[red_admiral]

I really liked this, and it's also why it's so hard to teach mathematics, which is part of my current job.

Most people think in a context-dependent way. If you ask, suppose Jane has three apples and John gives her two more apples, how many does she have - then most kids at the appropriate level will visualise apples and count to five. Give exactly the same problem but with "Jane has five McGuffins" and you'll get a confused stare followed by "what's a McGuffin?". Except of course for the one kid who has no problem with the math because they misheard it as McMuffin and could visualise that!

[Aerroon]

But the math we teach in school is context-sensitive. It matters what set your inputs are from and what set the output is supposed to be in. We usually don't mention that we're doing math on real numbers, we assume that based on the context.

22 + 8 = 6

This would be incorrect in an average math class, but when you're dealing with the clock then it's something understands. 8 hours after 10 pm is 6 am.

ab = ba

We teach the commutative property as though it is universal, but it isn't. With real numbers? Sure! Swap two matrices though and you're in trouble.

I don't think kids should necessarily be taught differently, but there is definitely (implicit) context involved in math. Even in geometry: the inner angles of a triangle add up to 180 degrees, right? But in spherical geometry the sum of the inner angles of a triangle can be larger.

[zozbot234]

Yes, abstracting away from the context only works if you can tell that the problem really is context-independent. This works well with apples, not so well with e.g. commutativity once you get to things like matrices. So abstraction (from "apples" to "numbers" to "matrices") can sometimes reintroduce context that had previosly been discarded.

[nicoburns]

I have a good friend who works as a high school physics teacher, and this is apparently exactly how they teach: they teach the intuition behind the problems so that the kids can visualise them.

[morelisp]

And this works up until about high school physics, but not too much further. Not only do many interesting mathematical objects lack some intuitive basis, some are interesting specifically because they behave counter-intuitively.

[pfdietz]

Mathematics builds abstractions, but at any level you first see the less abstract things that motivated the abstraction.

[xtiansimon]

One of my favorite _rudimentary ideas about mathematics_ comes from philosopher Cathy Legg describing the work of Charles Sanders Perice:

_"Perice had a hypothetical interpretation of mathematics. So mathematics doesn’t talk about what’s actual at all. Mathematics makes no positive claims. Mathematics just tells you if you make this hypothesis, then this must follow. So mathematics is the science that draws necessary conclusions."_

If you get your head around that, then apples and McGuffins are both permissible.

Episode 81: Cathy Legg discusses what Peirce’s categories can do for you https://elucidations.hum.uchicago.edu/Legg_WhatPeircesCatego...