[impendia]

Yeah, math professor here, this drove me crazy.

For example, I remember looking at the linear algebra book my department had used previously. Early on, it introduced the concept of the transpose of a matrix:

https://en.wikipedia.org/wiki/Transpose

Superficially, it looks like something good to introduce. It is fodder for easy homework exercises, and there is a satisfyingly long list of formal properties satisfied.

But why? What does the transpose mean? For what sort of problem would you want to compute it?

There are good answers to these questions (see the "transpose of a linear map" section of the Wikipedia article I linked), but they are not easy for a beginner to the subject to appreciate.