<p>“Can I jump in here with a less subjective example?”
Anecdotal evidence argument. D:</p>
<p>“Math in community college versus good colleges is vastly different, too. In fact, math in worse private university is vastly different in math from a better private university.
Proof-rigorous Linear Algebra, where the students are expected to be able to prove the basic theorems by the end of the course on their own steam, where part of the final is to be able to come up with proofs too vast to possibly memorize, where students are expected to have real understanding and are forced to demonstrate it in the form of problems that they could not possibly simply memorize a formula for, are more difficult than community college courses-- or again, courses from a worse 4-year university.”
Unless you’re using left r-modules over a ring instead of vector spaces over a field, linear algebra courses are pretty similar. </p>
<p>“My father took Ordinary Differential Equations at a not-so-hot 4-year university. I’m taking Ordinary Differential Equations at a difficult liberal arts college. Even he admits that there is no comparison. He says that he mostly learned memorization of formulas, algorithms for how to solve the problems, but that he really never gained a real understanding. My class simply doesn’t work that way; we need to formulate proofs, to solve such complex problems that memorization would never suffice.”
Most colleges have this thing called ODEs for engineers, which is a primarily applied class. Then there is the proof-based ODE class for math majors. </p>
<p>“More to the point, my mother is a professor at another fairly crappy 4-year university. She teaches Calculus; she has for 14 years, she still does. She follows about the same syllabus as all the other math professors. I took Calculus I and II at my current college. There were many problems, both on my homework and on projects, that my mother, Calculus professor, was not able to do. They were simply too difficult; she never presents her class with problems of anywhere near that difficulty. And she’s told me that.”
Either she’s completely unqualified to teach at a collegiate level, or you’re full of crap–professors most always have a PhD when teaching math, and at least a MA, which means she should have taken graduate analysis.</p>
<p>“Yes. This is not just community college versus four year colleges. This is not just subjective interpretations. At better (not better-ranked, but better) colleges, even math is very, very, VERY different. At worse colleges, it is more formulaic, more based on memorization, and it is much less proof-rigorous.”
There are not many restrictions forcing students to learn at a lower level than they feel comfortable with.</p>
<p>“Of course, at my small, difficult but not very highly-ranked liberal arts college, someone transferred from here to Cornell. The math at Cornell was way too easy for him, he found it a snap and never learned anything. So, he transferred back here for the difficult math.”
I don’t see your name on the list of Putnam top scores. I see a bunch of people from Cornell. LACs cannot compete with universities because they do not offer graduate-level material typically. If the “math” at Cornell was too easy, then I assume he’s exhausted all of the graduate courses of his concentration, solved all the Millennium Problems et cetera.</p>