Where does the "Visual Multiplication" technique originate from?

Question

There is a geometric technique to perform multiplication of numbers.

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But as the internet goes, it is hard to figure out who deserves the credit. What I've heard is

A mayan technique
From Vedic mathematics (possibly from the equally named book from Bharati Krishna Tirthaji)
Used in Japanese schools to teach kids about multiplication.

I would love it if somebody could shed some light on the origins of this technique.

This is more or less exactly the way most people learn to multiply, only it's with lines instead of numbers. If we look at the then's digit, for instance, we have $13$ intersections, but it's clearly visualized as $\color{red}{2} \times \color{gray}{4} + \color{green}{1}\times \color{pink}{5}$. Now, if you calculate it as you've learned to in school, then I'm willing to bet that you do the same calculation somewhere in there, albeit maybe hidden away. See for instance this video for more details. So IMO it's no more straight-forward than usual. — Arthur, Aug 03 '14 at 10:31
I didn't want to imply that the way we learn it at school, say, in Europe is more complicated, but sometimes visualising a technique can help certain students grok something better. — flq, Aug 03 '14 at 10:35
True. But seing how many think this is much easier and straight-forward without realising that it's the exact same thing always reminds me of how little the average person understands something so simple as "What is multiplication, really?" I see it as an indicator of how mathematics education all over the world must be wrong somehow if most people miss something like this, and it makes me angry at the world in general. I wrote what I wrote to vent out general anger, and it wasn't actually directed at you in any way. — Arthur, Aug 03 '14 at 10:42
It's difficult to come up with a sicker way to teach kids multiplication. — Christian Blatter, Aug 07 '14 at 09:52
Do you mean sicker or slicker? For a pair of digits it's a good visualisation. The significance of the groups' representing powers of ten, though, is obscure. If you try the method with four-digit numbers (or even some large digits in there) it is tricky to get your intersections to line up neatly, and very boring to draw. You shouldn't be counting the intersections anyway, once you've learned your tables. My vote is for sick, not slick. — HTFB, Aug 08 '14 at 13:35
I think that the hypothesis that it's a Mayan technique can be ruled out. The MacTutor page on Mayan mathematics is pretty informative, and contains the statement, "We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers." Some Mayan numerals do make use of sets of parallel horizontal lines, which may, at some point, have suggested to somebody a connection with visual multiplication, but I doubt there's anything to it. — Will Orrick, Aug 12 '14 at 16:58
@WillOrrick Your link to the story does not work anymore, what was the story? This was asked again, on hsm this time. Is there any non-anecdotal evidence that this was in used before 2006? Thank you. — Conifold, Aug 07 '19 at 23:48
@Conifold Foolish to rely on Google+ always being around. I looked into this back in 2014 and could find nothing on the internet prior to 2006, which makes me think that the method emerged around that time. The story was that the author of the 2006 MetaCafe video, who I believe is Indian, learned the method from his Chinese girlfriend who, in turn, learned the method from her schoolteacher. If I recall correctly, the MetaCafe author and his girlfriend were of the opinion that the method was an original invention of that schoolteacher. — Will Orrick, Aug 08 '19 at 01:46
@Conifold I'm not entirely sure what was in that Google+ post, but it may have been the original source for the story Bill Hart tells in a comment to this video by Vi Hart. Since I don't know how to get to that comment without scrolling for a very long time, here's an alternative link. Not sure how permanent this one will be. — Will Orrick, Aug 08 '19 at 01:50
Thank you again. I wrote an answer on hsm with credit to you and quotes from the linked comment. It is interesting that nobody mentions the original Chinese inventor's name. — Conifold, Aug 08 '19 at 05:47
@Conifold, since you wrote an answer in HSM, could you post an answer here pointing to HSM? I think it's time to get this question out of the Unanswered category. — brainjam, Aug 18 '20 at 20:49

score 2 · Accepted Answer · answered Oct 26 '20 at 20:08

There is an accepted answer at History of Science and Mathematics:

It is a fun method but it appears to be very recent. It is characterized as Chinese, Japanese, Korean, Indian, or even Mayan method in various internet posts, all of them recent, and without attribution, naturally. The "ancient origin" story is most likely made up ...

score 0 · Answer 2 · answered Aug 09 '14 at 02:09

0

Let's make a table:

        300 +   20 +   1
      ------------------      ----> 10000 x        6  = 60000
200 | 60000 + 4000 + 200             1000 x    (15+4) = 19000
 50 | 15000 + 1000 +  50              100 x (12+10+2) =  2400
  4 |  1200 +   80 +   4               10 x    ( 8+5) =   130
                                        1 x        4  =     4
                                    -------------------------
                                                        81434

This is a very organized way to use the FOIL identity:

$$ (100x + 10y + z)(100a + 10b + c) = 10^4a +10^3(ay+bx)+10^2(az+by+cx)+10(bz+cy)+cz$$

answered Aug 09 '14 at 02:09

cactus314

24,438

11

This is a history of math question. It's not asking why the method works. – Will Orrick Aug 09 '14 at 03:23
4

Considering that I did not ask for this but is a useful information nonetheless, would you be ok with me including it in the question (with proper credits) and y removing this answer? – flq Aug 13 '14 at 20:17
2

what is "FOIL"? – Rolazaro Azeveires Nov 29 '17 at 13:24
1

Also the Trachtenberg speed system puts an extra dot at intermediate places between each column to evaluate local square matrices.Believe it was from ancient Vedic maths; and there is also an unverified connection to the mayan civilization. – Narasimham Jan 31 '18 at 06:29
4

@RolazaroAzeveires it's an acronym for "First, Outside, Inside, Last" which is a mnemonic for multiplying two linear terms of the form $(ax+b)(cx+d)$ to make a quadratic which is taught in North American schools. It's basically a specialized case of the distributive rule for exactly this form which tends to confuse students the moments they try to multiply polynomials with more terms. – CyclotomicField May 15 '18 at 14:19
@CyclotomicField: thank you – Rolazaro Azeveires May 17 '18 at 18:50

Where does the "Visual Multiplication" technique originate from?

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