Cognitive Units, Connections and Compression in Mathematical Thinking1

This paper presents a theory of 'cognitive units' to describe mechanisms for compressing information into manageable mental units with powerful internal and external links. The first part of the paper addresses the fundamental ideas, relating them to other psychological concepts and to the neurophysiological activities that compress conceptual relationships into manageable cognitive structures. These include process-object duality, schema-object duality, cognitive networks, cognitive hierarchies and the varifocal theory of Skemp. We argue the necessity of having a more flexible view of the structures used in mathematical thinking to gain insight into why mathematics can be simple and powerful for some yet complex and difficult for others. In the second part of the paper we use the theory to analyze how different students cope with the standard contradiction proof that 2 is irrational and its generalization to the irrationality of 3. We identify three very different cognitive units that play fundamental roles and use them to reveal the contrast between the tight cognitive structure of those who grasp the meaning of the proof and the diffuse structure of those who do not. 1. Cognitive units and connections Mathematical thinking is handled by the biological structure of the human brain. As a multi-processing system, complex decision-making is reduced to manageable levels by suppressing inessential detail and focusing attention on important information. We begin with a piece of cognitive structure that can be held in the focus of attention all at one time. This might be a symbol, a specific fact such as '3+4 is 7', a general fact such as 'the sum of two even numbers is even', a relationship such as 'sin cos 22