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Comment by hemancuso | Hacker News Reader

This does not teach you multiplication as much as it teaches you a trick to get the result of multiplication. I doubt anyone is transferring this abstraction that results in the answer into something they can do in their head or extend on paper to larger numbers.

Teach kids to open the calculator app on their phone rather than to do this.

No fast way to learn multiplication other than to practice it.

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I desagree completely. It's not a trick. This is exactly how multiplication works. The idea behind 2x3=6 is

   |  |  | 
  -+--+--+-
  -+--+--+-
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In this method the number are decomposed using it's decimal representation, so 23x12 = (2 * 10+3) * (1 * 10+2) = 2 * 1 * 10^2+2 * 2 * 10 + 3 * 1 * 10 + 3 * 2 = 2 * 10^2+ (4+3) * 10 + 6 = 2 * 10^2+ 7 * 10 + 6 = 276

(I'd like to use bigger lines for the dozens.) This is exactly what happens in the method. See: http://imgur.com/S5nOh

If I had to use that in a class I would first use the "all graphical" representation, then the "mixed" representation and finally the "algebraic" representation. The lines are still there, but almost invisible.

I understand that most of the times nobody should use any hand method to multiply two four or five digit numbers. But some properties of the hand method are important, for example:

* Why to use approximated calculation you use the first digit and no the last digit?

* Why is possible to calculate the last digit of the result using only the original last digits of each number to multiply?

* How is this method related to polynomials multiplication?

* How is this related to the casting out nines check?

* Can you imagine the multiplication of a large number by 2 using this method? 3?

Most of these topics are not explored in a usual K-12 math course, and perhaps it's a good idea because some of them are a little tricky. But the proof of how this method works lies in the structure of the decimal representations of the numbers and the algebraic relations between the sum and multiplications. I think that for small children a graphic method like this one can give some insight of these properties, without all the details and formalizations.

It's also algorithmicly identical to the way people are taught to multiply in American schools now, except it introduces a geometric isomorphism that makes it easier for people to understand what's going on.

calculating it the western way (by multiplying and adding each column) is also a trick. it also can't be done in your head with larger numbers. These are just different approaches. when multiplying small numbers (probably most of the time for average person) this may be a faster approach.

This is exactly the same algorithm as the standard one we use : Multiply all possible combinations of digits and appropriately combine the results. This is just the graphical version of writing down numbers.

You could say the same thing about digital computers old and new. For example, for this (awesome) article http://horningtales.blogspot.com/2006/07/bit-serial-arithmet... describing an ancient bit-serial drum-memory computer, the author coins the acronym "AIGSA" to mean "as in grade school arithmetic".

But that just shows that we understand grade school multiplication. It doesn't mean that we know how best other people learn it.

This is how I was taught algebraic multiplication (using little plastic xs and ys and 1s instead of lines on paper) and I found it extremely helpful and it's still more or less how I visualize multiplication, dimensional analysis, etc in my head. I can see it making basic arithmetic easier to learn as well.

I was virtually immune to rote practice of intellectual tasks as a kid and mostly still am. I don't think I'm the only one, witness the near universal inability of US adults to perform long division, despite it being drilled into every schoolchild for hours on end.

I agree, I don't see how this actually teaches how multiplication actually works. It's a clever method to avoid solving the problem the classic way of using a memorized multiplication table and breaking down the problem into smaller segments. I don't even see how this saves you time once you get up to speed on either method. The only difference is that with my old school way I can explain to you why I get the answer I calculated while this method doesn't seem to offer that ability.

> I don't see how this actually teaches how multiplication actually works.

[snip]

> It's a clever method to avoid solving the problem the classic way of using a memorized multiplication table

Do you think multiplication actually works by means of a multiplication table?

No, if that's how I came across then that's not what I intended. Although, I feel you're a bit selective in your snips in an effort to make some kind of point. I know how multiplication works, but the memorization of the table speeds up the process. I was simply trying to say that this method was not much different, nor more superior, than table memorization.

My comment about not teaching how multiplication works is aimed at the statement made that this method is how Japanese students learn to multiply, as in the title of this thread.

I hate to disappoint you, but mathematics is nothing but a gigantic collection of tricks, rules, games and the like written in language impenetrable to outsiders.

If a mathematical paper had said something about graphical isomorphisms with a two-dimensional lattice, the average person would have been impressed, without understanding a word of it. But show them children doing it and suddenly it's a cheap trick.

Actually, thanks. That gives me an idea for the next time I have to explain higher math. I'll just find a way to show it to children first. They're usually easier to teach, anyhow.

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