Parity Theorem for Permutations - Pedagogical Observations

John O. Kiltinen

Since this approach to the Parity Theorem and the supporting demonstration software are new, I have had only one abstract algebra class in which to test it. This limited experience gives some pedagogical insight, however, which others might find helpful.

I used the program in the lecture in which I presented the Parity Theorem to the class. The students were given the software to use on their own computers as they studied the theorem. They were told they would be required to demonstrate their ability to use the algorithm on the mid-semester exam. I tested them by watching them carry out the algorithm with the demonstration software set in the “Computer First” mode, wherein the computer performs transpositions and the human user responds to produce a shorter sequence of transpositions.

The highlighting feature was turned off, so students had to know without the visual cue which box was the fixed First Box and which was the movable Tagged Box. The result was that all eight students in the course were able to demonstrate the ability to carry out the algorithm. Only two of them made a single mistake each, and each corrected the error on a second try.

The students were also told that on the final exam there would be a question asking them to explain why the shortening algorithm leads to a proof of the Parity Theorem. The results here confirm an experienced teacher’s insight that knowing how to perform an algorithm is a different thing from understanding its significance. The students’ responses varied from impressionistic discussions that focused too much attention on the reduction by two transpositions to ones that focused on the appropriate sequence of ideas but left out some important details.

I conclude that this proof offers no royal road to immediate understanding of the proof of the Parity Theorem. However, by giving them an approach that not only stays close to the definition of parity but also builds on hands-on experience, it lays a foundation for deeper understanding for those students who choose to return to the topic looking for mastery. This proof also has advantages for a lower division discrete mathematics course where there is less emphasis on proofs but where an appreciation for algorithms is important.