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Comment by semi-extrinsic

2 days ago

My two cents on this: I've done a lot of math, up to graduate courses in weird stuff like operator algebra. I've also read quite a bit of maths pedagogy.

I've come to understand that the key thing that determines success in math is ability to compress concepts.

When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concept in their minds. For them, taking the next step to multiplication is almost trivial.

When a college math student learns the triangle inequality, >99.99% understand it on a superficial level. But <0.01% compress it and play around with it in their minds, and can subsequently wield it like an elegant tool in surprising contexts. These are the people with "math minds".

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semi-extrinsic

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neom  2 days ago

wow.

I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))

I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.

  • semi-extrinsic  2 days ago

    You're welcome :)

    The foundations for these concepts were laid by Piaget and Brissiaud, but most of their work is in french. In English, "Young children reinvent arithmetic" by Kamii is an excellent and practically oriented book based on Piaget's theories, that you may find useful. Although it is 250 pages.

    This approach has become mainstream in maths teaching today, but unfortunately often misunderstood by teachers. The point of using different strategies to arrive at the same answer in arithmetics is NOT that children should memorize different strategies, but that they should be given as many tools as possible to increase the chance that they are able to play around with and compress the concept being learned.

    The clearest expression of the concept of compression is maybe in this paper, I don't know if it helps or if it's too academic.

    https://files.eric.ed.gov/fulltext/EJ780177.pdf

    • neom  2 days ago

      I should be able to chat with an llm about this paper, but my gut says you've given me the glimmer of where I need to go. This is something I've been deeply deeply frustrated about for 30 years now, I had really given up hope of ever being able to process mathematics (whatever they are) properly, it's a real task to figure out how to get someone to see how your brain work and then have them understand how to provide you with some framework to grasp what they know.

      Once again I wanted to thank you for slowing down and taking the time to leave this thoughtful comment, if everyone took 5 minutes to try to understand what the other person is saying to see if they can help, the world would be a considerably better place. Thank you.

  • Cyber_Mobius  2 days ago

    Just a tangent, but there's a nice trick for 8 x 12.

    In algebra, you learn that (a - b)(a + b) = a^2 - b^2. It's not too hard to spot this when it's all variables with a little practice but it's easy to overlook that you can apply this to arithmetic too anywhere that you can rewrite a problem as (a-b)(a+b). This happens when the difference between the two numbers you're trying to multiply is even.

    For a, take the halfway point between the two numbers, and for b, take half the difference between the numbers. So a = (8 + 12) / 2 = 10. b = (12 - 8) / 2 = 2.

    Here, 8 = 10 - 2 and 12 = 10 + 2. So you can do something like (10 - 2)(10 + 2) = 10^2 - 2^2 = 100 - 4 = 96.

    It's kind of a tossup if it's more useful on these smaller problems but it can be pretty fun to apply it to something like 17 x 23 which looks daunting on its own but 17 x 23 = (20-3)(20+3) = 20^2 - 3^2 = 400 - 9 = 391

  • eszed  2 days ago

    Calculating 8x12 in my head relies on a trick / technique - they call it "chunking", I believe, in the Common Core maths curriculum that US parents get so angry about - that (I'm also in my 40s) was never demonstrated in schools when we were kids. (They tried to make me memorize the 12x table, which I couldn't, so I calculated it my way instead; took a little longer, but not so much that anyone caught on that I wasn't doing what the teacher said.) I'd like to think I was smart enough to work it out for myself, but I suspect my dad showed it to me.

    I'll show it to you, but first: are you able to add 80 + 16 in your head? (There's another trick to learn for that.)