Learning Multiplication by Rote is a Disease

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Today a concerned reader took issue with what he understands my methods to be. (See comment #4 at Augends, Addends and the Commutative Law of Addition.)
Fair enough, but I think he may have misunderstood my methods.

That could, of course, be due to the way I communicate them (or miscommunicate them). First let me say that none of the algorithms (ways of solving math problems) I teach are “mine.” “Math Mojo” is the name of my attitude, not the methods. The methods have been either gleaned from better sources than me (and most are hundreds, if not thousands, of years older), or I have “re-invented” them. That is typical for most people’s alternative methods.

Now to the issue; the reader stated:

After all these years (30) of struggling to teach children math, I finally realize why it is so difficult. A brief perusal of some of the mathematical girations you go through to multiply two numbers together explains a lot of why kids are poor at math. Commutative and associative properties are more easily understood when you have the basic tools to work with without adding zeros then subtracting the number from your cousins name on your mother’s side of the family. Teach the basics by rote then progress to the more abstract. Simple to complex seems to work.

Professor Homunculus’ reply:

I’m sorry you’ve come to that conclusion. If you’ve been teaching math for 30 years, you surely have some insights. But I can’t see see how you’d say, “simple to complex” seems to work. May I ask where it seems to work? And if it does, why is it a struggle for you, and why is it so difficult? Have you been teaching with the “girations” (sic) you say I use to make it so frustrating?

I’m not quite sure I understand the logic of your position.

(Caveat - I feel very strongly about this issue. Please don’t take it personally. It’s a sickness.)

I teach hundreds of kids each year, in person, and I don’t know how many on the internet. The overwhelming reactions on both sides (theirs and mine) seem to be satisfaction and light bulbs going off left and right. Many children and their parents have actually cried in front of me in relief from “the basics” (times “tables”, multiplication “facts” and such.) Not everybody likes the methods I use (I wouldn’t expect them to) but so far, you are the first to blame me for the failure of your school system.

“Math Reform”

Please don’t get me wrong. I’m not representing some idiotic, airy-fairy “math reform” (as you call it). I probably want to throttle those people more than you do. Math Mojo is about discovering solid, meaningful relationships between numbers, not being a some new-age (rhymes with sewage) dunce. But neither is it about being a drone to a system that never really worked.

By the way, what is “math reform?” Is it the movement that resists the “tradition,” or is that movement that resists that? I’m never sure about that. (I resist both.)

As I’ve said in the Introduction to These Chronicles:

“Never accept ‘alternative’ as better until you have tested it. By the same reasoning, never accept the ‘accepted way’ until you have tested it, either.”

It also pays to pay attention to the results of that testing.

Come on, look at all people who actually excel at anything. Think Shakespeare was enthralled with writing book reports in which he’d get points taken off for bad grammar (”To who, my lord?” King Lear l.iv.24, V.iii. 249) ? Think Gauss liked adding up long columns and having to “show his work?” “Hey, Mozart, stop with that nonsense and go practice your scales!”

What? Not all students are exceptional? Gee, I wonder why anyone would think that. In one way or another, all of them are. That is not the common contradiction you might think it is. Read it again.

Simple Complexity? Complex Simplicity?

I don’t think there is anything more complex about the methods I teach than there is about what some people call “the basics.” I talk a lot about the theories behind them, but that’s only in order to explain it to those who are interested. You may be assuming otherwise, which is understandable, because sometimes I get so into it that I can’t shut the hell up. My bad.

Really, the only “dogma” I may subscribe to is that there is no one best way. And if there was, it certainly wouldn’t be the times tables.

“…and Let your Backbone Slip….”

If I can get most kids to know multiplication of one-digit numbers in less than five minutes with what you seem to think are “gyrations”, why on Earth would you devalue that in comparison to whatever “the basics” might be?

Here’s something that I’ve been wondering about. Maybe some interested reader has some thoughts on it:

If I gave a ten-year old child a car, and said, “Here are the keys. You put your foot on this pedal to make it go, and this one to make it stop. Turn the wheel to steer. Now you know how to drive. Be home by ten,” would that make any sense? I mean, yes, the kid would know, “the basics” but they’d be less than meaningless to him - they’d be frustrating. And they’d be dangerous to him and us.

How is that all that much different from giving a kid the “basics” (”Just shut up and learn them, or you’ll fail!”) of multiplication? He’s had no fun learning them, cannot relate the “facts” to anything else, and generally learns that this is exactly the point where you start hating school. Oh, boy!

Most children have no problem learning multiplication by 10. Imagine then saying to such a child, “Hey, did you ever notice that multiplication by nine is the same as multiplying by ten, then subtracting the number?”

Of course the child has. What kid hasn’t? Hmmmm, there is something they’ve noticed themselves. And now they are recognized as having seen that for themselves. Even if they hadn’t, they will see it now, and it will be interesting to them. (Imagine that!)

Now, with no difficulty, you’ve engaged the kid. You could treat the whole thing as a “math trick for nines”, and then go on pummeling them with the tables. Or you could have them try multiplying all the digits by nine that way to if it worked. Then try it with nine times a two-digit number. Did it work?

It did! Hotcha! It took them less than three minutes to learn the nines “tables,” and have a relationship with the number nine. Is that too complex?

Let me go out on a limb and try to put in clear terms what the heart of the problem is for most children learning multiplication:

We have a calcified system, which for some reason(s) finds it necessary to defend itself as ” the one best system”. So if the students don’t learn, it’s because they just won’t accept learning with “the one best system.”

Seems to me that if the kids aren’t learning with it, then the “one best system” isn’t working. Should we then blame it on a method that almost nobody in the public schools is teaching?

That doesn’t mean that we should Teach Extremely Rotten Crap instead. It means that we should search for ways that seem to work, which can rationally be shown to be meaningful, and try them. If they don’t work, try something else. Or is that too complex?

Progress can be complex. Yes, it’s simpler to keep failing, but I guess that just doesn’t float my boat. Most students and parents aren’t big fans of failure, either. And I’m betting that most teachers would say that they feel the same if they weren’t afraid of hearing the sound of NCLB (”No Child Left Behind”) jackboots in the stairwell.

Look, I’m frustrated with the crazy Ideas floating around, passing themselves off as “education,” especially when it comes to math, as you obviously are, too. I just think you are picking on the wrong target. Maybe much of my writing does not convey the absolute simplicity and elegance of the methods I endorse. Face it, I’m a hack, run-on-sentence kind of guy. ADD does that to you. (But I like it, heh, heh, heh…)

But if you experienced how quickly, and with what joy and relief most kids and adults learn from someone who really “gets” this stuff, I think you’d add some of it to your repertoire. Really, “just memorize it” only goes so far (and it doesn’t go anywhere that’s interesting).

Actually, let me say it now - Rote rots. It is a miserable and counterproductive method of inculcation disguised as teaching. It may work is some rare cases of children who have some mental handicaps, but it’s not the way to bet, even in those cases.

Almost every single person who has learned by rote could have learned more effectively by some other method. The only reason some people haven’t noticed this is because they have never tried anything else for any meaningful amount of time. It’s the “better- to-curse-the-darkness-than-light-a-single-candle” syndrome, and it is endemic to public schools.

Addendum:

Today someone sent in a comment about this post anonymously. I’m sorry, but I don’t accept anonymous comments. The commenter misunderstood almost everything I said, but he did bring up some valid points. Unfortunately, he erroneously assumed that I was against the points he was making. Maybe I should clear it up.

I do not think that the only thing teachers do is teach by rote. I can’t believe that anyone would be silly enough to think that’s what I was saying, but apparently the commenter did.

He assumed that I believed that “the traditional way” was devoid of teaching concepts. Of course the “traditional” way teaches concepts (albeit pedestrian ones). I’m just pointing out that rote memorization sucks, not everything about the “traditional” system sucks. Jeez.

Who’s Tradition is it Anyway?

By the way, it was the writer that kept using the term “traditional.” I don’t really like using that term broadly. I mean, what system? Who’s system? I, for one, am not ethnocentric enough to think that what’s taught in the schools in my (or any other) country is “the traditional system.”

The main premiss of the comment was, “The traditional method of teaching mathematics works.” Sure, and the Surge is working, and the check is in the mail, and Paul McCartney is dead.

Knowing by Heart

Aside from there being nothing offered to back up that dubious premiss, the commenter seemed to think that I was not for learning by heart.

Let me disabuse anyone of that idea. I am for knowing basic multiplication by heart. I an actually for learning basic math stone cold in your bones. Everyone should know up to 20 by 20 without having to think. 19 * 14 =6 for instance, should be as apparent as C-A-T spells cat. You shouldn’t have to think about it.

Here’s the big difference in our thinking, though: I know, that when you teach in person, it shouldn’t take more than a few weeks to get a child to have that stuff down so they can do it in their sleep, and none of it entails staring at meaningless “tables.”

Knowing by Heart vs. Learning by Rote

Rote is a method of learning (generally by brute, primitive memorization, without a system for memorizing). It is drudgery. On the other hand, “by heart” is a way of knowing. Let’s see if this can be made clear:

You know the names of your family members, sports stars, celebrities, teachers, colleagues, game rules, and myriad other things “by heart,” but you didn’t have to sit in front of a “table” and bore yourself to tears while the evil “Dr. Textbook” insisted that that was the only way to learn them, did you? Of course not. You learned them because you had a relationship with them that meant something to you. There was no resistance to learning them, either. Not only was it natural, but it was effective, and immediate. I’m here to tell you that you can do the same thing with numbers.

“Memories…”

In the next post, I think we should talk about memorization. One of the tools that many magicians use is mnemonics. Even a trivial ability with mnemonics beats the poop out of rote memorization. Here’s the catch, though - people who don’t use nor understand mnemonics have a really poor opinion of them. I can understand that. Being a magician, I’m also aware that a lot of people have really poor image of magicians as wise-guys seeking attention.

Digression:
Q: Why did God create mimes?
A: To give magicians someone they can look down on.

Ok, I’m back. It’s unfortunate that people will then generalize that magicians are jerks just because their weird Uncle Earnie used to force them to pick card after card for no reason. That’s not magic. Real magicians (Ricky Jay, Rene Levand, Cardini) have reached levels of art that Weird Earnie and trendy “street magicians” can not even dream of.

The same goes for mnemonics. If you want to experience an amazing example of math and mnemonics in action, google “Arthur Benjamin,” among others.

I have been using and teaching mnemonics for years, and have found that as soon as a skeptic tries it and learns it from a competent teacher (Harry Lorayne’s books being the best for laymen, I believe) those skeptics become evangelists for mnemonics. But don’t take my word for it.

More in the next post. (Don’t forget to check back!)

Calculators

Somewhere our commenter seemed to think that I was for calculators in elementary school classrooms. Odd, considering that one of my mantras is “Calculators were invented by vampires to suck your brains out.”

I was also accused of doing some kind of disgraceful disservice of generations of teachers. Get real. I’m trying to defend teachers (real ones, anyway) against the lock-step of NCLB, ill-conceived curriculums that don’t let teachers actually teach, standardized tests that take up teachers’ and students’ valuable time and only reward the testing services that lobby for them, and the tyranny of endless, meaningless paperwork teachers face every semester.

Don’t take my Word for it. (Tell ‘em, Albert!”)

According to this entry in Wickipedia,

“In his early teens, Albert attended the new and progressive Luitpold Gymnasium. His father intended for him to pursue electrical engineering, but Albert clashed with authorities and resented the school regimen. He later wrote that the spirit of learning and creative thought were lost in strict rote learning.”

This point is also frequently and erroneously ascribed to Math Mojo methods:

The commenter seems to think the methods I use are somehow part of the “improved math methods” that are screwing up math ed. I’m with the commenter on this one for part of his point. I do think that much of the know-nothing, feel-good crap that passes for education today is making math ed. even more miserable than it was when I was a kid (jurassic period when “tradition” reigned unquestioned).

The problem arises when one assumes that anything that isn’t traditional is part of that crap. It’s kind of pitiful to have people generalize that everything that isn’t their “one best method” is to be put in the same pile.

Here’s the poop:

The airy-fairy newage trash that some schools are falling for is awful. It is worse than awful - it’s tragic. They don’t actually engage the student’s minds in meaningful ways. Sure, their press kits say they do, and maybe some of them actually could, in the hands of a competent practitioner. But the schools never give their teachers enough training, or even time to learn the stuff, before they send them out to use it. Face it, most elementary teachers are so overworked as it is, they barely have to cover the minimum curriculum.

It’s sort of like saying, “Grasshopper, the best way to be is “Enlightened.” Now go out and show your students how to be enlightened,” without giving “Grasshopper” the time to learn and have that learning mature into a great love of Enlightement, which he can then first begin to share in a meaningful way.

That is why I write so much about Math Mojo methods. Although they mostly take minutes to learn, they are meaningless without learning some of the deeper concepts behind them. And by deeper, I don’t simply mean the algebra, or some algorithm. Learn the Math Mojo nuts and bolts, and think about the methods, and you can come up with many deeper thoughts that relate to numbers and patterns, as well as to life in general.

It beats the hell out of “just shut up and learn it, because I said it’s the best method, and I’ll fail you if you don’t, and it will all be your fault.”

Or not?

’nuff said,

Dr. Gregory House
Professor Homunculus