Understanding topological relationships through comparisons of similar knots

This paper examines an example of learning with artifacts using the commonplace materials of string and knots. Emphases include research into learning processes as well as construction of objects to assist learning. The inquiry concerns the development of mathematical thinking, topology in particular. The research methodology combines participant observation and clinical interview within a constructionist framework. The study was set in a self-styled, self-constructed environment that consisted of knots and a social substrate encouraging lively exchanges of ideas about them. Comparisons of certain knots helped to elicit conceptions of the fundamental topological relationships of neighborhood, continuity, and boundaries. The paper includes comments on the suitability of specific artifacts for specific kinds of thinking and learning, and emphasizes the importance for software design of considering different learning styles.