How should we teach kids multiplication?

That's funny, I seem to do it backwards...

I learned the 'shortcut' method in school (which worked just fine), but if for some reason I need to multiply two large numbers in my head, I tend to use the partial products method (or some close approximation thereof). It's just easier for me to keep track of offsets with simple multiplications than remembering what may seem to be random strings of numbers.

Don't know why, really. Maybe because they drilled us on the basic multiplication tables ad nauseum.

I've never heard of the 'partial products method' until about half-a-minute ago. What is it? How is it done? Why would it be a good teaching method?

I didn't actually get to the end of the video, but I think she explains it. If you study the diagram (partial products is on the right), you should get a good idea of how it works.

The basic justification for partial products is that the traditional method obscures what's going on. Why do you "carry" a number? Why are columns offset?

The partial products method is more transparent, demonstrating several properties of multiplication in its application. But it's definitely a lot slower!

That's funny. I seem to remember learning about place values before learning multiplication using the traditional method, so the carrying and offset columns just made intuitive sense. The partial products just looks way more complicated than it needs to be. I'll have to watch this video though.

"I think I'd prefer to see kids learn the partial products method first, then learn the "shortcut" later on."

I'd have to agree. I think that drilling does definitely have a place in learning math, but you absolutely need to understand the concepts first. It's not like you need to spend forever long on partial products, but it would be great to teach it first and follow it by the traditional method.

I also had not seen partial products until just now. And, I remember being a tiny bit confused with the principles of the traditional method when I first learned it (I got it within a few days, but if I had seen the partial products method first -- even once -- I would have been better off.)

I'm not impressed. For the last almost forty years math education has lurched from one revolution to another, the only constant being that new wave of students have been less prepared than the last. For Heaven's sake, you can't find a shop clerk able to make change anymore without the help of the cash register.

There is no Royal Road to mastery of arithmetic; drill is the only answer, or stop trying after the fundamentals have been taught, let them use calculators, and teach them to solve problems in math and logic. At least that way they will come away with something useful.

I always doubt the claims that "new wave of students have been less prepared than the last." The general gist always seems that kids now are dumber than when I was a kid. I think most people just seem to forget how dumb they actually were when they were a kid. That and the people in a position to make such comments were the smart kids in their classes.

I, too, am skeptical of DV82XL's statements. And I'm not sure exactly how "drill and practice" will help much with a complex subject such as multiple-digit multiplication. Sure everyone needs to know their times tables, but once you've mastered that, then learning to multiply multi-digit numbers involves learning a process, not memorization.

This doesn't seem too different, in principle, from the way kids have to spend forever calculating simple derivatives using limits, only to have some smart ass show everybody how you can just "bring down the exponent and subtract one."

As somebody who grew up being pretty successful in various state and national math competitions, I really like the partial sums methods, because hopefully it teaches kids WHAT they're doing instead of just how to poop out an answer to a multiplication problem.

So make them do it the hard way for a chapter or two, then let them move on to the shortcut. They'll be pissed enough at their teacher for wasting their time that they'll never forget the concepts of multiplications.

I became an English major when I found out my girlfriend at the time was doing some trumpet player the night before I was studying for a calc test my first semester of college. I swore off math forever. Sorta. It's a fun story, anyway.

Dave, thanks for the link. I've been here since CogDaily started, and I'm really happy to see that it has attracted a lot of fans.

As I mentioned here in a post about dual-process theory, it seems that reform-minded mathematics educators and administrators (and laypersons) favor the slower, more rigorous, conceptual S2 thinking over the quick, intuitive, superficial, algorithmic S1 thinking. One of the reasons for this is that, obviously, S1 thinking can get us in trouble in mathematics, which is in many ways counterintuitive (1/4 + 1/4 doesn't equal 2/8; the Monty Hall problem, etc.).

The partial products method is, in character at least, an outgrowth of this S2 favoritism. It is seen as a method that requires students to think about "what" they are doing, how multiplication works, etc.

That was hilarious. The TERC method she shows first [(20 + 5 + 1) x 31] is exactly how I do mental multiplication, if I'm forced to do it without paper. I couldn't have explained it better myself!

Of course if I have paper, I prefer to do it the traditional way. It is much faster.

I guess nether of my two critics did particularly well in reading comprehension ether.

I stated that the only way to develop a facility at computation was to practice - however I also said that if is no longer considered an important skill then we should just teach the basics and move on to the more important areas of application.

I grew up using log and trig tables and a slide-rule, most students have no idea how to use those tool, nor should they. I am just applying the same reasoning to this issue; beyond the basic mechanics why should they become skilled at dividing one mixed number by another?

As for the drop in math performance over the past several years, this has documented at length - even if it wasn't glaringly obvious.

You can't debate which styles are better without explicitly stating your criteria for what's good or bad. Those favoring the more conceptual, pure math approach are tacitly assuming that teaching kids rigorous, conceptual math is preferred to filling their minds up with as much utilitarian math as possible until they graduate high school.

I was once in the conceptual camp, but I deserted, and the reason is simple: because students vary considerably in IQ and personality traits, few students are even going to grasp what's going on conceptually (though they can parrot it back), and many of those who can are simply uninterested in pure math. Of course, if we separated the math nerdy brainiacs into a class of their own, then the abstract approach would be called for. So, it depends on the students you're teaching.

A previous comment derided the use of math by bookkeepers and engineers, but just ask students what they want to be when they grow up: what fraction will answer "mathematician" vs. any other field that uses calculators and concrete numbers? My stance is that math education is supposed to help the students fulfill their life goals -- not to make the teacher feel good and pure about their teaching style.

Take another example, and see how ridiculous the conceptual approach soon becomes (excepting the math nerdy brainiac classes). Back in pre-algebra, we all learned algoritms to see if a given number is divisible by 2, 3, 4, 5, 6, 8, 9, and 10. For example, to investigate divisibility by 3, you sum up the digits of the given number and see if that number is divisible by 3; if you can't tell after one application, you repeat until you can tell. The rule is similar for division by 9. Handy tricks, right?

Perhaps, but I guarantee that no one ever showed you why or how these algorithms work, as that would require a foray into number theory, in particular the fact that we use a base-10 system -- the algorithms would be different if we used base-6, for instance. And even if you were taught the rationale for some perverse reason, would you really care, assuming you weren't a math nerdy brainiac? No.

In sum, permitting exceptions for gifted math geek students, the other students should learn less conceptual math -- in their cases, it's either utilitarian math or no math retained at the end of high school, either because they can't grasp conceptual math or because they can but are by temperament inclined to shun the abstract. We also have regular and honors classes, so honors could use a little more conceptual work, but again in general, few students are ever going to benefit from a primarily conceptual approach, and we really should judge the worth of math education by how well it allows students to meet their goals in life.

I guess nether of my two critics did particularly well in reading comprehension ether.

We'll see about that.

I stated that the only way to develop a facility at computation was to practice

Actually you said "drill," but I think I got the idea. And I disputed it, suggesting that we're talking about a process, not rote memorization.

however I also said that if is no longer considered an important skill then we should just teach the basics and move on to the more important areas of application.

Certainly. But as I point out in the original post, it is not unimportant because it's necessary to be able to perform mental arithmetic in order to do algebra. What's more, the processes involved, if explained clearly, are directly applicable to algebra.

I grew up using log and trig tables and a slide-rule, most students have no idea how to use those tool, nor should they. I am just applying the same reasoning to this issue; beyond the basic mechanics why should they become skilled at dividing one mixed number by another?

But clearly you want your cashiers to be able to add and subtract, right?

As for the drop in math performance over the past several years, this has documented at length - even if it wasn't glaringly obvious.

I dispute that assertion, and I also dispute the implication that some sort of waffling over the mathematics curriculum is to blame for any alleged decline. Since you're the one who makes these claims, it's your burden to provide the evidence.

From: http://education.zdnet.com/?p=647

November 9, 2006
Educators address America's crisis in math performance

American school children lag far behind many other countries when it comes to math skills - a very worrisome trend to educators. That was the main topic of discussion recently at a forum of the National Mathematics Advisory Panel, reports the San Jose Mercury News.

Later:

Vern Williams, a math teacher said "...Americans need to dramatically transform their attitude toward math in order to improve learning. "In our country in general we've dumbed it down. We've taken algebra out of algebra. We need to understand that math involves hard work."

The bottom line is what Agnostic suggested, proper streaming is needed to make sure that those that will move forward in those fields that need a solid foundation in math be given it; and those that will not, be able to make change. And in this context 'making change' means enough practical mathematics to function in this culture.

And this is what concerns me the most. More than any other subject in math, an understanding of statistics, and probability has become of far greater importance to the average person, than knowing much of what is taught in even the less unchangingly courses.

I am not for one minuet suggesting that the 'old way was the best way', mathematics, as it is taught in schools has to change with the times - but nonsensical 'innovations', like teaching multiplication by partial products is not what is needed.

DV82XL:

I went and found the transcript to that session, and I didn't find that it supports your two assertions:

1) That U.S. math performance is declining, or
2) That this decline is due to "lurching" from one revolution to another

Instead, the speakers point to several fundamental differences between U.S. schools and schools in better-performing countries.

First, there is more stability and order in the school environment.

Second, teachers are able to more effectively focus students on complex mathematical problems (not drill and practice).

Third, related to the first two, is that in the U.S. there is significant difficulty retaining qualified teachers in the profession.

The researchers could find no single curricular program that led to superior achievement in other countries. The three items I list above are the factors that really matter.

Ya, I stand corrected there is nothing wrong with math education in the US.

Everything is just fine.

DV82XL:

It's fine for you to make sarcastic statements, but I still would like you to show me how shifting the math curriculum to demand more "practice" is going to help retain teachers, create stability, and focus students on complex problems.

Note: It's possible for U.S. math performance to not be declining and still lag behind other countries.

I've never seen the partial products method until right now, and honestly if you had shown a nine-year-old version of me that thing on the right I would probably cry. That would be the end of any interest in math I might have had.

Okay, I've looked over the comments and I think a bit of an apology is in order. If we're really lagging so far behind other countries -- and the evidence suggests we are (though I'm not convinced that part of this isn't due to the dropout rate/academic tracking in other countries), then whether or not U.S. performance is stable is beside the point.

But the larger point remains: Are curricular changes responsible for the U.S. decline? The evidence provided so far suggests not.

Re: US lagging -- that's mostly due to population heterogeneity for IQ, namely the substantial black and non-white hispanic populations in the US: the former average 1 SD below the white mean, while the latter, depending on which hispanic group it is, average 1/3 to 2/3 SD below the white mean. When you look only at whites in the US vs. mostly white countres like Sweden or the Netherlands, then the US ranks pretty highly in TIMSS. Asian-Americans rank higher than Asians in Japan or Taiwan.
http://www.arthurhu.com/index/timss.htm

But getting back to the topic, relying too much on conceptual approaches could even harm the math geek students. Pre-college math should prepare them for the things they'll encounter in science, math, & engineering in college. Because conceptual approaches take more time, and because there is so much math to learn, I'd prefer them to get as much under their belt before they get to college, at which point they can delve more into conceptual thinking.

It's just not very useful to think conceptually about trivial things like multiplying two numbers. If we eliminated all the superfluous parts of the standard math curriculum (like conic sections), and emphasized knowing how to do math (rather than why things work the way they do), someone who's good at math could get through the equivalent of 2 semesters of calculus, a semester of linear algebra, and a semester each of probability & statistics by the end of high school. Now that's a sufficiently diverse toolbox that they can think conceptually about interesting things.

I have always had a hard time in my high school and college math classes because I am a left to right adder. I do the biggest digits (most significant digits in computer speak) first. The teachers and professors teach right to left. As a result, I had to take each collage calculus classes 2 times, once to see where everything is going, and then to actually do well in the tests.

In my head, I can tell in in about 2 seconds that this multiplication is a million three something, but I wouldn't get the right answer as I loose track of ALL the carries. I also tend to clump digits into pairs as I can multiply 5 times 26 in my head - 100 and 30 so 130. Actually, it's 5x25 + 5x1 so 125 + 5. The next step is to add 5x30 + 5x1. The digits get visually shimmed over if that makes any sense, hence the brackets around the pairs of digits which don't line up to actual hundreds, thousands boundaries. This gets added together for 13155(00). The next step gets confusing. 3x25+3 (00)(0) gets added to prior sum but I don't visualize the (00) so it's 13155 + 780 = 13855 + 80 = 13935. and so forth, like I said, I tend to loose track of the carries in this large of a multiplication.

2631
X532
----
130 (00)(00)
155(00)
78(00)(0)
93(0)
48(00)
62

The good thing about this is that I can look at what the calculator gives me and "know" if it is right or wrong. Well, I can tell if the answer isn't close. The bad thing is that I am usually wrong, but very close when the calculator isn't near and I do it in my head.

Yikes, looking back at my process, no wonder I have trouble with higher level math. I don't suggest teaching this to anybody as it's a backwards view of doing math and makes learning other math much harder as it's coming at you backwards from how you think.

"But the larger point remains: Are curricular changes responsible for the U.S. decline?"

Bingo.

Of course not, conversely changes in the curriculum aren't going to make things any better. Yet this is what gets fiddled about with. Discipline issues, crumbling buildings, and indifferent parents, which would take some political courage to deal with are never addressed.

Dave, I'm sorry but Mr. Person and M.J. McDermott are both in bizarro world if they think the "partial products" method and the traditional method are pedagogically different. In fact, they're not different enough, in any sense, to warrant distinct names.

How can one look at Mr. Person's diagram, noticing the way the column on the right is broken up into "First Line," "Second Line," and "Third Line," and not see that to learn one is to learn the other? There is no operation that I can see in the right hand problem that is not implicit in the left.

McDermott's video is a shameless piece of propaganda, to be sure; just look at the way she demonstrates the first alternative method, drawing arrows around and around with no particular order. But if Mr. Person really wants to deflate her argument, wouldn't he do better just to show that, in fact, one of her altie methods IS the traditional method? (And then, like you say, kids should "learn the partial products method first, then learn the 'shortcut' later on...")

Yet this is what gets fiddled about with. Discipline issues, crumbling buildings, and indifferent parents, which would take some political courage to deal with are never addressed.

Yep. I'd add treatment of teachers to that list, but basically I agree with you. When the only way teachers are rewarded for good performance is to be transferred to less challenging positions, the system is broken.

And then there's lattice multiplication, which is apparently the "in" thing to teach these days, at least at my sister's HS. One number across the top and the other down the left side, each product in its own little box (ie, ones times tens in the second box in the ones column, tens times hundreds in the third box in the second column, etc), and then add. Same old method, different way of writing it. As far as I can tell, it's partial products without all the extra zeroes, and I think the idea is to refrain from intimidating kids - all the way through, they're dealing with the basic times tables and adding 2-digit numbers. Nothing long and scary.

Other than the little shortcuts that people who do a lot of mental arithmetic use, is there really any other way to do it?

Typical questions during a job interview for a top consulting firm or investment bank (at least a few years ago...):
- Guesstimate how many gas stations there are in China now
- Guesstimate the weight of a 747

Of course, no paper allowed.

The question should not be "what is more efficient" but "what is more effective" for each context, and then we can care about efficiency...

I'm with several other people on that this.

The partial products method is almost exactly how I do it when I have to do it mentally. I just do lines 3 to 1 depending on what resolution I need. I guess I just have bad working memory so I do what I can to get around that.

I have to say though, all mental arithmetic does is prepare you for menial jobs.

Teach people how to count properly, rather than a heap of tables by rote (or even historical facts/ formulae etc.) And teach them how to count in different bases, and what each column actually means. Then they will understand numbers.

While i agree that showing all the steps is a big improvement over the shortcut for teaching kids, there are other techniques that should also be taught. My son, in 4th grade, is being taught estimation, and checking addition with subtraction.

When i was in high school, i picked up a Japanese abacus (a Soroban) with two books on how to do it, and spent about three months doing it. Towards the end, the books introduced mental arithmetic. I went back and did all the examples that way. Two things struck me. First, i didn't make a single mistake. Second, it was really fast.

The examples included things like 147888 / 632 = 234. I was astonished that this could be done as mental arithmetic, and so easily. I was curious about how far this could go. I had a calculator (this was 1975, so this was brand new). I came up with examples where i'd multiply two nine digit numbers to get an eighteen digit answer. Also, i'd divide one nine digit number by another, and compute nine significant digits of answer. That took almost exactly one minute.

The Soroban is optimized for decimal. There are four beads below the bar, and one above. The single bead is worth 5, and the other four are worth one each. So, one rod can represent one digit - zero through nine.

So, i've taught my son how to add and subtact on the soroban. Then, i discovered that he wouldn't have one at school. So, i went back and taught him the same techniques but using fingers. The thumb is worth five. For a complete description, see my blog.

I should mention that i don't teach it the traditional way. Kojima has 20 cases, sort of exceptions, and when you've learned them all, you can add and subtract digits. I have eleven lessons for addition - step by step.

Why does it work? For one, carries (or borrows) are performed right away. There's nothing extra to remember. Secondly, addition and subtraction become mechanical. If there are two ways to do the next step, one will be impossible, so you do the other. Divides are really cool. If you guess wrong on how many 632's go into 147888 (i mean, you might, right?), it becomes clear when you subtract out the product. Well, you can always subtract an extra 632, or add one back. You don't have to back up and start from scratch. So, the guess can be 'wrong', and you still get the right answer. Since i dislike working with negative numbers, i'd try to guess low. Sometimes, i'd subtract an extra. No big deal.

The real problem with arithmetic, the thing that strikes fear in to the fearless, is that 2 + 3 = 5. If you get six, that isn't "close", it's just wrong. Fear of arithmetic is the fear of failure. The way that arithmetic is taught today, starting with addition and subtraction, is unreliable, and therefore inspires fear. I'd go with one of the other techniques first.

All of this said, multiplication is the easiest of the four operations to learn. Division is worst. Then addition, then subtraction, then multiplication. Why is addition harder than subtraction? Well, by the time you learn subtraction, you have built many skills learning addition, and you can just reverse them.

It really depends on how much is processed mentally and how much on paper. There really is no difference in the amount of effort. Mentally I would multiply the first number by 1000, a simple decimal shift, and divide by 2, in one step. Three steps followed by an addition would suffice. Some products are much easier than others to compute mentally, in order roughly, 0 and 1, 2 and 5 (10/2), 4 (2x2) and 9 (10-1), 3 6 (2x3) and 8 (2x2x2), 7. This is due to the decimal base. Not all steps are equal in difficulty.

Stephen: I remember seeing a young girl on a TV program here (Japan) years ago who was faster at multiplication/division of large numbers than anybody using a calculator. She explained that she learned on a soroban (which they still use in schools here, I think) and while she didn't use a physical soroban during the show, she used an imaginary one. They had a team of people with calculators on the show, and as they were tapping away you could see her fingers twitching as she calculated in her head.