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.