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Yes, yes, yes.

I have tried (fitfully) three times to sit down and learn what calculus actually means. I still don't get it. And I suspect that clients and families will push it ever further away.

Yes, give me real world puzzles and applications for learning new mental tools and I will probably love them. Make it abstract enough and I cannot see the value in learning it.

I was the same way :). I have an intuition-first guide to calculus, here:


I think the essentials on how to think with calculus can be conveyed in 1 minute. Let me know if the above helps :).

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 :-)

Great point, thanks for the comment.

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.

I loved the explanation of calculating area of a circle through an unwrapping of infinite rings. brilliant thank you

Try "The Hitchhiker's Guide to Calculus" by Michael Spivak. After reading that, it all just made sense and at the same time made me angry that they didn't teach it like that to us in high school.

Thanks for the suggestion!

For a good motivated treatment of calculus I'd recommend going back to Isaac Newton's Principia. It is surprisingly accessible, and really quite beautiful.

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