PRIME 2000 REPORT
A Mentor Teacher involved in PRIME – Her Perceptions about Mathematics and Mathematics Teaching and Learning
TERESA G. BANKER
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
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
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 5th period for 5 days, I believe…Coming into the school beforehand is excellent. A must! (interview)
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.
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 get everything really fast. They [student teachers] don’t have experience with student understanding. To them [Mary’s students] this is a hard course and I think somebody coming straight out of college, with all she’s been through, [thinks] why can’t they get this…that transition was pretty difficult. (interview)
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 5th period class. The 5th period class was obviously very receptive of her and looked forward to her coming back. They see it as a new face; they thought that was great. She got to know them and figured out all different things about them. I thought that was excellent. Excellent!! (interview)
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.
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.