I found this neat problem in Peter Winkler's excellent book, Mathematical Puzzles: A Connoisseur's Collection. I've dressed it up a little.
You have planned an expedition to travel in a 8000 mile loop around Antarctica. Your advance team has set up 20 fuel caches along the route, and has distributed 8000 miles worth of fuel among the caches. You know the amount of fuel at each cache, and the amount of fuel required to travel between any two consecutive caches. Prove that, regardless of the spacing of the caches or the amounts of fuel in each cache, you can complete the trip, assuming that you have an infinitely large fuel tank. Determine how to pick a cache you can start from.
(This might be an elementary problem, depending on how you look at it.)
Math and Sex
The late physicist Richard Feynman once said "Physics is like sex. Sure, it's useful but that's not why we do it." I think anyone who has been seduced by mathematics can appreciate that this applies to mathematics as well.
I was thinking that the analogy can be pushed a little further into mathematics education. I have been somewhat disheartened by the hostility that many in the mathematical research community have shown to the reform movement in K-12 mathematics education, particularly as regards discovery learning. As an example of this hostility, see the open letter to Secretary of Education Richard W. Riley that was signed by approximately 200 research mathematicians and scientists in 1999.
And yet, there is an example of the successful discovery learning technique in higher mathematics that whose success is, so far as I know, uncontroversial in higher mathematics circles. I refer to what is called the Moore Method, used by topologist R. L. Moore at the University of Texas. Mathematics graduate students were forbidden from reading about the subject and instead were led to rediscover the theory by themselves, given a set of axioms and very unobtrusive guidance. See here for an account by one of Moore's students. Based on the number and productivity of his Ph.D. students, Moore was one of the most successful mathematics thesis advisors of the first half of the 20th Century.
Granted, pedagogy that works for mathematics graduate students might not work for K-12 students. But, given the success of the Moore method, I wonder if the hostility to even considering the benefit of asking students to develop their own algorithms for arithmetic (for example) is akin to a parent's telling a teenager that sex is great, but only for when they are older. While such prudence may be sensible (if futile) with regards to advice regarding sexual experimentation, I doubt that it is helpful in developing a student's mathematical passion, or even ability.
I was thinking that the analogy can be pushed a little further into mathematics education. I have been somewhat disheartened by the hostility that many in the mathematical research community have shown to the reform movement in K-12 mathematics education, particularly as regards discovery learning. As an example of this hostility, see the open letter to Secretary of Education Richard W. Riley that was signed by approximately 200 research mathematicians and scientists in 1999.
And yet, there is an example of the successful discovery learning technique in higher mathematics that whose success is, so far as I know, uncontroversial in higher mathematics circles. I refer to what is called the Moore Method, used by topologist R. L. Moore at the University of Texas. Mathematics graduate students were forbidden from reading about the subject and instead were led to rediscover the theory by themselves, given a set of axioms and very unobtrusive guidance. See here for an account by one of Moore's students. Based on the number and productivity of his Ph.D. students, Moore was one of the most successful mathematics thesis advisors of the first half of the 20th Century.
Granted, pedagogy that works for mathematics graduate students might not work for K-12 students. But, given the success of the Moore method, I wonder if the hostility to even considering the benefit of asking students to develop their own algorithms for arithmetic (for example) is akin to a parent's telling a teenager that sex is great, but only for when they are older. While such prudence may be sensible (if futile) with regards to advice regarding sexual experimentation, I doubt that it is helpful in developing a student's mathematical passion, or even ability.
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