In this commentary, we make a case for the explicit inclusion of combinatorial topics in
mathematics curricula, where it is currently essentially absent. We suggest ways in which …

When solving counting problems, students often struggle with determining what they are
trying to count (and thus what problem type they are trying to solve and, ultimately, what …

Permutations and combinations are generally taught by requiring students to memorize
formulas and solve problems using the appropriate formula. Students who learn these topics …

Combinatorics is an area of mathematics with accessible, rich problems and applications in
a variety of fields. Combinatorial proof is an important topic within combinatorics that has …

The multiplication principle (MP) is foundational for combinatorial problem-solving. From a
units-coordination perspective, applying the MP with justification entails establishing unit …

In this paper, we present data from two iterative teaching experiments involving students'
constructions of four basic counting problems. The teaching experiments were designed to …

The multiplication principle (MP) is a fundamental aspect of combinatorial enumeration,
serving as an effective tool for solving counting problems and underlying many key …

Counting problems offer opportunities for rich mathematical thinking, and yet there is
evidence that students struggle to solve counting problems correctly. There is a need to …

Research has shown that solving counting problems correctly can be difficult for students at
all levels, and mathematics educators have sought to identify strategies and interventions to …

Combinatorial proof serves both as an important topic in combinatorics and as a type of
proof with certain properties and constraints. We report on a teaching experiment in which …