Two Olympiad Problems

[borisp]

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All calculus problems can be done without calculus.

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Yes, and all computer programs don't need computers to run them. I file this sort of thing under "true but irrelevant."

However, I do generally agree that a simple "stick in the sand" solution is preferred over one that mindlessly uses a technique. BUT, given that one understands the proof of why the technique works, the general method becomes more attractive. Lagrange multipliers (from another thread) is a good example, since the proof can be motivated very easily using geometry of graphs, tangent planes, etc. Of course, this is mostly speaking for advanced math students.

But this brings me to my question: DS, why are you so bent on explaining things in a way so that the average math student can understand them? I can understand the motivation to sell your publications, so if that is all, then so be it.

Beyond that, do you really think these simpler explanations really improve the average student's day to day decision making? If they weren't smart enough to solve a problem to begin with, what makes you think that they will be smart enough to recognize when they should be clever and actually employ what they have learned? I've found (in my own teaching) that the latter is really the most difficult hurdle. You can make people understand things, but you can't make them recognize when it is appropriate to apply what they have understood. That is, unless they are smart enough to understand more difficult and general techniques, and then we are back where we started.