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It's kind of funny, most people who major specifically in math or applied math will take something equivalent to a course in Real Analysis. RA is basically star

by gh02t 11y ago

It's kind of funny, most people who major specifically in math or applied math will take something equivalent to a course in Real Analysis. RA is basically starting over from scratch and reteaching you the exact same material as calculus, but in a much more formal and rigorous manner. Mostly it focuses on eliminating the not-so-rigorous concept of an infintesimal quantity and formalizing the notion of limits and convergence (albeit, there is a way to rigorously treat infintesimals called nonstandard analysis, but I digress). It involves a lot more open-ended thought than the endless memorization of earlier calculus classes. It's often dreaded by undergraduates as being very proof-heavy, but it was one of my favorites. It always felt like learning the "essence" of calculus, at least for me (who sucks at/dreads rote memorization).

JackFr 11y ago
That was my experience exactly. In the first class the professor told us that everything we learned in calculus was true, but not quite for the reason they taught us. We then re-proved everything withpoint set topology. But it felt like real math.
eli_gottlieb 11y ago
My real analysis textbook says on the back: Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.
- eru 11y ago
  Complex analysis is even neater. (But that's almost algebraic in its niceness.)
  - gh02t 11y ago
    I really enjoyed measure-theoretic probability theory myself.
    - eru 11y ago
      We did that as a precursor before probability theory. Really puts the latter on a sound footing.
hugh4 11y ago
What I wonder is whether it's necessary to teach it the wrong way before we teach it the right way. Neither answer to that question would surprise me. As an example, I remember that as a high school maths student I had real trouble believing that 0.9 repeater equals one. Nowadays, post university, it seems trivially true.
- gh02t 11y ago
  It depends I think. My experience has been that many people simply don't do well in "real" (i.e., formal and proof-based) math classes. Not to mention, a deep understanding of calculus isn't necessary for doing a lot of engineering, which provides most of the students in a lot of university calculus courses. On top of that, even if you start with an analysis-like approach of teaching calculus, you ultimately will have to spend some time plugging-and-chugging to get used to working with calculus on a practical level. On the other hand are people like me, who really don't do well with learn by memorization type classes, but does well in classes that are theory first, with a thick layer of application over that.

by gh02t 11y ago

JackFr 11y ago

That was my experience exactly. In the first class the professor told us that everything we learned in calculus was true, but not quite for the reason they taught us. We then re-proved everything withpoint set topology. But it felt like real math.

eli_gottlieb 11y ago

My real analysis textbook says on the back: Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician.

eru 11y ago
Complex analysis is even neater. (But that's almost algebraic in its niceness.)
- gh02t 11y ago
  I really enjoyed measure-theoretic probability theory myself.
  - eru 11y ago
    We did that as a precursor before probability theory. Really puts the latter on a sound footing.

hugh4 11y ago

What I wonder is whether it's necessary to teach it the wrong way before we teach it the right way. Neither answer to that question would surprise me. As an example, I remember that as a high school maths student I had real trouble believing that 0.9 repeater equals one. Nowadays, post university, it seems trivially true.

gh02t 11y ago
It depends I think. My experience has been that many people simply don't do well in "real" (i.e., formal and proof-based) math classes. Not to mention, a deep understanding of calculus isn't necessary for doing a lot of engineering, which provides most of the students in a lot of university calculus courses. On top of that, even if you start with an analysis-like approach of teaching calculus, you ultimately will have to spend some time plugging-and-chugging to get used to working with calculus on a practical level. On the other hand are people like me, who really don't do well with learn by memorization type classes, but does well in classes that are theory first, with a thick layer of application over that.