Mathematical Concepts as Complex Coherences

• In Event: Roundtable Session 15
  In Roundtable Session: What Is a Mathematical Concept?

Abstract

Theoretical Framework:

In this paper we draw on complexity research to make sense of a mathematical concept for the purposes of schooling. Within this frame, a complex phenomenon/entity is an emergent form – that is, it is a perceptible coherence that arises in the interactions of multiple agents/systems, manifests features and capacities that are not observed in previous agents/systems, maintains itself over some period of time,evolves in response to both internal and external dynamics.

Arguably, notions of complexity and emergence are already well represented in contemporary discussions of mathematics and mathematical concepts. For example, it is not uncommon to encounter suggestions that mathematics itself “is a living, breathing, changing organism …” (Burger & Starbird, 2005, p. xi) or that it “emerges as an autopoietic [i.e., self-creating and self-maintaining] system” (Sfard, 2008, p. 129). More pointedly, Foote (2007) has argued that mathematics is an adaptive, complex system that is approaching the limits of human verifiability.

Methods and Data:

In the full paper, the discussion is developed around one learner’s emergent interpretation of integer multiplication, as a number line-compression through the origin. This interpretation arose in a deliberate attempt to blend several instantiations of multiplication, and it is used to illustrate the assertion that, within a complexity frame, a mathematical concept might be construed as an ever-emergent coherence that interacts with many others in the ecosystem of mathematics.

This assertion has entailments in both macro and micro directions. Moving in a macro direction, the ecosystem of mathematics might be productively construed as a dynamic entity within a grander ecosystem of knowledge domains (and so on). Moving in a micro direction, on the personal level a concept might be seen to arise and evolve as instantiations (images, gestures, metaphors, metonymies, exemplars, applications, analogies, etc.) that collect and their entailments entangle with one another, enabling a cognitive coherence that is more than the mere sum of its constituents.

Scholarly Significance:

Notably, there is no claim of novelty in the suggestion that a mathematical concept is complex. Quite the contrary, it is acknowledged that this particular notion dates back centuries. For example, in the 1700s Kant described a concept as a structured cluster of representations that can be used to collect objects, scenarios or sequences of events or relations – a notion that directly informed Piaget’s (1953) account of a schema as a mental framework that is created as a child interacts with/in physical and social environments and integrates sensorimotor, symbolic, and operational instantiations. What complexity discourses offer, then, is not a new description, but interpretive devices that derive from the study of a great diversity of complex, emergent forms. These devices include
- nuanced accounts of key descriptors (e.g., functional sufficiency, recursive elaboration, structural coherence);
-a broad array of interpretative tools (e.g., decentralizated network structures, power law distributions); and
-pragmatic advice to trigger emergence and occasion transformation.

The balance of the full paper consists of explications of these sorts of devices as they might be used to inform the project of school mathematics.

Author

• Brent Davis, University of Calgary