Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic

Abstract

In this paper we consider the duality between process and concept in mathematics, in particular, using the same symbolism to represent both a process (such as the addition of two numbers 3 + 2) and the product of that process (the sum 3 + 2). The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema. Symbolism that inherently represents the amalgam of process/concept ambiguity we call a “procept.” We hypothesize that the successful mathematical thinker uses a mental structure that is manifest in the ability to think proceptually. We give empirical evidence from simple arithmetic to support the hypothesis that there is a qualitatively different kind of mathematical thought displayed by the more able thinker compared to that of the less able one. The less able are doing a more difficult form of mathematics, which eventually causes a divergence in performance between them and their more successful peers.

Students’ Concept Image and Its Impact on Reasoning towards the Concept of the Derivative

“…The general notion of the theorem or question is as a concept. Gray and Tall (1994) suggested the notion of the procept, which was taken to be characteristic of mathematical symbols. According to Gray and Tall, in a mathematical symbol contains two entities, symbol as process and symbol as a concept.…”

The cultural construction of subject discipline knowledge: comparing ‘abstraction’ in two international contexts

“…The second, type B, draws on the notion that mathematical ideas can be understood both as procedures and as objects (Gray and Tall, 1994; Gray et al, 1999; Sfard, 1991), and that effective mathematical thinkers can switch between these states, operating with both the procedure and the concept, and hence being able to think ‘proceptually’. A very simple example of this is addition, say 3 + 4 = 7.…”

Educational triage and ability-grouping in primary mathematics: a case-study of the impacts on low-attaining pupils

“…The literature suggests that consistently using small numbers and cubes forces pupils into using less sophisticated counting modelscounting out cubes and counting all the cubes togetherrather than having the opportunity to learn to work, for instance, with derived facts. This leads to pupils doing more, if unhelpful, mathematics, potentially restricting their mathematical development (Gray & Tall, 1994). Further, a low-level manipulative-led curriculum may have restricted pupils' mathematical gains through lack of preparation for the national tests.…”