I saw the word irrational a lot on Pi Day, and my friend Patrick Flynn recently invited me to join the Global Anti-Rationalization Foundation (GARF) on Facebook. Both of these things reminded me of the following story, which comes from a four-decade veteran of the classroom, John Benson of Evanston Township High School.
A student of John’s solved a problem and obtained a fractional answer that contained the square root of pi in the denominator. But then the student did a curious thing. He started to multiply the numerator and denominator by the square root of pi.
“What are you doing?” John asked.
“Rationalizing the denominator,” the student responded.
(If you don’t understand why that’s funny, you may be too young.)
Thanks to calculators, the process of rationalizing denominators is no longer necessary. One could make the argument that it wasn’t really necessary before the days of calculators, either, but there was at least some reason for doing it back then — converting a fractional result to a decimal was difficult if the denominator was irrational. If the denominator could be converted to a rational number, then the division would be less arduous.
When I worked for MathCounts, I was fortunate to spend two memorable years with John Benson. He served as a member of the MathCounts Question Writing Committee. John wrote phenomenal problems, had keen mathematical insight, and could put you on the path to a solution for every problem. But most importantly, John was inspirational. I once attended a presentation by John, and I loved the quote he displayed at the end of his talk: “You can’t get burnt out unless you were once on fire.” When he talked about his classroom, you realized what’s possible in math education. His philosophy has kept both he and his students inspired for nearly 40 years, and the following statement from John, which he claims to repeat often at parent-teacher conferences, will help you understand why:
I promise that I will try hard to make sure that something happens every day in class that cannot be replicated anywhere else. If a student is not in class, that student will never be able to recapture that teachable moment. They will have missed something.