THE NATURE OF MATHEMATICS AS VIEWED FROM COGNITIVE SCIENCE

The theory of embodied cognition proposes that the source of many ideas, including mathematical ones, lies in the common physical experience of human beings. Through mechanisms such as conceptual mappings and metaphors we are able to construct new ideas and understandings that are more abstract, in the sense of being less tied to physical experience. The purpose of this paper is to review work on the theory of embodied mathematics and to describe an application of the theory to understanding how transformation geometry is learned. My goal in this paper is to characterize one perspective on the nature of mathematics, drawing primarily from recent cognitive science and in particular the work of George Lakoff and Rafael Núñez (Lakoff & Núñez, 2000), and to offer an example of the application of this perspective in my own research. A theory of how human beings think and understand mathematically should be able to account for both the correct and consistent mathematical ideas of professional mathematicians, and for the variety of ideas that diverge from this body of shared knowledge. In addition, unless mathematics education wants to set mathematics aside as a special case, such a theory should draw on conceptual primitives and mechanisms that are not unique to mathematical thinking. This paper presents a perspective that addresses the nature of mathematics by drawing from work by a number of researchers in contemporary cognitive science, including Lakoff and Núñez (2002), Johnson (1987), Varela, Thompson and Rosch (1991), Dehaene (1997), Fauconnier and Turner (2002) and others. The theory of embodied mathematics The theory of embodied mathematics states in essence that: (1) mathematics emerges through the interaction of the mind with the world, and (2) that each of these elements (the mind and the world) offers constraints on what kind of mathematics human beings are able to be constructed. A specific aspect of the theory describes a conceptual mechanism that has been used to understand thought and language in areas outside of mathematics, that of conceptual metaphor. As Lakoff and Núñez state, Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III L. D. Edwards 2 (3) “A large number of the most basic, as well as the most sophisticated, mathematical ideas are metaphorical in nature” (Lakoff and Núñez, 2000, p. 364). What Lakoff and Núñez refer to as the theory of mind-based mathematics does not deny the objectivity of the material world; in fact, it accords to this world a constraining influence on the kind of mathematics that humans are able to construct. The world behaves consistently, fairly reliably, predictably when you put a collection of two objects together with another collection of two objects, you are reliably going to end up with a collection of four objects, not one of five or three. Human infants behave as if they have some awareness of this kind of predictable behavior, though in their case, this awareness extends only to the very smallest of collections, namely, single objects combined with single objects (cf., Dehaene, 1997). Evolutionarily, it seems to have been important for humans to be able to subitize, to keep track of small collections of objects, and this provides part of what the mind brings to the picture. There is a small amount of innate or virtually innate arithmetic that humans share with a few other kinds of animals, for example, chimps and parrots (Dehaene, 1997). But this starting point is very restricted, and one of the questions addressed by the theory of embodied mathematics has to do with identifying the conceptual mechanisms and building blocks that allow humans to go from this starting point to develop the incredible richness of contemporary and historical mathematics. Conceptual mechanisms and primitives In addressing the nature of mathematical thought, Lakoff and Núñez offer a set of mechanisms and cognitive primitives drawn from existing work in cognitive linguistics and embodied cognition. These primitives and mechanisms include prototypes, image schemas, aspectual concepts, conceptual metaphor, conceptual blends, and metonymy (for details, see Lakoff, 1987 and Lakoff & Núñez, 2000). Conceptual metaphor is an important (but not the only) mechanism proposed to account for how mathematical understandings are connected to the world and to each other. Conceptual metaphor is an unconscious mapping between a well-understood source domain to a less-well-understood target domain, one which carries with it the inferential structure of the source domain, thus allowing an understanding of the target domain to be constructed. A non-mathematical example might be the way in which research is conceptualized as a process of physical construction: we speak of someone’s work as having a firm foundation, or as extending previous work, even when there is clearly no actual process of physical building going on. Within mathematics, a simple example of conceptual metaphor would be the “arithmetic as object collection” metaphor, where our common, embodied experience of grouping or collecting objects serves as the source domain for constructing the arithmetic of natural numbers. Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III L. D. Edwards 3 It should be clarified that not all conceptual mappings draw from direct physical experience, or have to do with the manipulation of objects. In fact, only the most basic level of mathematics is constructed from metaphors that link to physical experience. The metaphors utilized at this level are called “grounding metaphors.” The majority of the concepts and processes of mathematics, according to the theory, are constructed through mappings between and among existing mathematical domains. An example of this kind of linking metaphor will be given later, when space is conceptualized in terms of sets of points. A third type of metaphor, not based on physical experience, is the redefinitional metaphor, defined as “metaphors that impose a technical understanding replacing ordinary concepts” (Lakoff & Núñez, 2000, p. 150). Conceptual metaphor is only one of the building blocks proposed in the theory of embodied mathematics. Conceptual blends are another. Blends consist of mappings that draw from more than one source domain or element to allow the construction of a target domain which is isomorphic to neither of the sources, but which draws from the inferential structure of each. An example would be the “numbers as points on a line” blend, where drawing from previously-constructed understandings of both numbers and lines, new entities, “number-points” are created that have characteristics of both. An embodied perspective on the learning of transformation geometry The theory of embodied cognition can provide a framework for understanding how both children and adults learn initial concepts within the domain of transformation geometry. In a series of studies beginning in 1989, I worked with eleven-to-fifteen year old students using a computer environment I designed for exploring transformation geometry (see Figure 1). Later, with Rina Zazkis, I investigated adult undergraduates’ learning of the same subject. In the first, most extensive study, I introduced three euclidean transformations, translation, rotation and reflection, to a whole class of sixth-grade (elevento twelve-year-old) students, utilizing sheets of overhead transparencies to illustrate them and to elicit the students’ own description of these motions. Then, I worked with twelve of these students for a period of five weeks, videotaping their interactions with the transformation geometry microworld (Edwards, 1991, 1992). A similar study was carried out two years later with a group of ten high school students (fifteen-year-olds). In this study, the students were introduced to the transformations in pairs, rather than in a whole group, using drawings on a sheet of paper (Edwards, 1997). Finally, Rina Zazkis and I examined the responses of fourteen college undergraduates, first to a paper-and-pencil task requiring them to predict the outcome of several transformations of the plane, and then, through videotapes of their use of a modified version of the microworld (Edwards & Zazkis, 1993). Although there were methodological differences across these studies, and some differences in specific results, overall, the responses of all of the participants, whether middle-school, high-school or adult, were remarkably Thematic Group 1 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III L. D. Edwards 4 similar. In particular, the students all seemed to have had the same initial expectations of how the transformations would work, and they made the same kinds of errors. This raises the question: why is there such consistency in how students, of various ages, learn transformation geometry? My initial expectation was that older students and adults would be less subject to “misconceptions” about geometric transformations and better able to carry out independent investigations, due to the more advanced state of development of their mathematical thinking. Yet this was not the case. The question of why students of various ages respond in a similar way to a “new” mathematical topic can, I believe, be productively investigated utilizing the theory of embodied mathematics. To do so will require examining the domain of transformation geometry from the perspective of contemporary mathematicians, and from the perspective of a learner who is meeting the domain, in a formal sense, for the first time. Figure 1: Transformations in the computer microworld Transformation geometry from the perspective of contemporary mathematics From the point of view of contemporary mathematicians, geometric transformations are mappings of the plane, and a “geometry” is “a study of the properties (expressed by postulate, definitions, and theorems) that are left invariant under a group of transformations” (Meserve, 1955, p. 21).