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I've got somewhat an issue: from time to time I have to teach some math to people who either avoided it or get trough by only knowing a few working algorithms.

I want to be sure that my lessons not only provide primitive formulas or concepts, but motivate students for learning more about the topic.

Doing that I usually find myself troubled by lacking proper examples of why a sophisticated theory can actually turn out to be benifitial for anyone who isn't pursuing a career of a scholar.

As for now I'm limited to usual stuff:

1) Showing how knowing analysis tricks can make computations easier (L'Hopital's rule and all that) or even possible (in case our only option for a function is approximating it with a series)

2) Show how grasping visual ideas can make it much easier to use all monstrous formulas — case of basic linear algebra.

3) Explaining how treating multiplication as area can allow to derive all neede polynomial formulas (including solving quadratics) without knowing anything else.

4) Refering to how going beyond the scope of a usual math course can make life easier while dealing with polynomials — e.g. knowing the Fundamental theorem and rules of polynomial division, rational roots test etc.

In a desperate search of inspiration I've watched some video courses (the best one yet is Revisiting Calculus by Herb Gross) as well as popular math youtubbers (Numberphile, Mathologer, 3Blue1Brown and others), but still feel like I can use a lot of help from people of the Math community who have way more experience in the field.

Thanks kindly for your insights and suggestions.

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    Perhaps this question is more appropriate on the Mathematics Educators site. – 2017-02-07
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    In any case, you should ask a more focused question. As it stands, what you wrote asks for examples of all subjects... – 2017-02-07
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    @pseudoeuclidean perhaps so, still I've seen a lot of questions like this here and most of the time answers were plain great – 2017-02-07
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    @MarianoSuárez-Álvarez well, since I'm not teaching anything beyond the usual set of topics (basic school stuff, trigonometry, calculus, basic linear algebra etc) I don't think there're many subjects to cover. But thanks for your input, I gotta be more precise. – 2017-02-07
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    You say they are not pursuing a career as a scholar, but what career *are* they pursuing? It should make a big difference whether they want to become a blacksmith or a financial broker. – 2017-02-07
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    @celtschk usually they have some pointless education in 'management' or 'economics', so they deal with math they hope to get comfortable with during the education. Otherwise they'd have some ordinary office jobs feeling that knowing some rigorous methods makes it easier to do stuff rather than relying on 'intuition'. – 2017-02-07
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    @user2057368, I suggest you browse a bit the Math Educators stackexhange site, see how they do things, and ask a new, better question there. Here it is mostly offtopic. – 2017-02-07
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    @user2057368, "basic school stuff, trigonometry, calculus, basic linear algebra" are four, somewhat immense, and rather different subjects ---in their contents, their requisites, and in several other respects. That you think that there are not many subjects to cover, even after my having dropped the "etc" is somewhat weird. – 2017-02-07
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    @MarianoSuárez-Álvarez thanks for your suggestion, I'll try asking it there (didn't know there's a special stackexchange for educators) – 2017-02-07

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