Comment by btilly | Hacker News Reader

Comment by btilly | Hacker News Reader

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This is absolutely wonderful advice for a high school student who wants to get a good foundation in STEM.

I'm not sure that it is applicable to an adult who needs a rough and ready understanding of Calculus.

I personally taught my brother enough Calculus to take a course that had it as a pre-requisite in under an hour. What did I focus on?

  1. The idea of approximations.
  2. The tangent line.
  3. How the tangent line connected to approximations.
  4. The derivative.
  5. The easiest formulas for differentiation and why they are true.  (All handwavy, heuristic big-O arguments.)
  6. That all possible max/min points can be found at the boundaries, or by finding where the derivative is 0 or non-existent.
  7. The Fundamental Theorem of Calculus aka why areas are the reverse of derivatives.
  8. The advice that if he had to actually calculate a derivative or integral, he should use a program like MAPLE.

Did he master the subject? Heck no!

Did he have to review his notes a bunch of times so it stuck? Of course!

But he went on to ace the course. And my guess is that he understood what makes Calculus tick better than most who took the course. (Sanity check. If you do not understand why the tangent line and derivative are connected, then you do not understand Calculus.)

Counting Differential Equations, I've had 5 semesters of calculus. (Calc I,II,III,IV, Diff Equ.) Until reading your comment, item 7, I had never heard this simple explanation of the meaning of the Fundamental Theorem of calculus ("why areas are the reverse of derivatives"). I was taught the fundamental theorem algebraically, and how to apply it, but none of my professors or textbooks ever explained what it meant.

Thank you for your honesty.

I wish it surprised me. But I suspect that your experience is more normal than not.

You never had someone in your introductory course show you some problem where you have e.g. a big water tank filling up, and you relate the flow rate with the volume of water in the tank, describing the relation as either sum(flow) = volume or diff(volume) = flow?

That’s quite depressing, since this is really the whole point of calculus.

* * *

Calculus really should be introduced with as much emphasis on physical modeling as possible. Differential equations are at the heart of the past several centuries of science, and understanding the basic ideas involved is crucial for everyone doing any kind of technical work, if not every citizen.

I like this version, which uses discrete computer simulation to cut down on some of the obscurantism of a formal heavily algebraic treatment, and lets students jump right into ideas which are significantly delayed in typical university math curricula. http://www.math.smith.edu/~callahan/intromine.html

That's really sad.

You might check out the sequence of activities in the Integral chapter of this book to get another perspective on the FTC.

I totally agree. It's the entire point of calculus, and it's what sets the entire field of Differential Equations in motion!

I guarantee you it was in your textbook and probably in lecture and you just glossed over and forgot. What does "applying it" even mean if not "finding the intrgral of a dervative or the derivative of an integral"? That's the entire point of the theorem, which is given the most powerful name mathematics pedagogy gives to important ideas, which should motivate you to glance at it more than once.

Or do you mean that you never connected "derivative and integral are inverses" with "integrals are areas" and "inverses are opposites"?

I think a critical ingredient for him was that he was motivated to get the exam passed. Otherwise new terms (unknown, not heard before, so scary and opaque) would overwhelm and attention would be lost.

How you did that introduction is very important. Every time you tell something new, it better be really small - or explained quickly and well, so the concept would stick before brain would get tired.

Thank you for listing your points.

I think that it really helped that back in grad school I read https://www.docdroid.net/z8ki/knuth.pdf and thought hard about Knuth's ideas. I even went so far as to make a course outline for a first course in Calculus based on his ideas.

The result is that I was able to break my exposition into one piece at a time, starting with ideas that were already accessible. More specifically I started with the idea that f(x) = approx(x) + error(x) where we want approx simple and error small. This motivates a language for describing what "small error" means, which motivates little-o for polynomials.

That's enough to do tangent lines and answer questions like, "Given the equation for position vs time, how fast is the rock going when it hits the bottom?"

Only after he could calculate tangent lines did I introduce the derivative.

Contrast to the usual approach where limits are an abstract concept with no obvious application, and then the derivative is introduced. It actually combines several ideas jumbled together. That's a big mental knot that almost nobody gets. And explaining it more carefully doesn't help because people keep getting to holding too many unintegrated ideas in their head at once.

And seriously, heuristic arguments that you can reproduce whenever you forget them are good. Take the product rule.

  f(x0+h) = f(x0) + f'(x0) h + o(h)
  g(x0+h) = g(x0) + g'(x0) h + o(h)

  (f*g)(x0+h)
    = (f(x0) + f'(x0) h + o(h)) * (g(x0) + g'(x0) h + o(h))
    = f(x0) * g(x0) + f(x0) * g'(x0) h + f(x0) * o(h) +
      f'(x0) h * g(x0) + f'(x0) h * g'(x0) h + f'(x0) h * o(h) +
      o(h) * g(x0) + o(h) * g'(x0) h + o(h) * o(h)
    = f(x0) * g(x0) + f(x0) * g'(x0) h + o(h) +
      f'(x0) * g(x0) h +                o(h) +           o(h) +
              o(h) +            o(h) +        o(h)
    = f(x0) * g(x0) + (f(x0)*g'(x0) + f'(x0)*g(x0)) h + o(h)

And we recognize the form of the tangent line and so the derivative of f*g is f' g + g f'.

I had him do that calculation with only minimal prompting. And I think that this is a calculation that I'd like every Calculus student to be able to do on demand. If you forget the rule and you know that argument, you can figure it out again.

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