Abstract:

The results presented here come from a study in which learning about topology was supported by working with knots. In particular, comparisons of similar knots became a useful way for people to see the elements of the configurations and come to understand relationships among the parts. This research is within the domain of mathematical thinking and is specifically concerned with how people develop understandings of the relationships of proximity that constitute topology. These relationships are characterized in terms of neighborhood, continuity, and boundaries. In combination with other basic conceptual structures - of classification, number, order, and seriations - the topological relationships are thought to give rise to understandings of mathematics.