I see that your examples of teaching fast (one minute, ten minutes) all use discrete models for continuous processes. For people who understand continuousness mathematically, going between discrete models to continuous models and vice versa is easy. However, there are indications that the other way around -- students who perceive these changing situations mathematically as intrinsically discrete -- going from discrete to continuous models might be extreme difficult (due to in-commensurability). Of course, students can learn procedures to make the switch from discrete to continuous models, but even then it is possible that this procedural experience is not build on deeper mathematical understanding. Tricky stuff, learning mathematics :-)
I think the key missing insight for me was that a continuous process and a discrete process can both point to the same result.
A pixellated word on a screen ("cat") conveys the same meaning as a perfectly smooth vector of the same word. The math idea, to me, is "can a discrete description/process" point at the same result that we get from a continuous one?
The idea of limits, is essentially figuring out when a discrete epsilon can still lead us to the true value of f(x) (if this trick works, we call it a continuous function, and can use calculus with it.)
Yes. You might be interested in Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (Vol. 11, pp. 153–166). Retrieved from http://dx.doi.org/10.1007/0-306-47203-1_10 and similar works. Volumes have been written about the limit concept, learning the limit concept, and different approaches to the limit. It is one of the more interesting mathematical concepts one encounters in high school, not in the least because of the ability to use it successfully without understanding the underlying concept and the philosophical implications of the concept.
Unfortunately, I doubt that these implications will be discussed in your average Mathematics classroom. Actually, my experiences with mathematics in primary, secondary and a lot of tertiary education, both as a student, teacher and observator, leads me to believe that conceptual discussion is rare, whilst procedural discussions (how to perform some calculation) are plenty and abound.
