PRIME 2000 REPORT
A Mentor Teacher involved in PRIME – His Perceptions about
Mathematics and Mathematics Teaching and Learning
TERESA G. BANKER
The Setting
Physical
The high school is located in a
large county in the state of Georgia. The campus is situated on several acres
of land and surrounded by affluent neighborhoods composed of diverse people
groups (e.g., Asian, Indian, African-American, and Caucasian). The school
building, itself, has been constructed in phases and connected to form a huge
complex. There is a large, multi-purpose commons area in the center of the
building which is used for serving lunch during the middle of the day and a
congregation area for students in the mornings before school begins. In the
afternoons after lunch, different classes that require a large, open space
periodically can be seen utilizing this area. Off this area in all directions
begins the maze of halls that run perpendicular to one another; there are two
levels in the building. There is also a large number of trailers which sit on
the back perimeter of the building. The amount of required space is
understandable when one considers the enrollment of over 3,300 students. For
the school year 2000 – 2001, the administration is working on a plan to provide
classes for the ninth and tenth grade students in one section of the building
while the eleventh and twelfth grade students will be taught in another section
of the building. This plan is designed to alleviate length of travel and
travel-time for the
student.
Research
The purpose of the PRIME project is to learn more about the nature
of the student teaching experience. PRIME researchers want to understand the
relationship between theory and practice; purportedly, there is a gap between
these two facets of the experience which motivates the research. This part of
the research, focusing on mentor teachers’ perceptions about mathematical
conversations with their student teachers, is guided by the following research
question:
1) What is the nature of
mathematical discussions between mentor teachers and student teachers?
a)
To what extent are the discussions about:
learning mathematics, mathematics,
teaching mathematics?
b)
How important are the conversations from the mentor’s perspective?
c)
What motivates or facilitates the conversation?
The following sections of the report
will describe the mentor teacher with respect to the research question. The
pseudonym chosen for the teacher is Tom.
The Case of Tom
Introduction
Tom is a veteran mathematics teacher with thirteen years of
classroom experience; he formerly coached girls’ basketball, girls’ and boys’
soccer, and boys’ cross country. Presently he teaches driver’s education after
school and on Saturdays, about 25 hours per week. He said the hours are much more
flexible and the money is better. He has earned a Master’s degree in
mathematics education and wants to complete the Specialist’s degree but has had
difficulties because of the lack of university math courses scheduled in the
evenings. He currently teaches one tech math II and four geometry classes. His
planning is fourth period, extra long because of the lunch schedule, during
which he is also assigned a lunch duty of 15 minutes.
Tom is an enthusiastic supporter of
PRIME and the procedures and schedules used with PRIME student teachers: “This
whole year, the whole process they’ve [student teachers] gone through has been
great for her [Tom’s student teacher]…I mean, her training is there and she’s
ready to go right to work now.” (interview) Tom has actively participated in
the meetings and activities involving all mentor teachers organized by PRIME as
well as those meetings that were specific to his school only. I have been in
attendance at most of these activities
and observed his participation.
I supervised four student teachers
at Tom’s school, including Tom’s student teacher. Through this supervisory
role, I became acquainted with Tom. My relationship with Tom is open and
mutually respectful. One of the mentor teachers from Tom’s school is a participant
as well as a researcher; therefore, I did not choose her as one of my
participants but chose two of the other teachers, including Tom, with whom I
had more access and a closer relationship. The fourth teacher was interviewed
by the participant/researcher because she had better access to this mentor
teacher and the data from her written survey.
Perspective
on Mathematical Conversations
The research question guiding this
part of the PRIME project focused on the mathematical conversations between
mentor teachers and their student teachers. As researchers we wanted to learn
what mentors classify as “mathematical conversations,” what the mentors hope to
accomplish with these conversations, and what motivates such conversations.
As I have talked with Tom, his views
about mathematics and the needs of his student teacher are based on the level
of mathematics required for teaching his classes. At no time has he discussed
mathematical content with his student teacher. He commented that he would only
talk about content if he saw her make a mistake, which, according to him, never
happened. He stated his conclusions this way:
|
…based
on what she was teaching, she, probably before she ever entered college, had
all the mathematical background she needed for the mathematics she was
actually teaching…The mathematics, I think, is covered quite sufficiently in
the college curriculum, as far as what you have to teach at the high school
level. (interview) |
To gain knowledge about mentor
teachers’ perceptions about the learning and teaching of mathematics,
researchers in the PRIME seminar (Spring, 2000) decided to ask the mentor
teachers to complete a written survey and then tape and transcribe follow-up
interviews of those mentor teachers chosen as participants in the study. The
written survey was designed to elicit information from the mentors about their
views concerning the learning and teaching of mathematics using similes: Learning mathematics is like: with a
list of possible choices and also a choice of Other. The same format was used
for A mathematics teacher is like a:
with more choices and the alternate Other. The remainder of the survey asked
information related to the observations by the mentor teachers of their student
teachers. This information specifically focused on mathematical conversations,
the teaching of mathematics, the knowledge of mathematics, knowledge domains
for training on the university campus and the high school campus, and
improvements of the student teacher which would help him or her facilitate
mathematical learning in the classroom.
Tom chose the following similes to
describe mathematics learning: learning
mathematics is like conducting an experiment; learning mathematics is not like
watching a movie. From these two statements and Tom’s explanations, I was
able to understand that Tom feels learning of mathematics requires the student
to take an active part in the process and that participation should never be
passive if learning is to be facilitated.
|
Conducting
an experiment – the student must be an active part of the process and
should feel comfortable to experiment in problem-solving. Watching
a movie – because the learner is not taking an active part of the
process. (survey) |
I
came to understand that Tom wants his students to take an active role where
they guide and direct what they are
learning while his role is to keep them on track with his goals for the class.
|
I
like them to make discoveries for themselves and basically apply what they
are learning to actual situations later on. I just facilitate the process and
keep them going in the right direction. (interview) |
To illustrate his views about
mathematics teaching, Tom said that a mathematics teacher is like a coach and not like a news broadcaster. He said the
role of a coach is “demonstrating, giving practice opportunities, supporting,
and encouraging,” while a news broadcaster “only provides information.”
(survey) After further probing in the interview, I found that Tom relies
heavily on the feedback he receives from his students to shape the direction
and speed with which he conducts his class.
|
…with
a news broadcaster you just sit there and watch them. It’s talking about a
mathematics teacher. All a newscaster does is talk and there’s no
interaction; you just sit there and watch and not interact. Teaching
mathematics should no be like that. Also, I don’t think it would be very
effective for the learner if the teacher is just delivering information and
not getting feedback from the students. The teacher needs to provide feedback
to the learner from the interactions with the students. I don’t think math
should be taught that way. Frequently I revise my progress for the day or how
far I get based on how the students are responding to me and how they’re
comprehending what we are doing. (interview) |
As
a mathematics teacher, Tom sees his role as a facilitator and guide who
directly benefits from the interactions with his students. The feedback from
these interactions allows him to gauge the progress of the class, not only for
the day’s lesson, but for a complete unit.
Disposition
Toward Mentoring
Tom specifically said that the
mathematical background of his student teacher is excellent. He thought the
college curriculum was very sufficient for the mathematical preparation of
student teachers, based on the mathematics required to teach in high school.
So, he thought his responsibilities as a mentor teacher were to share his
experience with the teaching and learning of mathematics. His main concerns
were with the pedagogy of teaching mathematics and how the students are going
to understand a concept and why they are not going to understand that
particular concept. Tom’s instruction to his student teacher was predominately
information of this nature.
|
It
was in a section she was heading into where they start off finding the sum of
all the interior angles of a regular polygon. That students can’t distinguish
when the directions say “find the measure of an angle vs. find the sum of all
the angles.” I was just explaining what to look for, you know, what was
probably going to happen when she got to this unit. I was pointing out to
her… that she might want to really stress that point. (interview) |
As the mentor teacher, Tom saw his
role as a guide for the student teacher. He provided guidance in the
application of school policy and in the extra duties expected of a teacher, not
to mention the intricacies of running a classroom and all that process entails.
The style Tom used with his student teacher could be labeled democratic. When
an issue arose, (e.g., how to set the point values on a test), he completely
withdrew from the decision-making process, forcing his student teacher to make
the decision for herself. He was dedicated to using his experience to help her
with teaching the mathematics, understanding student thinking and
misconceptions, and managing the behaviors exhibited by high school students. I
think he really enjoyed the experience of mentoring.
Mathematical
Discussions
At a meeting in Athens where all the
mentor teachers, student teachers, and college supervisors were gathered, the
topic of exploration was mathematical conversations. The college supervisors
circulated among the tables of student teachers and mentor teachers to listen
to discussions about mathematical conversations. In several cases the students
and teachers were discussing problems which involved some, really neat
mathematics and how one problem situation could be extended to draw upon a
large range of mathematical ideas and concepts. These discussions formed the basis
for the focus of the next part of the survey.
As I talked with Tom about
mathematical conversations, I understood that his definition was broader than
conversations involving pure mathematics, as we, the researchers, had posed in
Athens earlier in the semester. He thought it was unnecessary to have pure
mathematics conversations with his student teacher because of her excellent
background. Therefore, he classified conversations about the teaching and
learning of mathematics as mathematical conversations.
|
The
teaching of mathematics. I guess basically that was one of the things I
worked with her a lot throughout the process was that you learn a lot the
first year you teach a course. Just the experience and seeing where the
students will have difficulties. Of course, each year students are a little
different; the difficult concepts are different each year for students
because the students are different. But, in general, I just went over a lot
of things and expressed to her they might have difficulty with this idea or
concept. (interview) |
After I reviewed conversations with
Tom, the interview transcript, the written survey, and conversations with his
student teacher, it became clear to me that Tom’s focus and the focus of his
student teacher regarding mathematical conversations was preparation of lessons
and assessment instruments. Within this preparation, they considered both the
teaching and learning of mathematics of extreme importance.
|
We
discussed the difficulty students faced when finding each interior or
exterior angle of a regular polygon vs. finding the sum of all the angles and
why the students could not distinguish between similar concepts. (survey) |
Tom
used his experience and knowledge about the content of school mathematics and
his students’ strengths and weaknesses to guide his student teacher. He thought
it was important to share his experiences with her so she could better prepare
the lessons she would teach.
|
It
was maybe more a concept like this particular concept where the students will
have difficulty. Just explaining and elaborating on why they will have
difficulty with that concept, what she should be on the lookout for when she
is teaching. (interview) |
This
sharing of ideas was a daily activity which was motivated sometimes by Tom and
other times by his student teacher. These activities generally took place after
a class was completed before another class began, during planning period, or in
the morning or afternoon before or after classes, always in an informal
fashion.
Other
Domains for Development
Knowing mathematics content and the
pedagogical methods for teaching mathematics is only part of the story about
the teaching and learning of mathematics. Other factors can and do contribute
to the overall success of a mathematics teacher. Tom delineated two factors
which are of strong concern to him: having realistic expectations about student
performance and management issues.
Tom stated that the experience
during the first semester was invaluable to his student teacher. Because she
was able to observe classes during the fall, her preconceived ideas about
students and their abilities was greatly transformed.
|
I
know first semester when they [student teachers] came out to observe, she
shared with me how shocked she was as far as to what level the students
really were. In other words, they were not as far as she anticipated and she
really had to adjust her view of what the students in this class [geometry]
would be able to understand, what they would bring with them from the algebra
class, or how little they would actually bring with them from the algebra
class. So, maybe when she first came, her idea was that she could go right
into teaching some concept, what the section was focusing on without having
to go over the algebra behind it. And now I think she’s at the point that she
really realizes how important the review is and how, I mean, she can't just
simply hold them responsible for what we’re learning now. She’s got to look
back at work and catch them up to what we’re learning now. (interview) |
Tom
has alluded to the well-known mathematics education adage: “Start with them
where they are and take them where you want them to be.” Tom thought that his
student teacher had developed a good sense for where his students were and she
had learned that they were not going to come into her class prepared to go
where she wants to go without reviewing previously learned materials.
Another major factor about which Tom
was concerned is management issues. Regardless of the content, whether it is
mathematics, social studies, or languages, managing student behaviors has been
a top priority in classrooms everywhere. Tom
emphasized
the importance of controlling student behaviors to create a good learning
environment which accomplishes the goals of both teachers and students. He sees
it as a matter of assertiveness.
|
…when
she first started, I was concerned about her assertiveness, how she was going
to handle my classroom. In general, my students are very laid back and
easy-going, so it’s not likely to be a problem. But there are times when
something happens, like, for example, two weeks ago after lunch they all came
in very excited, pumped up. I was worried about how assertive she would be as
far as getting them calmed down and ready to go to work. She’s gotten a lot
better at that over the 10 weeks she’s been here. ..Some people probably
consider me to be too assertive sometimes when I handle discipline. So, I
think sometimes it’s almost a judgment call. It’s what your personal
preference is. (interview) |
Tom
admitted that management styles depend on personal preferences and student
teachers need to develop a style that works for them and does not jeopardize
the learning environment.
Conclusion
This is a case about a teacher who
served as a mentor for a student teacher. The case shows that this teacher
valued the mathematical conversations with the student teacher because those
conversations dealt directly with the teaching and learning of mathematics. The
teacher assumed the responsibility of guiding the student teacher by sharing
his experiences about teaching mathematics and teaching students. Both of these
factors are important in the training of student teachers. Also, this teacher
thought it important to share how students understand concepts and why they
often have misconceptions about topics in mathematics. Through this teacher’s
perceptions, the researcher understood that mathematical conversations have a
broad definition that encompasses pedagogical content knowledge and knowledge
of student understanding. The definition of mathematical conversations is not
limited to pure mathematics.
Appendix A
Copy of the written survey
completed by the teacher
Appendix B
List of specific data sources
1.
Written survey
2.
Transcribed interview
3.
Researcher’s journal