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Comment by somethingroma | Hacker News Reader

"This was the key. If you knew hhhhwwhy you didn’t have to memorize equations, or solve equations in the exact same way they did in the book..."

I feel like this is a valuable lesson a lot of people could benefit from.

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Knowing "why" is exteremely helpful in many cases. But let's not forget that, on the other hand, memorization plays a huge part in learning, and not only in mathematics. Learning "why" multiplication works will not help you to retain the multiplication table in your head. (Also, in many cases the answer to a question "why" you might get will be plain wrong - sometimes because the correct answer is too complicated, sometimes for other reasons.)

Memorizing multiplication tables is over rated. Seeing connections between those early multiples up to 12 is more interesting. Can be helpful I suppose in factorization.

Maybe I just don't trust my memory (as in how am I sure that 7*8 is 56) and why I despise rote memorization and rather retain memory from use and practice.

Regardless a good example of something to remember in math is the quadratic formula Still enjoyable to derive and 'see' why it works but also just used so much. That said, I wouldn't encourage memorizing it without first understanding it.

The best is the 9's table, how all the digits add up to 9. My mom's an elementary teacher and there's always a few third graders who figure that out on their own and love it.

I was legit angry that I didn't learn about the divisibility checks for 3 and 11 until college...

Additionally 9 * n = [n-1, 10-n] for n = 2-11; where n-1 is the digit in the 10's place and 10-n in the single place. This just an aesthetic curiosity. I know the pattern continues for larger n I've just never bothered to generalize it. Also never compared it to other bases.

> [n-1, 10-n] for n = 2-11

I think the usual 10 * n - n makes more sense and is much easier to remember.

It's not an aesthetic curiosity at all! 10 is equal to 1 mod 9... So suppose X is written in base 10 (a0 * 10^0 + a1 * 10^1 + a2 * 10^3 +...) and you want to find X mod 9. Then all of the 10^k's are just 1 (mod 9), so you just get the sum of the digits.

So if X is divisible by 9, then the sum of the digits (mod 9) is zero.

Same works for 3 (x is div by 3 iff the sum of the digits id divisible by 3). And 11 gets an /alternating/ sum of the digits, since 10 is -1 mod 11...

No. If you can’t multiply in your head, you are crippled in any quantitative reasoning. You have interjected too many steps in estimating, calculating, judging, etc.

This is not to say there aren’t useful, non-quantitative pursuits.

I don't understand what you're saying or objecting to.

Incase this is what you mean: remembering the multiplication table is the antithesis of multiplying in ones head.

> remembering the multiplication table is the antithesis of multiplying in ones head

But it's not! Because in order to be able to multiply numbers in your head efficiently you must remember the multiplication table.

No. By definition remembering an answer is not the same as doing.

Additionally my experience tells me otherwise.

I'm sorry but this seems patently wrong to me. There's only so much working stack space in your brain. If you're constantly having to think through multiple steps to multiply single digits then you're going to be at a serious disadvantage when you need to solve a bigger problem that involves more work than just single-digit multiplication.

No remembering the table is the ROM-table part. You need the algorithm part too.

I don't think that it's "memorization" that's the key: I think it's repetition. Learning anything, but especially mathematics, requires you to just sit down and solve problems over and over again. That is what builds the right connections in your mind, gives you a feel for the shape of the solution, and gives you good intuitions about problems.

Like: it's a pretty useless skill to have memorized what the anti-derivative of the cosecant function is. But if you solve enough integrals, and are familiar with the techniques, you can see the path to the solution of the problem, even without having memorized every single function and anti-derivative.

You can just do a lot of multiplications, and let your brain cache the results over time.

This. Although said tongue-in-cheek, is a basic principle. Once you understand the concept and the mechanics behind something, what rote learning buys you is to free cycles in the future so that you can take up the next-level task. Skipping this step is like expecting someone to play the piano just by learning how to read a music sheet and where each note is in the keyboard. You build up your muscle memory so that you're able to take on more complex pieces as you progress.

Spending time memorizing the multiplication table, then, is more efficient on both accounts: you will not need to do "a lot of multiplications" to begin with, and it takes less time for your brain to "cache the results".

Maybe that works best for some people, but as a kid I didn't do it. I used a printed multiplication table while tackling some more-interesting problems[0], and let the table soak in as a byproduct. It went quickly and did not turn me off on math for life. Paul Lockhart in Arithmetic also recommended this.

[0]: Multiplying two-digit numbers by one-digit, and that sort of thing. Lockhart had more artistic pursuits in mind.

(An obvious reason this might not apply: I was more talented than my grade-school peers. But most kids would learn arithmetic better if not forced to before they're ready, and then the talent difference would matter less.)

May I add my support to your opinion? Or, to put it more bluntly, the "why" is overrated. The "modern" math education (in the US) comes courtesy of people like Jean Piaget or Seymour Papert. These guys were true geniuses, but made a grave error. They generalized from their own experience ("to understand is to invent") to everyone else. Unfortunately, for 99% of the people, the style of learning by discovery is extremely inefficient. So we ended up with a system that sounds great in theory, but in practice you have kids in the fourth of fifth grade still going over addition and multiplication.

I agree as a generalization, but that does presume that the syllabus itself doesn't need rote interpretation.

  “No, we haven’t learned about some of the vectors yet, so for the sake of the problem, we just take this one out,”

Many many students self-combust when faced with garbage worded questions because they never learnt how to deconstruct them in class. When questions come with spelling errors and ambiguous variables it's tough.

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