Teaching Children to be Mathematicians Versus Teaching About Mathematics

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Abstract
The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child's work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is ‘Turtle Geometry’, which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a ‘cybernetic turtle’ controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical performance.