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I'm so sorry. I truly don't understand how anyone can have a functional use of math if they didn't at least learn basic arithmetic by rote. These alternate ways I see of doing addition, subtraction, division and multiplication out of common core are bonkers to me, because of how intensive they are in terms of the number of steps they require, or how much scratch paper you'd need for all the intermediate parts. They look more like academic proofs of how basic arithmetic works than how a person should be expected to functionally work with numbers in the spur of the moment.
I mean shit, just yesterday I was playing a game, figuring off the top of my head what the odds of a single 5 or 6 were off rolling a pair of dice. Came up with 20/36 in fairly short order. Although I will be marginally embarrassed if my off the top of the head work turns out to be wrong after all that.
I find this a little strange. Yes, rote memorization is a good idea. But every time I see someone criticize the common core methods it just seems like how I naturally learned to think about numbers? You definitely can truncate most of the steps, the point is spelling it all out. People will say the squares are pointless when you can just carry the one, but the whole point of the squares is to show how carrying the one works mechanically and how it works the same way with multiplication .
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I swear, if it wasn't for my late-Victorian educated granny teaching me how to do long division the old-fashioned way, I'd never have learned the way it was taught in school.
The Tom Lehrer (God rest the man) song is funny but acute if you're old enough to have gone through the process when schools were switching from the old way to the new way, and teachers weren't adequately trained yet in the new way.
same here, only it was my "learned calculus with a slide-rule" engineer dad who got so fed up with what the school system was trying to pull he just sat me down and long-handed it out with me.
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The base 10 "new math" in that song isn't THAT new (that is, it wasn't introduced with the post-Sputnik "New Math") -- that sort of subtraction (which I believe remained standard up to common core) dates back to 1821 in the US. The "old math" (for people under 35 who went to public school) in that song is indeed considerably older, and incidentally works better on a computer because the borrow only propagates one way. I don't know about the under-35 or private school variant.
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I got hit by the tail end of the new math. The way I was taught to do subtraction is definitely closer to Tom Lehrer's second method than the first; we would say that the two "borrowed" a one from the four, so the four got crossed out and replaced by a three while the two became twelve, and twelve minus three is nine.
I... like it? It makes a lot more sense than the first method. We didn't get taught that the four is actually four tens, since it's in the tens place, and that you are substracting ten from the forty and adding them to the two, but hearing him say it makes it obvious in retrospect that's why the algorithm works.
Tom Leher says "the important thing is to understand what you're doing rather than to get the right answer" like it's a joke, but I actually agree with that. As my calculus teacher said "your only advantage over the machine is your ability to think. Once you lose that, I prefer the machine. Calculators are faster and make no mistakes".
Anyway, we also did simple sets in elementary school. No non-decimal bases, though; I learned that on my own reading about computers, because binary and hexadecimal.
“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
Your ability to think matters because it enables you to get the right answer. The only problem with students who don’t understand is that they won’t be able to get the right answer in more general situations. An athlete doesn’t need to understand the physics of his sport or the biology behind his movements.
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I'm so bad at memorization that I never learned the multiplication table by heart. If someone asks me what 7 x 8 is, my mental process goes: Okay, I have no idea what 7 x 8 is, but that's the same as 14 x 4 (multiply the 7 by 2 and divide the 8 by 2). Then I can just:
Which only takes me a few seconds, even in my head.
Likewise, I never memorized most of the trigonometric identities. Instead, I memorized cos x + isin x = e^ix and rederive them at need. When I took the ABCTE math exam, I even practiced using Feynman's notation to make this faster. And the only reason I know the common derivatives is because of this song.
The one math quiz I totally bombed in high school was when our teacher gave us a list of squares and cubes to memorize and then deliberately did not give us enough time to calculate them, to check if we had indeed memorized them.
Personally I just remember the 10 times table and get everything from that.
7x8 is just 70 with a couple of 7s taken away, ie 70-14.
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See, that's the kind of 'innate understanding from first principles' that my brain just does not have for numbers. I learned my times tables and I'd be lost without them.
I look at that and go "but why pick 2? Why not multiply the 7 by 3 and divide the 8 by 4 if you're doing it that way?" Not getting the underlying patterns means I'm blind as to why "this number rather than that number, this of course is the quadrant of the circle for cos" etc. It's like trying to explain to someone tone-deaf that of course this note from hitting this key on the piano is not the same as this note hitting that key. (I'm bad at that as well, I love music but in music classes at school when we had to identify 'what note was that?' I bombed).
You have to pick the same number for the multiplication and division, but other than that 2 is picked just because it's a small, easy number to do division and multiplication with. (You could think of it as "taking" the number out of the one you're dividing and then "putting it back in" to the one you're multiplying, so the whole problem has the same numbers in total, just moved around.) Since 8 isn't divisible by 3, 3 isn't very useful - unless you really like fraction mathematics, I guess - but 4 works equally well:
Or, for the way I would do that last line in my head:
that still requires you to know that 4 x 7 = 28 and to me it's just as fast to learn all the times tables in that case.
I was just using it because you'd brought up "why not pick 4" - and, as demonstrated it's perfectly valid to pick 4! It would work fine. It's just that multiplying and dividing by 2 is usually easier for people, so that's what erwgv used.
Thank you for the explanation. It still seems a longer way round than just remembering the times tables, but if it works for people to understand the principles, I suppose that's the main point.
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