A cheat for the standard multiplication algorithm

Of course, I mean the one that looks like this (pick whatever format you prefer): 

The one on the left is the one that I learned in school. It was one of my great dislikes. I always made errors when I performed it. This post describes a work-around, a “cheat”, if you like, that drastically cut down on my errors. It's a method that I generally hid from my teachers and colleagues so they wouldn't know how weak I was in arithmetic. I have not seen it used elsewhere, but I found it so helpful that I would not be surprised to learn that others have discovered it. 

I have a curious mental glitch. For some reason my brain will occasionally swap digits. For example, when I see 432 on the page, it stays as 432 as long as I look at it, but if I were to avert my gaze in order to record the number elsewhere, I might write 342 and remain totally unaware that I had made an error.

I still make mistakes like this. I'm sure now that this memory quirk is the source of some of my difficulties with arithmetic. However, as a kid, it never crossed my mind that that I might have a twist in my neural circuits. I just accepted that I was not very good in arithmetic.

Here is how my memory quirk could affect me if I were to multiply 68 by 4 following the method that we were taught in elementary school: 

4 times 8 is 32. Write down the 2 and carry the 3. Keep the carry number in memory. Then 4 times 6 is 24, and 24 plus the carry number is 26 so the answer is 262. 

Although my addition skills are no great shakes, in this case I was not adding incorrectly—when I got 26 the addition was perfectly correct: 24 plus 2 is 26. What happened was that in my head I was swapping the memorized "carry number" (the 3) with the number I had just written down (the 2). 

When we were first learning the multiplication algorithm, instead of having to keep the carry number in our heads, we were permitted to write tiny numbers in the appropriate place above the multiplicand like so:

Writing down the carry numbers was not a cure. Even for single digit multipliers, it only helped a little bit. It did not help at all when the multiplier had more than one digit. The jumble of small carry numbers written above the multiplicand actually exacerbated the situation. 

By the time I started high school (grade eight) I had concocted a “cheat”. Instead of separating out the carry numbers either in my mind or on the worksheet, I wrote down the full product of the individual digits. The cheat was not perfect, but it substantially reduced my errors. Using the cheat, here is how to multiply 345 by 6.

The alignment is important. First the product of 6 and 5  is 30, so write 30 diagonally with the 3 in the ten’s column and the 0 in the units column but in the line below. Next, the product of of 6 and 4 is 24, so write 24 diagonally with 2 in the hundred’s column and the 4 in the ten’s column in the line below. This places the 4 directly below the 3, and both 4 and 3 are in the ten's column. Continue in this way, writing the product of 6 and 3 diagonally with the 1 in the thousand’s column and the 8 in the hundred’s column below the 2. Finally, add the columns vertically to get the answer. 

This can be adapted to multi-digit multipliers as shown below, but it starts to get messy.

To avoid the messiness, I usually multiply 345 separately by 4 and 6 on scrap paper as shown below on the right, and then transfer the answers to the proper position in the standard algorithm as shown on the left.  (I still do this when I have to multiply multi-digit numbers without a calculator.)

                                          

Why call this a “cheat”? Well, when I was a youngster, doing anything that was outside the rules was considered cheating, and, if nothing else, arithmetic was taught as a set of rules. So in that context, it was cheating.

* * *

You may notice that my cheat has a very strong resemblance to the lattice method of multiplication. Of course, back in my day they did not teach the lattice method, and I didn’t actually know about it until I taught a “math for elementary teachers” course. In such courses I avoided teaching the lattice method because I thought that writing things slantwise in one direction and adding things slantwise in the opposite direction followed by wrapping the answer around two sides of a square would screw up a student's understanding of place value. To be honest, I do not know if the lattice method really does lead to place value confusion. I also confess that I never taught students my cheat method, so, although I believe that the cheat's strong emphasis on the proper location of the digits with regard to place values would be beneficial, I really don't know if  that is the case. 

* * *

Regarding my memory quirk, I don't think it's dyslexia. If it is, then it’s extremely mild. It’s nowhere near as profound as what Toomai describes in his post about the dyslexic mathematician. What little I know about dyslexia is that it also causes troubles with algebra, and I never had difficulty with algebra. Surprisingly, my wonky memory regarding numbers doesn’t apply to letters.

Twelve-times

You shouldn't have to memorize the 12-times table. You can just multiply by 10 and by 2 and add the results. That was my very vocal argument to my grade three teacher many years ago. My friend, whom I'll call Tough-boy, joined the fray, but the  teacher was having none of it and she sent us outside the classroom to wait until the principal came by. 

That was sobering. In our young eyes, the principal was a vile humourless creature who enjoyed terrorizing schoolchildren. He regularly walked the corridors of the school, and as he walked he announced his whereabouts by bashing the hall lockers with a thick leather strap. As Tough-boy and I waited on the second floor, we could track his progress as he proceeded along the corridor on the floor below. I was very frightened. Tough-boy, who had been in this situation several times before, offered some practical advice: "Let your hand go limp just before the strap hits. It won't hurt nearly as much."  

We were strapped because we were being disruptive in class. It was intended to teach us to not argue with the teacher. It worked, but in my mind I believed that I was being punished because I couldn't master the 12-times table. 

Skip ahead a generation or two. A few months ago I asked my grandson (grade 5) if he had to memorize his multiplication tables. He said "Yes, of course." I asked him a few questions such as "What's 9 times 3? What's 4 times 7?" which he answered correctly, and then I told him that I always had trouble remembering 7 times 8. Without hesitation he said "It's 56." And then, without prompting, he said "You know how I remember it?  Because 6 times 8 is 48 and so 7 times 8 is 48 plus 8."  Apparently he was not disciplined for using this strategy. 

Here in Alberta there is dissatisfaction with the way elementary mathematics is being taught. A large number of people have signed a "back-to-basics" petition which they hope will eradicate strategies like the two that I have just described. I'm sure that my grade three teacher would approve, for the petition is really an endorsement of the way I was taught arithmetic: in those days "the basics" meant memorizing "facts" and following "rules", with such knowledge to be acquired almost exclusively by rote. 

For me, the basics did not work very well. As often as I tried to memorize the 12-times table, it never stuck with me for more than an hour or so. My ability to recall numbers was iffy, so I continued to use my "10 plus 2" method, and I hid this strategy from the teacher as best I could. I left grade three with a significant dislike of arithmetic. To borrow some words from the Alberta petition, I was "repulsed by math".

The back-to-basics crowd has persuaded our government to revise the curriculum, and students will now be required to recall multiplication facts up to 9 times 9. Actually, that's not an unreasonable demand. However it is not clear to me that students weren't already expected to do this anyway. But I wonder if my grandson's strategy for "memorizing" 7 times 8 will now be unacceptable. If so, I imagine that he might continue to use his strategy and just shut-up about it like I did.

The curriculum changes have not satisfied everyone, and there is a push for more.  (Check out some of the the articles curated in Egan Chernoff's Matthew Maddux Education  blog.) The argument is that intensive rote learning will improve the students' arithmetic skills which in turn will lead to a greater understanding of mathematics. 

Frankly, I don't buy it. The argument seems to be based more on opinion than evidence. For example, long multiplication is an application of the distributive law, and so one might conclude that mastery of the long multiplication algorithm in arithmetic would lead to ready understanding of the distributive law in algebra.  Sean Carter, who teaches grade 9 math in Australia, reports that this did not happen in his class. See his blog about it here.

Back in my day, we were told that there were good reasons for memorizing the 12 times table. Here are two justifications that I can remember: first, it would make it easier for us to comparison shop because things were often sold in dozens, and second, in order to convert feet to inches you had to multiply by twelve. Hmmph! At least we weren't told that we that needed to memorize the 12-times table on the off chance that we might some day visit Britain and have to convert shillings to pence.