Chapter 9 of Turing-biographer Andrew Hodges' new book One to Nine (which I discussed Monday) takes on mathematics education, which probably everyone agrees is pretty much a disaster. One of the aspects of math education that Hodges seems to be against is memorization:
Whether it is even essential to memorize the base-10 multiplication table, I doubt. It is more important to have a good general sense of what people call ballpark estimates; to know motes from beams, gnats from elephants, sledgehammers from nuts. (pg. 286)
Yet he also seems to think that students should be able to perform rudimentary calculations without a calculator:
You must know what numbers are all about. If you don't, you end up writing down answers a hundred times too big, or upside down. For this reason it is essential to learn about adding and multiplying without a calculator. (pg. 289)
For quite awhile in my teenage years, I knew there was something wrong with the way I multiplied certain numbers from the basic multiplication table. For most of the single-digit multiplications, the product popped immediately into my head. "What's 7 times 4?" would be followed by an unthinking "28."
But not always. There was a little section of the multiplication table where something different went on in my head, and as I began to analyze my thought processes, I understood what this difference was. Apparently at some point in early childhood before the entire multiplication grid was permanently etched upon my neurons, I had stopped memorizing when I realized I could instead mentally calculate the product. This "defect" applied to just three cells involving the numbers 6, 7, and 8. But not the squares: The squares of 6, 7, and 8 popped instantly into my head as 36, 49, and 64. It was the other combinations that I had to calculate "manually."
This is how I performed these three calculations in my head: For 7 times 8, I'd think "7 plus 7 is 14, times 2 is 28, times 2 is 56." Similarly, for 6 times 8 I'd think "6 and 6 are 12, times 2 is 24, times 2 is 48." The final problematic product was 6 times 7, and for that I'd think "7 times 3 is 21, times 2 is 42."
As I just now write down my mental process for these calculations, I realize that for 7 times 8, I could have started with 7 times 4, and similarly with 6 times 8 I could have started with 6 times 4, but for some reason I didn't. I don't know why, but maybe my consistency in how I did these multiplications is a key to the mental process: I had memorized the algorithm for these multiplications, but not the actual result! Moreover, despite the hundreds of times I must have performed these convoluted calculations, the products themselves never took root.
What had happened in my early education to truncate the memorization process before the entire multiplication table had been fully assimilated? Did I just find it easier at some point to double numbers a couple times in my head rather than to memorize these three products? I don't know. Nor do I know which approach is mentally healthier. But if you're going to multiply without a calculator (as Andrew Hodges evidently believes you should be able to do), then at some point you need to come up with the product of 7 and 8, and either you're actually figuring it out in your head the way I did, or you can skip that step and simply spit out the memorized value.
I think memorization is important. The quicker the simple sums and products come into your head, the easier you'll manage more complex calculations because you don't have to keep track of so many intermediate results. I think memorization should actually be emphasized much more than it is. I think most people would benefit from having memorized two-digit tens-complement calculations — for any two-digit number, to know what 100 minus that number is. In real life this is making change from a dollar in your head.
I went through my childhood and almost all my teenage years multiplying 6 times 7, 6 times 8, and 7 times 8 in the way that I described. But in the summer before I went to college, I figured I should finally do something about my multiplication problem. It seemed illicit to be attending an engineering and science school without having first completely memorizing the multiplication tables. I deliberately set out to memorize these three products. It didn't take long, of course, and from that time until today, I can retrieve them instantly. Now when I hear the numbers 7 and 8, I immediately think 56 even if I have no need to be actually multiplying those two digits.
Several years ago, while reading Andrew Hodges’ Alan Turing: The Enigma in the course of researching my book The Annotated Turing, I learned that Turing spent some time before beginning his studies at Cambridge in 1931 reading G. H. Hardy's classic A Course of Pure Mathematics (Hodges, pg. 58), first published in 1908 and still in print today in the 10th edition.
And there we have one of the many differences between Turing and Petzold: Turing prepared for college by reading G. H. Hardy's math textbook. Petzold prepared for college by learning his multiplication tables.
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