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## Main Question or Discussion Point

This is occasioned by the thread

https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/

but being somewhat personal I thought the sometimes rather sobersided mods could find it out of place there.

I was first interested by the proposition that (n + m) = (m + n) (1) - I hope it doesn't sound like boasting - but I remember realising this theorem in a flash of inspiration, aged about 5! Believe me or not it was in an abandoned cowshed! I won't say I really lived up to this early promise. Actually I realised at first a special case, that 6 + 4 gave the same as 4 + 6 = 10 (2). No one had told me this! I had learned the parts of that as separate bits of information which the teacher taught with home-made bits of red paper cutouts that we fitted together with 1, 2,... squares ruled on each bit, so 10 was a 3X3 square with one extra little square hanging on the left bottom corner. It was only a couple of days before I realised the full generalisation of equation (2) above. (Other slight fascinations of this thing were the word 'plus' which was in those days not used for anything else, it was a somehow satisfactory word like 'plug' and goes well with the exceptional symbol which stood out from ordinary letters by its squat character in many fonts.)

Of course I was not the first person in the world to realise it, nor did I attempt a rigorous proof and fortunately I was never asked for one. On the other hand I was never given one either.

For that matter I wonder how many of our teachers realised it. Certainly for multiplication they seemed not to. In those days multiplication was taught be a recited drill called 'times tables' that certainly took up a lot of school timetable. We had to learn 2 times numbers up to 12X2 the 3X... up to 12 X 12.* We could have done adequately with learning just over half the amount. But if commutativity was mentioned it was at a later stag and then only as an incidental trick for slickly simplifying calculations and not by name, so I question whether my teachers at least at elementary school realised the importance of commutativity.

I don't know how they get the ideas across these days, but I know a lot of parents think kids should be taught as I and they were. Although they are pretty imprinted in me I have my doubts. In the last year of 'O' level English exams a set book was 'The History of Mr. Polly' and H.G.Wells mentions (the novel is amongst other things a bit of a critique of English education of his time) that Polly could never remember whether it was six sevens or seven eights that made fifty six -

*There was even a half hearted attempt to teach 11X and 12X, which was feasible because there are some simplifying features though I think these I vaguely intuited feather than was shown. I think we did get to 13X but at that time I detached myself. I think I had a feeling that up to 12X12 was some real but rounded knowledge that seemed to be demanded and that all numbers you could have been called upon to multiply ended there - if they did not then there was no other place to stop.

https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/

but being somewhat personal I thought the sometimes rather sobersided mods could find it out of place there.

I was first interested by the proposition that (n + m) = (m + n) (1) - I hope it doesn't sound like boasting - but I remember realising this theorem in a flash of inspiration, aged about 5! Believe me or not it was in an abandoned cowshed! I won't say I really lived up to this early promise. Actually I realised at first a special case, that 6 + 4 gave the same as 4 + 6 = 10 (2). No one had told me this! I had learned the parts of that as separate bits of information which the teacher taught with home-made bits of red paper cutouts that we fitted together with 1, 2,... squares ruled on each bit, so 10 was a 3X3 square with one extra little square hanging on the left bottom corner. It was only a couple of days before I realised the full generalisation of equation (2) above. (Other slight fascinations of this thing were the word 'plus' which was in those days not used for anything else, it was a somehow satisfactory word like 'plug' and goes well with the exceptional symbol which stood out from ordinary letters by its squat character in many fonts.)

Of course I was not the first person in the world to realise it, nor did I attempt a rigorous proof and fortunately I was never asked for one. On the other hand I was never given one either.

For that matter I wonder how many of our teachers realised it. Certainly for multiplication they seemed not to. In those days multiplication was taught be a recited drill called 'times tables' that certainly took up a lot of school timetable. We had to learn 2 times numbers up to 12X2 the 3X... up to 12 X 12.* We could have done adequately with learning just over half the amount. But if commutativity was mentioned it was at a later stag and then only as an incidental trick for slickly simplifying calculations and not by name, so I question whether my teachers at least at elementary school realised the importance of commutativity.

I don't know how they get the ideas across these days, but I know a lot of parents think kids should be taught as I and they were. Although they are pretty imprinted in me I have my doubts. In the last year of 'O' level English exams a set book was 'The History of Mr. Polly' and H.G.Wells mentions (the novel is amongst other things a bit of a critique of English education of his time) that Polly could never remember whether it was six sevens or seven eights that made fifty six -

*and had no way of finding out*. That was an area of vague unreliability for me too though I am not as helpless as he. But if so I think it's my merit more than the school's!*There was even a half hearted attempt to teach 11X and 12X, which was feasible because there are some simplifying features though I think these I vaguely intuited feather than was shown. I think we did get to 13X but at that time I detached myself. I think I had a feeling that up to 12X12 was some real but rounded knowledge that seemed to be demanded and that all numbers you could have been called upon to multiply ended there - if they did not then there was no other place to stop.