Group Theory in School Mathematics? Teaching Permutation Cycles Through the 15-Puzzle

Abstract

Permutation cycles are generally associated with undergraduate abstract algebra. This exploratory study examined whether students in Grades 5-11 could construct and use cycle representations in the context of the 15-puzzle. After a 45-minute teacher-guided lesson moving from puzzle manipulation to arrow diagrams and cycle notation, 313 students analyzed one of two new configurations - one solvable and one unsolvable - and used a supplied rule to classify it. Of these students, $78.3\%$ constructed a correct cycle representation, and $67.7\%$ both constructed the representation correctly and reached the correct classification. Cycle-construction accuracy was similar for the two configurations, but classification was less often correct for the unsolvable configuration. The findings concern immediate, supported performance rather than full understanding of permutation cycles or group theory. Nevertheless, they show that many students could use cycle representations after brief instruction and that constructing the representation and using it to reach a conclusion were separate demands. One reason to teach permutations is that, as finite, discrete, non-formulaic functions, they can extend students' experience of functions beyond familiar formulae and continuous graphs.