The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics

Abstract

The canonical history of mathematics suggests that the late 19th-century "arithmetization" of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitivehistorical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice.

Mastering algebra retrains the visual system to perceive hierarchical structure in equations

“…When a mathematical expression is examined, eye movements respect the expression’s hierarchical structure, starting with the highest-precedence operation and moving sequentially to gradually lower-precedence operations (Landy, Jones, & Goldstone, 2008; Schneider, Maruyama, Dehaene, & Sigman, 2012). And while gestures can shape children’s early mathematical knowledge (Goldin-Meadow, Cook, & Mitchell, 2009), even experts gesture spontaneously to express their mathematical understanding (Marghetis & Núñez, 2013). A complete understanding of mathematical cognition requires that we study mathematics as it is actually accomplished, as an embodied practice: eyes darting across the blackboard, hands scribbling away.…”

Building Cognition: The Construction of Computational Representations for Scientific Discovery

“…In our account, the underlying mechanism in both these cases would be the gradual integration of the internal imagination process and the external model, and the implicit understanding of the system's behavior that emerges from this incorporation. This account of model‐based learning allows the use of DC as an analysis framework to understand learning situations involving manipulable models and novel digital media (Landy, Allen, & Zednik, ; Landy & Goldstone, ; Majumdar et al., ; Marghetis & Núnez, ; Ottmar, Landy, & Goldstone, ), and also extend learning frameworks based on modeling (such as Modeling Theory, Hestenes, ), thus taking the DC framework back to its original learning roots, as proposed by Pea () and Salomon ().…”

Conceptual Metaphors and Mathematical Practice: On Cognitive Studies of Historical Developments in Mathematics

“…Various approaches can be applied in the study of mathematical cognition. Introspective accounts (e.g., Hadamard, ), in vitro observations and experiments (e.g., Marghetis & Núñez, , this volume), and in vivo observations of actual mathematicians (e.g., Carter ), can all yield insights into mathematical thinking . Finally, the artifacts that result from or during mathematical activities, like notes, letters, articles, and books, can also be used as data for investigating the nature of the underlying and often implicit thought processes.…”