PRIME 2000 REPORT
A Mentor Teacher involved in PRIME – His 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 Tom.
The Case of Tom
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.
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.
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.
Copy of the written survey
completed by the teacher
List of specific data sources
1. Written survey
2. Transcribed interview
3. Researcher’s journal