Comment by ndriscoll
4 days ago
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.
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