Can 1/3 and 1/3 = 2/6? It seemed so • Hacker News Board

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

#211

Post by inimino » Tue, May 26, 2020, 6:29 PM UTC

Wowfunhappy wrote: post #187

forbiddenvoid wrote: post #174

Earlier quoted context omitted.

What they said wasn't wrong. Their mental model was absolutely correct. "One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.' Their written representation of the mental model was incorrect because their instruction was fo…

Separate from the discussion of what is or is not counterproductive / educative, the student's mental model was wrong. The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.

No. The mental model was exactly correct. Their assignment of notation to the model was non-standard when they wrote "+" for the operation of merging the wholes.

The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.

The confusion of the mental model with the notation is also a standard mistake.

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

#212

Post by [deleted] » Tue, May 26, 2020, 6:30 PM UTC

JadeNB wrote: post #206

lisper wrote: post #143

Earlier quoted context omitted.

> I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.

In both of these instances, I think that you are using composition to mean iteration . Iterating is repeated self-composition, so it is composition, but it seems likely that you meant the more specific term. (For example, "the successor function is not a composition of anything" is definitely false in the literal sense of composition. It's also false in the literal sense of iteration, but there it's clear what you me…

[deleted]

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

#213

Post by bsaul » Tue, May 26, 2020, 6:32 PM UTC

macspoofing wrote: post #65

bsaul wrote: post #58

Earlier quoted context omitted.

Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere. The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working. Much better (imho) than teaching math as a purely formal and transcendent topic that happen…

>We didn't invent fraction out of nowhere. But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy. >The moment where you hav…

"But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world"

This seems totally absurd to me. The concepts of proportion, ratio, etc preexisted any kind of formal definition of fraction.

They may have invented a definition of fraction, with the correct notation, and the correct set of rules after a long series of trial and errors (as with a lot of mathematical "rigourous definition"), but they always had in mind that this concept they tried to define should "work" when manipulating ratios / proportions, etc.

I may be wrong, but it seems to me that purely mathematical concepts spawning out of pure mathematical world exploration is a very modern (aka 19th century max) concept.

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

#214

Post by lisper » Tue, May 26, 2020, 6:34 PM UTC

JadeNB wrote: post #206

lisper wrote: post #143

Earlier quoted context omitted.

> I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.

In both of these instances, I think that you are using composition to mean iteration . Iterating is repeated self-composition, so it is composition, but it seems likely that you meant the more specific term. (For example, "the successor function is not a composition of anything" is definitely false in the literal sense of composition. It's also false in the literal sense of iteration, but there it's clear what you me…

> you meant the more specific term

No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it.

> "the successor function is not a composition of anything" is definitely false

That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?

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

#215

Post by mjevans » Tue, May 26, 2020, 6:34 PM UTC

DigitallyFidget wrote: post #150

Wowfunhappy wrote: post #111

Earlier quoted context omitted.

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.

Another way of labeling this:

  : = Out Of
  A = Table A
  B = Table B
  E = Every Table Added (A + B)
  
  1:3 * A + 1:3 * B = (1:3 + 1:3) * E
  2 out of 6 among Every Table Added

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

#216

Post by PaulDavisThe1st » Tue, May 26, 2020, 6:36 PM UTC

inimino wrote: post #208

Wowfunhappy wrote: post #111

Earlier quoted context omitted.

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime. Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do. Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing an…

there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves.

this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.

there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.

now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.

doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.

this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.

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

#217

Post by wwarner » Tue, May 26, 2020, 6:40 PM UTC

I would express this in a diagram with a dotted line around the set under consideration. When dealing with thirds, make a diagram containing three objects with their respective properties, and end by drawing a dotted line around the set. When switching to sixths, add three more objects, erase the line around the first three, and draw a new dotted line around the whole set of six. Then go on to sevenths by adding one more object, erasing the dotted line for the six and drawing it around all seven.

If I do it this way, my 4th grader can add fractions just like numbers without resorting to common denominators.

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

#218

Post by Geee » Tue, May 26, 2020, 6:41 PM UTC

I would have said that a whole table is 3/3 and that both tables together are 6/3 and not 6/6 (maybe they'll get it that 6/3 equals to 2). Probably a good idea to explain that the "whole" isn't all pens and bottles but the group of 3 or 6 or 12 or whatever.

Maybe start with explaining from the start what happens if there are 7 bottles, when bottles are counted in the groups of six-packs. How many six-packs can you make from 15 bottles?

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

#219

Post by inimino » Tue, May 26, 2020, 6:50 PM UTC

PaulDavisThe1st wrote: post #216

inimino wrote: post #208

Earlier quoted context omitted.

There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime. Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do. Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing an…

there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves. this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves. there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning…

There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this.

They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!

There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.

You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.

And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?

There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.

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

#220

Post by wcarey » Tue, May 26, 2020, 6:51 PM UTC

Phlogistique wrote: post #188

adrianmonk wrote: post #176

The difficulty isn't with fractions. It's about understanding what "+" means. Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie"). His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about t…

That's my favourite explanation so far. That's how I would put it to the student: Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples: * At your table, Bob is 1/3 of the table, and Sandra is…

We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us.

Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)