Mathematical Concepts as Devices

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

Abstract

Objectives:

A device is a machine that carves up and divides the material world into various configurations and distinct assemblages. In this presentation, we discuss examples of mathematical concepts – zero, square, even/odd parity, multiplication – for how they might be reconceived as devices. Using video data of classrooms to help support our argument, we outline a pedagogy of the concept as device that revisions concepts as generative or ontogenetic of new material configurations. We show how such an approach sustains student awareness of the vitality and indeterminism at the source of curricular concepts.

Theoretical Framework:

We propose a pedagogy of the concept that shifts the emphasis of theories of learning from the epistemological to the ontological, a shift that directs our attention to the materiality of mathematical concepts. A pedagogy of the concept aims to revise concepts as material and ontogenetic devices rather than logical conditions for possibility. A concept, according to Cutler and Mackenzie (2011), brings with it an entire “plane of immanence” on which we—as catalyzers of concepts,operate, explore and create. Concepts are active agents of change, mobilizing and being mobilized by way of the material assemblages they form with students, teachers and other agents in any given context. Indeed, the mobility of the concept is precisely what denies all claims to it being immaterial or ideal or a cognitive construct. In a compelling inversion of the usual aim for learning, Cutler and Mackenzie suggest that learning is that which sustains the mobility of concepts, in other words “that which resists determination as knowledge” (p. 68). We argue thus that the mathematical concept should not be considered an abstract universal detached from the particularities of its concrete instances.

Data and Methods:

Drawing on the work of the historian and philosopher of mathematics Gilles Châtelet (1993/2000), we will discuss several examples of mathematical concepts that can be thought of as devices that operate both logically and ontologically. In operating logically, they determine the conditions of what is possible, and in operating ontologically, they determine the actual configuration of the material world. It is this latter function that we dwell on, as we argue that it is under-theorized in current school curricula in the U.S. and Canada. In particular, we examine how (currently static) concepts in the mathematics curriculum might be mobilized, and how doing so might change the way we think of selecting and ordering concepts in a curriculum. We use two short videos from mathematics classrooms to ground our argument and to demonstrate how our new materialist approach sheds light on student learning.

Scholarly Significance:

We show how a pedagogy of the concept as device allows students and teachers to grasp how concepts are doing things – such as creating new entities, twisting old objects inside-out, or engendering new dimensions – rather than simply treating them as representations of or abstractions from a set of ‘concrete’ objects. This approach allows for a more creative approach to curriculum design and a more aesthetic approach to pedagogy.

Authors

• Elizabeth De Freitas, Adelphi University

• Nathalie Sinclair, Simon Fraser University