Epistemological obstacles in combinatorics as praxeological gaps among pre-service mathematics teachers

Abstract

Combinatorics is widely regarded as difficult, yet most studies stop at classifying student errors without explaining their internal structures. This study reframed epistemological obstacles in combinatorial counting principles as praxeological gaps. Drawing on Brousseau's notion of obstacle and Chevallard's four-component praxeology (task, technique, technology, theory), it asks not where a difficulty comes from but which part of a learner's knowledge has failed to develop. Within the prospective phase of Didactical Design Research, we collected written responses to two counting problems from 83 pre-service mathematics teachers who had completed a Discrete Mathematics course, interviewed 17 of them, and read their textbooks praxeologically. Triangulating these sources, we identified three recurring gaps: a technique whose justification had collapsed into a surface keyword rule ("and" means multiply, "or" means add); a technique fitted only to familiar problems; and a technique applied with no justification at all. Across all, the practical block was secure while the theoretical block remained thin, an asymmetry mirrored in the textbook, which defined the rules through keywords rather than independence and disjointness. Reading obstacles this way turns a general diagnosis into a specific one, giving teacher educators a tool for designing instruction that targets the precise failed component of reading.