The Wittig reaction is one of the most useful reactions in organic chemistry. Despite its prominence early in the organic chemistry curriculum, the exact mechanism of this reaction is still under debate, and this controversy is often neglected in the classroom. Introducing a simple computational study of the Wittig reaction illustrates the mechanistic uncertainty in this reaction and also demonstrates the use of computational methods in chemistry. This exercise is designed to be used with the Hyperchem software package, but can easily be adapted to other applications, both commercial and freeware. Learning outcomes focus on specific details of the Wittig reaction, the uncertainty of reaction mechanisms in general, and thermodynamic aspects of reactions.