Field Experiences as Opportunities for Mathematical Conversations
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Patricia S. Wilson University of Georgia pwilson@coe.uga.edu |
Christopher Drumm University of Georgia cdrumm@coe.uga.edu |
Field experiences
are a substantial part of secondary teacher preparation in mathematics, but it
is not clear what mathematical knowledge is gained from the experience. Some
argue that generic managerial functions are emphasized more than the learning
and teaching of specific content. This study investigated the nature of
mathematical discussions between mathematics teachers and student teachers in
six high schools and one middle school. Mathematics was discussed in the
context of the student teacher’s lesson, and the conversations focused on
pedagogical content knowledge. Mentor teachers offered advice on how they would
teach a specific topic or on student difficulties with the topic. Discussion of
content knowledge was motivated by student teachers’ questions or mathematical
difficulties. We found little evidence of collaborative mathematical
exploration between the mentor and student teacher.
Preservice teachers often claim that student teaching was one of the most valuable parts of their education. The opportunity to be in the context of a high school and to teach students seems to heighten their awareness and perhaps their urgency to learn. Field experiences provide an opportunity for both mentor teachers and student teachers to have significant mathematical conversations, but mathematics is only one area that receives attention in the field. We studied mathematical conversations, what motivates them, and related implications.
Purpose of Study
Despite the numerous opportunities for conversations that field experiences offer, researchers (Sudzina, Giebelhaus, & Coolican, 1997; Wideen, Mayer-Smith, & Moon, 1998) remind us that taking advantage of those opportunities is difficult. There are different goals between classroom teachers, student teachers and university supervisors. There are perceived gaps between theoretical constructs developed in teacher education programs and practical constructs valued by mentor teachers. In a recent study of four teachers mentoring preservice mathematics teachers, only one teacher was identified as spending a significant amount of time discussing mathematics or mathematical issues (Wilson, Anderson, Leatham, Lovin, & Sanchez, 1999). Mentors often spend time discussing managerial issues, school policies, and student behavior. This study was designed to gain insight into the following questions.
· What is the nature of mathematical conversations that occur in student teaching situations?
· What motivates these conversations?
· What can be concluded from these conversations?
Theoretical Framework
Our work with teachers has been
influenced by Charon (1998) who advocated symbolic interactionism and the power
of considering a person’s viewpoint in a given situation or event. Mentor
teachers, student teachers, and university supervisors have different roles
during field experiences, and they also bring different perspectives to teacher
development. There is a tension in preparing teachers between training teachers
to become good managers of the learning environment and educating teachers to
understand learning and learners (Goodman, 1984). Shulman (1986) argued that it
is not only important for teachers to gain general pedagogical knowledge but
also content knowledge and pedagogical content knowledge. Teachers need to know
mathematics, but they also need specialized
knowledge about how students learn mathematics and effective ways to teach
mathematics. We believe that mathematical
conversations and analysis of those conversations are important parts of
reflection on mathematics teaching, because the conversations add to the
knowledge that teachers use to connect mathematical ideas and make teaching
decisions.
Methodology and Data Sources
The Partnerships in Reform In Mathematics
Education (PRIME) project involves a collaborative effort between university
staff and mentor teachers who are working with preservice mathematics teachers
in six high schools and one middle school. In the
first year, the research focused on representing and understanding mentors'
views on the process of mentoring.
We found little evidence of mathematical conversations during the first year, but that was not the purpose of our data collection at that time. One teacher explained that student teachers learn their mathematics at the university, and they need to learn other lessons in the field prompting us to investigate mathematics during the second year. Research suggests that it is more common for mentor teachers to discuss management with student teachers than other issues (Goodman, 1984; Zeichner, 1985), but we wanted to know more about what, when and why mathematics was discussed.
During the second year, 32 mentors and 24 student teachers participated in the study. Mentors and student teachers completed an initial survey about student teaching and possible topics of discussion between mentors and students. We held a working session with mentor teachers and student teachers to discuss mathematical conversations and to engage in mathematical conversations in small groups. Twenty-two teachers completed a questionnaire about mathematical conversations and their views about the mathematical preparation of their student teachers. Similar data were collected from their student teachers. Fourteen teachers participated in follow-up interviews. Our data also included evaluation forms from student teachers and mentors, notes from mentor workshops, and field notes from supervisors. This paper focuses on the 14 mentors who participated in an interview and their student teachers.
As we analyzed our data, a working definition of mathematical conversations evolved. A conversation that was likely only to occur in a mathematics classroom was classified as mathematical. We recognize that our attention to these conversations does not mean that these conversations are necessarily typical.
We worked with self-report data and interviews in order to represent the perspectives of mentors and student teachers during the student teaching field experience. Classroom observations would add a viewpoint on mathematical conversations that is not represented in this paper.
Mentors’ Viewpoints
The initial
survey contained 53 topics, chosen from research studies, that mentors and
student teachers discuss. Mentors were asked to rate each on how
frequently they would discuss that topic and then select the five most
important and five least important topics. The three most popular were (1) classroom management, (2) strengths/weaknesses of the day’s
lesson, and (3) alternative ways of presenting mathematical content. Other
choices included student learning and understanding. The student teachers selected
(1) strengths/weaknesses of the day’s lesson and (2)
student understanding. Third place was shared by “alternative ways of
presenting mathematical content”, “classroom management”, “students’ learning
processes”, and “student motivation.” The agreement on classroom management,
student learning, teaching strategies, and performance critiques illustrated common expectations
for field work. This standard view of
field experiences did not focus on mathematics even though it may have been
part of the discussion. It is interesting that both teachers and mentors
selected the “nature of mathematics (i.e. what is mathematics)” as a least
important topic. Although it is a possibility that
the participants misunderstood the phrase, we have not identified epistemological
discussions between student teachers and mentors.
Views
of Mathematics Learning and Mathematics
Teaching
Most mentors viewed learning mathematics as an activity that required thinking and involvement of the participant. Half of the mentors chose a jigsaw puzzle explaining that learning mathematics is about putting the pieces together. Other characteristics included persistence, revelation, and the lack of memorization. Four mentors chose similes that emphasized beginning with a foundation and building complexity. One person noted that you must be active to do mathematics. This was echoed in the mentors’ choice of what was not like mathematics. Half of the mentors said mathematics is not like watching a movie because learning mathematics is not passive.
The mentors agreed that mathematics teaching is not telling mathematics but is nurturing and supporting students to achieve goals, similar to the roles of a coach or facilitator. Two of the mentors thought that teaching was like gardening. To them, the teacher provides nourishment, and the students’ mathematical abilities grow. Eight of the mentors indicated that teaching mathematics is not like news broadcasting because they are not reporting facts and must interact with their audience. Others felt that being an entertainer was a poor simile because although they hoped their students were enjoying their lessons, the goal of entertaining was secondary to helping them learn. Views on Mathematical Preparation of Student Teachers.
There are many topics vying for the attention of both the teacher and the student teacher. We know from our survey that mentors are interested in classroom management issues and also value a good knowledge of mathematics. Six of the mentors specifically stated that you need to know more mathematics than you are teaching. Two of those mentors explained that you can always learn more mathematics and implied that you should continue to learn mathematics throughout your career. One of the six mentors admitted that she did not often discuss mathematics with her student teacher because the student seemed proficient. In contrast, two mentors explained that student teachers really did not need to know much more than high school mathematics to teach what they taught. In general the mentors were satisfied with the mathematical preparation of the student teachers.
A good knowledge of mathematics is necessary for making connections, but it is probably not sufficient. Nine of the fourteen mentors discussed a need for connecting mathematics. Although they did not refer to the structure of mathematics, they noted that student teachers did not have a good knowledge of the “mathematics curriculum”. For example, they wanted student teachers to understand what ideas were prerequisite to what they were teaching. They also wanted their mentees to understand that the concepts they were currently teaching should build toward future topics. Some mentors also mentioned connecting mathematics to applications.
Mathematical Conversations
Half of the mentors reported that the described conversation originated with the student teacher. In 13 of the 14 conversations, the mentor appeared to be providing advice or information for the student teacher. We found that teaching episodes were excellent stimuli for the discussions. In general, mentors were preparing students to teach a lesson in the future (i.e. later that day, the next day, the next year). In some cases, mentors corrected a student teacher or supplied missing knowledge. Often mentors explained how they had taught the lesson in the past.
Mentors
reported having mathematical conversations at least once a week with 12 out of
14 reporting conversations at least 3 times a week. The student teachers
reported similar frequencies. Mentors were asked to describe a recent
mathematical conversation. Six conversations were classified as pedagogical
content knowledge (PCK), four as content knowledge (CK) and three were both. One
generic conversation about pupils’ tendencies toward misconceptions was
classified as pedagogical knowledge (PK).
Six of the conversations referred to mathematical skills (i.e. determining angles, simplifying radicals, defining absolute value, determining degrees of polynomials, factoring special products, graphing quadratic equations), and three involved mathematical concepts (i.e. exponential growth, logarithms, x0). Four mentors did not specify a topic but referred to modeling a lesson, misconceptions, critiquing a worksheet and discussing two-column proofs. Only one conversation appeared to be a mutual exploration of a mathematical topic (i.e. random numbers).
Four of the seven CK conversations addressed a deficient knowledge of the student teacher in order to prepare the student teacher to teach the topic at some point in the future. Two CK conversations discussed the value and placement of a particular concept (i.e. exponential growth, logarithms) in the curriculum. In the CK discussion about random numbers, the student teacher shared a calculator algorithm.
The PCK discussions addressed deficient knowledge on the part of the student teacher. Mentors were offering their advice to the student teachers in most cases. Five of the nine PCK conversations were about how to teach a particular topic based on the mentor’s experience. Three of the PCK conversations implied ways to teach based on the mentor’s knowledge about student learning. The remaining conversation focused on preparing a comprehensive worksheet.
This study suggests that mentors did discuss mathematics with their student teachers, but the conversations were focused on the teaching of mathematics and often on preparing for the next lesson. The typical conversations were centered on mentors providing advice on teaching the content. Advice was based on the mentors' experiences and motivated by the teaching of their mentees. A notable exception to our generalization was a discussion, dominated by the student teacher, on using a calculator to generate random numbers. Technology represents an area where student teachers felt competent and could draw from their experiences. Mathematical conversations may fail to meet their potential if they are used only as a tool for fixing urgent problems in a particular lesson.
We found evidence that mentors rely heavily on their experiences to guide their advice to preservice teachers. It is difficult to know to what extent a mentor's recall of experiences incorporates theories related to learning or teaching mathematics. Advice based on experience of what works is eagerly embraced by student teachers and may not be integrated with theory. This practice possibly perpetuates the status quo, but it also may lead to inappropriate application of the advice. In contrast, when teachers share the reasons that inform their practice, student teachers can move beyond imitation and begin to build their own theories needed to drive their own practice. Although field experiences provide an excellent context for such discussions, other responsibilities seemed to have a higher priority. Mentors’ attitudes toward theory may be influential. If the perceived gap between theory and practice can be minimized, the experiences student teachers gain in the field could complement and enhance their experiences at the university, creating a better understanding of learning and teaching mathematics.
References
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Goodman, J. (1984). Reflection and teacher education: A case study and theoretical analysis. Interchange, 15 (3), 9-26.
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Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Education Researcher, 15 (2), 4-14.
Sudzina, M., Giebelhaus, C., & Coolican, M. (1997).
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Wilson, P. S., Anderson, D., Leatham, K., Lovin, L. & Sanchez, W. (1999). Giving voice to mentor teachers. Proceedings of the Twenty First Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2, (pp. 811-817). Cuernavaca, Mexico: Universidad Autonoma del Estado de Morelos.
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