A computer-based approach to number theory learning : A review

Abstract

The visual presentation of the natural numbers within Number Worlds offers an alternate representation of the numbers compared to that of the more traditional number line. This alternate representation accentuates the patterns inherent in arithmetic sequences of multiples, as well as arithmetic sequences of non-multiples generated by ‘shifting’ the multiples by an integer value. It also allows users to see patterns, or lack thereof, in the structure of factors, square numbers, and primes. The second feature influencing the participants’ understanding of basic elementary number theory relationships lies in the ease with which the microworld accommodates experimentation. Participants were able to interact with Number Worlds in a way that allowed them the freedom to think both about the microworld and with it. These two features – visual representation and experimentation – are in fact closely interrelated. The crucial link between the two is the attention to patterns. Number patterns presented in a novel visual manner provoked aesthetic responses that invited experimentation, which generated new patterns, thereby reinforcing aesthetic engagement. Furthermore, the regularity of visual patterns led students to make conjectures, which could be in turn tested through experimentation and visual feedback. The participants’ conjectures show their perceptions of the relationships among the concepts of number theory, and the depiction of these concepts in the microworld.