At risk to be downvoted to death, I have to say that the new way of multiplication is actually _better_ than the old one, at least for kids starting to learn how to multiply.
All the new multiplication algorithm does is makes implicit additions explicit, so that you do not have to keep carry-over digit in memory. 144 is 120+24 and 720 is 600+120. Sure it trains your short memory, but we are talking about 7yr olds here. Once kid feels comfortable s/he will naturally move to the "old" way of multiplication.
And anyway... It is easy to nitpick on something we understand very well, but how about the topic? What would you do to make math more enjoyable in the classroom? I know what worked for me personally and that was not classroom drilling of multiplication, divisions, and stupid problems with two pipes filling the pool at different speeds.
What made me love math are Marin Gardner's books and other fun math books. It was much more interesting to solve crimes with what I later came to know as propositional logic, or cutting tori in different ways.
I think you've nailed it by considering the carry-overs.
With the new method there's less opportunity for error. The old way, there are two sets of carry digits: those arising in the initial multiplying, then more in the final adding up (though not in the example given). Sometimes I would try to remember them; sometimes I would write them down in various places; frequently I forgot or muddled them up somehow.
Another benefit is: the new way gives you a sense of the magnitude of the final answer more quickly. Approximations are useful. Contrast with the old method which begins by multiplying the least significant digits.
Either way is fine, I think, if you the child understands the underlying mechanism. I'm the father of a 3rd and 5th grader in US public schools and I think the math curriculum is very good. However, sitting down and explaining basic arithmetic is important. I also teach them to estimate and come up with a quick upper and lower bound before they start the problem.
I went to school in Germany and our methods of multiplication and especially that of long division were a little different. When I first looked at the way long division is done in the US I thought it was confusing and my first reaction was that my method was better. It took a little time to overcome my cultural bias but I'm all better now.
Still waiting for the US to switch to Celsius and metric, though...
I agree. For learning, the method shown is fine. Why? Because it teaches and honors place value, an extremely important concept in developing numeracy. I'm fine with any method that uses that.
I am not fine with something my kids call the "lattice method." The point of this method is simply to get the correct answer for multiplying large numbers. Is it clever? Yes. Does it work? Yes. But it does not teach math. If the educators' goal is simply to give a technique that works and gives the correct answer, teach them to use the calculator on their cell phones.
I'm not sure why you think this lattice method does not teach place value. The page you linked to doesn't explain (at all) why it works, sure, but the method is just a clever way of notating place. I would expect any good teacher to get his or her students to understand that, and not just mindlessly fill in numbers.
I would expect any good teacher to get his or her students to understand that
That's not how I've seen it used. I've seen it used as a procedure for getting the answer. Compare that to the method shown in the npr link, first line: the numerals are 2 and 3, but the numbers multiplied 20 and 30. That's what I mean by teaching place value.
In the the old-school method I learned, we respected place-value by indenting (from the right), optionally writing the zeros.
I suspect that the 'calculator age' has caused kids (and kids that have now grown up to be parents) and maybe educators to over-value an increase in significant digits in calculations. It's interesting that the method of multiplication illustrated in the article is referred to as 'new' since when you use a slide rule, you operate in this way anyway. And to echo Hans Bethe, this is often "good enough".
Surely that method of multiplication is not suited for larger numbers, to get 246 * 369 u do
200 * 300
200 * 60
200 * 9
40 * 300
40 * 60
40 * 9
6 * 300
6 * 60
6 * 9
So for x-digit multiplied by y-digit u need to write down x*y products and then sum it up, where as for the old method you need to only write down min(x,y) products and sum it up.
To be honest, I just know 36 times 24. I've spent enough time dealing with numbers with all their prime factors small that I can just see it's 864. It might also help that I know there are 24 hours in a day, 3600 seconds in an hour, and 86,400 seconds in a day -- and I probably have a lot of random facts like this somewhere in my head that I at least occasionally pin mental arithmetic problems on.
I don't really think the methods are important, what's important is whether students decide to learn more advanced math later (say at least up to Calculus), and more importantly whether they like doing so. Too many people say "I hate math", learning multiplication one way or another shouldn't affect that strongly.
The article has a few nice quotes I like: "Computers do arithmetic for us, Devlin says, but making computers do the things we want them to do requires algebraic thinking."
'"You cannot become good at algebra without a mastery of arithmetic," Devlin says, "but arithmetic itself is no longer the ultimate goal." Thus the emphasis in teaching mathematics today is on getting people to be sophisticated, algebraic thinkers.'
While I can do multi-digit multiplications just fine and have a high confidence in my answer, I'd much rather a calculator do it for me, allowing me a higher confidence in the answer and just being done faster to use the result for some other purpose.
I hope that I am not straying off topic, but certainly where I was schooled (Ireland), the main issue in the poor math performance falls on the "uncool" reputation which Maths has received in the past few years.
If you make it so that a child likes maths and feel's no peer pressure to just give up, the way in which one multiplies may become less significant.
Changing the way the multiplication algorithm is taught isn't going to fix anything.
There needs to be a fundamental change in the way mathematics is taught. Right now what most schools teach in math class resembles trivia more than mathematics. There is no appeal to intuition or understanding, because clueless symbol manipulation and memorization of formulas is apparently what really matters. It is not necessary that the student really understands what he is doing. Why is the product of two negative numbers a positive number? "That's just the rule." Why is dividing by a fraction the same as multiplying by the reciprocal? "That's just the rule." These are very simple concepts that can be both logically and intuitively presented to students, yet growing up those were the answers I received. Even high school geometry, which used to be taught decently, has now been replaced by curriculum that does not emphasize proofs or any kind of true understanding/intuition.
Unless students discover the beauty of mathematics independent of math class, or enjoy following arbitrary rules, it is no wonder they develop a dislike for mathematics.
Although the author's writing style is a little bit annoying at parts, this is a good read:
www.maa.org/devlin/LockhartsLament.pdf
I'm pretty math-challenged these days without a calculator (gone are the days I could run the numbers in my head faster than you could punch them into your calculator), but I tend to do weird things like break stuff down into primes and then multiply it out from there.
I have an 11 year old daughter who I help with math homework every week. Some of the new methods work for her, and some are terribly confusing to her. I think it's good to expose kids to a variety of methods in math. I am concerned, however, and the near-complete lack of emphasis on memorizing the basic math tables and doing calculations in your head. Doing that at a young age really helped me to understand numbers. Many of the techniques they teach in school are techniques I figured out on my own by exposing myself to large numbers of calculations.
This is the way I do multiplications mentally since being a child, but I never quite thought about that fact, that this is "new". How do you guys execute multiplications in mental math?
Aside from the 'new' way of multiplication (which doesn't strike me as too bad), I think asking kids to make a stem-and-leaf plot of the birthdays in their class is actually really cool. It shows them how the things they learn can be applied to real things and it could open the way for interesting discussions as well (though the birthday paradox might be a bit much for 5th graders perhaps).
After watching a video on Khan Academy on Lattice Multiplication, I assumed that it was the "new way" of doing multiplication by hand.
I certainly would use lattice multiplication in preference to traditional long multiplication methods if I ever had to multiply large numbers by hand. I wouldn't be able to say which method is better for a 7 year old learning the concepts though.
...Does anyone actually teach their kids with that retarded method? All it does it take up more space, and shift the burden from screwing up carries while multiplying to screwing up carries while adding.
And since it's so brain dead to implement, I can't see any significant amount of kids even reflecting on how 'the numbers work' based on that.
I think the argument is based around getting kids to realize that numbers are not monolithic things to put where X goes. They can be broken up and dealt with in multiple ways.
FWIW, I routinely find people who can't do simple single-times-double digit multiplications in their head, simply because it never occurs to them that they can turn 7 * 12 into (7 * 10) + (7 * 2), or 7 * 3 (=21) * 4 (4x2=8,4x1=4, =84). I figured out how to do so myself, school never taught it to me, and I seriously doubt more than a 5% left my high school knowing how to do this, much less actually using it.
As far as I know every child in England is currently being taught this method.
I'm not sure that I would leap to calling it retarded. I agree that it takes up more paper but paper isn't the most expensive thing in the world.
What it does do is make explicit why the old school method works. Look at the explanation given by the article for the old method:
'Answer = 720, Because: 2x36 = 72, with a 0 added in the ones place.'
No, the reason you get 720 isn't because you 'added' a 0 in the ones place (72+0 = 72). It's because you are multiplying by 20 and not by 2. The old method hides this fact and corrects for it by a mechanical process: first write down a zero and then shift the other entries to the left.
Your criticism is analogous to calling Assembly retarded because Haskell exists. They serve different purposes.
I still encounter people who try to teach long multiplication to kids by saying "add a zero in the ones place". And college students who fail to describe why they add those zeros (typically liberal arts majors (no offense. They just rarely minor in math, and many quit doing math entirely after they get the minimum to graduate)).
The way the article shows is essentially calculating:
(30 + 6) * (20 + 4)
Which is great and all, but I think it would make more sense to do this:
(24 * 6) * 1 + (24 * 3) * 10
This is a lot closer to how machines actually do multiplication, so if the goal is to conceptually understand multiplication so you can have a computer do it, that makes more sense to me.
My inclination would be to be much less computer like about it. First start with some observations about the numbers. 24 is one less than 25. 4 * 25 is 100. 36 is 9 * 4. Therefor the answer is (9 * 100)-36=864. Numbers have properties. You can use these properties to do less mental work or at least take up less mental temp space. Not sure how you teach people to do that.
I wish to hell I had learned the Soroban growing up. Now it's on my unending todo list. I'll probably make a go at it when my kid is ready for it. We can learn together :)
All the new multiplication algorithm does is makes implicit additions explicit, so that you do not have to keep carry-over digit in memory. 144 is 120+24 and 720 is 600+120. Sure it trains your short memory, but we are talking about 7yr olds here. Once kid feels comfortable s/he will naturally move to the "old" way of multiplication.
And anyway... It is easy to nitpick on something we understand very well, but how about the topic? What would you do to make math more enjoyable in the classroom? I know what worked for me personally and that was not classroom drilling of multiplication, divisions, and stupid problems with two pipes filling the pool at different speeds.
What made me love math are Marin Gardner's books and other fun math books. It was much more interesting to solve crimes with what I later came to know as propositional logic, or cutting tori in different ways.