[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.]
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.
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.
This is exactly why giving calculators to children who have not yet mastered any method of multiplication is bad.
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.
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.
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.
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.
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.
(1 + a)^2 = 1 + 2a + a^2
Thinking about approximations when a << 1, you can motivate why
(1 + a)^2 ~= 1 + 2a
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.
Teach kids to open the calculator app on their phone rather than to do this.
No fast way to learn multiplication other than to practice it.
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(I'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.
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.
But that just shows that we understand grade school multiplication. It doesn't mean that we know how best other people learn it.
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.
> 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?
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.
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.
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),
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,
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.
A critique of United States schools by Alex Reinhart that was posted here on HN soon after it was published
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?"
with actual video clips from the TIMSS study of classroom practices in various countries.
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.
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.
Google 'Vedic multiplication'. Its all good because it is unusual here in the UK and it looks interesting.
However, don't think it involves the drawing of lines like in the post.
it was more through a sing-songy multiplications table
..and so on..
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.
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 :-)
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.
seven  times  seven  equals  forty-nine  = 9
七 乘 七 如 四十九  = 7 (without using primary school mnemonics)
七 七 四十九  = 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.
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.
123 * 321
321 * 100 = 32100
321 * 20 = 6420
321 * 3 = 963
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.
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.
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.
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.
well, i generally do it using a 'covolution approach' like this:
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)
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.
(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
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."
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.
on unix/linux I just use bc -l
It's not even close.
If you trust your finances to that, you could get in trouble.
Runtime warning (func=(main), adr=18): non-zero scale in exponent
keith@xeon4:~$ bc -l
I wonder if this works in binary :D
4x + 2x^2 + 5 = 100