Wednesday, September 26, 2007

Looking Further at Multiplication

For some reason, the lattice method of multiplication really annoys some people. I wish I could understand what appears to be an irrational rejection of a perfectly sensible approach that is just as mathematically sound as the "traditional" way most Americans were supposed to learn to do multi-digit multiplications.

I recently had another fruitless "dialogue" (read: smash-your-head-against-a-brick-wall, you'll-get-further argument) with an entrenched foe of anything and everything viewed as tainted by the evil worm of fuzzy, reform thinking. I will try to edit it into something that some readers may find useful. What started this was the posting by another notorious reform opponent of a link to a YouTube video that shows a way to do multiplication by drawing a series of crossing lines. Of course, the underlying mathematics is the same as what makes the lattice method (and pretty much all methods I've seen taught) work. But the video appears to be presented more as a "neat trick" than as something to be taken seriously, and I know of no one who is teaching it to kids. It wouldn't be disastrous if someone were, but it seems a bit too much like a bit of showmanship than something anybody wants kids to learn.

The person who posted this to math-teach sarcastically called it "the lattice method made easy" or something like that. I commented that it WASN'T the lattice method, but that this video hardly constituted a reasoned critique of lattice multiplication or any other alternative model or algorithm. And once again into the breach stepped an old antagonist to insist that the Vedic multiplication really WAS the lattice method. What follows is comprised of some of the things I wrote in response to this anti-reform critic.

Why Not Teach Multiple Methods?

It's rather hard to get away from the fact that many students like and prefer the lattice method to the "traditional" one. So foes need to either "get over it," as the saying goes, or deal with the real issue: this and any method that makes mathematical sense should be taught "transparently": that is, the sense as well as the procedure needs to be investigated by teachers and students. Otherwise, it's no more or LESS of a trick than is ANY algorithm. If you deny that, you'd best have some logical argument to support why it's different to do what you learned in school, that the "traditional" method is "real" math, while the lattice method is not, etc. I can't help but insist that such a viewpoint has no merit, as I've yet to see any logical argument to persuade anyone who isn't simply entrenched in the Mathematically Correct/HOLD dogma

Who cares whether something is "standard" as long as it works and makes sense to the person(s) using it?

Of course, we don't expect teachers to teach EVERY single algorithm they've ever heard of or seen. If a KID were to present the algorithm from the video, it would be nice for the teacher to be able to recognize the underlying structure. But why identify it specifically with the lattice method? What you really seem not to get (though maybe you do and just can't acknowledge for political or religious reasons) is that ALL the algorithms one sees in EVERYDAY MATH and other books, reform or not, are based on the same idea: find a fast way to do repeated addition based on the fact that we have a place-value system.

Four Models Worth Exploring

Area Model

1) Note: the example illustrated above uses smaller numbers than the example I have explicated in the text, but the basic procedure and interpretation are the same.

Take a piece of decimal graph paper and find the product of 32 and 27 on it in the manner I will describe: write along the top of the page "32" and draw a line along the top boundary line of the graph (not of the paper itself) that runs across three blocks of ten and then two single squares in the next adjacent block of ten. Then write 27 down the left margin and draw a line down the left boundary line of the graph that runs down two blocks of then and then seven unit squares in the next adjacent block of ten.

Next, extend a line from to the left at the end of the line you just drew equal to the length of the line you drew at the top. Extend a line from the right end of the new line up to meet the end of the first line drawn.

The rest, which is easier to show than write, involves picking four colors and with markers, crayons, or colored pencils, color the 10 x 10 squares one color, the 10 x 7 rectangles a second color, the 2 x 10 rectangles a third color, and the 2 x 7 rectangle a fourth color. Counting up, you have six 10 x 10 squares + three 10 x 7 rectangles + two 2 x 10 rectangles + fourteen unit squares.

This gives us 6 x 100 + 3 x 70 + 2 x 20 + 14 = 600 + 210 + 40 + 14 = 864

Expanded notation

2) Compare this with the "expanded notation" model:

32 x 27 = 30 x 20 + 2 x 20 + 30 x 7 + 2 x 7 = 600 + 40 + 210 + 14 = 864.

The Lattice Method

3) Note: the example illustrated at the beginning of this entry uses different numbers. I am staying with the same two numbers throughout my explanations, but again, the directions and analysis remains the same.

Compare this with the "lattice method." You draw a 2 x 2 box and draw a diagonal in each box from the lower left to upper right corners.

You write 3 and 2 over the top boxes, respectively.

You write 2 and 7 down the right side next to the top right and bottom right boxes, respectively.

You multiply 2 x 7 and write 1 and 4 in the upper and lower compartments of the bottom right-hand box, respectively. You multiply 2 x 2 and write 0 and 4 in the upper and lower compartments of the top right-hand box, respectively. You multiply 3 x 2 and write 0 and 6 in the upper and lower compartments of the top left-hand box, respectively. And then you multiply 3 x 7 and write 2 and 1 in the upper and lower compartments of the lower left-hand box, respectively.

Now starting in the lower right hand corner, you do diagonal addition from "top to bottom," moving from right to left and writing one digit under each box, carrying any extra digit, if it occurs, to the next diagonal. You get in this example, from right to left, 4, 6, 8, and reading this from left to right as we normally do, we have the correct answer, 864.

Because of the numbers chosen for this example, there are no carries. I've found that they are the only stumbling block likely to emerge that isn't related to errors in the individual multiplications or final additions due to carelessness or lack of knowledge of one-digit math facts. After practice, most kids handle this carry issue with ease. The model extends to more digits and to decimal numbers as well.

Is this algorithm "better" than others? Not for me, personally, but then, I didn't learn it until a few years ago. Lots of students like it. It seems like a "neat" (in more senses than one) way to do the partial products.

What's clear is that it's really NOT significantly more time consuming, no matter what the nay-sayers insist upon arguing, and certainly not to the extent that they would have us believe. It's a compact algorithm, like all of them. The only one discussed so far that I do not present to teachers or kids as a way I actually would like to see them do their work for more than the purposes of THINKING about (and in this case VISUALIZING) the partial product <=> area similarities is, of course, the first one, because we really don't need to do a drawn out set of rectangles and coloring (which clearly DOES take a relatively long time by comparison to all the others) to get the answers. We only want to see that there is some underlying relationship between area and multiplication. For very visual kids, however, sometimes the coloring, which I present fourth, not first, really helps tie everything together, and for some it's a key breakthrough model, so it's definitely worth spending some time on it, at most part of a period, and then re-examining ALL the models one uses to see how they relate to one another.

Keep in mind, too, that we could use other tools here, including blocks, tiles, or other hands-on models. But I think we have enough for now to make the important points.

The "Standard" Algorithm

4) Compare this with the "standard algorithm"

32 x 27 = "2 x 7 = 14. Write down the 4, 'carry' the 1'; 3 x 7 = 21. Add that 1 you 'carried' and write down 22. Now move down to the next line underneath and write a 0 under the 4 (alternatively, "mentally shift one place to the left before writing down the next digit); 2 x 2 = 4. Write down the 4 (under the middle digit above). 3 x 2 = 6. Write down the 6 to the left of the 4. Now draw a line underneath what you just wrote and do the resulting column addition. 4 + 0 = 4; 2 + 4 = 6; 2 + 6 = 8. Read your answer from left to right: 864.

(Yes, I purposely mechanized the description of the last algorithm, but not unfairly so: that's what kids ARE taught to do, after all, and it's not all that hard to see some of the places that they can and often do go wrong. No algorithm is fool-proof, and any algorithm is subject to errors in the sub-calculations as well as in where one writes the partial results (here, the partial products). What's GLARINGLY missing from this particular algorithm is any conscious consideration of place-value. Except for that shifting, which is generally taught as a mindless step to be religiously observed, there's no acknowledgment that with the exception of the first multiplication, everything you say to yourself is a lie. You never multiply 3 x 7, but actually 30 times 7, and so forth. The compression process gains speed but loses information.)

Now, if every kid who was carrying out this or any other algorithm had a reasonably good grasp of what s/he was being asked to do, none of that "lying" would matter. But for far too many kids, that standard algorithm makes as much sense as voodoo (perhaps less). Is that the fault of the algorithm itself? Not really. It's a fact, however, that our standard algorithms for both multiplication and division are all about speed, and so we sacrifice some information (what the REAL partial products are in the case of multiplication; what we're subtracting in pieces from the dividend at each step of the division algorithm) without, we hope, loss of accuracy, compressing the repeated additions or repeated subtractions, respectively, because place value lets us do this.

But you REALLY have to think like a kid: not a kid who either doesn't give a rat's behind about comprehension, but is aces at following instructions and knows the basic number facts well enough to do these two algorithms with accuracy and minimal screw-ups, nor like a kid who REALLY gets what's going on, practices, and then can do the steps automatically, but rather a kid who has holes in his knowledge of the facts and/or is not adept at following a set of steps that make little or no sense to him. This kid is going to make repeated mistakes, either in the sub-calculations or in where he writes things, or in the additions or subtractions, or, likely, all over the place. And this kid will be so bloody confused about where he's going wrong that he likely will sink into deeper confusion, quite possibly convinced that either multiplication or long division (or both) is way beyond him, or that these are some pretty messed-up algorithms.

There really are lots of places to mess up this algorithm (and the division algorithm). So rather than scream that these are perfectly good algorithms that EVERY kids MUST learn and that all other models and/or algorithms are stupid, inefficient, impractical, dumbed-down, not "real" math, etc., why not accept the fact they are all grounded in perfectly solid mathematics, that some models will connect first for some kids, while others will connect first for other kids, and eventually a competent teacher will do her best to see to it that all kids have the chance to make the underlying connections among these models and algorithms. Then, it's a tad easier, in all likelihood, to make a truly convincing case for the standard algorithm as the fastest (because you don't have to write down as much), but it will STILL be up to the student to choose.

If your goal is understanding and competence, I see no other choice. If it's a fanatical and irrational opposition to alternatives, well, be my guest, but you'll never get me or thousands of math teachers who work with real kids to accept such bizarrely rigid thinking.

The actual lattice method makes sense if you take the time to think about it. I find it objectionable, however, that it, like the "standard" algorithm and most of the rest of grade school math, is generally taught with no eye towards why and how it works, only as a black box to be followed mindlessly.

My antagonist wrote: "Of course the actual lattice method makes sense if you think about it. No one questions its validity, rather the questioning is whether it has any real instructive value, or lasting value."

How are the terms "real instructive value" or "lasting value" meaningful? They're undefined and unexplicated terms here that are, I suspect, offered for rhetorical rather than logical impact.

What I find objectionable to the anti-reform approach to "analysis" is that it seems to care only about any straying from the "traditional" black boxes.

I'll simply state that any black box is undeniably a black box. If you look inside the box and show or learn how it works, then for you it's no longer a black box. I've repeatedly said that I'm incensed by any teacher who teaches ANY method as a "black box." The fact that this one gets taught that way infuriates me. As for efficiency and volume of paper, I think those are clearly red herrings, especially the paper issue, though the time one isn't much more relevant in practice. A kid who knows this method can whip through it with facility. Ditto all the above models EXCEPT for the area model. There, both time AND paper are definitely issues. But it's a model to be explored only for understanding, not as a long-term strategy.

I was asked, "In the classrooms you prefer, does a lattice method get taught in addition to a standard algorithm, or instead of the standard algorithm?"

I have NEVER seen anyone teach the lattice method by itself. Never. What I prefer here is irrelevant to what is actually done, but to be perfectly clear: I want at least the four methods I outlined above taught, for reasons I hope I've made clear. For my money, the partial-products method is really at the heart of ALL other methods, so I would like to see it taught as the first "efficient, compact" algorithm. However, I believe that students should first do some one digit times one digit problems as repeated addition and then at least one two digit times one digit problems (e.g., 13 x 7 and 7 x 13) in "both directions so that the idea that repeated addition is far too time-consuming and space-consuming to be either efficient or (as the numbers get bigger) terribly accurate. How easy is it to write too many or too few 7's or to miscalculate when doing just the example given?

The idea is to have students appreciate what is gained by compression, but also to see what could be lost in term of information and what we "say" to ourselves as we do these other algorithms.

My opponent wrote:

For example, from the California Grade 4 Standards:
" 3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multi digit number by a two-digit number and for dividing a multi digit number by a one-digit number; use relationships between them to simplify computations and to check results."

How is "demonstrate an understanding of" operationalized in any assessment you or the MC/HOLD crowd would accept? I know that "the ability to use" is operationalized on EVERY test I've ever seen.

He next wrote:

"There is still a use for teachers, Mike. And does your beloved lattice method makes it easier to detect 'understanding'?"

There's nothing more beloved about the lattice method, for me, than any of the others. They each have strengths and weaknesses, advantages and disadvantages. Any can be taught mindlessly. None SHOULD be taught that way.

"Demonstrating an understanding of ..." is not the same as 'mindlessly apply a method learned by pure rote drill and kill".

I agree

Then we're getting somewhere.

No, we're not. Because you still won't explain (and deliberately omitted my questions about) how you would assess the former in a way that you and I and Wayne and most reasonable, knowledgeable, intelligent people would accept as meaningful.

And you're still trying to denigrate all methods but one. I have no intention of doing anything of the kind, as I've explicitly outlined here and previously. I'm not trying to "sell" anything. I'm trying to get just one anti-reform person to finally admit that all these methods are grounded in real, valid mathematics, and that all can serve useful PEDAGOGICAL purposes. I don't really care which method kids choose to use, though I, too, would likely try to sell them on #2 and #4 over #1 (for sure) and #3 (though I have not objection to anyone who uses it if s/he uses it "well."

If your retort to all this is going to be more empty claims about time, paper, etc., leave me out of it. If you've got something non-rhetorical to offer, indicating that you've decided to drop your usual stance and engage in meaningful conversation, and if you want to answer that assessment issue and to operationalize the terms you used in your previous post on this topic, I'm all ears.


I wish I could report that the reply I received was useful. Unfortunately, as the reader likely has gleaned from just the dialogue in this entry, my opponent remains just that: a dedicated foe who has no interest in conceding anything positive about any "non-standard" approach to thinking about, teaching, or doing multiplication (or long division, or anything else in mathematics). Naturally, since I have a long history with him, I was not surprised. I no longer write on public lists in hopes of convincing those who are incensed about every and any reform idea or method with which they themselves were not taught, be it in mathematics, science, literacy, or any other educational area. I write to help clarify my own ideas and for those readers who are at least moderate or neutral or simply looking for varied viewpoints. When I mentioned that I would not continue with this fellow on these issues, but that I expected that what I presented about these models was likely useful to some readers, he retorted, "I doubt it." And of course, he really does.

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