Dynamic construals, static formalisms: Evidence from co-speech gesture during mathematical proving

This paper presents results from ongoing research using co-speech gesture to investigate the nature of mathematical concepts. Traditional accounts of the content of mathematical concepts have focused on formal language and definitions. In analysis, key concepts such as continuity are defined by means of static existential and universal quantifiers ranging over static numbers. Recent evidence from Cognitive Science, however, suggests that mathematical concepts are construed through various mechanisms of everyday cognition — such as conceptual metaphor and fictive motion — which are at odds with static conceptions. We analyse the co-speech gesture produced by graduate students while collaborating on a proof in analysis. The results support the claim that many mathematical concepts, which formally make use of static entities and relations, are, cognitively, inherently dynamic. 1 A previous version of this work was included in the Proceedings of the International Symposium on Mathematical Practice and Cognition, Alison Pease, Markus Guhe, and Alan Smaill (Eds.), at the AISB 2010 convention, 29 March – 1 April 2010, De Montfort University, Leicester, UK. INTRODUCTION Advanced mathematics is characterized by rigorous methods and symbolic manipulation. Traditionally, this has been taken as evidence for the abstract, formal nature of mathematical concepts. Contrary to this view, recent research from Cognitive Science suggests that nature of such concepts is largely embodied, and so mathematical concepts — such as infinity, continuity, number, and the rest — are not abstract and generated from formal definitions, but metaphorical and grounded in experience [17]. The evidence for the embodiment of mathematics has been drawn largely from mathematical language, using techniques in Cognitive Linguistics, although recent research has included some qualitative studies of gesture [21]. This paper reports on the results of an ongoing research project on the nature of mathematical proof. In particular, we focus on the co-speech gesture produced by graduate mathematics students while collaborating on a proof, and argue that the character of these gestures supports the claim that mathematical concepts are largely metaphorical and embodied and that their nature cannot be reduced to pure formalisms. This research extends earlier work that used gesture to adjudicate the cognitive reality of conceptual metaphor in mathematics [21, 22], moving beyond the pedagogical setting to consider advanced mathematical practice in a naturalistic setting. The paper is organized as follows. First, we briefly review the evidence for metaphor and fictive motion in mathematical cognition. Second, we introduce the study of gesture as a tool for investigating the cognitive reality of these phenomena during mathematical practice. Then we present some quantitative results on co-speech gesture production from a semi-structured case study of collaborative proof by mathematics graduate students at a large research university. Finally, we discuss implications for the nature of mathematical concepts. DYNAMISM IN MATHEMATICAL DISCOURSE How can we, as limited beings, understand abstract concepts that transcend our human experience? According to some approaches in Embodied Cognition, the answer lies largely in our shared embodiment, in the way our bodies — modulated by culture — structure thought and experience. Research in Cognitive Science suggests that abstract concepts are created and understood via a number of fundamental cognitive processes that extend the inferential structure of our bodily experiences. These processes include conceptual metaphor and metonymy [16], conceptual blending [4], and fictive motion [30]. Using the tools of Cognitive Linguistics, Lakoff and Núñez [17] argued that even mathematical concepts rely on these cognitive processes. While the details of the various construals underlying mathematical thought are beyond the scope of this article, in this section we will quickly review a few salient instances in which fictive motion lends dynamism to putatively static mathematical concepts. Consider the concept of a limit, a central notion in calculus. Formally, the limit of a function is defined by a chain of inequalities: Let a function f be defined on an open interval containing a, except possibly at a itself, and let L be a real number. Then limx 7→a f(x) = Lmeans that, for all ε > 0, there exists δ > 0, such that whenever 0 < |x− a| < δ, then |f(x)− L| < ε. Note that the limit notation includes a small arrow, which might suggest that the definition of a limit of a function would include some form of dynamism. The formal definition of a limit, however, refers only to static universal and existential quantifiers, static numbers, motion-less arithmetic difference and static inequalities. Nowhere in this definition is there any mention of movement. Mathematicians, on the other hand, speak of a function “tending to,” “moving toward” or “reaching” a limit — all of which, contra the formal definition, invoke a sense of motion [21]. These expressions are a form of fictive motion [30], the process by which we unconsciously conceptualize static entities in dynamic terms. Fictive motion construals always involve the motion of a trajector across a landscape. When we say, for instance, that “the Equator passes through Brazil,” the Equator — a purely imaginary static entity — is construed as a moving agent (trajector) dynamically crossing a country (landscape). Similarly, we can say that a fence stops at a tree or that a road runs along the coast — even though both fences and roads are completely stationary and thus incapable of stopping or running. This same cognitive mechanism of fictive motion injects dynamism into a wide range of statically defined mathematical entities. A function, for example, is formally defined as a static relation between two sets, the domain and the range, but mathematicians nevertheless describe functions dynamically as “reaching a limit,” “going down towards a minimum,” or “oscillating,” in each case evoking a construal in which an imaginary trajector travels along the path of the function [21]. Fictive motion is similarly at work when we say that sequences are “approaching,” “decreasing,” or “converging,” and when arithmetic is construed as motion along a number-line. Dynamism, therefore, is present throughout the language of mathematics — showing up in the discourse surrounding continuity, functions, and even arithmetic — and lends credence to the claim that mathematical thought itself is dynamic, metaphorical, and embodied. MATHEMATICAL GESTURE One objection to this line of reasoning is that these metaphorical construals are mandatory in mathematical discourse — and that, therefore, they are conventionalized, dead, stripped of any cognitive reality [21]. Consider the geometrical procedure of “reflection,” such as when a point in the Cartesian plane is reflected across the origin. The word “reflection” has roots in the Latin verb reflectere, meaning “to bend back.” There is no other way of describing the geometrical procedure, and therefore the same lexical item, “reflection,” is always used to describe the procedure — yet there is also no evidence suggesting that the concept of reflection involves an automatic construal of bending backwards. Certain aspects of mathematical discourse, 1 The number-line is itself a conceptual blend, the result of combining the mental spaces for number and space. then, are codified and devoid of cognitive significance. Could the dynamic discourse surrounding limits, functions, and sequences similarly involve conventionalized discourse? To address this objection, we must supplement corpus studies of mathematical discourse with additional lines of converging evidence. The cognitive reality of mathematical construals is supported by the study of gesture, that is, motor action co-produced with speech and thought in real time. Gesture is universal, unconscious, and essential to communication. Most importantly, gesture offers a “window into the mind” [8]. When co-produced with abstract thinking, gestures parallel the metaphorical mappings exhibited linguistically [19, 2, 26], and give us insight into the representation of mathematical concepts and solution strategies [1, 5]. In particular, Núñez [21, 22] demonstrated that mathematicians’ gestures in pedagogical contexts supply converging evidence for the metaphorical and embodied nature of mathematical concepts. Previous research on mathematical gesture, however, has dealt primarily with gesture production during pedagogy or in the context of elementary mathematical problem solving. In these settings, gestures were found to be dynamic — in line with the predictions of Cognitive Linguistics, and suggesting that mathematical concepts are metaphorical in those settings [22, 3]. But would we expect any other behavior? It is standard pedagogical practice to use “real world” examples of abstract concepts, to ground the abstruse in the everyday. Physics teachers might describe electricity, for instance, as “water running through a pipe,” effectively mapping intuitions about water volume and pressure onto the more abstract concepts of electrical current and voltage [6]. The use of this pedagogical scaffold, however, does not imply that electrical current is in reality the flow of water particles. Certainly, the expert physicist may call on such metaphors while instructing a naı̈ve student, or may subtly deploy these evocative images during heuristic reasoning, but all the while they might recognize that electrical current is fundamentally different from water flow. Expert practice requires the careful amendment of pedagogical metaphors. The learning of physics — one story goes — is marked by the gradual abandonment of these metaphorical construals, replacing such didactic scaffolds with genuine intuitions about basic physical phenomena [29]. Thus, while the evidence for dynamic gesture in mathematical pedagogy and communication is suggestive, it does not directly address the nature of mathematical practice — or of mathematics itself. Indeed, to date there is little research on co-speech gesture during the activities that are central to research mathematics, such as proving and communicating non-trivial results. When mathematicians are generating a proof or communicating with other expert mathematicians, do they deploy the same conceptual metaphors that are evidenced in corpus studies of mathematical discourse and in gesture studies of mathematical pedagogy? Or is the metaphorical content of these utterances an artifact of the pedagogical context? The current study uses the tools of Gesture Studies, Cognitive Linguistics, and Embodied Cognition to empirically investigate with quantitative methods the cognitive reality of fictive motion in mathematical practice. In a semi-controlled situation, we looked at the co-speech gesture of graduate mathematics students as they collaborated in pairs on a mathematical proof involving key concepts in analysis. If the meaning of such essential mathematical concepts is determined by their formal definition, then we should expect static co-speech gesture. If mathematical concepts are truly metaphorical and dynamic, on the other hand, we should expect the co-speech gesture of the graduate students to reflect this dynamism.