Loss of Dimension in the History of Calculus and in Student Reasoning

Abstract:Research indicates that calculus students often imagine objects losing a dimension entirely when a limit is taken, and that this image serves as an obstacle to their understanding of the fundamental theorem of calculus. Similar imagery, in the form of "indivisibles", was similarly unsupportive of the development of the fundamental theorem in the mid-1600s, unlike the more powerful subsequent imagery of infinitesimals. This parallel between student issues and historical issues suggests several implications for how to provide students with imagery that is more productive for understanding the fundamental theorem, such as the imagery of infinitesimals or the more modern quantitative limits approach, which relies heavily on quantitative reasoning.Keywords: Calculus; infinitesimals; history of Calculus; Fundamental Theorem of Calculus(ProQuest: ... denotes formulae omitted.)IntroductionRecently several colleagues told me they think that the fundamental theorem of calculus is not something that undergraduate students can really make sense of when they take a course in calculus, although it may snap into focus later in their education1. The implication was that we should not bother spending very much time teaching it. This is a bit too cynical for me. I don't think it's too much to ask that students come out of a calculus class understanding the fondamental idea of calculus, why there is a deep connection between integrals and derivatives. Nonetheless, research has shown that there are some common mental images that students use to make sense of derivatives and integrals, images that can serve as obstacles to their learning of the fundamental theorem of calculus.The purpose of this article is to explore one of the most common of these mental images: the loss of dimension. First I describe this mental image and some of its manifestations in student reasoning, and discuss its limitations for promoting understanding of the fundamental theorem of calculus. Then I discuss some historical occurrences of pre-calculus ideas that also involve the loss of dimension, in the form of "indivisibles," and how these indivisibles were likewise limited in promoting the understanding of the fundamental theorem of calculus. I describe how the approach of infinitesimals, which replaced indivisibles, led to the successful conceptualization of calculus. This suggests that such imagery of infinitesimals might be useful for calculus students as well. Finally, I discuss the more modern approach of focusing just on limits of quantities, and why this is also more successful than approaches that involve the loss of dimension.It is important to clarify what I mean by "mental image" in this article. One way to interpret a mental image is loosely as Vinner does, as a set of mental pictures that a person uses when considering a mathematical term like "limit"-pictures that may not necessarily, if somehow externalized, satisfy the mathematical definition of the term (e.g. Tall & Vinner 1981, Vinner 1989, 1991, Vinner & Dreyfus, 1989). While this notion of mental image captures the spirit of the idea I will be using, a Piagetian interpretation of mental image clarifies a few aspects of this. Piaget generally speaks of a mental image as expressing an anticipated outcome of an action taken on an object (Piaget 1967). These actions and objects can certainly be products of reflective abstraction, and so may be many steps removed from actual physical actions and perceived objects, but nonetheless the idea of mental image is ultimately grounded in physical action and attention (Thompson 1994). This means that a mental image (a) need not be entirely visual, (b) need not directly correspond with "real-world" objects or operations, and (c) need not directly correspond with things that have mathematical definitions. The Piagetian interpretation of mental image helps us keep in mind how such an image functions in the learning, rather than simply noting how it may or may not differ from standard mathematical treatments or definitions. …