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Author Topic: "People Who Understand Math" and "How Math is Taught"  (Read 8439 times)

Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #60 on: March 21, 2013, 06:36:19 pm »

Wait, reading the OP, did you just say that (2/3)*(1/3)=8?
I think I might have read that wrong.

No, he said that (2+2/3)/(1/3) is eight.

Graknorke

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #61 on: March 21, 2013, 06:39:08 pm »

Wait, reading the OP, did you just say that (2/3)*(1/3)=8?
I think I might have read that wrong.

No, he said that (2+2/3)/(1/3) is eight.
Sometimes I wonder why I trust my own ability to read words.
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #62 on: March 21, 2013, 06:41:12 pm »

Actually, why do they spend any time on dividing fractions at all?

x/(y/z) = x*(z/y). This is always true, and makes it far easier to work with.

So, (2+2/3)/(1/3)? How about... 3(2+2/3)?

Graknorke

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #63 on: March 21, 2013, 06:46:10 pm »

There's other ways to work out dividing fractions?
I've just always multiplied by the inverse. It's actually the only way I know how to.
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lorb

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #64 on: March 21, 2013, 07:08:14 pm »

The question this thread raises is: do you understand why multiplication with the inverse is the same as dividing by a fraction?
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #65 on: March 21, 2013, 07:16:00 pm »

The question this thread raises is: do you understand why multiplication with the inverse is the same as dividing by a fraction?

Well, yeah. Multiplying is literally the inverse of division. xy = x/y-1.

LordBucket

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #66 on: March 21, 2013, 07:18:27 pm »

There's other ways to work out dividing fractions?
I've just always multiplied by the inverse. It's actually the only way I know how to.

Sure there are other ways. You could count by increments of the fraction until you reach the target. For example, let's say you were given the following word problem:

A recipe calls for 2 and two thirds cups of sugar. You only have a third cup measuring cup. How many times do you need to fill the measuring cup to get the right amount of sugar?

This is essentially the same division by a fraction question posed in the OP, but rather than multiplying by the inverse it's viscerally obvious that it takes three third-cups of sugar to make one cup. So all you really need to do is add 3+3+2 to get 8. Thinking of it this way is likely to be faster than thinking of it "2 and two thirds times three" which would involve multiplying two by three, then multiplying three by two thirds, and adding the two results.

3+3+2 is easier than 6+((2/3)*3)



In this case, however, I believe the method being taught by the education system is this horribly complicated mess:

Walking through the steps...

2 2/3 / 1/3

8/3 / 1/3

8/3 * 3/1

8*3 = 24
3*1 = 3

24/3 = 8
_________
2 2/3/ 1/3 = 8


That's just needlessly complicated. There's no reason to write out 6 lines of text when it's enough to simply realize that there are three thirds in a whole, and therefore all you need to do is count out three, then six, plus two is eight.

Skyrunner

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #67 on: March 21, 2013, 07:26:08 pm »

I would do that as...

(2+2/3) / (1/3)
(8/3) / (1/3)
8

... Probably the exact same way. But I've been doing this long enough to subconsciously cancel out >_>
Only the ugly whole+fraction notation trips me up...

Am I hopeless? XD

There's other ways to work out dividing fractions?
I've just always multiplied by the inverse. It's actually the only way I know how to.
Tis true.
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palsch

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #68 on: March 21, 2013, 07:36:35 pm »

In this case, however, I believe the method being taught by the education system is this horribly complicated mess:

Walking through the steps...
...
That's just needlessly complicated. There's no reason to write out 6 lines of text when it's enough to simply realize that there are three thirds in a whole, and therefore all you need to do is count out three, then six, plus two is eight.
Except that that's a general method for any fraction division. In the neat and trivial case you are looking at it's over complicated. But give me a similarly quick solution to (9 + 17/23)/(3 + 8/27).

Learning the most general methods for classes of problems is the only thing that makes sense, even when shortcuts exist. Those shortcuts usually only apply to a subset of problems, and relying on them too much is setting you up for failure when they aren't around. I'd almost say that teaching the problems where the shortcuts exist is worse than just diving into the deep end, because it creates the illusion that the full methods are overly complex and torturous for those students who recognise the quicker methods.

As it is, I'd let people get away with using the quicker methods when available, but it does hurt learning the methods that most need practice and drilling to actually fix in your mind. There are whole classes of problem I'm sketchy about solving because I didn't run through enough example problems that didn't have neat and cute solutions that let me skip the whole handle turning exercise. The result is I'm not confident enough to turn that handle without constant outside reference.
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #69 on: March 21, 2013, 07:51:49 pm »

In this case, however, I believe the method being taught by the education system is this horribly complicated mess:

Walking through the steps...
...
That's just needlessly complicated. There's no reason to write out 6 lines of text when it's enough to simply realize that there are three thirds in a whole, and therefore all you need to do is count out three, then six, plus two is eight.
But give me a similarly quick solution to (9 + 17/23)/(3 + 8/27).

(3+8/27)-1 = 27/89

(27/89)(23/23) = 621/2047

(224/23)(89/89) = 20203/2047

(20203*621)/2047 = 12546063/2047 = 6129

The answer is 6129

EDIT: I reread and then I think "what did I just write"
« Last Edit: March 21, 2013, 08:17:00 pm by Putnam »
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Frumple

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #70 on: March 21, 2013, 08:14:36 pm »

You should read Lockharts Lament (linked in the top quote), if you haven't by now.
I actually think I have before, as I read the first bits, but I don't remember it. So I've read it again, and prepared a response! Which I'll spoiler 'cause it's kinda' long and poorly edited. Agreed with a lot of what Lockhart wrote, even if there were bits that irk me and I sorta' disagree with. But here, have a spoiler:

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palsch

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #71 on: March 21, 2013, 08:37:43 pm »

...
Caught your edit. I'll leave the problem up as an exercise for anyone interested for now. Some notes on it though;

I deliberately chose the numbers to involve all large primes and numbers that don't share prime factors so you can't simplify the fractions down. It's very much not a question that you would usually see given in a class teaching these methods just because it doesn't give a nice neat cute answer that is obviously right or wrong. On the other hand it does have an easy sanity check you can make to see if the answer is close (9 and a bit / 3 and a smaller bit ~ ?).

It also requires that you can actually solve absolutely any problem of this sort without falling back on special properties of a particular set of numbers. Which is kinda my point. I get the feeling that if more questions were of this sort, people who 'just get it' would hate them (because they are purely about the mechanics and not intuitive or otherwise insightful observations) but would be better with the fundamentals after they got used to them. It's not like you lose the insights or your intuition for the problems where those apply, but you aren't completely lost at sea when they get taken away by a bastard like me.
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Dutchling

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #72 on: March 21, 2013, 08:47:30 pm »

ptw
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Putnam

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #73 on: March 21, 2013, 08:53:30 pm »

...
Caught your edit. I'll leave the problem up as an exercise for anyone interested for now. Some notes on it though;


And I'll leave my answer up as an example of what not to do :P

LordBucket

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Re: "People Who Understand Math" and "How Math is Taught"
« Reply #74 on: March 21, 2013, 09:24:12 pm »

give me a similarly quick solution to (9 + 17/23)/(3 + 8/27).

1) Round off to nine and three quarters divided by three and a third
2) Turn that into 975 / 333
3) Immediately realize that the answer is "just a little bit less than three"
4) Subtract 975 from 1000 to get 25, and round off 333 to 325
5) Divide 325 by 25 and conclude that the amount of "less than three" is about 1/13th
6) Subtract 1/13th from 3 and get "about 2.93"

Answer is correct to within less than one percent, and can be done in your head in a few seconds.

Give me a real life situation in which such a problem would come up for which that level of precision was inadequate, but for some reason a calculator or computer was not available.

Quote
As it is, I'd let people get away with using the quicker methods when available,
but it does hurt learning the methods that most need practice and drilling to actually fix in your mind.

Would you rather be the person who can sit down with pen and paper and come up with the correct answer of 6048/2047 in two or three minutes...or the person who can sit down with a pen and paper and come with up with the correct answer of 6048/2047 in two or three minutes AND can casually look at it for a few seconds and tell you that it's just a bit less than three?

Quote
It also requires that you can actually solve absolutely any problem of this sort without falling back on special properties of a particular set of numbers. Which is kinda my point. I get the feeling that if more questions were of this sort, people who 'just get it' would hate them (because they are purely about the mechanics and not intuitive or otherwise insightful observations) but would be better with the fundamentals after they got used to them. It's not like you lose the insights or your intuition for the problems where those apply, but you aren't completely lost at sea when they get taken away by a bastard like me.

I don't see anyone suggesting that long forms not be taught. The topic is general non-comprehension of math. In my own personal life, I run into far more people facing problems like "how many third-cups of sugar do I need" and who are unable to do them than I run into cases like yours that are resistant to non-longform methods.
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