**PRIME 2000 REPORT**

A Mentor Teacher involved in PRIME – Her 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 Mary.

The Case of Mary

__Introduction__

Mary is a veteran mathematics teacher with ten years of classroom
experience. She also serves as Junior Class Sponsor in the high school where
she teaches. She has earned a Bachelors degree and plans to begin work on her
Masters degree in the summer of 2000. When I first visited her room, I was
struck by the number of inspirational and motivational posters she has placed
around the room. She also has a corner with a collage of family pictures which
sends the message, “I’m a real person!” She currently teaches two tech math I
and two algebra I classes. However, one section of algebra I is a block class
which meets for two hours, back to back. This class was created to help a group
of ninth graders, who failed first semester, stay on track with other
classmates because of their math potential. Her planning is sixth period.

Mary uses an overhead projector to
present examples and demonstrate the mathematical concepts of the day. The
projector allows her to face her students so that she can gauge the students’
understanding and promote interaction with them to help her evaluate their
progress. She generally allows 10 to 15 minutes at the end of class for the
students to work individually while she walks around the room checking on
individual progress and understanding.

Mary is an enthusiastic supporter of
PRIME and the procedures and schedules used with PRIME student teachers.

The
students came and observed the three of us, David, Kathy, and me. Once they
chose which of us would be their mentor, they came back and observed us more.
I thought that was really good that they observed us more. They even came and
taught one of our classes for a few days—my student teacher taught 5 |

Mary told me about her experiences during
student teaching to compare and contrast with the PRIME student teachers. She
wholeheartedly supports the program because of her own experiences.

When
I think back to my student teaching, it was awful. My supervisor was an elementary person. She told us from day 1
that she did not know math, it wouldn’t matter what I wrote up there, she
wouldn’t understand it. I got no help at all…I walked into the school the
first day of student teaching. (interview) |

Mary
has actively participated in the meetings and activities involving mentor
teachers, both at the university and her high school; she has, however, had
conflicts with some meetings.

I supervised four student teachers
at Mary’s school, including Mary’s student teacher. Through this supervisory
role, I became acquainted with Mary. My relationship with Mary is open and
mutually respectful; I feel a stronger bond with her than with the other mentor
teachers in her school. One of the mentor teachers from Mary’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 Mary, 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.

Talking with Mary and reviewing the
data from the survey and interview have given me a perspective on Mary’s views
about mathematics and how she perceives the needs of her student teacher. These
perceptions are based on the level of mathematics required for teaching her
classes.

To
me, in her case [the student teacher], her math background is very good. She
doesn’t need me to explain how to do that part…We were on basic topics that
she knows how to do well. (interview) |

Mary
never discussed mathematical content with her student teacher because the
necessity did not present itself during the time the student teacher worked in
Mary’s classes.

In
these courses her knowledge and everything about the math is there.
(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.

To describe mathematics learning,
Mary chose the similes: *learning
mathematics is like working a jigsaw puzzle; learning mathematics is not like
watching a movie.* From these statements and Mary’s explanations I came to
understand Mary thinks that learning mathematics is an active process for
students and learning requires exploration to gain the knowledge which is the
end product of this process.

I
choose working a jigsaw puzzle—in math you are always working towards an end
product. You have to keep working and searching until you get it right. I
choose watching a movie. In no way is mathematics passive – like watching a
movie. (survey) |

To
facilitate the students’ learning, Mary stated that individual teachers will approach
teaching in different ways, but her goal is to promote interactions with the
students which facilitate their learning.

Watching
a movie is passive, so watching a movie, to me, absolutely relates nothing to
what I do. I mean there are times the kids are watching me, but, hopefully,
it is interactive, watching me work an example, learning, and then working
along with me. (interview) |

To describe the mathematics teacher,
Mary stated that a mathematics teacher is like a news broadcaster and an
entertainer but not a missionary. She said there are certain concepts in
mathematics which must be presented as a broadcaster presents the news, but she
added the entertainer in her makes the presentation “innovative and
interesting.” (survey) When I probed Mary about the meaning she ascribed to the
missionary image, she commented that this type of teaching is totally
different, but it is still a form of teaching.

__Disposition
Toward Mentoring__

Mary specifically described the
mathematical background of her student teacher as very knowledgeable. (survey)
She thought the college curriculum was sufficient for the mathematical
preparation of student teachers, based on the mathematics required to teach in
high school.

…again,
and this is me, this is specific to what I teach. Going back to the same
thing, if they’ve [student teachers] gotten this far, the mathematics part of
ninth grade math is a given, pretty much, for somebody that’s getting ready
to graduate from college. To me the math teaching and teaching in general WAY
outweighs the math. (interview) |

As
a mentor teacher, Mary saw her role and responsibilities as one of guiding her
student teacher with respect to the teaching and learning of mathematics. She
felt strongly about sharing her experience with the student teacher, sharing
“some of my tricks that help them [Mary’s students] learn better.” Her main
concerns were with the pedagogy of teaching mathematics and how the students
are going to understand a concept and why that concept might not be easy for
them. (Mary’s classes are ninth grade level and she feels there are specific
and different approaches for this level student.) Mary’s instruction to her
student teacher predominately took this form. She related a sample conversation
in which she and the student teacher discussed simplifying radicals.

We
were talking about ninth graders learning to simplify radicals for the first
time, without a calculator. I explained to her, I feel like the way
that I teach that is very elementary because we make factor trees. I was
explaining to her my approach to take the guess work out of simplifying
radicals. She may assume that it’s
the square root of 50 and they will see a 25 in there. The kids, seeing it
for the first time, don’t. I
explained to her why I make factor trees, why I write all the prime factors
under the radical. And I go through and I circle pairs. That has had to
happen a lot where I have to explain to her “Yes, the square root of 9 is
easy, we know that , but they’ve never done that before.” You have to keep in
perspective where they are coming from. (interview) |

The
style Mary used with her student teacher was different from the other mentor
teachers in this high school. She decided to use a spiral notebook to write
comments about the teaching of her student teacher. Sometimes these comments
were in the form of statements and sometimes questions. The student teacher
would then write a response to Mary’s comments. If Mary was unclear about the
student teacher’s response, then they would discuss the particular item
further. Her student teacher found this method to be very helpful. Although
there were some rough spots that Mary and I had to work through with her
student teacher, I think she thoroughly 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. (This meeting was one of
those times when Mary had a conflict.)

Talking with Mary led me to
understand that her definition of mathematical conversations was broader than
the definition the PRIME researchers first stipulated, i.e., involving pure
mathematics. Mary thought that her observations dictated guiding her student
teacher about student understanding and pedagogical matters rather than the
mathematics the student teacher needed for teaching Mary’s classes.

…That
had to happen a lot…I think as far as a college student that’s one of the
hardest things. To understand why
they [Mary’s students] don’t just |

Mary
said that “most, if not all, mathematical conversations” were on these terms.
Her goal was to help her student teacher grasp the level of understanding
Mary’s students could achieve and what approaches, “tricks,” and methods would better facilitate the
students’ learning so her students would “be where I want them to be.”

The mathematical conversations
shared between Mary and her student teacher focused on the lessons,
preparation, and student understanding. This focus was characterized by aspects
relating to the teaching and learning of mathematics.

We
would talk about the lessons, the prepping, materials, activities to use, and
how long to spend on certain concepts. I would have to help her with what my
students needed to know and share with her those things I do to help them
learn and understand. (interview) |

Mary
shared her experience to assure good preparation for her students because Mary
always puts her students first. As a mentor teacher, however, she wanted to share
her experience with her student teacher to benefit the preparation gained from
the field experience. This sharing of ideas and approaches occurred about three
or four times a week in a very informal fashion, during planning periods or
other breaks in the schedule. Mary was pleased with the attitude of her student
teacher, “She took my direction well.”

__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.

Mary
outlined two factors which she thinks are very important: enthusiasm for
teaching and knowing the students.

At the beginning of the experience,
Mary’s student teacher was preoccupied with a personal problem. The student
teacher allowed this problem to color and flavor her actions in the classroom
and her relationship with others. This situation was very disturbing to Mary
and myself. However, with some counseling and direction, the student teacher
changed her attitude and met our expectations.

[Student
teacher] has really been doing great since you talked with her. We never really discussed the
meeting--except she sent me an email from home that night apologizing. She has had some great lessons and has
been very enthusiastic. I really
appreciate the change. I feel she
will be a great teacher. (personal communication, 31 January 2000) |

After
talking with Mary, I came to understand that she feels a teacher needs a
positive attitude which exudes enthusiasm for teaching. She said it this way:

…that
get up and go that says, “Let me go
to school and do the VERY BEST I can.” (interview) |

To Mary, knowing the students is of
extreme importance. This *knowing *not
only relates to how students understand mathematics and the misconceptions they
may have, but also how their personal lives can be an important way to motivate
them. Mary was extremely pleased with her student teacher first semester
because she took such an interest in her “future classes.”

…Once
the decision [the choice of mentor teacher] was made, when we got each of
ours, I thought it was really good that they observed us more, they got to
work with our students, they got to teach, they got to know…like [student
teacher] really got to know my 5 |

Mary
wants to know as much about her students as she can because this knowledge
enables her to help them learn mathematics. Mathematics education research has
shown that context plays an extremely important role in effective mathematics
learning. Mary is yet another example of a teacher who has come to this
understanding through her experience in the mathematics classroom.

Conclusion

This is a case about a teacher who
served as a mentor for a student teacher.

The
case shows that the mathematical conversations shared between the teacher and
the student teacher were valued because the mentor saw her role with the
student teacher as that of a guide. The guidance she provided from her
experiences of teaching mathematics and teaching students predominately
involved the teaching and learning of mathematics. Additionally, she thought it
was important to share her knowledge of student understanding and misconceptions.
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.