LINIER ALGEBRA

When I started teaching the subject I found three kinds of texts. There were applied mathematics books that avoid proofs and covered the linear algebra only as needed for their applications. There were advanced books that assumed students could understand their elegant proofs and also understand how to answer the homework questions having seen only one or two examples. And, there were books that spent a good part of the semester doing elementary things such as multiplying matrices and computing determinants, only to suddenly change level to working with definitions and proofs.

Each of these three types was a problem in my classroom. The applications were interesting but I wanted to focus on the linear algebra. The advanced books were beautiful but my students were not ready for them. And, the level-switching books resulted in a great deal of grief.
I took a level-switching book as an undergraduate, so I understood the struggle my students had with this. At the start of the semester they thought that these were like calculus books, where material labelled `proof' should be skipped in favor of the computational examples. Then, when the level switched, no amount of discussion on my part could convince students to switch with it, and the semester ended unhappily.
That is, while I wish I could say that my students now perform at the level of the advanced books, I cannot. However, as a teacher I can work steadily to bring them up to it over the course of our undergraduate program. This means stepping back from focussing on rote computations in favor of focussing on an understanding of the mathematics. It means proving things and having students understand, e.g., that matrix multiplication is the application of a linear function. But it means also avoiding an approach that is too advanced for the students: the presentation must emphasize motivation, must have many illustrative examples, and must include exercises with many of the medium-difficult questions that are a challenge to a learner without being overwhelming. And, it means communicating to my students that the change of focus is what we are up to, right from the start.