and

its Manifestation in Teaching Practices

*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)

INTRODUCTION

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.

CONCEPTUAL FRAMEWORK

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.

METHODS

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).

Participant

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.

Limitations

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.

FINDINGS

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.

DISCUSSIONS

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.

CONCLUSIONS

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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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?

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?

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.