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Can 1/3 and 1/3 = 2/6? It seemed so

marilynburnsmathblog.com

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#271

Post by TeMPOraL »

kortilla wrote: post #269
TeMPOraL wrote: post #258

Earlier quoted context omitted.

Not sure if this is at the level of comprehension of these kids, but I'd explain it this way: "kids at table A" and "kids at table B" are both "kids", but different amount of kids. You can treat them as the same only if you have a conversion factor. So, if you know that there are 12 kids at table A, and 20 kids at table B, you can multiply your variables by these amounts, and now both expressions have the unit "kids"…

> you can multiply your variables by these amounts, and now both expressions have the unit In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.

Since numbers are usually introduced to children in terms of quantities of things (e.g. 3 apples), I think it might suffice to stress the importance of keeping these things always attached to those numbers. In general, numbers very rarely go alone.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#272

Post by DavidSJ »

lisper wrote: post #264
DavidSJ wrote: post #263

Earlier quoted context omitted.

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”. Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said. As for multiplication being “composed” addition, addition being “compos…

> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. Th…

You seem to be jumping around in terms of what position you're taking. Is times(a, b) function composition because a and b are functions, or is times(a, b) function composition because it's composition of the addition function, or is times(a, b) function composition because multiplication is a function? Those are very different assertions, with very different implications to the original discussion, and so far as I can tell you’ve made all three of them. Maybe they're all true, maybe one or two of them is true, maybe it's a matter of perspective which are true. But I don't think you're being clear or consistent in which assertion your position is based on.

And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall).

At this point I feel we're going around in circles, so I'll bow out.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#273

Post by matvore »

I think the problem is the teacher moved from one type of "fraction problem" to another kind.

The example of drinking 4 bottles of water to only have 2/6 left of the original pack is doing an integral problem and wrapping up the answer as a fraction. Same deal with the pack of 12 pencils.

The students can grasp those problems as e.g. (1+1)/6 rather than (1/6 + 1/6). In other words, there is only one "whole" in the problem.

When you're adding the fractions of desks filled with students, the fraction is counter-intuitive because both desks have to have the same number of students for it to make sense. (1/3 of a 5000-student round table is different from a 3-student table). And to say "the units are wrong" is kind of a limited way of explaining this. The units also need to have the table capacity as part of the unit identity (i.e. the units would have to be 3-student-table). That's a pretty sophisticated way of thinking about units.

I think that after the pencil/water examples, transitioning from the pencil/water pack examples to a more "pure" fraction example would be better. e.g. one group of students eats 2/3 of a pizza, and another group eats 2/3 of a pizza. Now you can throw away one of the original pizza boxes and put the two remaining 1/3 in a single box which is 2/3 full. Now the "whole" for each fraction is no longer arbitrary (like 6 bottles of water or 12 pencils).

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#274

Post by jiggawatts »

wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

Allow me to introduce modern theoretical physics, where comments like "X is obvious in context" never is, except to the person that wrote it. Don't worry though, they helpfully add that "When X isn't what it seems in context, this will be called out". Okay. "Sometimes it may not be." Wat?

Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated.

If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group.

I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#275

Post by lisper »

DavidSJ wrote: post #272
lisper wrote: post #264

Earlier quoted context omitted.

> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. Th…

You seem to be jumping around in terms of what position you're taking. Is times(a, b) function composition because a and b are functions, or is times(a, b) function composition because it's composition of the addition function, or is times(a, b) function composition because multiplication is a function? Those are very different assertions, with very different implications to the original discussion, and so far as I c…

> Those are very different assertions

But they are not mutually exclusive. That's the whole point.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#276

Post by app4soft »

forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

> The units are missing

> The student came up and wrote 1/3 + 1/3 = 2/6.

Units in your example are unclear. Also results interpretation depend on question.

Here is my example:

Student should wrote 2 articles, but wrote 1 of 3 pages for the 1st article and 1 of 3 pages for the 2nd article.

Q: How many pages student wrote? - A: 2/6.

Q: What part of the whole task is done? - A: 1/3.

Q: How many tasks partially completed? - A: 2/2.

Q: How many tasks fully completed? - A: 0/2.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#277

Post by httpsterio »

forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

If you are omitting the units, it would be a fair assumption that the values all belong to the same unit, unless stated otherwise. You're representing the fractions related to each other and you can't do that if the fractions are of two separate things.

I could say that 1+1 = 2 or 1+1 = 10 but it wouldn't be right to say that 1+1=2 && 1+1=10 because while both are true if we're talking about decimals and binary, we're omitting the units and everything loses its' meaning if we do that.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#278

Post by renewiltord »

wcarey wrote: post #146
gowld wrote: post #110

Earlier quoted context omitted.

Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.

Is that commonly done?

It is not. Though it wouldn't be too hard to adapt, I think, if it was noted at the beginning of a text. For it to take, it would have to power some influential result, sort of like how the summation convention in tensor notation works (something that infuriated me as a kid).

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#280

Post by nyanpasu64 »

ColinWright wrote: post #17
wcarey wrote: post #10

Earlier quoted context omitted.

The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced…

As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...

Excerpt from the retraction:

> In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true.

A sad story :(

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