Multiplication, Algorithms, Tricks, and “The One Best Method”

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I’ve just been perusing a very interesting blog (and a great resource for teachers in public schools). It’s called MathNotations.

This post intrigued and annoyed me, though. (Hey, maybe that’s a sign that it is a good blog!) It’s a poll about which method should be used to teach multidigit multiplication, like 48*73, for example. (If you do go to the link, make sure you scroll down and read the comment on Jan 30th by Michael Paul Goldenberg. It is excellent.)

Unfortunately, this poll is guilty of the same myopia as the American school system in general. It’s about creating a “standard.” Standard is just another word for limitation for people who really don’t know how to excel.

In the case of this poll, it is about choosing (out of an artificially limited group of choices - which is the logical fallacy of “false dichotomies”) how multidigit multiplication should be taught.

The wording of the poll is:

“Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5:”

Um, here are my options? I think not.

One of the great problems in (at least) American education today is that we’re firmly locked, sealed, and vacuum-packed into the box of pedagogical dogma.

Standard algorithm, partial products, lattice method, indeed! The myth of the “one best method” is still so rampant in our “developed” nation.

Education is not about inculcation of any algorithm. It is about students gaining insight, knowledge and lasting value. You can’t do that with “just shut up and learn this method,” just as you can’t do it with, “I’ll shut up and let you teach yourself.” Those are the ultimate false dichotomy in education of our time.

If teachers don’t know at least ten methods of how to multiply, they shouldn’t be teaching multiplication to more than nine students. It’s easy and important to understand multiplication in depth if you are entrusted to teach it to young minds. Go to the library, get “Calculator’s Cunning” by Karl Menninger, and get some chops.

And that doesn’t mean “tricks.” God, how I hate tricks. They trivialize anything they are attached to. How can I say this? I can because I am a professional magician. We (at least the good ones) hate “tricks”. One thing magicians know, is that as soon as you teach the “trick,” the magic is gone. It takes all the appreciation out of the effect.

You never show anyone how to do anything until they are ready to appreciate the thought and effort behind it. You would only teach a trick to another magician, or serious student of magic. You wouldn’t even teach it to one of them unless they’ve demonstrated that they are ready for it, and have a firm basis in the other magical concepts and skills that they need to pursue the trick you are teaching.

One of the dangers of teaching “tricks” is that you, as the teacher, might actually think that you are seeing a light bulb go off when the child says, “Oh, I get it!” But that is the same false light bulb that we magicians see every time a person says “Oh, I see how he did it now!” when someone tells him how a particular magic effect is done. They only know the most superficial part of the method. They can’t actually do the effect to any worthwhile degree, they only “know how it’s done!”

It’s like the hip jazz musician who meets the suburban musicologist, and says about him, “Yeah man, that cat knows where it’s at, too bad he doesn’t know what it is.”

The same goes for teaching multiplication. You must teach the reasons that the method works. If the child isn’t ready to understand the reason, s/he is not ready to use the “trick.” In other words, it shouldn’t be a trivial trick, it should be a meaningful method. And that meaningful method should be based on the distributive property.

How do you do this?

You have to get to know the child, and where s/he is with math so far. What so many pedagogues forget, is that education is about the student, not about the material. If the child struggles with addition, take a step back and cover that until the child understands it in his bones.

It doesn’t matter that you have to cover curriculum. It doesn’t matter that you are on the multiplication unit in school at this point. Clearly the pupil is not. You are a teacher, you know this. The administrators don’t, I know, I know. This is a problem. You can please them, or you can teach math. You can’t do both. If you can’t fight a bad system that you’re in, you are the system.

So you figure out if the child is ready to learn what you plan to teach them. If they struggle with “the tables,” and you are about to teach them the standard algorithm, you must get get them up to speed until the real light bulb goes off in their head - until they understand that “times” (with whole numbers) means “groups of”.

Then you must make sure they understand the distributive property in order for them to learn what they are multiplying when they multiply multidigit numbers. Have you explained that to them well enough? Do you understand it yourself?

It doesn’t matter which of the typical methods are taught in schools if they are going to be taught as “tricks” or taught as “show-and-tell” of “how to do it.” None of them will have any meaning.

And by meaning, I don’t mean, “grades went up.” You can get great grades with “tricks.” It makes teacher’s work easier. But it doesn’t teach anything valuable in the long run. If you teach for understanding, you get lasting value. If you teach with tricks and games, you are teaching that math is only good if it is not about the math. Great lesson, huh?

Education is not about inculcation of any algorithm. It is about students gaining insight, knowledge and lasting value. You can’t do that with “just shut up and learn this method,” just as you can’t do it with, “I’ll shut up and let you teach yourself.” Those are the ultimate false dichotomy in education of our time.

As usual, please take my views cum grano salis (with a grain of salt.) I’d love your comments, input, thoughts, rebuttals, etc. Math Mojo improves more from your input than from mine.

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