This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms filled by students. More often than not, I use the so-called problem method in the courses I teach, and I advocate a particular philosophy of student-centered teaching. Yet, when I evaluate myself, something bothers me. As a professional mathematics educator, the best I can do is to help my students to learn the concepts and the techniques of the course **internally**, i.e. bounded to the syllabus of the course.

What if I could see beyond the course? What if I was an active mathematician who indeed works with those concepts and techniques, and knows a more advanced and perhaps more general version of those ideas? I was faced with these questions years ago when people started to compare my teaching with the teaching of a mathematician who is indeed an excellent "traditional" lecturer. To my own view, in a sense he could give to his students "more", since he could also **see beyond the course**. I had forgotten the whole issue until the current term; for the first time I am teaching a course in **linear algebra and matrices** for mathematics undergraduate students. That excellent colleague of mine is not around now (!), but the question is badly with me:

If I could see beyond the course what ideas (concepts, techniques, theorems, proofs, problems) would I stress more?

To keep the question suitable for MO, please do not "argue", and just give one piece of concrete advice to a person who now teach to potentially some of your future colleagues!

PS. In this paper (Moore and Less; PRIMUS) you may find the story of the course that the comparison mentioned above started with.

drawing a pictureaspect of curve sketching is less important than theunderstanding how derivatives workaspect. $\endgroup$ – Nik Weaver Oct 8 '13 at 21:17Linear Algebra Done Rightby Sheldon Axler, andLinear Algebra and Its Applicationsby Gilbert Strang. Speaking of applications, the use of singular value decomposition and related techniques to extract information from large datasets is currently a fairly hot topic;Understanding Complex Datasetsby David Skillicorn is one entry point into the subject. $\endgroup$ – Timothy Chow Oct 9 '13 at 2:11