I’ve read your link to John Leslie with both curiosity and bafflement.
17 x 24 is not perhaps the best example of a question for which no answer comes immediately to mind. Seventeen has the curious property that 17 x 6 = 102. (The recurring decimal 1⁄6 = 0.166666… hints to us that 17 x 6 = 102 is just the first of a series of near misses on a round number, 167 x 6 = 1002, 1667 x 6 = 10002, etc). So multiplying 17 by any small multiple of 6 is no harder than the two times table. In particular 17 x 24 = 17 x (6 x 4) = (17 x 6) x 4 = 102 x 4 = 408.
17 x 23 might have served better, were it not for the curious symmetry around the number 20, with 17 = 20 − 3 while 23 = 20 + 3. One is reminded of the identity (x + y)(x—y) = x^2 - y^2 which is often useful in arithmetic and tells us at once that 17 x 23 = 20 x 20 − 3 x 3 = 400 − 9 = 391.
17 x 25 has a different defect as an example, because one can hardly avoid apprehending 25 as one quarter of 100, which stimulates the observation that 17 = 16 + 1 and 16 is full of yummy fourness.
17 x 25 = (16 + 1) x 25 = (4 x 4 + 1) x 25 = 4 x 4 x 25 + 1 x 25 = 4 x 100 + 25 = 425.
17 x 26 is a better example. Nature has its little jokes. 7 x 3 = 21 therefore 17 x 13 = (1 + 7) x (1 + 3) = (1 + 1) + 7 x 3 = 2 + 21 = 221. We get the correct answer by outrageously bogus reasoning. And we are surely puzzled. Why does 21 show up in 17 x 13? Aren’t larger products always messed up and nasty? (This is connected to 7 + 3 = 10). Any-one who is in on the joke will immediately say 17 x 26 = 17 x (13 x 2) = (17 x 13) x 2 = 221 x 2 = 442. But few people are.
Some people advocate cultivating a friendship with the integers. Learning the multiplication table, up to 25 times 25, by the means exemplified above, is part of what they mean by this.
Others, full of sullen resentment at the practical usefulness of arithmetic, advocate memorizing ones times tables by the grimly efficient deployment of general purpose techniques of rote memorization such as the Anki deck. But who in this second camp sees any need to go beyond ten times ten?
Does John Leslie have a foot in both camps? Does he set the twenty-five times table as the goal and also indicate rote memorization as the means?
I’m not sure exactly what he had in mind, but learning the multiplication tables using Anki isn’t exactly rote.
Now, this may not be the case for others, but when I see a new problem like 17 x 24, I don’t just keep reading off the answer until I remember it when the note comes back around. Instead, I try to answer it using mental arithmetic, no matter how long it takes. I do this by breaking the problem into easier problems (perhaps by multiplying 17 x 20 and then adding that to 17 x 4). Sooner or later my brain will simply present the answers to the intermediate steps for me to add together and only much later do those steps fade away completely and the final answer is immediately retrievable.
Doing things this way, simply as a matter of course, you develop somewhat of a feel for how certain numbers multiply and develop a kind of “friendship with the integers.” Er, at least, that’s what it feels like from the inside.
I’ve read your link to John Leslie with both curiosity and bafflement.
17 x 24 is not perhaps the best example of a question for which no answer comes immediately to mind. Seventeen has the curious property that 17 x 6 = 102. (The recurring decimal 1⁄6 = 0.166666… hints to us that 17 x 6 = 102 is just the first of a series of near misses on a round number, 167 x 6 = 1002, 1667 x 6 = 10002, etc). So multiplying 17 by any small multiple of 6 is no harder than the two times table. In particular 17 x 24 = 17 x (6 x 4) = (17 x 6) x 4 = 102 x 4 = 408.
17 x 23 might have served better, were it not for the curious symmetry around the number 20, with 17 = 20 − 3 while 23 = 20 + 3. One is reminded of the identity (x + y)(x—y) = x^2 - y^2 which is often useful in arithmetic and tells us at once that 17 x 23 = 20 x 20 − 3 x 3 = 400 − 9 = 391.
17 x 25 has a different defect as an example, because one can hardly avoid apprehending 25 as one quarter of 100, which stimulates the observation that 17 = 16 + 1 and 16 is full of yummy fourness. 17 x 25 = (16 + 1) x 25 = (4 x 4 + 1) x 25 = 4 x 4 x 25 + 1 x 25 = 4 x 100 + 25 = 425.
17 x 26 is a better example. Nature has its little jokes. 7 x 3 = 21 therefore 17 x 13 = (1 + 7) x (1 + 3) = (1 + 1) + 7 x 3 = 2 + 21 = 221. We get the correct answer by outrageously bogus reasoning. And we are surely puzzled. Why does 21 show up in 17 x 13? Aren’t larger products always messed up and nasty? (This is connected to 7 + 3 = 10). Any-one who is in on the joke will immediately say 17 x 26 = 17 x (13 x 2) = (17 x 13) x 2 = 221 x 2 = 442. But few people are.
Some people advocate cultivating a friendship with the integers. Learning the multiplication table, up to 25 times 25, by the means exemplified above, is part of what they mean by this.
Others, full of sullen resentment at the practical usefulness of arithmetic, advocate memorizing ones times tables by the grimly efficient deployment of general purpose techniques of rote memorization such as the Anki deck. But who in this second camp sees any need to go beyond ten times ten?
Does John Leslie have a foot in both camps? Does he set the twenty-five times table as the goal and also indicate rote memorization as the means?
I’m not sure exactly what he had in mind, but learning the multiplication tables using Anki isn’t exactly rote.
Now, this may not be the case for others, but when I see a new problem like 17 x 24, I don’t just keep reading off the answer until I remember it when the note comes back around. Instead, I try to answer it using mental arithmetic, no matter how long it takes. I do this by breaking the problem into easier problems (perhaps by multiplying 17 x 20 and then adding that to 17 x 4). Sooner or later my brain will simply present the answers to the intermediate steps for me to add together and only much later do those steps fade away completely and the final answer is immediately retrievable.
Doing things this way, simply as a matter of course, you develop somewhat of a feel for how certain numbers multiply and develop a kind of “friendship with the integers.” Er, at least, that’s what it feels like from the inside.