Comment by vel0city
4 days ago
> unless they are in the middle of constructing the rationals and need to make a distinction between integers and equivalence classes of pairs of integers
Finally, you're starting to understand the context of the question at hand.
I'm also happy you're starting to show you do understand there's a notational difference between 6/2 and 3. That the values are the same the notation is quite different, thus there are some differences. Not functionally, true, but notationally.
The notational difference was the point of this lesson. You may think it'll only be a barrier in the future to point it out like that (maybe it is!), but the notational difference was the lesson.
> Fluidity in switching through and following different notations
If you don't really have an understanding of the notations, you're going to have a hard time being fluid switching between them.
> An intro proofs professor
An intro proofs professor wasn't leading the lesson, it was probably an elementary or middle school math teacher. The point of the lesson is different, the context of the lesson is different.
Given that most math teachers haven't studied algebra/likely haven't seen the definition of any of these things, and the distinction is not relevant when discussing rounding, I highly doubt that the teacher was making that distinction, or even aware it exists. More likely, the teacher was making a distinction that does not exist, which only confuses students.
In any context that a child is working in, 6/2 and 3.0 are a whole numbers. If the teacher says otherwise, they are wrong. Just because the teacher wants to teach a lesson doesn't mean that lesson is actually correct. The teacher is just confused.
If they weren't confused, it would be highly inappropriate to go into that level of detail with anyone other than a curious gifted kid that's asking questions that are years ahead of a normal curriculum. So much so that it's beyond the level of knowledge expected of a schoolteacher.
You also wouldn't mark it wrong because the entire point is to define things in a way that makes the distinction go away. Even after that distinction has been presented and is front-of-mind, you still generally write down whatever representative is convenient.
It's either literally wrong, philosophically/pedagogically wrong, or both.
> most math teachers haven't studied algebra
Pretty sure all teachers I had even in elementary school studied at least high school level algebra. In middle school and above they all had masters or better in mathematics.
> the distinction is not relevant
> I highly doubt that the teacher was making that distinction, or even aware it exists
> the teacher was making a distinction that does not exist
The distinction both exists and does not exist. Incredible.
> it would be highly inappropriate to go into that level of detail
The detail of a thing that does not exist, right?
> it's beyond the level of knowledge expected of a schoolteacher.
Right, the teacher is wrong because you wouldn't expect the schoolteacher to be smart enough to be right about it.
Yes, the distinction both exists and it doesn't. When defining things, you might start off by saying "the natural numbers are von Neumann ordinals". Then you construct the integers as certain infinite sets of pairs of natural numbers, and you say "actually when I say natural numbers I mean integers that contain a pair where the 2nd number is 0". Then you define rationals as certain infinite sets of pairs of integers, and say "actually when I say integers I mean rationals that contain a pair where the 2nd number is 1", and so on. So for a brief moment during the construction of the next step, there is a distinction. Then you immediately retcon your definition and get rid of it. No one ever uses the intermediate definitions again.
There's similar logical snags when trying to define real numbers because technically you'll need distances which have to be rational because you don't have real numbers yet, but really you'd like distances to be real. It's not actually an issue though, and as far as everyone is concerned, distances are real.
Or you define things only up to unique isomorphism by their properties and wash your hands of the whole ordeal. The construction is merely to show that some object with those properties exists.
The teacher is wrong because if they are being pedantic about it to a child, they're a bad teacher. And they're missing the point.