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How Japanese Kids Learn To Multiply

magicalmaths.org

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Re: How Japanese Kids Learn To Multiply

#81

Post by mturmon »

ianb wrote: post #60
Someone wrote: post #14

This is a variant of http://en.wikipedia.org/wiki/Lattice_multiplication that does not make it as easy to carry tens a column to the left as that method does. I also think lattice multiplication makes it easier to understand for pupils why the trick actually performs a multiplication. So, I would teach them lattice multiplication instead.

Another approach: http://en.wikipedia.org/wiki/Grid_method_multiplication Of course it's all multiplication, and all essentially the same thing. I find the grid method the least notation-focused of at least these three, and so I think more accessible to assimilation. Maybe I'm biased, I didn't learn multiplication this way but found myself naturally doing it like this in my head and then was pleasantly surprised to f…

Thanks for the link. The grid is also helpful as a way to visualize things like:

  (1 + a)^2 = 1 + 2a + a^2
There is a square with side 1+a, and the grid consists of four pieces, sizes 1, a, a, and a^2.

Thinking about approximations when a 

  (1 + a)^2 ~= 1 + 2a
really easily by imagining the grid you refer to.

In other words, keeping this grid picture in mind can be helpful to more mature mathematicians/physicists/engineers...probably more useful than the standard multiplication algorithm is.

Re: How Japanese Kids Learn To Multiply

#82

Post by evincarofautumn »

Someone wrote: post #14

This is a variant of http://en.wikipedia.org/wiki/Lattice_multiplication that does not make it as easy to carry tens a column to the left as that method does. I also think lattice multiplication makes it easier to understand for pupils why the trick actually performs a multiplication. So, I would teach them lattice multiplication instead.

It’s also easy to do in your head, and works pleasantly for numbers in any base, including polynomials.

Re: How Japanese Kids Learn To Multiply

#83

Post by shabble »

solox3 wrote: post #51
talmand wrote: post #28

Earlier quoted context omitted.

It isn't that different, in my American elementary school I had to memorize the multiplication table up to 12x12. But I don't recall anything about the English language being a hindrance so your example is interesting. There doesn't seem to be that much fewer syllables in 7x7=49 in English but I'm assuming that as the numbers increase then the syllables get worse in Japanese as opposed to English? For instance, 148x3…

Using your example 7 x 7 = 49 seven [2] times [1] seven [2] equals [1] forty-nine [3] = 9 七[1] 乘[1] 七[1] 如[1] 四十九 [3] = 7 (without using primary school mnemonics) 七[1] 七[1] 四十九 [3] = 5 (using primary school mnemonics) Even at that primitive level, a change of language can give you a 23% and 45% speed-up of basic arithmetic operations, respectively.

A similar rote recitation exercise I recall from (a british) school was of the form '[Seven] [Seven][s] [is|are] [Forty-Nine]', of which only the 'is' is redundant, and is a single short syllable.

Re: How Japanese Kids Learn To Multiply

#84

Post by SiVal »

tokenadult wrote: post #9

The more important difference between mathematics education in Japan and mathematics education in (say) the United States is how hard the problems are and the encouragement to pupils in Japan to try to figure things out for themselves. I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils'…

Tokenadult, I always enjoy your well-informed comments on topics such as this. I think you may have misinterpreted Stigler's 1999 "The Teaching Gap", though, as many of us did. In that book, he reports on a study of math teaching in the US, Japan, and Germany, and finds Japan's results to be far superior to the others and their teaching methods very different, and different in exactly the way you describe.

But he did a followup study involving more countries to see if most or all high-performing countries used the Japanese approach. It turned out that they did not. Some were more like the US than the US.

Here (http://timssvideo.com/sites/default/files/Closing%20the%20Te...) is one place where Stigler reports his updated findings and attempts to debunk your (and my) initial conclusion---a conclusion that seemed strongly justified by his 1999 book---that, as you posted above, "The more important difference between mathematics education in Japan...and the US is...the encouragement to pupils in Japan to try to figure things out for themselves."

He points out that another high-performing country in his second study, Hong Kong, was more US-like and less Japan-like on this spectrum than the US itself. On the dimension you're calling "more important", the low-performing US is between the high-performing Hong Kong and the high-performing Japan.

His conclusion was that the main factor was not having kids figure things out for themselves but having teachers carefully teach kids the relationships among things. It didn't matter if the US kids spent time practicing procedures. The Chinese kids spent MORE time practicing procedures and did better, but then the Chinese teachers spent time directly pointing out important relationships, which the US teachers didn't do much of. The Japanese kids had to spend a lot of time figuring things out for themselves, but then the teachers would gather them together and carefully lead them to see relationships that they hadn't seen when working by themselves. The US teachers would tell kids to figure things out for themselves and basically leave their learning to whatever they managed to figure out.

Given equal IQ, time on task, etc., it's the effectiveness with which mathematical relationships are made clear to the students (part of which requires significant procedural drill, which Japanese kids do after school) that matters most. A lot of time is wasted in the US doing procedural drill with no conceptual understanding, with even more wasted on constructivist "discovery" methods whereby kids are supposed to somehow teach themselves and each other the mathematical relationships, and all of this led by teachers who aren't required by their union to even know anything about mathematical relationships much less teach them.

Re: How Japanese Kids Learn To Multiply

#85

Post by marshray »

sks wrote: post #62
hemancuso wrote: post #10

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.

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.

Re: How Japanese Kids Learn To Multiply

#86

Post by tokenadult »

SiVal wrote: post #84
tokenadult wrote: post #9

The more important difference between mathematics education in Japan and mathematics education in (say) the United States is how hard the problems are and the encouragement to pupils in Japan to try to figure things out for themselves. I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils'…

Tokenadult, I always enjoy your well-informed comments on topics such as this. I think you may have misinterpreted Stigler's 1999 "The Teaching Gap", though, as many of us did. In that book, he reports on a study of math teaching in the US, Japan, and Germany, and finds Japan's results to be far superior to the others and their teaching methods very different, and different in exactly the way you describe. But he did…

Thank you, SiVal. I didn't see contact information in your user profile (and indeed the contact information in my user profile is rather subtle until I do a personal website update). So here I will say thanks for your comment. I'll be revising some FAQs based on what you wrote. Feel free to contact me off-site if you'd like to discuss these issues more. (Much of today I am updating my personal website on its seventeenth birthday, and then I'll have to finish a revised FAQ promised to another participant here a few days ago, a response to a link that shows up too often in discussions on the topic of international educational comparisons.)

Re: How Japanese Kids Learn To Multiply

#87

Post by bcoates »

hemancuso wrote: post #10

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.

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.

Re: How Japanese Kids Learn To Multiply

#88

Post by aghull »

How convenient that all the examples use digits in the 1-3 range. This looks painful for normal numbers with a mix of large and small digits.

This method always made more sense to me: http://www.ehow.com/video_12244670_solve-multiplication-prob... as it combines the idea of the diagonal lines in the "japanese" method with the mnemonics of the [1-9]x[1-9] multiplication tables that everyone should also learn.

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