It's very clear you're out of your element on this, and you have multiple people with an actual math background telling you the objection is somewhere between meaningless and wrong.

The takeaway from trying to really nail down a definition of "integers" (or anything, really) is going to be something along the lines of "if it quacks like a duck up to unique isomorphism, it's a duck". The encoding is not important and one frequently swaps among encodings when convenient. In any case, no one who knows any math is going to say to a child that 3 and 3.0 aren't interchangable outside of some extremely specific contexts. In fact that's not even encoding: it's notation. They can be literally equal, not just equivalent. Those particular contexts aren't ordained, and e.g. propagation of uncertainty is "better" than significant figures if you're doing engineering anyway.

Writing something like '10/3=3' is likely to trigger the mathematicians because lots of people get confused about what '=' is supposed to mean (and often use it to mean something like "next step indicator"). '3=3.0' not so much.

> outside of some extremely specific contexts.

The exact context was given. They wanted only whole numbers.

> Writing something like '10/3=3' is likely to trigger the mathematicians

Sure, when lacking the context of all answers should be rounded to the nearest whole number. But that was the context, and it's astounding so many people with alleged math backgrounds arguing things like intergers aren't a thing to understand.

Assuming you want to be able to make statements like ℕ⊆ℚ⊆ℝ (as one normally does), 3.0 is a whole number, 3 is a real number, and 3.0=3=2.9999999...

Being equal to 3, 2.9999... is also a whole number.

Teaching to use '=' in a statement like '10/3=3' is an example of where teachers don't know math in depth and make errors about details that are actually important/later cause confusion. 10/3 is not equal to 3. '=' doesn't mean "answer". Then not accepting 3.0 which is equal to 3 just layers on that confusion. '=' is transitive. If a=b and b=c, then a=c.

Saying 3.0≠3 is a subtlety you really only get into in math when defining these things, and then you immediately redefine them so that 3.0=3 and you don't have to think about it again.

You're continuing to focus on the functional similarities and functional equivalences of 3, 3.0, 6/2, 2.999... etc. You're right, from an arithmetic standpoint, these are all the same value. I've definitely always agreed with this and fully understand it.

But the question wasn't testing that you knew how to divide and round. The question was testing if you understood what the teacher was trying to teach about whole numbers, integers, rational numbers, real numbers, etc.

6/2 as written is not an integer. It is not a whole number. The value it represents can be written as a whole number, I fully agree, but as written it itself is not a whole number. Whole numbers are the set of numbers Z including -3, -2, -1, 0, 1, 2, 3, 4,... I doubt any math teacher, upon teaching "what is a whole number", draws a number line and proceeds to label it -3.999, -6/2, -12/6, -5/5, 0, 5/5, 12/6, 6/2, 3.999..., and on.

The notation was the key part of the question and was a key part of the answer.

The teacher wasn't looking for a value (which is what you're so focused on looking at), they were looking for a notation, a format.

Then the teacher is teaching something that someone more knowledgeable in the subject will later have to unteach. I'm focusing on functional equivalences because that's how math works as practiced by mathematicians. The functional equivalences are the point, and you may not notice it, but you're also relying on those equivalences, which is why you can write "6/2" in the first place. Integers are already equivalence classes of pairs of natural numbers (which is why 2-3=3-4). Rationals are equivalence classes of pairs of integers (which is why 3/1 = 6/2). If you actually try to define any of this stuff in a coherent way, you're immediately forced to deal with equivalence as a central idea.

6/2 is a whole number. 6/2 = 3. 3 is a whole number. They are equal. Usually, they are the exact same mathematical object. It is not merely that they share properties. They are literally definitionally the exact same thing (the same set in ZFC). "n is a whole number" is a proposition. It is true for n=6/2.

If a teacher is teaching that 6/2 is not an integer, 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, then they are wrong. The very first thing you do after you're forced to make that distinction is you make it go away. They shouldn't be teaching the student to hyperfocus on a specific notation or format. That's a bad lesson to teach, and is something a real teacher will need to fix later. Actual mathematics professors are happy to let you write "let <christmas tree>∈ℝ". An intro proofs professor will definitely put something like "-3.999..., -6/2, -12/6, -5/5, 0, 5/5, 12/6, 6/2, 3.999..." on a number line to illustrate the point that these are just different ways to write the same thing. Fluidity in switching through and following different notations without getting distracted is a centrally important mathematical skill.

> 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.