This research examines one high achieving preservice teacher’s conceptions of the function concept and her line of reasoning as she after a calculus course tries to make clear what constitutes a function. A group of twenty-five preservice teachers specializing in mathematics and science participated in the study. One preservice teacher, called Emma, who completed the calculus course with “high pass” is the research subject for the case study in this presentation. The concepts of function and equation are closely linked for the preservice teacher in question. In the advancement of the preservice teacher’s reasoning on functions, feedback from evoked concept images to the concept definition is central, and leads her to a more developed understanding of what constitutes a function. The findings of the research call attention to one component of importance in the development of a concept image, namely, the feedback-process from evoked concept images to the concept definition. Based on the case study results, preservice teachers seemingly need to be exposed to problem formulations concerning the relationships of the function concept to other mathematical concepts (this conclusion is also supported by Hansson, 2006). It was made clear during the interview with Emma that she is not experienced in working with such problems. During an individual’s reasoning, the concept image will almost always be evoked, whereas the concept definition will remain inactive or even be forgotten (Tall & Vinner, 1981). The study indicates that one significant component in the development of preservice teachers’ understanding of the function concept could be to stimulate a feedback of evoked concept images to the concept definition in various contexts, as when Emma comes to a better understanding in the interview. The results of the study also support Vinner’s (1991, 1992) suggestion that this type of reconnection is primarily possible with problems that are not of the standard variety.

References

Hansson, Ö. (2006). Studying the views of preservice teachers on the concept of function(LTU doctoral thesis series No. 2006:22). Luleå: Luleå University of Technology. Tall, D., & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D.Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer.Vinner, S. (1992). The function concept as prototype for problems in mathematics. In G.Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 195-213). Washington, DC: Mathematical Association of America.