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Comment by carapace | Hacker News Reader

This is hella cool.

"Knowing when the math is hard is way harder than the math itself"

But then maybe the math is hard only because it's not being explained well? (I hope it's uncontroversial to suggest that our current methods of teaching math are not the best of all possible worlds.)

I get that this problem came up in the context of of a math puzzle contest, and that some people enjoy solving puzzles. I am questioning their utility as an educational device.

I kinda think that we should teach math as fast as we can so that we can concentrate on the stuff that's really hard, not just apparently hard because someone is being coy with the easy routes.

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Are you trying to say that math is hard because the teachers are withholding all the tricks?

> I am questioning their utility as an educational device.

Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?

> Are you trying to say that math is hard because the teachers are withholding all the tricks?

Kind of, although I don't think they do it deliberately.

Things like teaching logarithms without a slide rule.

> Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?

You're right. Let me try again.

Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.

I think that most math seems hard to most people only because we are not creative in the ways that it is presented. We should use science to figure out how to present math so that people get it as fast as they can, in part so that we can find and concentrate on the actually hard math problems.

E.g. Alan Kay using Smalltalk to teach calculus to little kids in the context of modelling falling objects, to me kinda proves that it shouldn't take a whole semester to teach calculus to teenagers.

> Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.

Yes, those references demonstrate that some mathematical facts can be demonstrated easily, given the right presentation.

I agree that there is large gap between current mathematical presentations and the optimal presentation.

I am curious how effective the optimal presentation is. How easy can we make math? We have to be careful of the trap that the 3b1b video warns us against - when we understand something it is very difficult to put yourself in the shoes of a beginner. Something that looks like an elegant and clear presentation may seem like gibberish to the beginner (look at the YouTube comments on the 3b1b video).

I tried to google "Alan kay teaching calculus smalltalk" and didn't find anything. I am curious to see how much he actually taught the kids. It's clear that the typical college student, after taking a typical calculus class, doesn't really get the point of calculus. They may be able to follow some algorithms for differentiating and integrating but I don't think they understand when they can apply calculus.

I went and looked the Alan Kay thing up. It's not calculus, just acceleration. Sorry! It's described in "The Real Computer Revolution Hasn’t Happened Yet" http://www.vpri.org/pdf/m2007007a_revolution.pdf in the section titled "Children Discover, Measure, and Mathematically Model Galilean Gravity"

I think an important piece of the puzzle (no pun intended) is customized presentations with feedback for individually-tailored education. But then it occurs to me that group activity is also crucial for learning, eh? I'm not an expert.

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