Guiding Students to Discover Counting Formulas - With a Modeling Twist

Introduction

In undergraduate mathematics classes, counting is often taught by presenting
formulas for students to memorize so they can solve a standard set
of problems.

We provide an alternative to this approach in the form of an activity that
is appropriate to use in any course that covers counting. It is designed to
allow students to discover counting formulas by engaging in data-gathering
simulations, with each simulation representing one of 12 mathematicallydistinct
counting problems. Ultimately, students may discover the entirety
of Gian-Carlo Rota and Joel Spencer’s “Twelvefold Way,” which we
present in the Appendix.