> Taking Calculus at not-Stanford may have led to a very different result.
I doubt it. As my professor used to say, math is math. Plus, intro-level undergrad math courses are pretty standardized across the US. Similar textbooks, similar problem sets, and similar past exams.
There's nothing wrong to feel calculus is hard even if you're the top of your high school class. Most people hit a wall at certain abstraction level. Malcolm wrote in his book about smart people simply not being able to grok organic chemistry. Some of my smart classmates could not really understand the concept of limit intuitively. Some of them dropped out of abstract algebra course. I myself dropped out of model theory as I couldn't get why I should even bother with the course. None of these failures prevent people from finding success in other areas.
Unfortunately many ostensibly good schools also have notoriously poorly taught introductory math and science courses.
Stanford's introductory CS sequence has a good reputation however. Consider this course (for example) which teaches students bare-metal programming (and real-world interfacing) on a raspberry pi: https://cs107e.github.io/schedule/
Advanced freshman math, the kind for the top few students, varies extremely widely. As far as I can tell, if you made it all the way through the series at Harvard or Stanford, then you are an excellent student regardless of whether you get a good grade. I’m sure a couple other schools are like that. Many schools that you might expect to have similar classes don’t.
Sadly the universities with courses like this seem to me mediocre at advertising that, no, if you are not specifically very very good at math (and very very good at following a very fast paced course and you don’t have a strong background), then that’s not the class for you and you should be in the other honors math class.
Do you have any actual empirical evidence? In my undergrad, even different STEM departments (e.g., math, EE) had differing rigor and workload based on the average scores they required of admissions. Almost all grading was done relative to other students anyway, so even with the exact same course, things would be very different.
As an example, we had to prove all continuous functions on a closed interval are monotonically continuous on that interval, in Calculus I. Which we had in our first semester with no choice. I doubt this level of rigor is at all common.
I used limit as an example. It could be other concepts, simple or sophisticated.
I don’t think it’s about how professors teach, but about that people simply can’t get some abstractions. Have you seen kids in high school who can’t do algebra even if you find the best teachers for them?
1. How do you even express the idea of continuity if not through limits? (Sure, there's the topological definition and the epsilon-delta criterion but those are merely an abstraction or a different way of phrasing the same idea.)
2. The key difference between the rationals and the reals is that the latter are Cauchy-complete. How do you even define (in an intuitive way) what that means without referring to limits?
3. The reals are typically constructed as (equivalence classes of) Cauchy sequences, i.e. by their very definition they are the limits of their Cauchy sequences.
i think, for the purpose of introducing the continuity of the reals, for the purpose of introducing limits, one could skip any discussion that involves constructing the rationals or building the reals from them, at least to start.
instead go directly from integers to reals by introducing countable vs. uncountable infinities.
sure. and despite all the trouble people have, their definition is quite simple. the crux of the problem is understanding continuity. if you truly understand the continuity of the reals (which is the "real" invention), limits can be derived almost effortlessly.
in most calculus teaching, continuity is quickly stated and then there's a bunch of rules and these seemingly contradictory discussions involving delta and epsilon. they're not contradictory if continuity is understood and internalized.
Do you prefer the Dedekind cut construction of the real numbers over the identification of Cauchy sequences of rational numbers with common limit? In ZFC, they coincide, but in constructive mathematics, one can build a model wherein they differ (see https://mathoverflow.net/questions/128569/a-model-where-dede... ).
I think a contributing factor for organic chemistry is not everyone teaches the mindset needed to do well. Good teachers will help students explicitly build a heuristic mental model for what the electrons might do in a particular bonding environment. Not everyone thinks naturally that way, but I think it’s a lot more effective than the raw pattern matching approach many people seem to bring to organic chemistry
I doubt it. As my professor used to say, math is math. Plus, intro-level undergrad math courses are pretty standardized across the US. Similar textbooks, similar problem sets, and similar past exams.
There's nothing wrong to feel calculus is hard even if you're the top of your high school class. Most people hit a wall at certain abstraction level. Malcolm wrote in his book about smart people simply not being able to grok organic chemistry. Some of my smart classmates could not really understand the concept of limit intuitively. Some of them dropped out of abstract algebra course. I myself dropped out of model theory as I couldn't get why I should even bother with the course. None of these failures prevent people from finding success in other areas.