That's what happens when you try to "build an intuition" for a concept instead of defining it.
Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!!
The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)."
Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)
As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult.
I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.
About a third work their way through it fine.
About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2
...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.
A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.
I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.
Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!!
The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)."
Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)