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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 got trough by only knowing a few working algorithms. The only thing that unites all those people: they are sure that math is extremely boring and there's no sense an learning anything except for how to solve the tasks they're given.

I want to be sure that my lessons not only provide primitive formulas or concepts, but motivate students for learning more about the topic, thus I need to show how all that 'pointless knowledge' may turn out to be not so pointless.

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, programmer, nuclear physicist etc.

As for now I'm limited to usual stuff:

1) Showing how some 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 (determinants, linear transformations, matrix multiplication etc.)

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.

I don't teach anything fancy: basic school stuff, some trigonometry, calculus, basic linear algebra and all that (no probability theory so far)

Thanks kindly for your insights and suggestions.

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    $\begingroup$ My sense is that this question is overly broad, as the kind of example you need will necessarily vary from topic to topic. However in general I think it is always possibly to justify the position that "remembering stuff" is almost always less reliable than "understanding stuff". Memories fail; formulas and mnemonics are easily mis-remembered when we need them. Once we understand something, on the other hand, it tends to stay understood. $\endgroup$ – mweiss Feb 7 '17 at 22:50
  • $\begingroup$ @mweiss thanks, you got my idea exactly — understanding demands less memory a proves to be more sustainable in the long term. However, I'm kinda lost in terms of advices. For example, should I tell them to study some group theory, will it help them in their tasks? Any use in studying all the rigorous theorems of classical analysis (with all the points and segments)? Perhaps suggesting basic stuff from topology could come in handy? Will simmetry and transofrmation groups somehow be useful in working with typical figures in ${\mathbb R}^2$ and ${\mathbb R}^3$? $\endgroup$ – user2057368 Feb 8 '17 at 7:22
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    $\begingroup$ Are you talking about people who are resistant to theory of all kinds, or only to theory in mathematics? If it's the latter, you can leverage their appreciation of theory in some other field. If it's the former, you need to be working with the people who are teaching the students in other subjects. I don't think it's particularly helpful to see this as an issue isolated to mathematics. $\endgroup$ – Alexander Woo Feb 10 '17 at 4:23
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    $\begingroup$ @user2057368 - there is theory in the humanities also - in fact in every field of academic study. What kinds of theory are they getting in their other studies? Do they care about that either? If they don't care about ideas and thinking at all, why are they in school? $\endgroup$ – Alexander Woo Feb 10 '17 at 16:13
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    $\begingroup$ I do feel this question is too broad (voting to close; no answers after a bout a week; voting to close). Perhaps if you searched/asked questions about one particular subject at a time? $\endgroup$ – Daniel R. Collins Feb 14 '17 at 0:28

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