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June 13, 2008
Roscoe, N.Y.
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:
Yet he also seems to think that students should be able to perform rudimentary calculations without a calculator:
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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(c) Copyright Charles Petzold
www.charlespetzold.com
Comments:
Learning by rote is a disease.
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.
To be able to remember something "in your bones" does not require a single minute of rote memorization.
For a detailed explanation of this, check out:
http://mathmojo.com/chronicles/2008/03/29/learning-multiplication-by-rote-is-a-disease/
If your school had encouraged you to keep learning with the engaged, intelligent methods you came up with yourself, you'd know the "tables" inside out.
Hotcha!
Brian at MathMojo.com
— Brian Foley (a.k.a. Professor Homunculus), Sat, 14 Jun 2008 01:50:48 -0400 (EDT)
Interesting read :-) I realize that I, too, used to have a couple of holes in my multiplication table, although for 7 times 8 I'd come up with 64-8, which in turn is 64-10+2, a stepwise refinement which works well with much larger numbers and let you find decent ballpark answers faster than most people can reach their pocket calculator. A bit of modular arithmetic is quite useful in the mix too, I’d suggest to start with mod(12) congruence, if only to translate the 24h system.
As far as mathematics are concerned (or “higher math” as you say in the US), I believe that understanding is a lot more important than memorizing. Of course you have to memorize the theorems and definitions, but it’s also most important to understand the proofs, or at least to get a good grasp about how they work, before you can master anything.
— Axel, Sat, 14 Jun 2008 06:00:52 -0400 (EDT)
I had exactly the same experience. But I never did memorize those products and still think of 7 x 8 as 7x7=49 49+7=56.
For a long time I suspected that there were two different kinds of people: people who were good at memorization and people who were not so good but could figure things out. Those who are good at memorization remember lots more stuff, others just remember enough to figure out the answer.
My favorite example is converting between centigrade and Fahrenheit. I think some people (of those who actually ever do this conversion) remember the formula, whatever that is, and others, like me, who just figure it out when it comes up.
BTW, I think this is part of the reason I don't find programming as exciting as I used to. Back in the "good ol' days" we had a programming language and simple libraries and had to think up solutions to all but the most basic problems. Now there are libraries and packages to do almost anything and programing has become more and more an exercise in memorizing products and APIs.
— Walter Williams, Sat, 14 Jun 2008 17:58:01 -0400 (EDT)
I have some holes in the multiplication table, too. The 7, 8, 9 range is a bit sketchy, but (like Walter) I find myself working from memorized values. E.g., for 8x7 I do 7x7 or 8x8 then add or subtract as needed.
And even for mental subtraction and addition I oftentimes do something similar. When summing 47 and 36 I'll "simplify" it to 50+33, which is a heckuva lot easier to add in the head, IMO.
My wife and I are planning on home schooling our children, so it will be interesting to read what others have to say about math education, discover our own ways, and observe how our children best learn. I've got a cousin who's a really sharp fellow who home schools his daughter, and he uses a lot of geometric constructs to illustrate pre-algebra type lessons. For example, using construction paper and scissors to show (a+b)^2 = a^2 + 2ab + b^2.
— Scott Mitchell, Sat, 14 Jun 2008 23:39:02 -0400 (EDT)
learning by rote is not a disease.
A touch typist doesn't merely "understand" the keyboard, he must have lots of practice in order to type quickly. If you have to think where the 'A' is on your keyboard, you will be frustrated every time you want to type up a report.
Multiplication should be just as instantaneous so that you won't be ripped off at the grocery store.
It's really not so hard. I went to http://www.mathsisfun.com/numbers/math-trainer-multiply.html and taught myself the 12x12 table very quickly. No skip count, no silly songs. Just do it and challenge yourself just like you did when you learned how to type and it will be part of you.
— ari-freedom, Sun, 15 Jun 2008 12:25:11 -0400 (EDT)
Now thats very interesting. I thought it was just me who had problems with 7 * 8. For a long time, I would do it 7 * 7 and then add 7. But then at one point of time, I stopped doing it that way and whenever I came across 7 * 8 I would realize that its the difficult product whose answer is 56.
— gazzal, Mon, 16 Jun 2008 03:51:28 -0400 (EDT)
This is really interesting to me, because I have always calculated these values rather than memorizing them. 6 * 8 I would always think "5 * 8 is 40, plus 8 is 48", and so on. Eventually, repetition taught me the answers, but for a few values as I reply from memory I still double-check through arithmetic. I've always wondered what's special about 6, 7, and 8, and now more so.
— Owen Pellegrin, Mon, 16 Jun 2008 13:14:25 -0400 (EDT)
The nice thing about the algorithmic approach that you described is that you get good at keeping track of intermediate values. This is particularly helpful whenever you want to do an operation that requires this. For example figuring out how much change to ask for when you take a cab, typically $20 minus ((fare + (fare * 15%)) rounded up to the nearest dollar).
Another example of operations that require intermediate values came up recently when I was watching “Are You Smarter Than a Fifth Grader” and this question was asked. “If a pound of peanuts costs 48 cents, how much does 5/8 of a pound cost?” After reading at the question I did the following mental calculations. First I simplified the 48/8 by reducing 48/8 to 24/4. I again simplified by reducing 24/4 to 12/2. I could have continued to reduce 12/2 to 6 but for some reason I didn’t. Perhaps more than two reductions is too difficult for me to remember so instead I multiplied 12 * 5. Whenever I multiply a large number by 5 I divide by 2 and then multiple by 10. In this case 6 * 10= 60. I then divided 60 by 2 and got the answer 30. So my steps were this. 48/2, 24/2, 8/4, 12/2, 6*10, 60/2 which was 6 steps. This probably took me less than 10 seconds.
In case you are wondering what happened on the show the adult chose to “peek” at the fifth grader’s answer of 40 and accepted it as the correct answer and then lost. Which means both where likely unable to do a process that required intermediate steps. They also failed to estimate that 40 was too high, which for me also requires multiple steps (48 / 2 = 24, 40 – 24 = 16, 1/8 of 48 is not equal to 16 because 16 * 8 > 80 and 80 > 48)
— eff five, Mon, 16 Jun 2008 13:18:51 -0400 (EDT)
I also always figured 7x8 = 7x7 + 7, it is very interesting that this is so common and for the same numbers. I think memorizing would have been better. But when I realized I was doing this, I consciously started applying the same principle (compute a similar easy problem then add the difference back in later) to much larger numbers, which I still do today, e.g.: 348 + 873 = 350 + 850 ... - 2 + 23.
— Brad, Mon, 16 Jun 2008 19:41:54 -0400 (EDT)
Msr. Eff five's way of finding the answer to “If a pound of peanuts costs 48 cents, how much does 5/8 of a pound cost?” is a little roundabout. 48/8 is 6. 6 * 5 is 30. Done!
I had my kids learn the 12x12 times tables and even go beyond to 13's. Familiarity with these products speeds up working on all sorts of math problems.
— Andy, Wed, 18 Jun 2008 07:08:19 -0400 (EDT)
Andy-
You are correct that I should have "seen" that 48/8 is 6, but my point was that the algorithmic approach makes you proficient at dealing with intermediate values.
For instance if I bump up the price from 48 to $1.48, or doubled the denominator to 16 the problem should be more challenging but to me it’s the same. Is it more challenging for you?
This isn’t to say that knowing N X N times tables isn't useful but knowing the times tables doesn't help you when you need to remember intermediate values.
— eff Five, Fri, 27 Jun 2008 17:57:26 -0400 (EDT)
To soften up the first simplify 148/8 to 27/2 before switching over to the algorithmic method.
The second is immediate, as 48/16=3.
An example in which numbers have no common factors would take that away.
The times tables and other familiar values are a mental "compiler optimization" used to both speed up calculation and cut down the number of intermediate values that one has to hold.
For example: a friend told me he was asked on a phone interview to compute 2 to the 18th power in his head without any aids.
There are those who could dispose of that immediately as they know 2 to many powers. Or once worked on an 36 bit architecture and know the max value of a a 36 bit word, and divide that in half. But not me, nor my friend.
I looked for a power of 2 that divided 18 and found 2**6 is 2**2 * 2**2 * 2**2 which is 4*4*4 =16*4 = 64.
2**18 = 2**6 * 2**6 * 2**6
So we changed the problem to having to calc 64*64*64.
64*64 is (60+4) * (60+4)
= 60*60 + 2*60*4+16
= 3600+480+16= 4096.
Time to hold intermediate values - to multiply 4096 * 64.
This is 4 * 64 *1000 + 9 * 64 *10 + 6* 64
= 2*128*1000 + (10*64 -64) *10 + 360+24
= 256000 + 5760 +384
= 262144.
Andy
— Andy, Wed, 16 Jul 2008 07:03:36 -0400 (EDT)