Practice makes perfect
Now go practice
Obviously, calculus teachers have calculus mastered. How do they do it? Some people say that it's because they've been teaching calculus for years, but keep in mind that plenty of calculus teachers already seem to have the calculus mastered right before they start teaching. Seems to me like taking Calculus 1, 2, and 3 over and again (basically repeating the courses a bunch of times) is the only way to get that level of mastery. But I doubt anyone's going to do that. So how do the teachers know calculus so well? Do they just wake up their first day of teaching calculus and magically know everything?
Ask yourself, how did Newton and Leibniz do it? They didn't have textbooks, or calculus classes, and most of calculus hadn't even been discovered before then.
Btw, Tucson AZ is also my hometown.
My 2 cents.
Well, my Mathematical Physics professor told us that you need to see/learn the material about 5 - 7 times before you master it. (Rather distressing for someone taking the course in one semester!) There is also the concept that when you teach a class you really learn it for the first time. Bear in mind that your professors probably have taught it a number of times as a TA while they were in college.
-Dan
Also keep in mind that your math teacher probably took more advaced math classes when he was in college than just calculus, and so built on the fundamentals of calculus over time. Consider: now that you are taking calculus you are probably much better at basic algebra than you were when you took algebra in school. In fact you could probably teach algebra and the students would marvel at how well you know the material.
let's take another example (an easy one) to see how this practice works:
when you are young (perhaps say, 7) you might learn that: 2 + 3 = 5. when asked why, you might respond "teacher said so!"
perhaps later, you might reply (at age 10, perhaps): well 2 + 3 = 1 + 1 + 1 + 1 + 1 = 5 ones = 5 (you are starting to think things out for yourself).
much later, you might learn Peano's axioms (perhaps you are 22, now), and know:
2 + 3 = s(s(0)) + s(s(s(0))) = s(s(s(0)) + s(s(0))) = s(s[s(s(0)) + s(0)]) = s(s(s(s(s(0))))) = s(s(s(s(1)))) = s(s(s(2))) = s(s(3)) = s(4) = 5
that is, rigorously and logically, you realize that 5 is the 3rd counting number after 2 (this gives a way to compute 2+3 in case you forget how to add).
now, when teaching a 7-year old, you can "do it on your fingers" to make sure you instruct them correctly (ancient accountants used something like an abacus to check their "sums", more versatile than the human hand).
my point being: when you understand the "theory behind the theorems" you see WHY they HAVE to be true, which is a deeper level of understanding then knowing how to "solve a (particular) problem". it's more efficient, too, you can keep "more knowledge in your memory with less information". if you know algebra inside and out, you can pretty much forget most of your arithmetic.
i no longer need to remember if 121 is or isn't divisible by 3, i can "figure it out" (and it takes about 3 seconds).
well the simplest explanation is demonstration: one and one is two (often people use something like fingers, or oranges to do this).
the trouble with defining things, is, of course: defining our definitions. at some point we have to give up, and just hope people "get it" (that is: there are concepts we understand, but cannot define, in a linguistic sense). given this difficulty, i find it quite amazing people learned language at all! we seem to have an innate faculty for "equivalencizing", identifying distinct objects as "the same" for our intents and purposes.
or, as the devil would say:
1+1 cannot possibly be 2, there is no 2 in 1+1, only 1 and +.
My experience was almost magical.
I did okay in college, got my A's and B's in my major (Math).
Then I started teaching.
For example, I was to teach the Law of Sines the next day.
I reviewed the derivation in the current textbook, taking careful notes.
About halfway through, I said, "OMG, this is so simple!"
The steps were so logical, I didn't need notes.
This happened with 99% of the topics I taught.
Everything fell into place . . . piece of cake!
Question: Why wasn't all this equally clear when I first learned it?
I took notes, memorized formulas, did practice problems,
. . and still worried when I took an exam.
And now I wonder how I could get only a "B" on such simple stuff.
I assume it has a lot to do with Maturity.
By the time I began teaching, I had had graduate-level courses:
. . Projective Geometry, Topology, etc.
Perhaps my newly acquired "height" gave a better perspective
. . on the "lower" topics?
That's my guess.
The same happened with Calculus.
Once I had to teach a topic, it was forever simple.
(I can still derive the Product Formula
from the definition of the derivative.)
I assume that many of you have had the same experience
while tutoring. .Once you explain a concept to someone
the whole idea seems to crystalize in the brain, and you
have an ah-ha! moment. .And your reaction is "I see!" ...
or better yet, "Of course!" .And it's yours forever.