Examples play a significant role in the teaching and learning of mathematics. Often it is with a carefully formulated example that subtleties in a definition or an algorithmic process can be detected. Examples make it possible to consider generalities, such as describing the behavior of graphs of quadratic equations by exploring specific instances using a computer software package like GeoGebra. Examples can also limit one’s mathematical perspective. We know, for instance, that students will take note when geometric shapes are always oriented in the same way. Such observations can lead students to argue for unnecessary restrictions on a triangle. Ideally, mathematical learning experiences provide opportunities to develop a rich array of examples that contribute to students’ problem solving skills and understanding.

Interestingly, what we know about the role of examples in teaching and learning mathematics is not adequately informed by the mathematician’s perspective. Few studies on examples or related topics have mathematician participants. There is evidence that mathematicians frequently generate examples in the process of validating a proof given by a peer (Weber, 2008). In addition, given the number of proofs without words publications, it is reasonable to assume that mathematicians value opportunities to “see the general in the specific” (Mason & Pimm, 1984). That is, mathematicians sometimes use a generic example to make a proof. However, we have little evidence that speaks to the mathematicians’ perspective on the use of examples in teaching or how examples support their own mathematical work. We suspect that having this information would contribute significantly to thinking about teaching and learning undergraduate mathematics. We attempt to address this need by attending to the ways in which university mathematicians talked about the role of examples in their teaching and in their own professional work.

Watson and Mason’s work (2005) has been particularly useful for our purposes. They noted that the word example is used to mean different things. For instance, example may refer to a mathematical object (example of) or a problem that has been solved (worked example), among other things. We also find their notion of example space relevant in that it provides a metaphor for the cognitive structure at play related to a learner’s use of examples. This is closely related to Tall and Vinner’s (1981) notion of personal concept image, which includes all that one may think of related to a concept. Concept image has been shown to be particularly resilient even in the face of examples that are in direct conflict.

This study is part of a broader research project designed to understand how mathematicians make sense of definitions. Initially, we used a grounded theory approach to conduct a preliminary analysis of interview data, which revealed an examples-related theme (Author). In this follow-up study we revisited the data to conduct a microanalysis using an analytic framework based on Watson and Mason’s work (2005). This microanalysis was guided by the following questions: 1) What are the different meanings for “example” referenced by the mathematicians? 2) How do the mathematicians’ descriptions of their use of examples relate to Watson and Mason’s notion of example spaces?

Eight mathematicians employed at a large university in the northwestern region of the United States volunteered to participate in a single interview that lasted about an hour. Five of the mathematicians were engaged in mathematical research at the time of the interviews; three of them worked in multiple areas. All of the mathematicians were responsible for teaching at both the undergraduate and graduate levels. Their teaching experience ranged from five to more than twenty years.

During the interviews, which were audiotaped, the mathematicians were first asked to describe how they make sense of new mathematical definitions and to provide a recent instance of doing so. The second interview question asked participants to share how they support students’ work with definitions. The mathematicians were then asked to engage in an example-generation activity, and finally were given a formal definition and asked to share how they would go about making sense of the definition. For this final task, we chose an algebraic definition of formal power series, anticipating that the mathematicians would be less familiar, and therefore providing insight into their sense making processes.

In the proposed session, we will highlight the major findings of this microanalysis. In brief, mathematicians spoke about examples as key to developing understanding. In particular, they noted that examples should be chosen carefully so that they serve to draw attention to important aspects. Creating or considering non-examples was considered an essential component of understanding; as articulated by one participant “…the only way to get there is [to] look at concrete examples and look at concrete non-examples.” The use of examples and non-examples seemed critical for the mathematicians’ own understanding and how they went about supporting their students’ understanding.

One way the mathematicians described using examples was to create shared experiences with a particular concept for their students. This seemed to be an attempt to engage students in a process of exploration similar to what some of the mathematicians described for themselves. While the mathematicians did not speak of example spaces, the process they described certainly attended to the development of personal example spaces for the purpose of reflection upon the concept and upon aspects of a formal definition.

The mathematicians also spoke of using non-examples to develop precision. A recurring observation was that it was not enough to know that an object was not a representative of a defined collection. They expected their students (and themselves in their own work) to know which aspect of the definition was not satisfied. In this sense, knowing what is not may be an important part of a personal example space for mathematicians.

Finally, references to examples were primarily but not exclusively of the example of type. Some of the mathematicians referred to worked examples as an important part of their instruction. Continuing analysis will focus on how mathematicians perceive the role of worked examples in students’ learning.

References

Author.

Mason, J. & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277-289.

Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.

Watson, A. & Mason, J. (2005). Mathematics as a Constructive Activity: Learners Generating Examples. Mahwah, N.J.: Lawrence Erlbaum Associates.

Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431-459.