5 ms·

I don't know if there's too much calculus, but there's definitely too much of the wrong type of calculus. Congratulations, you have just learned to integrate.

by hugh4 11y ago

I don't know if there's too much calculus, but there's definitely too much of the wrong type of calculus. Congratulations, you have just learned to integrate. Let us now spend the next three years running through several thousand different examples of analytic integration of one-dimensional continuous functions. More dimensions? Path integrals? Numerical integration methods that you'll almost certainly have to apply in the real world because most functions don't have analytic integrals? Differential equations? Never heard of them. Now let's all integrate (x^2 - sin(x))/coth(x).

Gibbon1 11y ago
I think there is too much focus on the mechanics and not enough on the big picture behind what you're trying to do. Mostly it's a lot of. Step one recognize the problem. Step two apply the solutionator+ and turn zee crank. Step three receive kibble. +Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.
- 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.
- vitd 11y ago
  I concur. What's worse was when they finally did get to an example of how you would use something, it was always so contrived as to be pointless. For example, you've learned about dealing with rates of change. Now lets talk about the rate at which we pour water into a bucket. For the love of god! Stop making problems about pouring things into other things! This has never come up in my life! And as an example, it doesn't help me relate to anything!
  - QuercusMax 11y ago
    I remember the semester I was taking Calc II, I was also playing a lot of the game Terminus. (This was in 2000.) The game prided itself on having realistic space flight physics, so you had to spend the same amount of time accelerating and decelerating when traveling between locations. I remember realizing I could actually use the calculus I was learning to figure out how long / how far it would take to decelerate, and so plan my thruster burns accordingly. Haven't used it since, though... It's a pity.
- agumonkey 11y ago
  As a kid I was very gifted at swallowing mechanics, and misled myself into thinking I was good at math. Like an ability to parse musical score very fast, but unable to play any music. Getting the concept is far more important, but far harder to do in class with quantifiable results (unless we measure the average grin on students face, maybe).
jimhefferon 11y ago
I teach these courses. I can tell you that everything in them is asked for by a client department. Not saying that change wouldn't be good, but these things are there because we are asked to put them there. In fact, we are asked to put many more things there that don't fit in the course.

by hugh4 11y ago

Gibbon1 11y ago

I think there is too much focus on the mechanics and not enough on the big picture behind what you're trying to do. Mostly it's a lot of. Step one recognize the problem. Step two apply the solutionator+ and turn zee crank. Step three receive kibble. +Solutionator: Bunch of mechanical steps that the student understands in the exact same way a trained monkey understands an organ grinder.

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.
vitd 11y ago
I concur. What's worse was when they finally did get to an example of how you would use something, it was always so contrived as to be pointless. For example, you've learned about dealing with rates of change. Now lets talk about the rate at which we pour water into a bucket. For the love of god! Stop making problems about pouring things into other things! This has never come up in my life! And as an example, it doesn't help me relate to anything!
- QuercusMax 11y ago
  I remember the semester I was taking Calc II, I was also playing a lot of the game Terminus. (This was in 2000.) The game prided itself on having realistic space flight physics, so you had to spend the same amount of time accelerating and decelerating when traveling between locations. I remember realizing I could actually use the calculus I was learning to figure out how long / how far it would take to decelerate, and so plan my thruster burns accordingly. Haven't used it since, though... It's a pity.
agumonkey 11y ago
As a kid I was very gifted at swallowing mechanics, and misled myself into thinking I was good at math. Like an ability to parse musical score very fast, but unable to play any music. Getting the concept is far more important, but far harder to do in class with quantifiable results (unless we measure the average grin on students face, maybe).

jimhefferon 11y ago

I teach these courses. I can tell you that everything in them is asked for by a client department. Not saying that change wouldn't be good, but these things are there because we are asked to put them there. In fact, we are asked to put many more things there that don't fit in the course.