First-Year Undergraduate Students’ Ways of Thinking in Combinatorics

Abstract

The fundamental counting principle and counting problems, jointly called combinatorics in this study, commence in the last year of secondary school and proceeds in undergraduate studies and beyond. The South African secondary school situation reflect the same worldwide scenario where counting principles is regarded as difficult and poorly performed in the final national examinations. The purpose of this study was to explore undergraduate students’ mental constructions in solving counting problems against the backdrop of Grade 12 fundamental counting principles. The Action-Process-Object-Schema theory was used to describe undergraduate students’ thinking ways in combinatorics. A single case of a first-year class of 182 students was considered in this study, whereby they all wrote a task on combinatorics and a seven were further interviewed. The findings revealed that students were skilled at solving problems involving the counting principles, which was mainly step-by-step application of the formulae. This conception is at action level according to APOS theory, but the goal of teaching is to guide students to attain the object mental conception. Object conception allows for solving of diverse real-world counting problems and promotes mathematical thinking skills.