Conceptual change and mathematics learning

Although some historians of mathematics find the Kuhnian conceptual change approach particularly fruitful in the case of mathematics (e.g., Corry, 1993; Dauben, 1984; Kitcher, 1992), there is a general reluctance in philosophy and history of science circles to apply the conceptual change approach to mathematics. That mathematics is based on deductive proof and not on experiment, that is proven to be extremely tolerant of anomalies, and it does not display the radical incommensurability of theory before and after revolution, are some of the reasons why Thomas Kuhn himself exempted mathematics from the pattern of scientific development and change presented in the Scientific Revolutions (see Mahoney, 1997). According to Mahoney (1997: p. 2) ‘‘synthetic geometry, invariant theory, or quaternions may lose interest for mathematicians, subjects may be judged obsolescent or fruitless, but they do not seem to cease to be mathematics in the way that Aristotle’s mechanics ceased to be mechanics, or Galen’s physiology ceased to be physiology, or phlogistron chemistry ceased to be chemistry’’1. Unlike science, the formulation of a new theory in mathematics usually carries mathematics to a more general level of analysis and enables a wider perspective that makes possible solutions that have been impossible to formulate before (Corry, 1993; Dauben, 1984). Similar differences between the natural sciences and mathematics are also noted by some of the authors in the present issue.

We find this discussion particularly interesting because it helps to illuminate some of the debates that have taken place in the conceptual change approach literature as it applies to learning situations. More specifically, a number of researchers have pointed out that even in the case of the natural sciences conceptual change should not be seen in terms of the replacement of students’ naı¨ve physics with the ‘‘correct’’ scientific theory but in terms of enabling students to develop multiple perspectives and/or more abstract explanatory frameworks with greater generality and power (Driver et al., 1994; Spada, 1994). It thus appears that the theory replacement issue (which represents a significant difference in the historical development of the natural sciences compared to mathematics) may not be an issue in the case of learning and instruction.

In fact, students are confronted with similar situations when they learn mathematics and science. As it is the case that students developa naı¨ve physics on the basis of everyday experience, they also develop a ‘‘naı¨ve mathematics’’, which appears to be neurologically based (developed through a long process of evolution),

and that consists of certain core principles or presuppositions (such as the presupposition of discreteness in the number concept) that facilitate some kinds of learning but inhibit others (Dehaene, 1998; Gelman, 2000; Lipton & Spelke, 2003).

Such similarities support the argument that the conceptual change approach can be fruitfully applied in the case of learning mathematics.