Intuitive Mathematics: Theoretical and Educational Implications

What kinds of intuitions do people have for solving problems in a formal logic system? Studies on intuitive physics have shown that people hold a set of naive beliefs Caramazza, and Green (1980) found that when people were asked to draw the path of a moving object shot through a curved tube, they believed that the object would move along a curved (instead of a straight) path even in the absence of external forces. Such an Aristotelian conceptualization of motion, although mistaken, may be based in part on forming an analogy to real-life examples, such as the Earth's circular movement around the sun (one does not " see " the forces that sustain such a movement). Does there exist a similar body of knowledge that we can refer to as " intuitive mathematics? " That is, can we identify a set of naive beliefs that are applied to solving abstract mathematics problems? If so, how do these intuitions hinder or facilitate problem solving? The answers to these questions have implications for both psychology and education. By examining the nature of intuitive mathematics we could help (a) improve our understanding of people's formal-and informal-reasoning skills and (b) create more effective instructional materials. The focus of much research, to date, has been on the development of early mathematical cognition. A prime example comes from Rochel Gelman and colleagues' (e. work on implicit counting principles that enable preschool children to understand and to perform addition. In contrast, there has been much less emphasis on intuitive understanding that develops as a result of learning mathematical procedures in later years. The largest lament on the part of the education community has been that an emphasis on learning procedures can lead to rote execution of problem-solving steps, resulting in lack of correct intuition for these procedures. For example, students often learn how to execute a procedure, such as multicolumn subtraction, without Mathematical Intuition 3 understanding its underlying teleology (VanLehn, 1990). This finding has led to a proliferation of educational programs that emphasize the conceptual over the procedural (e.g., NCTM, 1989). In this chapter, we will examine the nature and origin of what we term as symbolic intuition, or the intuitive understanding of mathematical symbols that develops as a result of experience with formal and abstract school-based procedures. Before we define more formally what we mean by intuition, in general, and symbolic intuition, in particular, we would …