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[Putnam]

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]

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.

[Putnam]

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]

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.

[lorb]

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

[Putnam]

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]

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]

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.

[palsch]

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.

[Putnam]

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"

[Frumple]

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:

Spoiler: Been a while since I've done something like this. Book review? Dunno. Was fun. Some vulgarity. Another bit of wall o'text, meh. (click to show/hide)

Stream of consciousness LL as I read it:

Page 2: Man, I think I've read this before... Also, parallels with the state of English education? How dogdamn surprising >_>

Page 3: Someone who says there's a difference between a poetic dreamer and a rational thinker needs to be bludgeoned with logic books until they speak in sentential. There's no goddamn difference between two people able to speak eloquently in their respective languages. There is only the eloquent and the not.

Page 3: My philosopher sidings twitch. Bastard needs to step off my turf, damnit. Or at least stop calling dibs on what my homeboys started. We were doing patterns from ideas when we made your bloody field of study Lockhart! *fist shake*

Page 5: Now I'm thinking "fanfiction is the mathematics of the written language" *pseudotroll*

Page 5: Math not being the truth, but the explanation... kinda' like grammar is not a book. An expression of a language is different from the language itself, perhaps.

Page 5: Art of explanation... just like all the other arts. Which include everything. What isn't art? I guess what already exists before manipulation? I prefer the "language of addition" myself, as that actually differentiates it from all the other arts, which are themselves just languages of different sort. Light, sound, reason, the physical world, etc., so forth, so on. So an Art of Explanation, perhaps. It's a crowded pantheon.

Page 6: Difference between mathematics and other arts (all of the arts, all of the everything) my arse. Though I guess it was talking cultural view of it, which yeah, true enough. People look at math classes a lot like they look at english classes. Wonder why... [/rhetorical]

Page 7-8: Simp/salv thing fairly spot on, really. We would kinda' benefit from more and better art language classes taught in school. They say that each language you speak is another soul you own, and if one soul is the greatest of things, then surely several are even better?

Page 8: Hear hear on the math textbooks. Shitty pieces of shit. I occasionally run into fairly decent works in other fields -- good anthologies, good summaries with good source lists and excerpts, stuff that makes things inviting and interesting to read. The math texts I was dealt during school weren't them.

Page 8: Oy. You stepping again bro. Logic is the music of reason. Kinda' the language of reason, too. I'll let it slide, you seem to be on a roll.

Page 9: Depends on which infinity you're talking about! There's more than one sort. Though even the formulaic sort isn't really a number, per se, iirc.

Page 9: Technique, context. Word up. Struggle and frustration... maybe. It's harder with some of the unrelated issues I mentioned in my first post in th'thread, but presumably separating functional real-life stuff and actual math into separate classes might help the burden, I'unno. There's some social/psychological things going on there, beyond what LL seems to get to. Which is fine, an essay doesn't have to be all inclusive.

Page 11: Replace "painting and poetry" with "communicating in any language". The statement remains the same!

Page 11: Teaching is open conversation with a person! Miracle of miracles! Congratulations, you've figured out the secret good teachers already know~ Now juggle a couple hundred open conversations a day <3 _{good luck}

Page 11: Simplicio can now officially suck my rhetoricock. mumblemutter

Page 12: Salvati brother, you're basically saying the same thing I've been saying about philosophy for a few years now. Not exactly surprising considering which lead to the other *trollface* Though if they were siblings, philosophy would only be a little older, I guess.

Page 12: Body of mathematical facts... heh. A parallel to classical literature, I wonder...? Or the great masterpieces of music and the more visual arts?

Page 13: Yes, bloody well yes the musician speaks another language! You don't have to be literate to speak! You don't need flawless grammar to communicate! It can make it prettier, but it's not a requirement, not for plain communication nor for beautiful communication! Salvi buddy I think I love you but now you're kinda' shitting on linguistics and the sublime fundamental beauty of communication and I think maybe we should talk about our relationship.

Page 15: I've occasionally said to others folks would have an easier time getting into mathematics if someone would translate the shit into goddamn English. LL's basically saying the same thing, I'd say. Anyone worth two shits knows better than to use technical language when you're initially teaching things (and if they don't know better, I know a couple dozen teachers who'd be happy to hit them with sticks until they get the point. There's a reason many of my philosophy classes would, for example, use epistemology and the study of knowledge interchangeably throughout the semester for the introductory courses.). That comes later, when they need to discuss things rapidly between people already in the know.

Page 17-18: Teach 'em to speak the language, not just phrases from it. Word up.

Page 18: Brother speaks truth on proofs. Once you can read it (and I'm bit rusty right now, but I wasn't for a short period) logic proofs are kinda beautiful when they're done well. Sometimes they're even funny, heh. You can make a damn good joke with one of those things, providing you can actually read it. [/language parallel]

Page 20: Paradox, formal logic, word up. And hey, yeah, proofs can be translated into English! It's just another way of saying something.

Page 22: No idea what they're saying: For all they know, they're saying that the orange flies at midnight, not asking where the bathroom is. It just so happens that "orange flies at midnight" is how the language they're learning refers to the bathroom for some convoluted historical/cultural reason.

Page 22: Definitions matter. Yesh. The assumptions you make are a fundamental part of creating proofs <3

Last bits: Makes me damn glad I never got to trig or beyond >_> I actually do sorta' appreciate math nowadays, even if it's kinda' how I appreciate French. Beautiful but fairly incomprehensible 'cause I've never really learned more of the language. Can communicate enough to get by, but my artistic/linguistic leanings have gone in other directions and... effort

[palsch]

...

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.

[Dutchling]

ptw

[Putnam]

...

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

[LordBucket]

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.

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?

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.