A Mean-Based Approach for Teaching the Concept of Integration

The Riemann sum definition of integration is ubiquitous in college calculus courses. This, however, is not the only possible way to define an integral. Here, we present an alternative mean-based definition of integration. We conjecture that this definition is more accessible to students. In support of this proposition we first present a theoretical argument, pertaining to how human beings think about infinity, followed by a discussion of results from a small-scale teaching experiment. This teaching experiment was conducted with a small group of pre-service elementary school teachers all of whom had no prior calculus experience. These students have less exposure to mathematical content than typical students entering college calculus. However, they were successful in developing the mean-based formulation of integration. Additionally, our analysis of the struggles, conceptual stumbling blocks, and successes the group encountered is used to inform future teaching utilizing this alternative, mean-based definition of integration.