This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Withou

This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Without these, there are no tangible points of reference, aside from derivations from known equations. But mere transformations rarely offer insight without adding some sort of context. I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.