A Teacher's Belief about the Nature of Mathematics
its Manifestation in Teaching Practices

Kyungsoon Jeon



This study investigated an expert teacher's belief about the nature of mathematics and its manifestation in teaching in a high school mathematics classroom. The data included an in-depth interview and field observations, and were analyzed by using the constant comparative method from qualitative research. The teacher's generalist approach to mathematics as a learner, the importance of self-concept in supporting the learning process in both the teacher and students, and teaching as providing assistance to the students were found. This study contributed to current research on teachers' beliefs system, and brought a need to complete a path that one travels from novice to expert in mathematics teaching.

"It is very hard to make silk purse out of sow's ear... You lead a horse to water, but you can't make him drink." (A quote from the participant teacher, 1997)

Teachers' beliefs about mathematics and its teaching play a significant, albeit subtle, role in shaping their behavior (Thompson, 1984). To understand how this interactive relationship between teachers' beliefs and teaching practices interacts and functions has been an interest to many mathematics educators. In this paper, I examined a mathematics teacher's belief about the nature of mathematics (how does the teacher understand mathematics?), and its manifestation in the teacher's teaching practices (how does that view affect the teacher's teaching practices in classroom?).
I realized that there had been a quite amount of research done about teachers' beliefs system. But, most of the research were done with beginning teachers or preservice teachers at universities, so called "novices". For example, Barry Shealy (1994) studied how two preservice mathematics teachers shaped their beliefs about mathematics, and how the beliefs affected their teaching of mathematics. I thought this bias in sample selection had limited the scopes of understanding about mathematics teachers' beliefs. For another example, in a study done by Livingston and Borko (1990), they explained that novices' cognitive schemata for content and pedagogy in mathematics teaching were less elaborated, interconnected, and accessible than those of experts. They also tried to provide comparisons of novices and experts' teaching. But, the study did not give a detailed analysis from the perspective of the experts, and didn't show the characteristics that were likely to be uniquely possessed by the experts. Therefore, I ended up with little information about how experts perform in teaching mathematics, and why they are different. I thought this kind of information would be critical in bridging the gap in the studies of novices and experts, their beliefs about mathematics, and their teaching practices. In addition, this would contribute to the betterment of teacher education programs in mathematics education. Therefore, I decided to examine an expert teacher's case in this study.
The purpose of this study was to identify an experienced high school mathematics teacher's belief about the nature of mathematics. Two research questions were posed based upon the purpose of this study. First, I wanted to know what factors helped the teacher in shaping the current belief system about mathematics. Second, I wanted to look at how the teacher's belief system was manifested in classroom teaching episodes. The meanings she described about the nature of mathematics, and their manifestation in her classroom teaching will be discussed.

In general, researchers in mathematics education have understood teaching as a complex cognitive skill so far, and naturally their studies have focused on the individual analysis with a child or a teacher from the perspectives of the cognitive psychology. In fact, they have concentrated on a child's learning, and studies about teaching and teachers have been ignored in this framework of individual cognitive psychology. This fact was very problematic to me because I believed teaching and learning of mathematics could not simply be made on the level of the cognitive domain of individuals. Rather, I thought the understanding about teaching and learning of mathematics should be made on the level of socio-cultural domain (in particular, societies, schools, classrooms, among peers, and individuals all together). I wanted to understand teaching and study teachers with a different framework. Therefore, I took the usefulness of selected aspects of Vygotsky (1978)'s approaches to teaching situated in many different contexts, and the socio-cultural theory (Bishop, 1988; Cobb and Bauersfeld, 1995) influenced by Vygotsky. In particular, Vygotsky's notion of the "zone of proximal development (ZPD)" were taken as a theoretical framework of this study:
"The ZPD is defined as the difference between a learner's assisted and unassisted performance on a task (Vygotsky, 1978)."
Also, a socio-cultural theory was applied to this study in a microscopic way. Brown et al.(1996, p. 65) explained that the socio-cultural theory analyzed classroom-based teaching and learning in two ways. One was macroscopic where more socially oriented traditions located learning within the larger socio-cultural institution of schooling. And the other was microscopic where more local interactions occurred between teacher and students in the classroom, as well as within the individual learner. In this paper, I only examined the local interactions occurring between a teacher and students in a classroom. But, I suggest further studies be able to expand the span both to the macroscopic level of the larger socio-cultural institution of schooling and to the microscopic level of the within-individual learner. By doing so, understanding about teaching of mathematics will have a more clear picture encompassing various dimensions of teaching and learning situation.
As a framework to describe and understand the microscopic perspective between teacher and students, I took Tharp and Gallimore (1988)'s vision of education and their general redefinition of teaching as assistance of performance through the ZPD:
"Teaching consists in assisting performance through the ZPD. Teaching can be said to occur when assistance is offered at points in the ZPD at which performance requires assistance (1988, p. 31)."
I focused on Tharp and Gallimore's claim that teaching should involve assisting the performance of learners. Taking Tharp and Gallimore's explanation about teaching was very important for me to continue this study. With this explanation, I started looking at teaching as constructive and I was able to be free from the conventional framework of cognitive psychology where learners constructed their own learning.
Having these frameworks helped me clearly identify what I was looking for throughout the whole process of this study. Most of all, they guided me to find features of my research questions while making me tied to the purpose of this study.

Research Design
First of all, I decided to take a research design from the qualitative research framework:
Qualitative research should "provide perspective rather than truth, empirical assessment of local decision makers' theories of action rather than generation and verification of universal theories, and context-bound explorations rather than generalizations" (Patton, 1990, p. 491).
Let me assume that I took a method from quantitative research design which was frequently used in research in mathematics education. Then, probably I would have needed a "hypothesis" and variables for organizing and conceptualizing my research at the onset of this study. In Eisenhart's words (1988, p. 102), I needed a "system" that must be grasped before my research goals, questions, and methods. But, I had a question: "how could I define them in advance of the actual study of the participant?" It was impossible for me to set up the "system" without any investigation because I wanted to study a teacher's mind, activities, and construction of the relationships between the two. I expected for them to be emerging in the process of this study. So, doing a qualitative research made more sense to me.
After deciding to do qualitative research, I considered which method among the five traditions (e.g., biography, phenomenology, grounded theory, ethnography, and case study, Creswell, 1998; Merriam, 1998) would be the most appropriate one to pursue the goal of this study. I knew that I was interested in a particular person and activities of the person. In one of my readings, Eisenhart (1988, p. 101) mentioned that most educational researchers had been trained in the tradition of experimental psychology, their constructs had been used across people, settings, and time to obtain consistent measures of development. But, I also knew that I did not expect to find a general explanation about the human being. I wanted to have a holistic understanding about my study. This is why I took a case study (Creswell, 1998 & Merriam, 1998 ) among the five traditions of qualitative research: A qualitative case study is an intensive, holistic description and analysis of a single instance, phenomenon, or social unit (Merriam, p. 27).
As Merriam (1998, p. 19) said about a case study, my interest was in process rather than outcomes, in context rather than a specific variable, in discovery rather than confirmation. Therefore, choosing a case study was appropriate in gaining an in-depth understanding of the situation and meaning for the involved and in seeing through the researcher's eyes (Merriam, 1998, p. 238).

The participant of this study was a mathematics teacher working in a public high school at a university town in the state of Georgia. Her name was Jessie (a pseudonym for the participant) and was in her 50's. I did purposeful sampling (Merriam, 1998; Creswell, 1998): "Purposeful sampling is based on the assumption that the investigator wants to discover, understand, and gain insight and therefore must select a sample from which the most can be learned (p. 61)". and criterion-based selection (LeCompte & Preissle, 1993):"In criterion-based selection you "create a list of the attributes essential" to your study and then "proceed to find or locate a unit matching the list" (p. 70)." I chose these two selection methods because I needed a participant who was an experienced mathematics teacher and also had a reputation for excellence in teaching. I also admit that it was a convenience selection (Goetze & LeCompte, 1984, pp. 72-73). I was aware of the limitations of this study, and took advantage of rapport which had been established by my previous experiences with Jessie. Even though it was convenience sampling, I had a rationale for choosing the participant. The selection was decided since it was appropriate to my initial research questions regardless of the sampling issues.
Jessie was a type of teacher who opened her classroom to researchers and always wanted to take some constructive criticisms from them. She characterized herself in the following terms while describing her experiences in her own education:
"I have to tell you that I am just a learning nut... If you can teach me something new, I love it."
A consideration about the participant's personal characteristics might have been brought in so that analysis about the participant's belief system and manifestation in teaching could have been understood in a different way. I, as a researcher was aware that personal characteristics of each individual teacher was an important factor possibly even in this study. But, this consideration was not dealt with in this study, and would be left to future studies due to the limitation of this study.

Data Collection
The investigation reported in this paper adopted two qualitative research data collection methods, an interview and a classroom observation. At the beginning of the study, I actually planned an open structured interview, but it turned out to be difficult for me to get to the points that I wanted to examine with my research questions. So, I started to devise a semistructured interview and made an interview protocol while doing literature review (See Appendix A for the interview protocol). Finally, I conducted a semistructured in-depth interview for 90 minutes. The interview protocol became deeply tied to the purpose of this study, the research questions, and particularly the conceptual framework of this study. While interviewing, I didn't follow the questioning order as I had planned on the interview protocol. Rather, I naturally followed the participant's responses, and made connected series of questioning. After the interview was ended, I found I had covered most of the questions on the interview protocol. The practices with my group members in ERS 799 class must have been helpful. And, it was a surprising and nice experience as a beginner in interviewing.
The observation of a mathematics class out of Jessie's 5 classes was conducted one week after the interview. The class was the 4th period, Advanced Algebra 3/Trigonometry class. The selection of this class was made by Jessie's request during a contact after the interview. I was a complete observer. The class was audiotaped in case I didn't follow the teacher's words while making field notes. I also prepared an observation protocol before the observation (See Appendix B for the observation protocol), and it came from the ongoing initial analysis of the interview data. Therefore, it was a focused observation. But, I did not entirely depend upon the observation protocol. Rather, I tried to observe as much emerging things as possible.
T he interview sought the participant's own descriptions. It was tape-recorded and transcribed by myself (See Appendix C for the interview transcript). The interview transcript provided information about understanding the teacher's belief system. The field notes from the observation acted as detailed instructional records of the participant. The data were complementary and supportive of each other later in the analysis. Also, memos were constantly made whenever I hit upon any ideas about the data, and were used as another form of data.

Data Analysis
applied the modified constant comparative method (Creswell, 1998; Merriam, 1998; Corbin and Strauss, 1990) since this method fit the inductive and concept-building orientation of this study. The continuous comparison of the participant's remarks and verbal and physical expressions of the participant with each other were made. The most important process of my data analysis was an open coding (Creswell, p. 57) process where I tried to search for initial categories of information that Jessie expressed in the scattered forms during the interview and the observation. This was a long and repetitive process and continued until I felt comfortable with that I identified all the possible categories. I made categories in emic terms. Then, I started axial coding (Creswell, p. 57), and emic terms were added in this process. By the axial coding, I explored conditions explaining Jessie's structuring of belief system about mathematics. I also delineated the consequences regarding the manifestation of her belief in teaching practices. Reflective notes (Creswell, p. 140) from the observation were helpful in giving insights on the process of finding emerging themes.
This process of analysis was the most fun part of conducting this study. At the beginning, I was uncertain about what would ultimately be meaningful from the data. But, some categories started forming from the initial process of the interview data analysis, and kept being backed up by further analysis of the interview data through repetitive visits. Results from the observation were adding to my categories and building them to a more sensible status. I was experiencing the function of successive layers of inferential glue mentioned by Miles and Huberman (1994, p. 261).

Quality Issues
To maintain the rigor in the whole process of study, several methods were taken (e.g., repetitive visits to the previous data, using personal information about the participant, keeping good record of data, doing peer reviews, and so on). The quality issues in terms of enhancing validity and reliability were considered throughout this study.
For internal validity, triangulations from different sources of data were done. The two methods, interview and observation were one of the examples of the method triangulation. This method triangulation aimed towards a holistic understanding about Jessie's case, and also aimed towards a plausible explanation about the case being studied. I did member checks while analyzing the data from the participant. This was especially crucial in acknowledging discrepancies between the participant and me. For example, when I questioned Jessie about my interpretation of her remarks about her previous educational experiences at a university as being negative (which might have affected the analysis about her belief system), she corrected my interpretation. In fact, this member check turned out to be more helpful. It allowed me to gain additional information about how Jessie looked at learning as a learner herself. Also, peer review was done throughout this study. This peer review provided me with ideas that how I could revisit my data and data analysis in different people's eyes.
Most of all, I want to stress the importance of deciding and stating of my conceptual framework in keeping the strong internal validity for this study. At the initial stage of this study, in particular, during the reading of lots of literature related to this study I had a difficulty in deciding my conceptual framework. A lot of interesting studies were done by many people. It seemed that there was too much information going on. I should admit that I was likely to be lost especially during the data analysis process without this conceptual framework. But, once I took the framework from Vygotsky and Tharp and Gallimore, it kept bringing me back to my research question. I was always asking myself "Are you verifying what you want to verify?". The strong internal validity was kept by the virtue of qualitative research.
The triangulation and the peer review which enhanced the internal validity played an important role in enhancing reliability, too. Throughout the whole study, I was consistent by these two methods. For example, I was consistent in finding results from the data collected. Also, an audit trail was used to enhance reliability. All of the written documentations were kept for the audit trail.
External validity was maintained by doing thick description (Creswell, p. 185). By thick description, I tried to describe what actually happened to the participant rather than just to describe what I heard and saw. I also provided as many narratives from the participant as possible in findings. The findings would not be generalizable to a general population, say all mathematics teachers. But, I left the generalizability to readers, in particular, to those interested in the study of mathematics teachers' belief and their teaching practices.
Consistent efforts to maintain ethics were pursued during the whole process of this study. All data were collected under the consent of the participant, and kept safely to be protected by any possible harm. All the recorded data will be destroyed as soon as this study is completed.
After the observation, I struggled with an idea that I might have had a wrong data, which might have a big impact on my findings. As I mentioned earlier, the Algebra 3/Trigonometry class was chosen by the request of the participant out of her five different mathematics classes that she was teaching. She might have reacted to me with the selected group of advanced students in the Algebra 3/Trigonometry class. For example, if I had observed her 5th period, Geometry class where the students' motivation and achievement level is lower compared to that of the Algebra 3/Trigonometry, I might have come up with different data showing the manifestation of her belief system differently.
I thought this problem was caused by doing the convenience sampling. I chose my participant who had a very close relationship with me. I found myself being trapped in her bias to show me good sides of her students learning. Consequently, I limited myself to only the teaching with those motivated and advanced group of students. If I conduct a study in the future, I will be careful in protecting it from participant's biases as much as possible as well as from my bias. Of course, the importance of more data and triangulations will be pursued in the future study. This was an important lesson in conducting this study related to the sampling issues.

Belief about mathematics : "Generalist Approach" supported by "Self Concept"
Jessie viewed mathematics as something approachable by anybody; I interpreted this as a generalist approach. But, this notion of anybody was exalted to somebody who was interested in learning mathematics and was not afraid of challenging new things; This was elaborated as a "generalist approach" supported by "self concept" . The word, generalist, is an emic term and was consistently found in the data from Jessie. I decided to keep this emic term for describing my emerging theme about Jessie's belief system. The following is Jessie's description about mathematics and her belief about mathematics:
The (high school) math is fine. The math is elegant and lovely... and very interesting to me for most of the time. But, I do not pretend myself... it is like math or nothing. It could be math, it could be biology, it could be English because I like it all... If I put my mind on it (mathematics), I will get interested in it.

Not only did she take the generalist approach in understanding mathematics, but also she took the generalist approach in understanding other subjects. Therefore, she was able to give equal importance to other subject matters such as science, arts, humanities, and so on.
Jessie did not believe that mathematics was a difficult subject which only a few people could do. Rather she believed that she could get access to it any time when she felt a need. For example, she went back to a university to get a master's degree in mathematics education because she needed it to teach her students with more confidence in the subject matter, and later to get for a specialist degree for the same reason. When she was asked if she wanted to pursue another degree in mathematics, she said "no" because that was not what she needed for her work with students. But, she said she wanted to learn calculus because she wanted to teach an advanced level course before she retired. She could start doing the mathematics any time when she was motivated to learn it. In Jessie's words, she was a learning nut in almost everything.
Jessie believed that she was a good teacher. Not only did she feel comfortable with the subject matter, but also she felt comfortable with her students. While describing her belief about mathematics, she showed how the belief affected her as a mathematics teacher:
I can think... I feel like a math professional. Now, this is one thing that I have learned. I am a really good teacher. I am just wonderful with this age group. I think... and in general with all the teaching task. But, you know what. I have a passion for the teaching and the engagement of helping people... young people to learn something. As shown in the above, the belief system constructed a confidence in teaching, working with the student, and other tasks in school.

Manifestation of belief system in teaching: Teacher as an "Assistant" in students' learning
Jessie valued students' own mathematics. She said that mathematics should never be thought in separation to students. For example, any preference to so-called pure mathematics among mathematicians was not found. Rather, she considered high school mathematics as much important as that in university or among mathematicians. She appreciated students' interest and willingly helped students learn mathematics. She tried to give help to all students, but the help were excellerated when the students showed her an interest in their learning mathematics. She was excited when she mentioned one of her students. The student was successful in learning mathematics and finally went to a medical school in a prestigious university by the combination of the student's belief in learning (the student's belief in school) and her assistance despite of difficult environments surrounding him:
It's wonderful. We just wish there were more... because you see... with our teaching faculty here there are many of us who can take the kids and just fly with them if they would let us... But, they have to believe that we can help them move from here to here and that somehow... They will put everything into him. We can help him. We can make a big difference.

Jessie showed strong confidence in dealing with mathematics as a learner herself. This fact naturally lead her to have strong confidence in teaching mathematics in her classroom. She claimed that she was a good teacher. In particular, she said that she was a better teacher when her students were initiators of their learning and she was acting on it. Once they showed her interest and enthusiasm in mathematics learning, she believed that she could take them to the point that they wanted:
The kid has to take it and wants to keep it... and do something with that. I will say we always want it. That's our role. But, you just cannot... another way we say that you lead a horse to a water, but you can't make him drink. But, constantly trying, but the students have interacted with us and trying to inspire our love of learning. And children who really don't have that it's very difficult.
In teaching practices, Jessie enjoyed surprises from her students and took challenges as a joy of teaching mathematics. She said that one of things that she was strongest with was that she really was not intimidated by students' questions or something that came as a surprise while teaching. In fact, during the observation, she had 11 questions from different individual students, and she nicely responded to most of the questions. There was a moment when the problem was not immediately solved by both her and her students. She said to the student to do some more explorations together, and they finally got the answer for the problem. It was an investigation with graphing calculators in which the whole class was involved. Her role was an assistant when the students initiated their problems where her explanations could be critical roles in solving the problems. The following was Jessie's comment after the observation: I don't mind repeating the problems as long as they showed me interests and paid attention to me.

She knew that her students could not follow everything without her consistent help. But, she believed that they could learn if they worked with her with their own interest in learning mathematics. So, she did not mind how much repetitive assistance she should provide to her students as long as the students' self concept existed.
In summary, Jessie believed mathematics was a subject that could be taught and learned by a person who was interested in it. She approached mathematics with the generalist approach. This generalist approach was reflected in the form of confidence in her teaching practices and her belief about students' learning. She believed that her students could learn mathematics if they were supported by their self concept in learning mathematics. Once the students were armed with this self concept, she played a role of a good assistant in the students' learning of mathematics.

There has been a controversial debate in defining the nature of mathematics. And people have defined it in many different ways reflecting their own philosophies related to teaching and learning of mathematics. The teachers and students described mathematics as correct answers, exact and predetermined solutions, learned procedures, and no gray areas. But, when the debate was boiled down to teachers and students, they seemed to finish the debate and reach a common agreement. Research done by Borasi (1990) revealed that mathematics was defined as a rigid set of formulae by both teachers and students. I think this is why this study with Jessie was interesting. To Jessie, mathematics was not a rigid set of formulae. Frequently reported "mathematics anxiety" as a learner did not appear in the data. More over, she was showing a confidence in both the learning and teaching of mathematics. I think a further study about the process of structuring of Jessie's belief system should be done to understand how Jessie became an expert from a novice. This will be an effort to understand how novices can become experts.
It was also interesting to see from her descriptions about her students the notion of the ZPD by Vygotsky (1978). Jessie believed that she could help her students to make a difference. And, Jessie's students actually developed the ZPD by getting help from their teacher. But, one thing that I want to stress is that the ZPD was developed at different levels according to the level of each student's interest in learning. An example was the story about the successful student in her previous teaching experiences as described above in the paper. While he was having success in his learning with Jessie, there were a lot of students who did not go through the same way as the student did with the same teacher. So, I had a question: "Why was the ZPD developed at different levels?". This question may sound trivial knowing that everybody learns differently. But, I think this is why studies about teaching and learning can not be done separately. In this sense, I think studies about the role of a teacher should be done in conjunction with the examination of the role of students. Again, this will be an effort for mathematics educators to escape from the individual cognitive frameworks into a more socio-cultural understanding. I think this will provide more base knowledge about the teaching and learning of mathematics to mathematics education in a broader context.

One of the assumptions that I had for this study was the equalization of being "an experienced teacher" and being "an expert teacher" when I explained the participant's case. I assumed that Jessie was an expert teacher without any proof except for that she was an experienced teacher and had some reputation of excellence. I acknowledge that it is not true in reality and I should have studied more to make the equalization. It was because of my perspective, and of course, because of other limitations of this study such as time, and participant selection issue related to IRB approval. If a future study is conducted, I want to include various perspectives from her colleagues, her school principal, and her students to make that judgment of experts in mathematics teaching. In particular, the perspectives from students will be important in a sense of understanding mathematics education from both the teacher and students sides.
This study, unlike other studies on novice teachers cited in this paper, tried to reveal the characteristics of an individual who is near the extremes on the continuum of expertise .
It is premature to offer recommendations for policy and practice until we understand more completely "the path that one travels from novice to expert" (Berliner, 1988, p. 1). It is certain that this process of complete understanding about the path will need a long period of time and special efforts. But, a method from qualitative research will be able to contribute to the process by its descriptive nature and its pursuit of holistic understanding.


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Appendix A - Interview Protocol

1. Could you tell me about your educational background (high school & college years, graduate study)?
Where did you grow up? Were you a good student in mathematics in your high school?
2. When did you decide to be a mathematics teacher? (any special moment)
3. What does "teaching" mean to you?
4. What do you think the role of a teacher would be?(content & pedagogy)
What kind of things from your students do you most value? How do you understand students who have real difficulty in even easy mathematics?
What do you think of the conditions of being a good mathematics teacher (mathematical ability, honesty and hard work, skills in managing school work and classroom)?
5. How many classes are you teaching now?
Which class do you like the most? & Why?
6. Could you compare the knowledge that you got at the university and the knowledge that you got from your teaching experiences?
Have you ever exposed to any kind of learning theories (constructivism...)? Was there anything that you found it valuable in shaping your belief about teaching?
7. Do you think that students are active learners and create their own knowledge?
8. Who do you think should provide a motivation to learn mathematics?
9. When did you start using calculators in your teaching?
What is the most benefit of calculators in teaching and learning mathematics? What about computers?
10. Could you tell me about a student who you remember the most?
What was so special about him or her?

Appendix B - Observation Protocol

Focus: How are her belief about mathematics identified in the analysis of the interview manifested in real teaching practices?
1. Does she attempt to involve the entire class in her lesson?
2. Does the class end after she completes everything she planned?
3. Does she cover essential material?
4. Does she ask more questions and less demonstrations while teaching?
5. Are her explanations accurate and clear?
6. Are her explanations summarized and contrasted to one another?
7. Does she experience confusion in instances when she attempts to respond to unanticipated students' questions?
8. Does she balance easily responsiveness with comprehensiveness?
9. Does she draw effective graphs or diagrams to help students visualize the problems to be solved?
10. Does she experience difficulty generating examples and providing explanations for unexpected students' questions?
11. Does she realize the importance for students' understanding of highlighting specific concepts or of communicating the big picture?

Appendix C - Observation Notes

It was 11: 15 am on a Wednesday when I arrived at my participant teacher's classroom in a High school, and she was teaching the Algebra 3/Trig class. Since I was supposed to observe the 4th period, Advanced Algebra 3/Trig starting at 11:40 am, I waited at the outside of the classroom. The classroom door was open, and I was able to hear the teacher's voice. The students in the class were making quite big noises, and the teacher frequently said "shee" to make them quite. I unconsciously started counting the number of "shee" from the teacher for the remaining 15 minutes of the 3rd period. The counting ended up with 5. At 11:30, the 3rd period was over and the students went out from the classroom. I went into the classroom and said hello to the teacher. The students of the 4th period started getting in while I was talking to the teacher. I asked the teacher about today's lesson plan. She told me she was going to review a test that student took yesterday. The square-shaped classroom was filled with 28 students (17 boys, 11 girls) and had no empty seat remaining. Even the teacher's desk was taken by a boy. The class was big. But, the students all were focused and waiting for the teacher to distribute the test back to them. There were many posters on the walls of the classroom including 3 Einstein's pictures, a picture of Native Americans, a world map, a big paper of tessellation, a poster about Fibonacci numbers, an art poster, and a poster about "Nature". Looking at these posters, one might have an idea that this classroom is a Science classroom, not a mathematics classroom. One Macintosh computer was in front of the classroom. Just right above the computer, there was a TV, a VCR, and an American flag attached to the TV. The teacher was standing in front of the classroom and started working on the problems in the text on a overhead projector. The teacher introduced me to the students as a young Ph.D. student in Mathematics Education. A boy asked me if I could take a make-up for him. The lesson began at 11:49 and lasted till 12:39. Every student had a graphic calculator, and so did the teacher. They worked on odd and even functions, inverse functions, ellipse and finding foci, translations of functions, and graphing some high degree functions. The teacher often used drawings on the overhead to explain some important concepts, and it was effective in a sense that the students were responsive to the teacher's explanations and questions. It seemed like that the class had a norm among the teacher and the students; everybody is expected to be responsive to make proceedings. The class continued making a notable sequence of interactions which is "question initiated by the teacher-students' answer-the teacher's explanation-another question. The teacher never went on to a new problem unless the students were responsive to her question. For example, the teacher seldom completed a sentence by herself. Rather, the students completed the sentence, then the teacher started a different problem. The students paid attention to the teacher throughout the whole period. In fact, there were only 3 incidences of saying "shee" from the teacher to make them quite. The bell rang at 12:35, and some of students stood up to leave the classroom. But the teacher didn't stop explaining some problems. She went to a student and worked on a problem together. There were two other students who didn't leave the classroom and they had some time to work with the teacher individually. Finally, everybody was gone. The teacher and I left the classroom together and headed to the mathematics department.

Quotes from Observation
You must plan to study and do on your own.
Every time I am the doer you're the seer. But, that doesn't always happen.
Inverse is a big thing in mathematics (OC: fragmentation of mathematics)
You've got to practice before you get to here.
We studied, then you don't study it... then you don't get it. I am tired of doing that.
Tell me what to write.
You must understand this is what you're aiming for.
You have to have the general idea to do this.
You tell me what I should write.
You don't know if you memorize it.
Be thinking and checking. Then you will really understand it.
Graphing is a good idea.
Grandma's getting tired of this. How many time did we do this?
Isn't this cool? (A student responded, "Cool! x-axis can be a line of symmetry!)
Which is which? That's a good question (OC: encouragement for students)
I don't mind repeating it as long as they pay attention to me and want to learn it.