Wow, I had no idea this blog existed - I loved Marilyn Burns's children's math books as a kid, awesome to see her still in action.
Can 1/3 and 1/3 = 2/6? It seemed so
221–230 of 308 posts
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#222This 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…
So, have been any sufficient effort in design a new syntax of maths that could help the regular folk? If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
There are some gentle efforts in that direction (tau radians for example), but mathematical notation is old (except the parts that are new). And it's really hard to strike the right balance between concision and ambiguity.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#223Earlier 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.
Somewhat aside, but in grade school I recall learning that ratios were written [group-A]:[group-B], not [group-A]:[total]. So the girl-boy ratio at the table would be 1:2. Different for you?
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#224Earlier quoted context omitted.
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 thou…
> 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.
> the point I was trying to make, which is that infix notation is a Really Bad Idea
Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection?
> What is successor a composition of?
It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'.
Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.)
If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.)
By the way, you quoted (https://news.ycombinator.com/item?id=23314776):
> > you meant the more specific term
Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#225This 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.
If you meant "function", would you generally write 𝑓(3) instead of f(3)?
(I have no mathematics background past K-12)
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#226Earlier quoted context omitted.
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.
Somewhat aside, but in grade school I recall learning that ratios were written [group-A]:[group-B], not [group-A]:[total]. So the girl-boy ratio at the table would be 1:2. Different for you?
It really depends on the two things being compared. The ratio of boys to girls is 1:2, but the ratio of boys (implicitly: to the total) is 1:3.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#227You're taking a weighted average, not adding the fractions.
It takes a while to learn what that is used for.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#228This 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…
Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)). I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always. Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant…
When actually working with trigonometric identities, using parentheses typically gets unwieldily very quickly. Same reason sin(x) often becomes sin x.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#229Earlier 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.
A good example of this can be found in an incredible algorithm with a terrible name: Fast inverse square root . It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0] Related to this: the way…
That one is clear to me, although it might be for a reason you find displeasing: if the thing being computed was an "inverse square root" in the sense of a functional inverse, then you'd be computing a square and it would make much more sense to call it that instead.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#230Earlier quoted context omitted.
Absolutely terrifying. How did he communicate with other peer mathematicians?
Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
…but that's not a fraction of pi?