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

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

#51

Post by quietbritishjim »

didgeoridoo wrote: post #13
CydeWeys wrote: post #2

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surpris…

Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2. You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof…

I think this just adds to the confusion. Is "adding" ratios really the same as averaging them? (I would just say that adding ratios is simply not defined.) Can you only average things by moving from fractions to ratios and then back again?

I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel free to make up a different symbol for the averaging operator, just let the kids know that it's not standard.

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

#52

Post by wcarey »

Spivak wrote: post #46
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.

What isn't really true?

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

#54

Post by ColinWright »

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.

There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}.

Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead.

Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be.

I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.

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

#55

Post by siberianbear »

ValleZ wrote: post #22

I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.

One of the things I've realized in life is the best teachers are people who were not very good at something and then became very good at it through sheer practice and grit.

People who are naturally gifted at something because they can't understand the reason for someone else's failure can't teach well.

I worked for a year of work study as a math tutor at a community college. "How to do integration by parts[1]?" one student asked? That's easy to explain. My hardest day was when I was trying to help this one woman understand that when you multiply two negative numbers you get a positive numbers. "Wouldn't that make it more negative?" she asked.

I decided I would have been a terrible math teacher even though I was very good at math. And reading this article was interesting to me because it explained to me why I would be a terrible math teacher.

[1] https://en.wikipedia.org/wiki/Integration_by_parts

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

#56

Post by abnry »

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.

But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.

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

#57

Post by wcarey »

ColinWright wrote: post #54
twic wrote: post #18

Earlier quoted context omitted.

> 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.

There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}. Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead. Now I'm not saying it's good notation, but some of the fields I've worked in, the n…

> I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.

Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.

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

#58

Post by bsaul »

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…

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 happens to apply to real-world problems, and start with axiomatic definitions (which is the way maths are often taught).

Note : i'm sure you're not advocating for that as well, and i don't mean to contradict you. Just that i think it's better to start with a partially broken metaphor, then fix it using formal definition, than not try at all.

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

#59

Post by macspoofing »

twic wrote: post #45
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.

>Fractions are like groups of pencils or tables of girls and boys.

No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1?

>Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.

I think you have it reversed. The formal definition is the pure definition, which is ancillary to the application to the real-world but covers much much more.

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

#60

Post by qubex »

wycy wrote: post #47
mkl wrote: post #7

Earlier quoted context omitted.

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.

This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f(x)=fx=f×x while f[x] means “apply the function f to the argument x”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.
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