Topic: How they should start teaching math
I honestly believe in the lower grades, they should give people a top-down view of mathematics. Not even just for "practical applications", which in my opinion is a failed fucking experiment because it's a lie and kids don't give a shit about "practical" applications of mathematics where they're not actually practical or normally applied, what I think should be done is a general overview of mathematics as a field. That would have interested me as a kid *way* more than anything they ever did for me. They're trying to obscure the truth, show everything in a practical light, generate interest in math, all this fucking bullshit, but all they really need to do is not lie. Have an overview course very early on about the grander theories in math, maybe a little history. Sort of like a survey course for fucking any other course sequence.
Only math majors end up getting to take these sorts of survey courses at my Uni. Why? It doesn't have to be the nitty gritty details, just create a survey course that explains how components in lower maths ends up directly tying together with extremely complex higher math. Teachers always tell you that math "builds on itself", but no kid knows what that means in a visceral way. They don't give a shit if it builds on itself, because they simply don't know where the fuck it's building to. Why should they care about complex mathematics at the end of the road? They probably won't even end up doing any of that math, anyway. Since they don't care about any of that, why give a fuck about any of the lower levels?
There should be much more done in this arena. If in Algebra in middle school I knew how many of the constructs I'm learning directly translated into the foundations of much more complex abstractions for very interesting and sexy looking geometric patterns and conceptual calculus-like limit stuff, let alone any of the more advanced concepts that even now I don't know much about... I'd be way, way more interested. It was like they put fucking horse blinders on me in school and said "learn this because learn it". "It's practical" when everyone can clearly see it's not practical for non-mathematicians. "It's interesting when you really think about it" when no kid in any of those classes has the requisite knowledge to understand *why* it's interesting or what it's supposed to build up to...
Shit bothers me. The more I got into discrete math the more pissed I was at my shitty math teachers for neglecting any actual insight into where math goes, at all. Not a single word was mentioned about geometric shapes in math or concepts of infinitely finite lines or complex multidimensional planes. Euclidean space can be fascinating from an intuitive conceptual standpoint.
And you know what really gets me? Even in University a lot of the professors neglect to say anything about how a lot of this can directly be used in mapping the physical world, or at least a conceptual version of it. And that's the whole fucking point... The most fun I ever had in mathematics was when we got my teacher to go off on a tangent about euclidean space curves, and why the bounds of a given plane are even existing. This happened because a kid asked why one can't just cross-over towards another thing within the problem. IIRC. I have it written down. She gave a semi-ok answer but I ended up coming up with an even better answer while going through it a bunch of times in my head. It made real intuitive sense. That was the most interesting thing to me. Why they don't try to generate this kind of interest and link it to lower maths is beyond me.