Wednesday, October 16, 2013

Teaching Multiplication to Kids


When I was a kid, learning to multiply in our heads was called “learning our times tables.”  Perhaps British kids have to “learn off” their times tables.  At my kid’s school everybody calls them “math facts.”  Whatever you call them, they’re a drag for kids.  I suffered learning them, and my brothers suffered, and my kids have suffered.  I’m just finishing up helping my younger daughter Lindsay master them.

Lately, I’ve stopped to really think about this process, using my adult, educated mind, and have come to a couple of conclusions:  one, the most difficult thing about learning times tables is the intimidation factor; and two, this intimidation factor can be greatly reduced.  In this post I’ll share my brothers’ and my childhood experience with times tables (mainly so you can laugh at us), and some tips—complete with visual aids—for lessening the anxiety of your kid (or your nephew or niece or whomever).

My times tables ordeal

Ages ago, my third grade teacher gave our class a lecture about how multiplication works, or is supposed to work.  I remember nothing about the lecture, other than the fact of it and its having been very brief.  Then we were given a test.  I crashed and burned on the first problem, 3 x 2, because I didn’t know the first thing about multiplication.  (If somebody had said, “‘Three times two’ basically means ‘two three times, or 2+2+2,’” I’d have gotten it.  But either nobody said this, or I wasn’t listening.)  So I simply added the two numbers instead.  Maybe I was hoping for partial credit but probably I had no plan.  (I was actually executing advice on writing which I’d get decades later:  “Write what you know.”  Needless to say this doesn’t apply to math.)

Eventually I figured out this was just a memorization game.  Of course I knew my 2s, and for some reason had had no problem, in kindergarten or first grade, memorizing (probably through some kind of eerie chant), “5 , 10, 15, 20, 25, 30, 35, 40, 45, 50!”  (Stephen King could do something horrifying with this, I’m sure.)  But when it came time to memorize my 3s, I felt I was in the grip of an invincible foe.  Quiz after quiz after quiz I failed.  (“Fail” meant getting at least one problem wrong, or not finishing the quiz in the short time allotted.)

My elementary school was of the “Child Left Behind” philosophy.  No kid was going to be slowed down just because he’d outpaced his stupid classmates.  As soon as a kid aced his multiply-by-3s quiz, he moved on to his 4s, and so forth, so a slow kid like me could feel the full brunt of being stuck in the 3s while some other kid (maybe even—gasp—a girl!) was on her 7s.  It was essentially a shame-based system.

I well remember the day, weeks into this struggle, when I finally aced my 3s quiz.  My friend John walked over, put his hand out, and said, “Welcome to the 4s!”  I was so dazed I didn’t realize he was talking about math.  I thought he said “Welcome to The Force” (i.e.,  a “Star Wars” reference, this being 1978).  Then I realized I’d joined him in the 4s, where he’d been stuck for a good while.  Recalling how much harder the 3s had been than the 2s, I naturally assumed the 4s would be that much harder than the 3s, and I still had the 6s, 7s, 8s, and 9s to go.  I almost died of despair.

Notably, nobody explained to me that the 4s are easier than the 3s, and that far from having just started, I was actually close to a third of the way done.  (More on this later.)  All I got was the implicit message, “You better pull your head out of the grease bucket, you dunce.”  But I have no recollection of struggling in the 4s, nor of the process of learning the rest of my times tables.  It was like finishing the 3s was the real battle, and the rest was just gravy.  Only recently have I discovered why this was the case.

My brothers’ times tables ordeal

I think I was actually done with my times tables more or less on time, because otherwise I’d have been chewed out by my dad, which I’d certainly remember, just like I remember my dad chewing out my brothers.  Our failure must have been hard for our old man, who was so good at math he memorized all the logarithm tables.  (Remember that giant table, squeezed so small it was practically microfiche, in the back of your math book?  No, probably you don’t, and you may not even remember or have heard of microfiche.  Suffice to say, times tables are a grain of sand compared to the endless beach of log tables.)  My dad memorized these tables so he could multiply giant, multi-digit numbers in his head (according to some arcane trick I may have learned once but forgot almost instantly).  For my dad’s worthless children to fail to memorize their lowly times tables must have been a real blow to him.

My brother Bryan remembers “the [verbal] beat-down from Dad” this way: 
“Dad, can I have a bicycle?”
“Do you know your times tables yet?”
“What’s six times seven?”
“Um... 43?”

I take issue with this recollection, for a couple of reasons.  One, none of us would have dared ask our dad for so much as a mechanical pencil, much less a bike.  Two, it implies that had we answered correctly, we could be rewarded with something cool like a bike, which is false (though our dad did buy us awesome ten-speeds for our ninth birthdays).  But I well remember my dad quizzing my brothers and being visibly disgusted when they answered incorrectly, as they always did.  And I remember him complaining to our mom, “These boys don’t even know their times tables!”

Our dad became so desperate, he finally resorted to something that he almost never did:  he threw money at the problem.  He went out and bought the QuizKid Racers.  These were calculator-like devices made by National Semiconductor.  (As a parent, I feel humbled by this.  I’ve never bought anything for my kids made by National Semiconductor.)  The QuizKids could drill a kid on his arithmetic, but even better, they could be linked together so two kids could race.  Where the educational system and my brothers had failed, sibling rivalry succeeded.  It probably helped that I could go toe-to-toe with my older brothers, a humiliation they were determined to put behind them.  It seems like we all had those times tables down in no time.  (It couldn’t have been long, because the QuizKid Racers actually outlived their usefulness instead of being lost or destroyed.  That’s saying a lot.)

The intimidation factor

How could the QuizKids have so suddenly broken this a logjam?  I think it’s simply because they focused us.  The feedback loop was very tight:  a good beep and a green light rewarded us, and a bad beep and a red light punished us.  We became as lockstep with the QuizKid as a hamster hammering a paddle to get a pellet of food or cocaine.  For once, we were just doing the work instead of gnashing our teeth, rending our garments, and having fits of despair over the futility of the enterprise.

So is the answer simply to bring back the QuizKid?  No; as I’ll discuss later, that method of focusing a kid has its own problems.  I think before you even start quizzing your kid, you should explain a few things that will—right off the bat—reduce some of the anxiety.

To start, you can explain that the 3s are actually the hardest set of times tables to learn.  Why?  Well, for one thing, some of the 3s multiplication problems have some answers that are even, but others are odd.  This isn’t true for the 4s, 6s, or 8s.  For another thing, to learn your 3s you have to memorize six math facts, but the 4s require memorizing only 5 math facts; the 5s you already know; and the 6s only comprise 4 math facts, the 7s only comprise 3 math facts, and so on.

To illustrate this to my younger daughter, I created some tables.  To start, here’s the full table of times tables from one through ten, which suggests that 100 (i.e., 10 x 10) facts must be memorized:

But of course every kid already knows his or her 1s, which wipes out 19 math facts right off the bat:

And the 10s are totally self-evident, as every kid knows you just add a zero to the number.  So that eliminates another 17 math facts.  See, we’re down what looks like 64 facts and we haven’t even started multiplying yet!

The facts keep tumbling down because by third grade every kid alive already knows his 2s; after all, counting by 2s is easy and fun.  Boom, another 15 math facts are gone.

And remember the 5s, with Stephen King and the spooky chanting?  We can knock off another 13 math facts.  That’s right, 13 fewer things to memorize, leaving us with a total of what looks like just 36 total math facts, and we still haven’t done our first multiplication problem.

Above, we can count the non-shaded values in the 3s column, which is how you show your kid there are really only six math facts to memorize before the 3s are done.  Sure, learning the 3s is still hard, but six facts doesn’t seem like an insurmountable task.  When the 3s are (finally!) done, the remaining table looks like this:

Point out to your kid how cool it is that mastering the 3s knocked out not just a column, but a row of the table as well.  So you’ve got only five math facts to go before the 4s are complete.  And then, after the 4s, you’re down to what looks like just 16 math facts left.  It’s like magic!  Look at what remains after the 4s are done:

Your kid already knew the 5s, and to master the 6s requires just four more math facts.  Now, what’s that business with the color-coded cells?  Well, it shows the phenomenon that makes each series of math facts less cumbersome than the one before it:  most math facts are repeated.  Look at the result of 6 x 7, the 42 shown in yellow.  It’s the same as 7 x 6, also a yellow 42 in the next column.  So when you learn 6 x 7, you’ve automatically learned 7 x 6, which is why learning the 7s is only three new math facts.  And when you learned 6 x 8, in orange, you knocked off 8 x 6 as well, which is why learning the 8s is only two new math facts, and when you memorized 6 x 9, that pale green 54 did double duty as 9 x 6 later.

Above you can easily see how the 7s is just 3 new math facts, two of which (the blue 56 and the pink 63) kill off future memorization tasks.  Learning your 9s means memorizing exactly one new math fact:  9 x 9.  All the other 9s were picked up along the way.

You know what’s really encouraging for your kid?  Step him or her through the actual number of total math facts that must be learned.  At first it looks like 100:  you know, 10 x 10.  After helping Lindsay grasp that the 1s, 10s, 2s, and 5s are “free” (i.e., she already knew them), I asked her how many math facts that left.  She did some crunching and came up with 8 x 6:  that is, eight facts each for the 3s, 4s, 6s, 7s, 8s, and 9s.  Seems like a good guess (since she didn’t know to account for the replication of facts).  And if you thought you had 100 facts to learn, getting it down to 48 seems pretty cool.  But the actual number is far less, as Lindsay was delighted to learn.  That actual total number of discrete math facts?  Only 21.  Seems impossible, doesn’t it?

Here’s how it works.  Learning the 3s is six math facts, as I’ve discussed.  The 4s is only five facts, the 5s every kid already knows, the 6s is only four facts, the 7s is three, the 8s two, and the 9s only one.  The actual number of discrete math facts to memorize is therefore 6+5+4+3+2+1 = 21.  A far cry from 100, or from the 48 my daughter thought she had to learn.

This is even more exciting when you show the momentum that seems to build up.  After you’ve managed to memorize the six math facts you need for the 3s, you have just fifteen math facts left, total.  Once you’ve mastered the 4s (which aren’t so bad because the answers are always even, and there are only five facts to memorize), you have just ten math facts left!  With all this in mind, it’s no long any wonder I don’t remember learning my 4s through 10s.  It went fast, with each series shorter than the last.

In short, the seemingly vast array of math facts appears much less intimidating when your kid grasps that there are only 21 facts total, and that the process of memorizing each set gets easier as you go along.  Without all that anxiety, the actual learning can be more easily carried out.

How to drill your kid

Okay, so the anxiety has been mitigated and the kid is ready to learn.  Now is it time to look for some QuizKids on eBay?  I’ll argue not.  In the case of my family, I’m not sure the QuizKid technique was purely for the good.  For one thing, sibling rivalry—though a powerful motivator—can be toxic to families when used recklessly.  Giving my older brothers a run for their money, via the QuizKids, might have ended up bringing much trouble down on me later.  (Something sure did.)  For many years, the rivalry among my brothers and me was so intense it was like we were all on different teams.  It was almost unheard of for one brother to support the other in anything.  Was the modest achievement of finally learning our times tables worth this much rivalry?  Perhaps not.

Meanwhile, once my brother Bryan had achieved dominance on the QuizKid, he challenged his friend, whom we’ll call John, to a competition.  John was better looking than any of us, a better athlete, more popular, and more confident, and above all he was a rich kid.  He couldn’t resist the impressive technology of the QuizKid, and probably overestimated his chances in the competition, so he accepted the challenge.  And Bryan straight-out whupped him!  It was a glorious moment for Bryan, the achievement of “the holy grail of intelligence!” as he put it.  Naturally, this gave John some serious sour grapes; Bryan recalls, “Of course after his whupping, [John] declared that times tables were stupid.”  Being beaten by somebody as historically unimpressive as my brother surely took a toll, and years later I heard John was arrested for driving around in his muscle car with a dirtbag friend shooting horses with a BB gun.  Was this a result of QuizKid-induced trauma?  Hard to say, but I’m not going to take any chances with competition-based learning methods.

So with Lindsay I tried good old fashioned flash cards, but as it turns out these have a serious Achilles’ heel:   they’re too easy to lose, especially in the hands of a child.  By the second time I used them, a third of the cards were missing.  Where do they go?  It’s like with socks in the laundry … I have no idea!  But each card lost is like two math facts your child might never learn.  (Well, with duplicates, maybe not that many … but you get the idea.)

So I tried this PC game where there’s a stick of dynamite and you have to enter the math answer before the fuse runs out.  Get the answer right, and some wacky cartoon face pops up saying “Right on!” or “Amazing!”  But this game proved very stressful for Lindsay, what with the hissing of the fuse and occasional explosions, and it had the additional irritation of not working right after the first few games.  It stopped accepting mouse input so I had to type in the answers, which spoiled the dream of outsourcing the process to my kid so I could go do something else.  (I think I was supposed to fork out some money to maintain the full functionality, but I refuse to give money to terrorists.)  What’s worse, the website cartoon art was really lame and I got good and sick of looking at it.  If the images had some style (along the lines of an Edward Gorey or Roz Chast drawing) maybe I wouldn’t have gotten so sick of them.  Besides, my kid will be sitting in front of screens all her adult life … why rush her into it?

So I ended up creating simple one-page quizzes, double-sided with complete facts (e.g., 3 x 7 = 21) on one side and problems only (e.g., 3 x 7 = ?) on the other.  They’re just to  look at, flash card style, since writing answers wastes time.  Lindsay can study the facts for awhile, then drill herself, and then have me drill her verbally. 

What’s really notable is that when I drill her verbally, she invariably walks around in a circle while answering.  During one session she rolled around on an exercise ball.  The physical activity seems to help her think.  Educators are studying the link between motion and learning; one classroom at my kids’ elementary school swapped out half their chairs for exercise balls to study this.  I’m reminded of something I read about Bill Gates years ago:  “While he is working, he rocks … his upper body rocks down to an almost forty-five-degree angle, rocks back up, rocks down again….  He rocks at different levels of intensity according to his mood.  Sometimes people who are in the meetings begin to rock with him.”  Just in case this kind of motion helps my daughter learn her math, I’ll stick to the oral drills instead of sitting her in front of a computer or asking her to manipulate a QuizKid-type app on the smartphone or tablet she doesn’t own.


I’d be happy to e-mail my quiz sheets to anybody who wants them.  They’re nothing fancy but I could save you the time of creating your own.  Just e-mail me.  Exercise ball not included.

A final note:  if you’d like to read about how I tutored a fifth-grader on his math, click here.

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