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How Japanese Kids Learn To Multiply (magicalmaths.org)
272 points by TomAnthony on Jan 14, 2013 | hide | past | web | favorite | 99 comments



This is a method that is occasionally taught in Japanese classrooms, but one could also say that of chunking or the traditional long multiplication algorithm. Want to know the real super-secret-special Asian magic math sauce? Practice. (But if calling it kungfu math lets you teach it is a US classroom and get precious instructional hours by calling math cultural studies then by all means Orientalize away.)

[Edit: I have a pair of chips on my shoulder about this, one on my right shoulder and one on my left, as it were, and that's why I sort of blew up here. I really don't think I'm wrong re: the irritant, though. The SMSC referenced, for example, is Spiritual Moral Social Cultural (skills), and it is pretty much exactly as froufrou as you are guessing. You'll also note that every sentence in the post mentions "Japanese" and it's at the most rage-inducing Stuff White People Like level of surface-learning one could imagine, too.)

[Edit: Please up vote tokenadult's comment before this one, because while multiplication is taught by drill2kill here, that comment is better than mine re: math pedagogy generally.]


I tutored a student who counted on his fingers. He was 21.

Within 6 lessons, using Khan Academy, I had him adding four digit numbers in his head, multiplying large numbers, and doing algebra.

He would learn a technique from me or the video, then apply it in practice drills, then review at the next lesson.

Total instruction time was 12-20 hours. I think he did a bit on his own.

I concluded that no one had actually checked to see if he could understand math. North American students hardly try calculations now; they're given calculators at a very early age.


Not trying to disagree with you about the crappiness of this student's education, or how sad it is that he counted with his fingers, but computation != mathematics.

I HATE the fact that American schools force students to do rote computation over and over again, with very little focus on the concepts behind the computations. I hated "math" until my 6th grade homeroom (not math) teacher took the initiative of writing high school level algebra word problems on the board, and not telling us how to solve them until after we attempted them. "A train going from St. Louis is headed toward L.A., and another train from L.A. is headed to St. Louis. they are going 40 mph, when will they pass each other" etc.

Suddenly, when the computation becomes a tool to satisfy an end, rather than the end itself, math becomes interesting.


... computation != mathematics.

This is exactly why giving calculators to children who have not yet mastered any method of multiplication is bad.


Its kind of subjective though. In my schooling, when we moved from basic arithmetic to "word problems" as these were called (i.e which required you to formulate the mathematics yourself), I was completely lost and intensely dreaded the process. It seems a bit silly saying it, but at that time I didn't understand that I had to actually understand what the problem was asking of me. Before then, learning math had been "if you see A, do B" and I would frantically search for "patterns" to similar problems. It took me a long time to realize that the problem itself was telling me how to formulate the solution.

There is an amusing end to this story. When I entered college, I realized that this pattern-finding approach to solve math was used (in vain for the most part) by many of my classmates to try to game exams. It seems that it had worked well for them through high-school, but failed miserably at college.


> I concluded that no one had actually checked to see if he could understand math. North American students hardly try calculations now; they're given calculators at a very early age.

That was something that caught me off guard a few years ago. In 2008 I was considering becoming a teacher (high school math/technology). For various reasons I chose a different path, but I sat in on some courses at a friend's school to get a 'day in the life of' feel and an opportunity to talk to other teachers. One class caught me by surprise. It was a 7th grade math course, essentially pre-algebra. However, the students did everything with calculators. The lesson that day was on interest, simple and complex, a way to try and tie the concepts down to real world examples. The students did everything with calculators, not one was using pencil and paper to work out the math (though they all recorded the steps they'd done on the calculators). Talking to the teachers, that was how their curriculum was setup, not just a one off case.

That wasn't what turned me away from teaching, but left me confused about how things had changed so much in the 15 years or so since I'd been in those seats.


I think calculators for interest calculations makes a lot of sense. I'd be pretty impressed by students doing P(1 + r/M)^N without a calculator, though I suppose you could do repeated tedious multiplications of P*(long decimal number). We definitely used calculators for that 20 years ago.


Well, that particular exercise was something like: You have $P and the interest rate is I% per month, what would you have after 1 month, 2 months etc. It was easily doable at the level that exercise required by hand. However yes, if the formula had been more complicated then calculators would've made sense. From talking to the teachers though, the sense I got at the time was that calculators were never restricted regardless of the difficulty or ease of the exercise.

EDIT: Did mention complex interest, that part made sense to use the calculator for. My observation was merely attempting to point out the ubiquity of the calculator in the classroom. I'd be interested to see the capability of students after a curriculum like that at doing mental and hand calculations, my understanding is that here in Georgia the various classwide exams they take also permit calculators so the students would have little occasion to demonstrate that skillset.


Can't really blame you. I know next to nothing about teaching or Japanese culture, but this article gave me a sick feeling.


How widespread are abacuses (Soraban)? I saw the below article [1] (and listened to the corresponding podcast) and the speed mental abacus calculations seem incredible.

[1] http://www.guardian.co.uk/science/alexs-adventures-in-number...


(n.b. They're called soroban here -- mentioned so that folks would be able to look it up more easily.) One is generally exposed to them as a curiosity in math classes, but if you want to get good with them, you go to an after-school cram school. There exist many, even in Ogaki, which specialize in nothing other than abaci. The secret to good abacus use is, again, to drill the "'()#'% out of it.


I'm partial to vihart's perspective on this at http://www.youtube.com/watch?v=a-e8fzqv3CE It's moderately interesting, but ultimately just a way of doing the exact same thing. Multiplication can be reduced entirely or partially to counting, that's worth understanding. And, if explained correctly, this can give insight into multiplying polynomials. Also, showing that counting the intersections of 8 lines crossing 9 lines is tedious may present times tables as convenient time savers rather than teacher inflicted torture.


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 find it had a name. It's also amenable to estimation.


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, you can motivate why

  (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.


I learned the lattice method in the 5th grade in 1981 here in a US public school. I loved it and used it for years afterward. I didn't realize until I was in high school that it wasn't part of the curriculum but only something my teacher thought was useful.


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


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.


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

   |  |  | 
  -+--+--+-
  -+--+--+-
   |  |  | 
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 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.


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.


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 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.


> 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.


The videos only show the technique using small digits. For the larger digits this method quickly becomes cumbersome.

Latttice multiplication is much better for large numbers and deals with a lot of the carrying of overflowing digits which occurs in both normal and "Japanese" multiplication.

http://www.youtube.com/watch?v=cijuPxHgZAA

edit: spelling


yes, there are no '57 * 86' examples, which would be counting intersections for quite some time.


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' families come from multiple countries in Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over the United States also enroll in my classes. See my user profile for more specifics.) Simply by benefit of a better-designed set of instructional materials (formerly English translations of Russian textbooks, with reference to the Singapore textbooks, and now the Prealgebra textbook from the Art of Problem Solving),

http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

the pupils in my classes can make big jumps in mathematics level (as verified by various standardized tests they take in their schools of regular enrollment, and by their participation in the AMC mathematics tests) and gains in confidence and delight in solving unfamiliar problems. More schools in the United States could do this, if only they would. The experience of Singapore shows that a rethinking of the entire national education system is desirable for best results,

http://www.merga.net.au/documents/RP182006.pdf

but an immediate implementation of the best English-language textbooks, rarely used in United States schools, would be one helpful way to start improving mathematics instruction in the United States. There are accessible descriptions for teachers in the United States of the system in Singapore, for example, "Beyond Singapore's Mathematics Textbooks: Focused and Flexible Supports for Teaching and Learning" American Educator, Winter 2009-2010 pages 28-38.

http://www.aft.org/pdfs/americaneducator/winter2009/wang-ive...

A critique of United States schools by Alex Reinhart that was posted here on HN soon after it was published

http://www.refsmmat.com/articles/unreasonable-math.html

begins with the statement "In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades." This implicitly mentions a key difference between United States schools and schools in countries with better performance: American teachers show a method and then expect students to repeat applying the method to very similar exercises, while teachers in high-performing countries show an open-ended problem first, and have the students grapple with how to solve it and what method would be useful in related but not identical problems. From The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999):

"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x

"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77

A great video on the differences in teaching approaches can be found at "What if Khan Academy was made in Japan?"

http://www.youtube.com/watch?v=CHoXRvGTtAQ

with actual video clips from the TIMSS study of classroom practices in various countries.


I know that there was a previous link on HN about the math education approach and materials used by Phillips Exeter which very much followed this approach. Where the students are allowed to discover the relevant mathematics in a structured way so that by the time they learn the "standard" way of expressing a line they understand intuitively why it must be so.

It was very interesting, and actually quite similar to the way I was taught physics in university tutorials - I think I've commented on HN before about the way we "discovered" an approximation similar to Born-Oppenheimer as part of a cleverly designed series of small group tutorials.

edit: I just noticed that the previous link was in fact posted by you.


Timothy Gowers (mathematician, fields medalist) actually wrote about this phenomena quite recently as well: http://gowers.wordpress.com/2012/11/20/what-maths-a-level-do...


Regarding the "dark arts" quote: not my experience. I'm not present at my kids lessons but the homework indicates a wide variety of approaches, methods and applications are used. The math textbooks are much more interesting and varied than the ones I had as a kid.


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.


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.)


Teachers in the U.S. who expect students to struggle with math problems instead of just giving them answers after the student puts in the most minimal effort can get in big trouble for doing so. (Source: My math teaching career.)


32 year old Japanese here. I can unequivocally say I've never seen anything like this in my life. Possibly the younger generations, but I somehow doubt it, as people are pretty slow to adopt completely new teaching methods. Pretty cool though...


I have seen students from India use this method for small numbers and a partial sum method for larger numbers.

http://wsgfl2.westsussex.gov.uk/Aplaws/maths/multicultural/V...

Google 'Vedic multiplication'. Its all good because it is unusual here in the UK and it looks interesting.


Yes, now that I recall, I believe this was a very popular topic in Japan a few years ago (I wasn't in Japan to see it first hand). It was referred to as "India-style multiplication," and some guy(s) sold millions of instructional books. http://math.hoge2.info/kakezan/pr04.html

However, don't think it involves the drawing of lines like in the post.


ditto. i am japanese and did not learn multiplication this way.

it was more through a sing-songy multiplications table nininga shi nisanga roku ..and so on..


This is cool but not a part of any standard curriculum in Japan despite the title.


Agreed. I went to Japanese elementary school (in college now, so within the last 15 years) and didn't learn this lines method ever.

What I found greatest about the Japanese method of learning multiplication was actually the method for learning single-digit products. The Japanese use a system called "kuku" (translated, "9 by 9") which involves memorizing a rhythmic chant that goes through the entire multiplication table, with each product being concisely expressed in a few syllables. This is made possible by the various ways in which a number can be pronounced in Japanese.

I think the method is made necessary because to say the full products takes a lot of syllables in Japanese (e.g. 7x7=49 would be "nana kakeru nana wa yonjyuu-kyuu"). So perhaps it doesn't differ too much from the way you learn multiplication tables at an English-speaking school, but a cool method nonetheless.

Video here: http://www.youtube.com/watch?v=1hLyzXM53IE


My three daughters go to doyogako (Saturday Japanese language school) which follows the standard curriculum which is why I knew. I haven't seen them use the kuku though. I'm actually working with the youngest on flashcards right now.

I've always thought it very valuable that they do math both there and at their regular US school in the hope that it will mean they aren't dramatically slowed down when math appears in a conversation like second language learners are. They certainly are crazy fluent in both languages otherwise compared to any second language learner, but I recently discovered my oldest tends to think of multiplication products in Japanese first even when in an English context. The brain is tricky :-)


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, 148x385=56980?


Yes, there are examples where the syllable discrepancy is a bit larger between English and Japanese, but I think you're right -- perhaps the syllable difference is not too big of a deal. Now that I think more about it, perhaps the actual advantage may be in the rhythmic nature of the mnemonic. By modifying syllable numbers for various products, the entire recitation gets a kind of rhythmic flow. (watch the video to see what I mean)

All I can say is that in my personal experience learning the multiplication tables in both languages around the same time, it was dramatically easier in Japanese. Perhaps the small reduction in syllables caused that, and perhaps the "rhythmification" was more responsible.

I do recall seeing a study somewhere that showed people who spoke Chinese were able to remember more numbers, ostensibly because each digit takes less syllables to speak in Chinese so the total syllables to remember in one's head is smaller.


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.


Eh, one example isn't much for statistics. Looking at the average for 1, 2, and 3 digit numbers (ie, [0,9] * [0,9], [0,99] * [0,99], ...), using the sentence "<x> times <y> is <x*y>":

English, 1 digits: 6.5 syllables; English, 2 digits: 14.9 syllables; English, 3 digits: 25.4 syllables.

I'll let other people figure out the numbers for their own languages, but I'm guessing most languages will have similar lengths. Numbers tend to be short in any language.


The podcast I tangentially mentioned above talks about Soraban, and Japanese people learning Kuku:

http://www.bbc.co.uk/programmes/b01nkxkv


Isn't that the same as seven times seven equals forty-nine? In portuguese, sete vezes sete igual a quarenta e nove. Japanese doesn't look especially lengthy.


I first thought of "multiply" in the biblical sense when I read the title.


Still quicker to do the math in your head:

    123 * 321

    321 * 100 = 32100
    321 * 20  =  6420
    321 * 3   =   963
                -----
                39483
You just need to know the times table and keep a little stack in your head :)


Keeping the stack has always been the problem for me. At least for larger numbers. Luckily, there's RealCalc on Android which is by far the best calculator app I've ever seen.


I'm a fan of Droid48. It emulates a HP-48, so it displays the stack on-screen and has advanced capabilities like graphing if you need them.


Interesting in that it shows how geometry can related to mathematics, but very laborious when applied to larger numbers. Amounts to adding up the contents of each place in the numeric result one-by-one. The work involved is the sum of the output digits.


> how geometry can related to mathematics

Geometry is part of mathematics!

More generally, it's sad to see this attitude all over the place, as if geometry and drawing were somehow "less mathematical" than writing and symbols. Both, algebra/analysis as well as geometry, have their place in mathematics, and complement each other very well.

The same holds for creativity versus stringency, by the way. A theorem has to be proved (stringency), but how did find that proof? By aternating of stringency and creativity back and forth! And why did you ask the question answered by the theorem in the first place? Creativity and real-world problems (e.g. physics or economics)!

Note that the distinction between geometry/algebra is totally unrelated to the distinction between creativity/stringency. However, more often that not media associate math only with the algebra+stringency part, glossing over the other equally important aspects. This leads to a totally distorted image of the wonderful field of mathematics.


I think that it is important to note that without good instruction no method is "good". It is really really important for anybody learning anything to understand fundamentally what is going on. This is the problem with most people that have math phobia's (I don't believe there are people that are not good at math). They are taught to memorize their times tables with only a fleeting mention that this really a summation. On this weak foundation people add more concepts, and BOOM they are math phobic. The thing I do like about the "line method" is it is a very visual way to show children what is going on. The chunks, if you will, are literally the place of the digit.


I wouldn't learn how to multiply from the Japanese. They are clearly pretty bad at it. http://en.wikipedia.org/wiki/File:Bdrates_of_Japan_since_195...


This is quite a nice mathematical trick, but nothing more. It may allow children to be able to perform large multiplication at a young age, but it is also setting them up to be confused later on by providing them with a method which abstracts from the actual arithmetic. What on earth is going to happen when they run into algebraic multiplication? I think it is much more useful to teach them to split up the parts of the numbers and multiply those, if they are too difficult to be multiplied mentally. For example instead of 11x22 do 10x22 + 20 + 2.


A good way to introduce this starter is to put up a map of the world and get learners to point out Japan. As a teacher you can then move into how Japanese Pupils learn to multiply.

Shouldn't the focus be on teaching kids to multiply, rather than teaching them that "all Japanese people do it this way, look how strange"? How do the Germans do math? What about Indians?

It really frustrated me that the author chose to illustrate a potentially very helpful teaching aide by focusing so much of the apparent strangeness of Japanese culture. Not exactly setting a very good example.


Every time this comes up it's shot down. It isn't part of their curriculum.


This method is also referred to as Mayan Multiplication which isn't necessarily exclusive to Japan or a technique the Japanese invented. It's not Mayan either, but just a neat technique to make multiplying numbers easier. Although it might be hard to perform in your head this way.

The babylonians, egyptians, and romans had their own techniques because multiplication by hand is not easy, and it's time consuming. Techniques like these enabled those civilizations to build wonderful engineering marvels. The point is these techniques don't violate math or subvert teaching math because it's not the same thing.

This subject was originally called arithmetic when I was in school which was different than math. Arithmetic was the rudimentary techniques for how numbers were added/subtracted/multiplied/divided where math was word problems that required you know arithmetic to solve but also required logic and abstraction. So yes arithmetic != math, but that doesn't mean teaching arithmetic is NOT useful. They are different concepts that are related, but not the same.

I think this is perfectly fine to teach someone this. The whole computation isn't math argument is a misplaced. We're talking about 2nd or 3rd grade students. They'll learn math at the higher levels again.


By the way, if you or you children need a quick and easy way to practice mental calculations, find an implementation of the game http://en.wikipedia.org/wiki/Des_chiffres_et_des_lettres on your phone.


Thanks for bringing back memories of ski vacations in the French Alps. We couldn't understand most of the stuff on TV but loved this show.


Good morning, that's a nice tnettenba.


I never understood why this is easier than just manually doing the FOIL. 1312 = (10 + 3)(10 + 2) = 1010 + 102 + 310 + 32. Drawing all those lines and counting intersections seems tedious.


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.


Nice trick, but it would be better if explained in correlation to the traditional algorithm, showing you can separate work by units and that they're really the same at the core.


What's more impressive is flash anzan. The ability to visualize an abacus and then with it do additions of large sums rapidly and accurately.


Indeed. This short video shows the above in action, nothing short of amazing.

https://www.youtube.com/watch?v=6m6s-ulE6LY


It's a neat trick, but the method doesn't do much to reveal why multiplication works as it does. Most students will just sit there drawing lines and counting intersections without understanding why it gives them the right answer. Memorizing a routine without understanding why it works is just one step above memorizing a table.


My sister-in-law is a Japanese school teacher. I showed this too her and she said she does not recognize it at all. My mom has a master's in math education and teaches at the local junior college. I'll let you know if she has any comment on this.


This is indeed very interesting technique, but I don't think it is very practical and fast.


My son's home from school when I see this. We try out a few examples, starting with those in the videos. Then we do a few more...and then we do: 13 x 23. I'm obviously missing something hidden in this method, as our result, 416...fail ;(


it's hardly practical. Yes, interesting, but I would not use it at all. The way I know how to do multiplication "manually" is superior than this. You line up numbers on top of each other. Then you start multiplying first digit of the bottom number with all the digits of above number, from right to left and adding any carry over. You start writing the result on the same column you are multiplying. Then you add them all.

    123
    123 
  x______
    369
   246
  123
+________

  15129
That's what I learned in elementary school and have been using since then. It's interesting to know how others to do it.


> That's what I learned in elementary school and have been using since then. It's interesting to know how others to do it.

well, i generally do it using a 'covolution approach' like this:

     45
   x 86
   ------
       0 (5*6 == 30) -> take the units place, and put '3' as carry over
      4  (6*4+8*5+3 == 47) -> take the units place, and put '4' as carry over
    36   (8*4+4 == 36)
-------------

   3640
-------------

with sufficient practice, you can multiply arbitrary 3 digits in approx 10-15 seconds or so ;)

[edit-1]: my formatting sucks, cannot seem to align stuff nicely at all.


This method does not convey any understanding of the problem. I'm not sure it's a good idea in the long run. It'll create a class of people who know how to solve a particular mathematical problem, but don't understand why.


Great. Now do it for 79*86.


I'd rather 86*80-86 == 6880-86 == 6794


I do that too, round to the nearest multiple of 5 or 10 and fix it at the end :D


realistically all this or any other process is doing is expansion and taking advantage of the commutative nature of multiplication.

(70 + 9)x(80 + 6)

70x80 + 9x80 + 6x70 + 6x9

56(00) + 72(0) + 42(0) + 54

Guess what - long story short you need to know your times tables. I don't see how the lines method is any easier than long multiplication, which in itself is not hard with a little time and explanation. If you can use the lines, or any other method, as a way of conveying what I did above - then you've succeeded.

Edit: Stars go to italics and I don't know the escape character


* It's not backslash!

I think you can only do * with an unpaired * terminator * ?

I think you can only do * with an unpaired * terminator ?

I think you can only do xx with an unpaired xx terminator xx ?

[that line is this with no spaces around :I think you can only do x * x with an unpaired x * x terminator x * x ?

Nope, looks like as long as it's spaced from the prior/next character then it doesn't cause italicisation.

Help says: "Text surrounded by asterisks is italicized, if the character after the first asterisk isn't whitespace."


> I don't see how the lines method is any easier than long multiplication

You probably learned multiplication using something similar to the lines method. If you had 3 x 4, you'd draw out 3 circles and then put 4 more circles inside and count up all the small ones. This is just a logical extension of that.


Interesting indeed. Reminds me of this [Chinese suanpan](http://en.wikipedia.org/wiki/Suanpan).


Cool but impractical - hard to do in your head and cumbersome when drawing something more than 3 lines. But indeed magical, no argument here :)


I see the video, but how does this work?

on unix/linux I just use bc -l


bc is a pretty misleading calculator, without specifying rounding it thinks 2^(40/1.5) is 67108864.

It's not even close.

If you trust your finances to that, you could get in trouble.


The man speaks the truth, but I do get a warning

   2.0^(40.0/1.5)
   Runtime warning (func=(main), adr=18): non-zero scale in exponent
   67108864.00000000000000000000
Apparently the ^ function only works with integer powers. The exponential and natural log functions work as expected

http://www.johndcook.com/blog/2010/07/14/bc-math-library/

   keith@xeon4:~$ bc -l
   e(l(2)*(40.0/1.5))
   106528681.30999083085437360438
nice example.


Don't try to use this method to do mind calculation, it may cause senile dementia.


nice trick for numbers with each digit smaller than 5


I'm guessing you use a 5s digit like Roman numerals and the abacus do.


I think only smaller than 4...

I wonder if this works in binary :D


That was my first thought, so I checked, and yes, it does. You just change the carrying rules. But by the time you've accounted for the number of digits you'd need to do anything useful, it's fairly useless except from a theoretical computational perspective - even when converted to logic circuits, I'm guessing it's not the most efficient solution out there.


Actually, this method is one of the most efficient ways to multiply in software. It's called the "Comba" method in that context, and is efficient for multiplying 32x32 words or smaller, due to cache effects and function overhead required by more complicated algorithms. Over about 32 words on many architectures, the algorithmic advantage of the Karatsuba method wins out.


It doesn't work for 25x25, so what's the point?


It does work for 25x25.


You're right, I see it now, it represents 400 + 20*10 + 25 which is indeed 625. They just avoided those cases in the article and only showed those where the number of crossing neatly formed the digits.


Now, using this new technique, attempt to solve this problem

4x + 2x^2 + 5 = 100

:)




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