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How Japanese Kids Learn To Multiply

272 points by TomAnthony 14y ago

cincinnatus 14y ago
This is cool but not a part of any standard curriculum in Japan despite the title.
- gklitt 14y ago
  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 http://www.youtube.com/watch?v=1hLyzXM53IE
  - cincinnatus 14y ago
    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 :-)
  - talmand 14y ago
    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?
    - solox3 14y ago
      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.
      - saalweachter 14y ago
        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.
      - shabble 14y ago
        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.
    - gklitt 14y ago
      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.
  - ricardobeat 14y ago
    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.
  - gadders 14y ago
    The podcast I tangentially mentioned above talks about Soraban, and Japanese people learning Kuku: http://www.bbc.co.uk/programmes/b01nkxkv http://www.bbc.co.uk/programmes/b01nkxkv
- benatkin 14y ago
  I first thought of "multiply" in the biblical sense when I read the title.
rossjudson 14y ago
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.
- vog 14y ago
  > 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.
agumonkey 14y ago
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.
symmetricsaurus 14y ago
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 http://www.youtube.com/watch?v=cijuPxHgZAA edit: spelling
- boredguy8 14y ago
  yes, there are no '57 * 86' examples, which would be counting intersections for quite some time.
  - DrDreams 14y ago
    You guys are exactly right. This method is bullshit snake oil for weeaboos as it involves counting up to the product of every single product, which is tedious and error-prone. It is safe to label anyone who praises this method as a math-hipster or pretender who only likes the warm fuzzy feelings of simulated simplicity rather than actual, real, pragmatic, useful value. They should be fucked gently with a chainsaw.
patio11 14y ago
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.]
- bradleyland 14y ago
  Can't really blame you. I know next to nothing about teaching or Japanese culture, but this article gave me a sick feeling.
- graeme 14y ago
  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.
  - Jtsummers 14y ago
    > 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.
tuananh 14y ago
Interesting indeed. Reminds me of this [Chinese suanpan](http://en.wikipedia.org/wiki/Suanpan http://en.wikipedia.org/wiki/Suanpan).
tokenadult 14y ago
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=prealgebra 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 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-iverson.pdf 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 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 http://www.youtube.com/watch?v=CHoXRvGTtAQ with actual video clips from the TIMSS study of classroom practices in various countries.
hemancuso 14y ago
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.
- talmand 14y ago
  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.
  - derleth 14y ago
    > 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?
    - talmand 14y ago
      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.
  - Natsu 14y ago
    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.
Jabbles 14y ago
Great. Now do it for 79*86.
- deleted 14y ago
  [deleted]
- est 14y ago
  I'd rather 86*80-86 == 6880-86 == 6794
  - ricardobeat 14y ago
    I do that too, round to the nearest multiple of 5 or 10 and fix it at the end :D
- Ntrails 14y ago
  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
  - saraid216 14y ago
    > 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.
  - pbhjpbhj 14y ago
    * 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."
esalazar 14y ago
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.
est 14y ago
nice trick for numbers with each digit smaller than 5
- Aardwolf 14y ago
  I think only smaller than 4... I wonder if this works in binary :D
  - 3chelon 14y ago
    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.
    - tuxdev 14y ago
      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.
- bcoates 14y ago
  I'm guessing you use a 5s digit like Roman numerals and the abacus do.
Someone 14y ago
This is a variant of http://en.wikipedia.org/wiki/Lattice_multiplication 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.
- ianb 14y ago
  Another approach: http://en.wikipedia.org/wiki/Grid_method_multiplication 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.
  - mturmon 14y ago
    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.
- weirdkid 14y ago
  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.
- evincarofautumn 14y ago
  It’s also easy to do in your head, and works pleasantly for numbers in any base, including polynomials.
nikolakirev 14y ago
This is indeed very interesting technique, but I don't think it is very practical and fast.
badatmath 14y ago
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...
- keithpeter 14y ago
  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/VedicMultIntro.htm 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.
  - badatmath 14y ago
    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 http://math.hoge2.info/kakezan/pr04.html However, don't think it involves the drawing of lines like in the post.
- asdf333 14y ago
  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..
missing_cipher 14y ago
Every time this comes up it's shot down. It isn't part of their curriculum.
acomjean 14y ago
I see the video, but how does this work? on unix/linux I just use bc -l
- Aardwolf 14y ago
  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.
  - keithpeter 14y ago
    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/ 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.
Aardwolf 14y ago
It doesn't work for 25x25, so what's the point?
- leddt 14y ago
  It does work for 25x25.
  - Aardwolf 14y ago
    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.
antiterra 14y ago
I'm partial to vihart's perspective on this at http://www.youtube.com/watch?v=a-e8fzqv3CE 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.
ricardobeat 14y ago
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 :)
- tsahyt 14y ago
  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.
  - jff 14y ago
    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.
muloka 14y ago
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.
- partomniscient 14y ago
  Indeed. This short video shows the above in action, nothing short of amazing. https://www.youtube.com/watch?v=6m6s-ulE6LY https://www.youtube.com/watch?v=6m6s-ulE6LY
sean-duffy 14y ago
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.
dweekly 14y ago
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_1950.svg http://en.wikipedia.org/wiki/File:Bdrates_of_Japan_since_195...
JohnFromBuffalo 14y ago
My revelation came while taking Linear Algebra and Discrete Math in college as an Engineer/CS student. It really abstractly told me why these things were happening and how to deal with them plus giving me a real survey to mathematic techniques. The technique is a bit gimmick, but works for that part of world vs. what we do - which is also gimmick on adds/subtracts on multiple numbers. Its less about what we are teaching our kids, and how to handle it efficiently. Anyway, drawing lines is nothing more than adding numbers like what we do. I see nothing special about this.
prawks 14y ago
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.
frouaix 14y ago
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 http://en.wikipedia.org/wiki/Des_chiffres_et_des_lettres on your phone.
- zwieback 14y ago
  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.
- bitwize 14y ago
  Good morning, that's a nice tnettenba.
schultz9999 14y ago
Cool but impractical - hard to do in your head and cumbersome when drawing something more than 3 lines. But indeed magical, no argument here :)
tylermauthe 14y ago
Now, using this new technique, attempt to solve this problem 4x + 2x^2 + 5 = 100 :)
mumrah 14y ago
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.
chubbard 14y ago
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.
aghull 14y ago
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-problem-using-diagonal-lines.html 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.
niels_olson 14y ago
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.
shn 14y ago
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.
- signa11 14y ago
  > 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.
wilhow 14y ago
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.
sinker 14y ago
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.
notemouse 14y ago
Don't try to use this method to do mind calculation, it may cause senile dementia.
jhancock 14y ago
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 ;(

272 points by TomAnthony 14y ago

cincinnatus 14y ago

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

gklitt 14y ago
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 http://www.youtube.com/watch?v=1hLyzXM53IE
- cincinnatus 14y ago
  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 :-)
- talmand 14y ago
  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?
  - solox3 14y ago
    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.
    - saalweachter 14y ago
      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.
    - shabble 14y ago
      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.
  - gklitt 14y ago
    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.
- ricardobeat 14y ago
  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.
- gadders 14y ago
  The podcast I tangentially mentioned above talks about Soraban, and Japanese people learning Kuku: http://www.bbc.co.uk/programmes/b01nkxkv http://www.bbc.co.uk/programmes/b01nkxkv
benatkin 14y ago
I first thought of "multiply" in the biblical sense when I read the title.

rossjudson 14y ago

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.

vog 14y ago
> 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.

agumonkey 14y ago

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.

symmetricsaurus 14y ago

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

boredguy8 14y ago
yes, there are no '57 * 86' examples, which would be counting intersections for quite some time.
- DrDreams 14y ago
  You guys are exactly right. This method is bullshit snake oil for weeaboos as it involves counting up to the product of every single product, which is tedious and error-prone. It is safe to label anyone who praises this method as a math-hipster or pretender who only likes the warm fuzzy feelings of simulated simplicity rather than actual, real, pragmatic, useful value. They should be fucked gently with a chainsaw.

patio11 14y ago

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.]

bradleyland 14y ago
Can't really blame you. I know next to nothing about teaching or Japanese culture, but this article gave me a sick feeling.
graeme 14y ago
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.
- Jtsummers 14y ago
  > 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.

tuananh 14y ago

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

tokenadult 14y ago

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=prealgebra 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 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-iverson.pdf 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 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 http://www.youtube.com/watch?v=CHoXRvGTtAQ with actual video clips from the TIMSS study of classroom practices in various countries.

hemancuso 14y ago

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.

talmand 14y ago
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.
- derleth 14y ago
  > 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?
  - talmand 14y ago
    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.
- Natsu 14y ago
  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.

Jabbles 14y ago

Great. Now do it for 79*86.

deleted 14y ago
[deleted]
est 14y ago
I'd rather 86*80-86 == 6880-86 == 6794
- ricardobeat 14y ago
  I do that too, round to the nearest multiple of 5 or 10 and fix it at the end :D
Ntrails 14y ago
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
- saraid216 14y ago
  > 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.
- pbhjpbhj 14y ago
  * 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."

esalazar 14y ago

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.

est 14y ago

nice trick for numbers with each digit smaller than 5

Aardwolf 14y ago
I think only smaller than 4... I wonder if this works in binary :D
- 3chelon 14y ago
  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.
  - tuxdev 14y ago
    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.
bcoates 14y ago
I'm guessing you use a 5s digit like Roman numerals and the abacus do.

Someone 14y ago

This is a variant of http://en.wikipedia.org/wiki/Lattice_multiplication 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.

ianb 14y ago
Another approach: http://en.wikipedia.org/wiki/Grid_method_multiplication 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.
- mturmon 14y ago
  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.
weirdkid 14y ago
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.
evincarofautumn 14y ago
It’s also easy to do in your head, and works pleasantly for numbers in any base, including polynomials.

nikolakirev 14y ago

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

badatmath 14y ago

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...

keithpeter 14y ago
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/VedicMultIntro.htm 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.
- badatmath 14y ago
  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 http://math.hoge2.info/kakezan/pr04.html However, don't think it involves the drawing of lines like in the post.
asdf333 14y ago
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..

missing_cipher 14y ago

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

acomjean 14y ago

I see the video, but how does this work? on unix/linux I just use bc -l

Aardwolf 14y ago
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.
- keithpeter 14y ago
  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/ 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.

Aardwolf 14y ago

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

leddt 14y ago
It does work for 25x25.
- Aardwolf 14y ago
  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.

antiterra 14y ago

I'm partial to vihart's perspective on this at http://www.youtube.com/watch?v=a-e8fzqv3CE 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.

ricardobeat 14y ago

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 :)

tsahyt 14y ago
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.
- jff 14y ago
  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.

muloka 14y ago

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.

partomniscient 14y ago
Indeed. This short video shows the above in action, nothing short of amazing. https://www.youtube.com/watch?v=6m6s-ulE6LY https://www.youtube.com/watch?v=6m6s-ulE6LY

sean-duffy 14y ago

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.

dweekly 14y ago

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_1950.svg http://en.wikipedia.org/wiki/File:Bdrates_of_Japan_since_195...

JohnFromBuffalo 14y ago

My revelation came while taking Linear Algebra and Discrete Math in college as an Engineer/CS student. It really abstractly told me why these things were happening and how to deal with them plus giving me a real survey to mathematic techniques. The technique is a bit gimmick, but works for that part of world vs. what we do - which is also gimmick on adds/subtracts on multiple numbers. Its less about what we are teaching our kids, and how to handle it efficiently. Anyway, drawing lines is nothing more than adding numbers like what we do. I see nothing special about this.

prawks 14y ago

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.

frouaix 14y ago

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 http://en.wikipedia.org/wiki/Des_chiffres_et_des_lettres on your phone.

zwieback 14y ago
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.
bitwize 14y ago
Good morning, that's a nice tnettenba.

schultz9999 14y ago

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

tylermauthe 14y ago

Now, using this new technique, attempt to solve this problem 4x + 2x^2 + 5 = 100 :)

mumrah 14y ago

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.

chubbard 14y ago

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.

aghull 14y ago

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-problem-using-diagonal-lines.html 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.

niels_olson 14y ago

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.

shn 14y ago

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.

signa11 14y ago
> 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.

wilhow 14y ago

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.

sinker 14y ago

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.

notemouse 14y ago

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

jhancock 14y ago

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 ;(