LEARNING MATHEMATICS WHILE LEARNING TO TEACH: MATHEMATICAL INSIGHTS PROSPECTIVE TEACHERS EXPERIENCE WHEN WORKING WITH STUDENTS

Sandra Crespo

Michigan State University

crespo@msu.edu

Abstract: The question of how preservice teachers might learn more mathematics while they learn to teach is explored in this paper. It is proposed that in the context of field-related experiences during their mathematics methods courses, preservice teachers face and learn from the puzzling mathematical situations that arise in their work with students. The contexts in which these situation arise—(a) selecting and designing mathematical tasks; (b) analyzing students' work and (c) offering mathematical explanations to their students—are described and illustrated. Two contrasting orientations—inquirying and intimidated—towards learning mathematics while teaching are also discussed.

Introduction

The typical structure of teacher education programs seems to assume that learning mathematics occurs prior to learning mathematics pedagogy. In most teacher preparation programs, mathematics courses are typically taught independently of pedagogical concerns. Mathematics education courses, in turn, tend to place mathematical inquiry in the background while focusing on the theories and methods of teaching and learning. Research, however, suggests that neither of these structural arrangements have been very successful in helping preservice teachers construct interconnected knowledge of mathematics and mathematical pedagogy (see Brown & Borko, 1992; Lappan & Theule-Lubienski, 1994). That is, even though preservice teachers may extend their mathematical understandings, they do not necessarily translate this understanding into pedagogical practice. The construction of mathematical understanding, on the other hand, is very difficult to promote in courses which focus on instructional methods and practices (Simon, 1994).

Such results or lack thereof, however, are not surprising when considered from the perspective of learning in authentic contexts of practice (see Brown, Collins, & Duguid, 1989; Lampert, 1985; Schon, 1983). Current teacher education practices, however, have become responsive to the idea of learning through engagement in authentic teaching activity. Course-related field experiences have, for instance, become more popular and often take the form of classroom observations, interviews with students, and even teaching episodes. These field-related assignments, however, are often thought of as opportunities to develop prospective teachers' pedagogical content knowledge. A reasonable question to ask, in light of the issues discussed, is how can such experiences be also construed as opportunities to study and investigate subject matter?

Data Sources and Analysis

To explore the question of how field-related experiences could become occasions for prospective teachers to investigate mathematics I draw upon data of my own teaching of elementary mathematics methods. For the past five years I have incorporated different kinds of field-related experiences into the mathematics methods courses I teach. In this paper I focus on three types of interactive field-related experiences I have offered at one time or another: (a) a mathematics letter writing exchange, (b) a mathematical interview, and (c) a teaching session. For each of these interactive activities preservice teachers are asked to identify and write about an interesting, surprising, puzzling event. These written reflections are used to describe the nature of preservice teachers' developing mathematical understanding and dispositions towards mathematical inquiry. Furthermore, the data of preservice teachers with contrasting experiences are used to describe and analyze typical orientations towards the learning of mathematics while teaching. In addition, I draw upon data from a “teaching experiment” in my most recent methods course to further examine the question raised in this paper. In this assignment I explicitly asked my students to keep a journal of “mathematical insights” throughout the semester. This journal was defined as a notebook for collecting insights experienced during regular on-campus classes, field visits, work with students, readings, and independent study.

Results

In the context of their interactive experiences with students, I have noticed that prospective teachers in my mathematics methods courses have engaged in mathematical explorations of their own. These explorations have typically occurred in the following contexts: (a) when selecting and designing mathematical tasks; (b) when analyzing students' work, particularly when dealing with students' incorrect work; and (c) when offering mathematical explanations to their students. I will illustrate these with examples from each of the three interactive teaching activities mentioned.

Designing Mathematical Tasks

There are several ways in which preservice teachers engage in mathematical inquiry in this context. Some examples include figuring out the mathematical demands of the tasks and questions they are going to pose to their students. Other explorations take place when adapting or re-scaling mathematical tasks to make them more accessible to students of differing abilities and different grade levels. Preservice teachers also engage in mathematical inquiry when they design extensions and related questions to the tasks they offer to students. An example can be found in “Camilla's” work. She chose to re-scale one of the problems we had worked on in our class by changing its fractional numbers from thirds to halves. By doing so, she made an interesting discovery, that is, that her students were able to obtain the correct answer to the new version of the problem while using an erroneous solution method. This unexpected outcome launched Camilla into her own investigation as to why such method worked for halves and not thirds, and into the reasons why such a “minor” change could alter the original problem.

Analyzing Student's Work

This is another context that provides preservice teachers with multiple opportunities for mathematical inquiry. Some examples include learning from students their different strategies for thinking about a particular problem or concept. Other opportunities arise when in the process of planning or having actual interactions with students they have to assess the validity and generalizability of students' methods. Furthermore, in this context they often need to provide counter scenarios or examples to disprove or challenge a student's erroneous conceptualization. This happened to “Daniela” when interviewing a 2^nd Grader about her strategies for sharing cookies among different number of people. She found her student theorizing that if the number of cookies was even, it could be shared “evenly” among people, and that if the number of cookies was odd, it could not. Daniela, therefore, found herself in a position to challenge her student's theory and have the student realize that this theory does not work for all cases.

Providing Mathematical Explanations

When working with young students, preservice teachers often find themselves providing definitions for seemingly simple ideas, or ideas they have forgotten and taken for granted. For instance, when interviewing students about their mental strategies for doubling whole numbers, and for sharing different numbers of cookies between even and odd number of people, many of my students had to define what these two terms meant. Another example from “Thea” in the context of her letter exchanges with a fourth grader will better illustrate. Thea found herself having to formulate a mathematical explanation when her student was trying to figure out “how to tell whether or not one fraction is bigger than another.” Initially her student thought that the fraction with the bigger denominator was the biggest fraction. Thea devised an explanation using drawings (as opposed to symbolic manipulation such as finding a common denominator) to show the student that this was not the case. Her student, however used the drawings to show that the fraction with the largest denominator was actually bigger by drawing two different sized wholes for the given fractions. Faced with this puzzling work, Thea realized that in her explanation it was assumed that the two fractions were parts of the same or equal sized wholes. Further analysis of this led Thea to ask an important mathematical question, What happens when you compare fractions that belong to unequal-sized wholes? This work led Thea to later make an important mathematical discovery, that “when you use common denominators to determine which fraction is bigger you are ensuring that each ‘whole’ is the same size.”

Preservice Teachers' Reactions to Puzzling Mathematical Situations

The above examples show preservice teachers who have a common orientation towards the puzzling mathematical situations they face while teaching. Their orientation can be characterized as an “inquirying or adventurous” approach towards learning mathematics from their experiences with students. Other preservice teachers, however, have not reacted with such open mind. In contrast to this inquirying orientation, other preservice teachers tend to have an “intimidated or evasive” response to the mathematics that arise in their work with students. An example from Terry, who also received some very interesting responses in her student's letters, will illustrate. Terry asked her student to use a page with eight 4x6 (dotted grid) rectangles to draw 1/4 in as many ways as possible. Her student replied by sending not only a few typical drawings for 1/4, she also sent unusual partitions (equal sizes but different shapes) for that fraction. In addition, the student asked whether such uneven shapes were allowed. Terry's reaction to her student's interesting work, however was quite different from Thea. While Thea explicitly raised questions about her student's work and continued to investigate it, Terry made no further attempts to investigate the questions the student raised and did not provide a response to her student's query in her response letters.

Discussion

From the teacher development perspective (see Brown and Borko, 1992; Kagan, 1992) the explanation for why prospective teachers like Camilla, Daniela, and Thea seem to be more naturally inclined to investigate puzzling episodes in their teaching have to do with their level of development and concern as learners of teaching. According to developmental theories of knowledge growth of teachers, prospective teachers move through stages of development and concerns which explains why they attend to certain things and ignore others. It may be possible that concerns for their own understanding of the subject matter is not high in the priority list of preservice teachers like Terry. A different perspective, however, which takes into consideration the context and structure of the task these preservice teachers were asked to perform, in turn, suggests that the explanation may instead lie in the inexplicit nature of these tasks. Studies of structural constraints in tasks posed to prospective teachers often report this to be the case (e.g., Richert, 1992; Grossman, 1992). It is plausible that if a more explicit structure for investigating mathematical insights were provided, everyone would have shown an “inclination” to do so.

The teaching experiment I mentioned earlier offers some insight into this issue. Preliminary analysis of the data suggests that while field related experiences may offer preservice teachers multiple opportunities to explore mathematical ideas, it is necessary to provide the structure and the expectation to do so. Many of the self-initiated entries preservice teachers made in their “Math Insights” journal focused initially on pedagogical concerns such as issues of gender, calculator usage, and on students' abilities and behaviors. Yet with prodding and specific prompts their journal entries eventually began to focus on the mathematical questions they encountered when working with students. In a class of 27 preservice teachers more than half of their entries focused on self-initiated explorations that arose during their fieldwork with students. Most importantly, this exercise brought about some important realizations for prospective teachers about the role and the importance of their mathematical dispositions and understandings for their learning to teach. In short, this experiment highlights the importance of making the learning of mathematics an explicit and important goal of pedagogical courses. If this is not emphasized, preservice teachers are not likely to focus their attention onto (or be able to recognize and learn from) the mathematics that arise in the context of their teaching.

References

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