Martin Gardner, 1914-2010

Martin Gardner passed away last week at the age of 95. He was a pioneer of “recreational mathematics”, wrote a column called “Mathematical Games” in Scientific American for many years, and published many books filled with mathematical puzzles.

Here’s a puzzle from one of Gardner’s books that was posted on the New York Times site on the occasion of his 95th birthday:

Two missiles speed directly toward each other, one at 9,000 miles per hour and the other at 21,000 miles per hour. They start 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they collide.

(Feel free to post your solution as a comment)

Now, missiles don’t speed directly toward each other at 21,000 mph in the real world; they follow parabolic paths (if air resistance is neglected) and move at half that speed at most.  Even if missiles did move in straight lines at realistic speeds, who cares about missile collisions aside from a few scientists and engineers who work on missile defense systems?  There is no doubt that this is a contrived problem, and many would question whether students can learn math by solving such problems.  This is a criticism often repeated by proponents of “reform” math programs like Jo Boaler or the person who recently sent an anonymous message to the SCASD Elementary Math Program Review Committee: “[A] good curriculum will focus on opportunities for application to real-world problems, because this is what makes learning “stick”.  A good curriculum, or implementation should minimize the use  of “contrived” problems (Two trains leave New York and Chicago at the same time…).”

I disagree with the implication that relevance to the student’s life equates to the potential for learning.  I think that such contrived problems – if they are thoughtfully devised – can foster good mathematical thinking by encouraging the solver to think in unfamiliar ways.  The problem above is a good example. Most people don’t associate math with creativity, but they are wrong about that, and abstract problems can be useful in promoting creative math.  As Douglas Hofstadter put it:

Martin’s columns and writings radiate a profound exuberance in the constant novelty of human thought. What comes through, even if it’s never explicitly expressed, is a kind of informal version of Gödel’s theorem for human thinking—a sense that creative minds will always one-up the pedestrian expectations generated by unimaginative, logic-bound thinking. There is an exultation in the breakout from expected patterns, the violation of seemingly ironclad laws, the making of wildly unexpected connections, the revelation that two seemingly identical properties are really quite different, and the counterexamples that make it all blindingly clear (at least for a moment—then you forget how it worked!)…. If nothing else, reading Martin Gardner should convince you that the human mind’s pathways of finding truths are as diverse and unpredictable as the pathways of evolution itself.