action as related to belief emerged. The collaborative dialogue negotiated and defined what the written statements could not.

The teachers had a vested interest in communicating their true beliefs for the following reasons. (1) They sought implications for improved practice. (2) The professional setting and conversation demanded collegial respect and the amount of trust necessary to reveal oneself to another. (3) The teachers indicated a need to enter into the inquiry, itself as self-study researchers. (4) Their requests for feedback, advice, and open communication, over the course of our year together, suggested a professional working relationship.

The construction of theoretical beliefs derived from classroom observations and my part in the reflective discussions gave me a participatory role in the study (Spradley, 1980). This had the effect of reducing the power relationship. The reflective dialogues provided a place to share our insights in regard to the teaching and learning of mathematics.

CONCLUSION

The study described here focuses on methods of unpacking teachers' beliefs in the pedagogical content area of mathematics. Both the Videoand Theory- Reflection methods provide a setting and opportunity for teachers' self-analysis of beliefs and practices in mathematics. Results imply that collaborative reflective dialogue (ly help teachers to discover beliefs that affect their practice. In the process, (2) teachers also change practices deemed not in alignment with their true beliefs. (3) The selfreflective nature of the process provides an opportunity for personalized professional development. Teachers appreciate the collaborative nature of dialogue and may assume responsibility for their own change pre^cass.

The data present change as non-linear. Teachers move between beliefs till negotiation and change occur. It suggests that change is not a single move forward, but a continual ongoing process. This negotiation process implies that teachers' beliefs and ideals may invariably predate practice just as thought races ahead of speech or as circumstances "get in the an v of goals. The dilemma-ridden nature of teaching may mask or even prevent the direct alignment of practice and belief. Teacher beliefs represent the ideal while classroom practices embody a foreshadowing and striving for that goal.

The study suggests the need for further exploring a variety of reflection methods. It may also imply a need fri- a redefinition of what is meant by teacher change. When moved to the context of preservice education. implications include: a need to understand the teaching candidate's current beliefs concerning mathematics pedagogy, a need for the preservice teacher to express his/her beliefs and their changes, and his/her need to try out multiple pedagogy methods in schools with real children. It also suggests that the videotaping and viewing of these tapes may promote teacher professioal development.

Futher research in this area may lead to a deeper understanding of reflection-- at the belief level. Additional studies are needed to study teachers who become more aware of their tacit beliefs. Do they gradually close the gap between theory and practice? Do they show signs of reform in their teaching of mathematics? Do they continue to reflect on their reasons for choosing content, method, and manner in mathematics instruction?

1 See Kagan, 1990 and Munby, 1984 for other methods of eliciting teacher beliefs.

2. The year of the study was her fifth year teaching in relation to Bob's 22 years.

3. By constructivist pedagogy, I refer to a classroom situation where the center of activity does not revolve solely around the teacher, where authority, responsibility for learning, and the classroom agenda is shared.

4. Theoretical beliefs that were presented to the teachers on cards are printed in uppercase lettering.

5. The following is an example of a "Toothpick Problem" (A figure is shown of one large square that has been subdivided into fourths resulting in four smaller squares. Altogether 12 toothpicks have been used.) Build the figure with toothpicks. Using all the toothpicks. move 3 toothpicks to create 3 equal squares.

7. See Florio-Ruane, 1991.

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