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One is often drawn to offbeat mathematical ideas and how they could revolutionize mathematics or at least make maths more easy to learn.

Current examples are:

Of course, such innovation can be seen in a historical perspective where all math concepts were at some time radical (irrational numbers, logarithms or hyperbolic geometry).

Also, these innovations can be seen to reduce cognitive load (division in base 12 is easier), simplify notation (replace 2 \pi with \tau), enrich our vocabulary (sets + morphisms = category) or clarify important concepts (triangle geometry vs circle geometry).

But is it worth pursuing these ideas and their actual use in the classroom? Or do they just get in the way and we should just accept things for what they are?

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    $\begingroup$ To me, the five bullet-pointed items look like a grab bag of things that are of very different characters and relevance to education. Tau versus pi is just a trivial notational matter. Base 12 is impractical educationally and socially. Non-standard calculus just adds justification to how calculus has always been practiced by scientists and engineers. Category theory is completely mainstream for professional mathematicians, but the question is whether it can or should filter down into the educational system. I don't see any reasonable way to discuss all of these things on the same footing. $\endgroup$ – Ben Crowell Jul 8 '17 at 12:53
  • $\begingroup$ @Ben Crowell + others : I agree this is a grab bag of ideas on the spectrum of crank to mainstream. Each needs to be looked at independently. Rational trigonometry seems compelling and its founder seems intent on pushing its educational benefits see youtube.com/watch?v=K0ll4_xOtzQ. $\endgroup$ – pdmclean Jul 8 '17 at 23:07
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It may or not be worth pursuing them yourself (often not). But even if worth it for yourself, it is a very bad idea for the students. these are kids who are new to normal trigonometry. It is enough of rush of strangeness just to cover that. And it is hard for them. And it has high utility in other math courses and in physics. Concentrate on the immediate task, which is plenty hard enough, to teach normal math.

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    $\begingroup$ +1 ... Category theory and nonstandard calculus may be taught in more advanced contexts for specialized students. The others listed may be done as "recreational" or "enrichment" topics. But (as guest said) they will not help students in basic courses. $\endgroup$ – Gerald Edgar Jul 8 '17 at 12:49
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I would strongly advise against teaching any of these topics except possibly non-standard calculus. (Even then you should spend some time to mention limits.) A large part of early math education is learning the usual language so that your students can communicate with others. Using radically non-standard methods robs them of the enormous benefits of communication without a benefit of compensating size. (i.e. Base-12 arithmetic buys you easier division by 3, which is a pretty mild benefit. More usefully, learn base-10 to communicate with people and base-2 to communicate with computers.)

Non-standard calculus should, however, be fine. After all, it looks exactly like standard calculus other than in theorems involving limits.

Category theory is also deserves its own separate mention. It just doesn't have any obvious benefit unless the objects you are dealing with are fairly abstract.

P.S. Are your completely sure that rational trig is even true? That website seems a bit ... off.

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    $\begingroup$ Limits are already a main topic of nonstandard calculus; what the infinitesimals replace is the ϵ-δ argument form. $\endgroup$ – user797 Jul 8 '17 at 20:19
  • $\begingroup$ @Hurkyl My comment claimed that the theorems which involved limits looked different, not that non-standard calculus did not have the concept of limits. $\endgroup$ – Adam Jul 8 '17 at 20:30
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    $\begingroup$ Regarding the PS, I've looked rational trig in the past. My ultimate impression is that it's legitimate, but interpolates poorly between classical trig and vector algebra: rational trig falls well short of the algebraic simplicity of vector algebra, and doesn't share with classical trig the reasons one might actually prefer trig over vector algebra. $\endgroup$ – user797 Jul 8 '17 at 20:44

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