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

#41

Post by ColinWright » Tue, May 26, 2020, 1:39 PM UTC

twic wrote: post #35

ColinWright wrote: post #17

Earlier quoted context omitted.

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...

It's a bit more complicated than "debunked/retracted", as that second document explains: > Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor. The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to…

Agreed.

I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.

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

#42

Post by jacobush » Tue, May 26, 2020, 1:40 PM UTC

wcarey wrote: post #23

mkl wrote: post #21

Earlier quoted context omitted.

Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.

And so it is. edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use. But I think we often thrown them in unintentio…

You used too little context when arguing with an audience who isn't used to that style of writing...

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

#43

Post by macspoofing » Tue, May 26, 2020, 1:42 PM UTC

Spivak wrote: post #24

I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express. The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the…

>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled."

I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.

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

#44

Post by twic » Tue, May 26, 2020, 1:42 PM UTC

JoeAltmaier wrote: post #32

No cat has nine tails. Every cat has one more tail than no cat. Ergo, every cat has 10 tails.

Half a loaf is greater than nothing. Nothing is greater than God. Therefore, half a loaf is greater than God.

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

#45

Post by twic » Tue, May 26, 2020, 1:44 PM UTC

macspoofing wrote: post #28

This is a great example of the downside of over-relying on metaphors and analogies to teach mathematical concepts. The reality is that fractions are not like groups of pencils or tables of girls and boys. They are rigorously defined mathematical constructs that occasionally can be mapped to real-world things (and usually with severe constraints). 1/3 + 1/3 isn't 2/6 because it doesn't follow from the underlying axiom…

Fractions are like groups of pencils or tables of girls and boys. Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.

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

#46

Post by Spivak » Tue, May 26, 2020, 1:44 PM UTC

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…

I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time.

When you see something like.

    f(x) = summation(x^n, n=0, 10)

We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.

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

#47

Post by wycy » Tue, May 26, 2020, 1:44 PM UTC

mkl wrote: post #7

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…

I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.

I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.

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

#48

Post by DavidSJ » Tue, May 26, 2020, 1:44 PM UTC

twic wrote: post #18

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…

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

They’re both inverses: one with respect to multiplication, the other with respect to function composition.

In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value.

There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.

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

#49

Post by mywittyname » Tue, May 26, 2020, 1:45 PM UTC

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…

This was what I came up with in the moment, but with drawing. Draw two circles on the board, each divided into thirds and compare it with a single circle divided into sixths. It demonstrates that the total has grown.

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

#50

Post by wcarey » Tue, May 26, 2020, 1:45 PM UTC

jessermeyer wrote: post #34

Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality. From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions co…

Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?