Academic priorities 247

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I'm not sure this was an ordinary public school, but that's sort of irrelevant.

Sorry I didn't finish answering your question there.

I'd expect smoothness to be understood in college, maybe high school for the truly precious. I think its hard to understand what C infinity really is without a good course in math.

Riemann integration can be taught in HS, so a student might learn some of the limits of Riemann integration by the end of high school, but I wouldn't expect them to be able to decide whether a given function is Riemann integrable or not, rather recognize some of the special cases (all cts. functions are integrable, certain weird ones are not).

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software handling

To be honest, I don't really have much experience with physics or any other subject that uses a lot of computation. I will ask around and see how this is done.

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If you think intuition and theory are worthless for engineering, then I think you should re-buttess your opinion. Theory and intuition can often speed up computations immeasurably, by at least giving you a clue where to look and examine or by giving you insight as to which approach to take on a computation.

For example, it is possible to approximate any continuous function by either a polynomial or Cesaro sum (in fact with Cesaro sums you can do any L2 function), because they are more or less equivalent. Now, theory gives you two options and provides you with measures of how accurate they are and how fast they converge, which then allows the intelligent and knowledgeable user to pick the method that is fastest for a given situation. It is often better to work smart than work fast. That being said, it's also important to understand how theory diverges from reality, some theories sound great on paper, but are clearly not viable from an engineering perspective.

As another example, try computing the sine integral over 0 to infinity. There are two (at least) ways of computing it. One is to rely on some beautiful complex analysis results, and makes the answer relatively straight forward. Another is to do this by semi-brute force, and that is really really ugly (and it still relies on some useful theory about even and odd functions) and time consuming.

I'm sure that such environments exist, but I'm also certain there are environments and situations where theory is essential to a project or idea, see above for more details. You really need a balance of the two.

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When would you propose to teach about logarithms and exponentiation?

David