misogynisticfeminist said:
the invention of calculus made physics all the more "difficult".
Actually it made it much simpler, I think.
I honestly believe that the reason that most people believe that calculus (and math in general) is difficult is due to the archaic pedagogic styles of educational formalism more than anything else.
There are three really huge problem with educational formalism (in all areas, not just mathematics):
1. It is not geared toward education, instead it's geared toward competition for a degree.
2. It requires the students to learn far too many trivial superfluous topics which wastes the student's valuable time.
3. It is far too slow to in updating and improving pedagogic styles.
I'm 55 years old and I just re-took calculus I, II and III, as a refresher. It was like déjà vu from the last time I took it almost 30 years ago! They haven't changed a thing (except they use graphing calculators now). I mean, I realize that the math hasn't changed, but I would have thought that the style of pedagogy would have changed. It looks to me like its been on hold for at least 30 years!
If I could hire instructors to teach me in whatever manner I chose I would design the courses to compliment each other more directly. In other words, I'd have the math courses directly compliment the physics courses that I was talking concurrently. I'd also hire math instructors that aren't afraid to get a little concrete in their examples. So many mathematicians seem to have a phobia of concrete examples. They are so bent on abstraction that they are afraid to get specific. Personally I have always learned much better from doing specific examples first and then moving on to abstractions from there. Give me something I can
apply and I'll take the abstraction from there. I
promise that it won't hold me back!
On the contrary, being continuously fed nothing but abstract examples without ever being given a concrete example can indeed hold me back.
I could probably add one more thing,..
4. I personally believe that educational institutions could do much better by melding some subjects together into better-organized curricula.
I vividly remember when I was in high school I could not understand abstract algebra. To me it was just a bunch of meaningless rules that required memory. I just couldn't remember all the rules, or why they were valid.
After high school I went to a vocational school for electronics. By using Ohm's law and all the other circuit equations I quickly learned how to manipulate equations because I could
see the relationships between the quantities. So I learned algebra from a phenomenological point of view. After that I was able to move on to more abstract forms of algebra because I understood the foundation of why it works.
Also, I have taught many courses on electronics over the years. I've had many students that would shiver at the word "algebra". But after showing them how quantities relate to each other using electronic circuits they were soon doing algebra with no problem. For some reason educational institutions seem to have never caught-on to the value of this phenomenological approach to teaching mathematics. I personally don't see this as limiting the abstraction at all. On the contrary I think it enhances it.
Going back to the idea of calculus. The biggest problem for most students is really the algebra. The calculus itself is relatively easy. It's the algebraic manipulations that most students have problems with. And, I believe, the reason they have such problems is because they never really learned algebra intuitively, instead they just learned it as a bunch of rules that they quickly forgot how to apply.
