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> One big problem with the plan, which a lot of people don't realize, is that your typical university math degree isn't that much more challenging than doing we

> One big problem with the plan, which a lot of people don't realize, is that your typical university math degree isn't that much more challenging than doing well in high school math. That wasn't my experience. (I went to a top liberal arts college - it didn't give minors, but I took enough math courses that it would've been a minor at other schools.) In my experience, post-calculus college-level math is qualitatively different from high-school math in focusing on proofs and abstractions over problem-solving, techniques & algorithms. I had an intuitive sense for high-school math. College math kicked my ass, because I was asked to prove everything rigorously, and the proofs just kept coming, dozens of them per homework assignment. That required a very different way of thinking than high-school math did: you needed to remember this large body of theorems and combine them in creative ways to derive new proofs, you couldn't just memorize recipes and techniques that will get you the answer. I found that physics was more like the continuation of high-school math, except that then your courses had all this lab work and experiment design and fundamental physical principles. One of my friends (a math/physics double major) described it like this: "In physics vectors are objects that live in a vector space, while in math a vector space is a set of axioms from which the concept of a vector and rules for manipulating them naturally fall out." Edit: hmm, just looked at the username. I know you, and I also know you're better at math than I am. I'm curious just what kind of math you did in high school and what you did in college where the college math was no harder than high school. Either your high school had some really great math courses or your college had some pretty shitty ones. Maybe it's just where we went to school.

fyp 7y ago
If he was doing olympiads in high school, he was already doing proof-based math then. I think Terry Tao describes the stages of mathematical maturity really well: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...
lacker 7y ago
Well, I should be more specific. I think once you get into the sort of number theory where you prove things about modular arithmetic, and the sort of analysis where you are talking about abstract spaces rather than n dimensions of real numbers, those courses are notably different from high school courses, and all the focus on proofs is intellectually stimulating to advanced kids. The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university. Also, there’s a big difference between colleges. Not because of the curriculum as much as the other students who will set the pace of the class. Often the same kid who could attend Ohio State at 14 could also attend Princeton at 18, and that makes a difference even if the course titles are the same.
- nostrademons 7y ago
  Ah, I'd agree with that. Calculus/probability/statistics/combinatorics were usually freshman courses at Amherst, and you could often place out of them with APs. Linear algebra was taught proof-based; it and DiffEq were usually sophomore-level courses. The rest of the major was all proof-based.
- JackFr 7y ago
  >The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university. That's just not true. That's the freshman/sophomore curriculum. And there are big differences between the diff eq, stats & linear algebra course which are which are 'proof' vs. those which are 'practical'.
- chongli 7y ago
  I think once you get into the sort of number theory where you prove things about modular arithmetic That’s one small part of the intro algebra course at my university, which all math and CS students take in the fall of first year. The first year calculus courses are considerably proof-based as well, and lots of first years end up failing despite being a high 90s student at their high school. A more typical high school student who got 80s in their sleeves and calculus courses would fail everything and be asked to withdraw.
bigred100 7y ago
I’m not sure how else you’d describe a vector than something that lives in a vector space. The approach of a mathematician might be different but I do believe the idea of scaling and adding things (and then stuff like inner products) is a lot more important than axioms. And at least to me the rules for manipulating them fall out after a huge amount of technical work and details.

fyp 7y ago

If he was doing olympiads in high school, he was already doing proof-based math then. I think Terry Tao describes the stages of mathematical maturity really well: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...

lacker 7y ago

Well, I should be more specific. I think once you get into the sort of number theory where you prove things about modular arithmetic, and the sort of analysis where you are talking about abstract spaces rather than n dimensions of real numbers, those courses are notably different from high school courses, and all the focus on proofs is intellectually stimulating to advanced kids. The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university. Also, there’s a big difference between colleges. Not because of the curriculum as much as the other students who will set the pace of the class. Often the same kid who could attend Ohio State at 14 could also attend Princeton at 18, and that makes a difference even if the course titles are the same.

nostrademons 7y ago
Ah, I'd agree with that. Calculus/probability/statistics/combinatorics were usually freshman courses at Amherst, and you could often place out of them with APs. Linear algebra was taught proof-based; it and DiffEq were usually sophomore-level courses. The rest of the major was all proof-based.
JackFr 7y ago
>The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university. That's just not true. That's the freshman/sophomore curriculum. And there are big differences between the diff eq, stats & linear algebra course which are which are 'proof' vs. those which are 'practical'.
chongli 7y ago
I think once you get into the sort of number theory where you prove things about modular arithmetic That’s one small part of the intro algebra course at my university, which all math and CS students take in the fall of first year. The first year calculus courses are considerably proof-based as well, and lots of first years end up failing despite being a high 90s student at their high school. A more typical high school student who got 80s in their sleeves and calculus courses would fail everything and be asked to withdraw.

bigred100 7y ago

I’m not sure how else you’d describe a vector than something that lives in a vector space. The approach of a mathematician might be different but I do believe the idea of scaling and adding things (and then stuff like inner products) is a lot more important than axioms. And at least to me the rules for manipulating them fall out after a huge amount of technical work and details.