Author:

Keywords:

Isis problem, Proof and proving, Routine and adaptive expertise, Mathematical representations, Dimensionality, proof, proving

Abstract:

The Isis problem, so named because of a mystical connection with the cult of the Egyptian goddess Isis, is as follows: Which rectangles, with whole-number sides (in some unit), have the property that the area and the rectangle are numerically) equal? For example, a 2 × 5 rectangle has area 10 and perimeter 14, so it does not have the property. In the fist part of this workshop, participants will be invited to work in groups on the Isis problem. More concretely, we will ask them to answer the following three questions: 1. Find all rectangles with whole-number sides have area = perimeter. 2. Prove what you find (i.e. proof that this are the only ones). 3. Look for alternative proofs. Participants will experience that it is the kind of problem that can be solved by clear thinking without needing advanced mathematical techniques. On the other hand, it is not immediately obvious how routine methods learnt in school can be applied. In the second part of this workshop, we will discuss the different proofs given by the participants (supplied with some proofs that were found by a group of Flemish pre-service mathematics teachers). Comparison of a variety of proofs is a good context for discussing with students and teachers ideas such as elegance and transparency of proofs, and the contrast between proofs that can be seen to be logically sound, and those that give a feeling of getting to the heart of the problem and explaining it. Finally, we will discuss some interesting extensions of the Isis probem and its relation to fundamental aspects of dimensionality.