Where is balance Academic priorities

OK smoothness, continuity and analyticity are all nice. BUT when do you start teaching people that functions with even C0 are the *rare* exception in any function space and what some of the implications are of that? On the other hand, one can get so lost in drawing pretty pictures of fractals and computing iterative maps as to loose sight of the fact that while amusing diversions these things also have applicability to real world problems and computation. And even if one balances these two *theoretical* idea-sets, somewhen one must get the students to understand that while the coastline of England is fractal, that considering it at scales of 1cm or lower is not merely unnecessary precision, but, in fact, rather without meaning.

The concepts without both the tools of computation and real example problems to provide context are sterile. The computational tools without concepts end up being rote memorization and without real example problems are typically without interest to most students. And real world example problems without tools of computation are unsolvable, and without concepts the solutions, and often the problems as well, are incomprehensible to most students.

Just as the oft-perceived tension between analytic tools and numeric tools is artifactual -- really good design and analysis requires availability of both -- so is any perception of tension between concept, (good) examples and tools-methods in teaching an illusion -- without all three most students are operating under good, old "garbage in, garbage out".