Re: [math-fun] multiplication in school
I agree with whoever said that understanding WHY arithmetic works, is helpful, not just memorizing how to do it. I did memorize the mul-table, and while I do not think my memory is very good, I also do not think I had problems with memorizing this table. Obviously the mul-table is easier to memorize than a same-size table of random numbers because if you forget an entry, you can re-obtain it from an adjacent one. So it is like self-sealing rubber, gaps in your memory self-repair in this application. IF you understand that. Which I did, although I do not think any teacher ever told me this. Now the real reason schoolboy multiplication works is the distributive law, which again I do not think any teacher ever told us. But I started to appreciate that about age 10, I think (?), which for me was something of an "aha!" moment. Understanding what was going on. Versus being a robot. Now I do not think I ever, or anyhow not for a long time, learned how to do long division, but I always managed nevertheless because I used some sort of crude iterative correction technique instead to get the right answer. (I haven't long-divided by hand in years, can I even still do it?) Actually, the way schoolboy long-division works IS an iterative correction technique, when you break it down, depending again on distributive law, and I think I was unable to do long division until I came to understand that, which was when I was a pretty old schoolboy. My father apparently got taught in school to take square roots via a paper & pencil procedure, but I was never taught that in school but again was able to use iterative correction techniques when I needed that. If all we do is churn out children who can do arithmetic robotically without understanding why it works, I do not think that is a worthwhile educational accomplishment in the modern era. Understanding WHY procedures work, is worthwhile and also will cause your roboticism to be better. I also think one of my school teachers thought a/b + c/d = (a+b)/(c+d) and taught this to the class, which was not helpful...
"And that is not all. There is a new development that is, so far, top secret and which, strictly speaking, I ought not to mention. Still - we may have made a break-through on the square root front.” — The Feeling of Power, Isaac Asimov http://downlode.org/Etext/power.html
On Feb 13, 2015, at 1:18 PM, Warren D Smith <warren.wds@gmail.com> wrote:
I agree with whoever said that understanding WHY arithmetic works, is helpful, not just memorizing how to do it.
I did memorize the mul-table, and while I do not think my memory is very good, I also do not think I had problems with memorizing this table. Obviously the mul-table is easier to memorize than a same-size table of random numbers because if you forget an entry, you can re-obtain it from an adjacent one. So it is like self-sealing rubber, gaps in your memory self-repair in this application.
IF you understand that. Which I did, although I do not think any teacher ever told me this.
Now the real reason schoolboy multiplication works is the distributive law, which again I do not think any teacher ever told us. But I started to appreciate that about age 10, I think (?), which for me was something of an "aha!" moment. Understanding what was going on. Versus being a robot.
Now I do not think I ever, or anyhow not for a long time, learned how to do long division, but I always managed nevertheless because I used some sort of crude iterative correction technique instead to get the right answer. (I haven't long-divided by hand in years, can I even still do it?) Actually, the way schoolboy long-division works IS an iterative correction technique, when you break it down, depending again on distributive law, and I think I was unable to do long division until I came to understand that, which was when I was a pretty old schoolboy.
My father apparently got taught in school to take square roots via a paper & pencil procedure, but I was never taught that in school but again was able to use iterative correction techniques when I needed that.
If all we do is churn out children who can do arithmetic robotically without understanding why it works, I do not think that is a worthwhile educational accomplishment in the modern era. Understanding WHY procedures work, is worthwhile and also will cause your roboticism to be better.
I also think one of my school teachers thought a/b + c/d = (a+b)/(c+d) and taught this to the class, which was not helpful...
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Good old Myron Aub. --Dan
On Feb 13, 2015, at 10:28 AM, Tom Knight <tk@mit.edu> wrote:
"And that is not all. There is a new development that is, so far, top secret and which, strictly speaking, I ought not to mention. Still - we may have made a break-through on the square root front.” — The Feeling of Power, Isaac Asimov
participants (3)
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Dan Asimov -
Tom Knight -
Warren D Smith