Third-grade students’ strategies for presenting permutations of three or four objects

Abstract

The basic notions of combinatorics, including systematic listing and permutations, are important topics of mathematics that are recommended to be learned eventually from elementary schools. However, there is little research in mathematics education regarding elementary students’ combinatorial strategies. The purpose of this research is to investigate the third-grade students’ strategies for presenting permutations of three or four objects. In a semi-structured interview, 21 third-grade students answered questions related to writing 3-digit and 4-digit numbers and arranging coloured dominoes. At a first glance, many of the students’ solutions seemed apparently random, but we designed a coding system and showed each student’s response in the form of a matrix and discovered some hidden strategies. We also used the levels of children’s combinatorial thinking presented by English (1992) (non-planning, transitional and odometer) as a basic framework to analyse the solution strategies of the participants. We found nine different strategies (mono-solution, incomplete-random, complete-random, cyclical, incomplete-difference-place, complete-difference-place, incomplete-odometer, irregular-odometer and regular-odometer) that students used to present the permutations of numbers and dominoes. We discussed these strategies accompanied by students’ responses in detail.