| Abstract: | In the course of a research project now in progress, three successive division problems were presented to students in Grades 7–12. The first problem concerned a geometrical line segment, while the other two dealt with material substances (copper wire and water). All three problems involved the same process: successive division. Two of the problems (line segment and copper wire) were also figurally similar. Our data indicate that the similarity in the process had a profound effect on students' responses. The effect of the similarity in process suggests that the repeated process of division has a coercive effect, imposing itself on students' responses and encouraging then to view successive division processes as finite or infinite regardless of the content of the problem. It is possible to trace out, step by step, a more or less parallel process of development for the ideas of points and continuity and those dealing with atoms and physical objects in the child's conception of the ideal world. The only difference between these two processes is that to the child's way of thinking physical points or atoms still possess surface and volume, whereas mathematical points tend to lose all extension (though during the stages of development which concerns us here, this remains only a tendency.) (Piaget & Inhelder, 1948, pp. 126). Our first naive impression of nature and matter is that of continuity. Be it a piece of matter or a volume of liquid we invariably conceive it as divisible into infinity, and even so small a part of it appears to us to possess the same properties as the whole. (Hilbert, 1925, pp. 162). |
