The other morning, while at my local diner for pre-golf breakfast, I sat at a rocky table. I was talking with a man I knew at the next table, and he grabbed an unused paper placemat and started folding it. After the customary 5 or 6 folds (I don't think he got to seven folds), he handed it to me and I stuck it under the table leg and the rockiness was solved.
But, aha!, that was not the end! I, being the dastardly nasty math teacher type, asked him a simple question: If you folded just like you have been folding, but do it 100 times, how thick would it be?
The immediate response was "100 times the thickness of the paper". At that point I said "not even close, you've got to go higher". He asked how high, so rather than telling him I took out my cell phone calculator, punched in a guesstimate for the placemat thickness (I took it for about 300 to the inch) and multiplied it by 2 to the hundredth power. He agreed that exponents could be used for repeated doubling, so we calculated it out. The number is not relevant, but its magnitude (about 11 billion light-years).
He made the usual "no way!" response. Then his tablemates got into the conversation. One of them said "you must be a math teacher", and I said I had been. That's okay.
What befuddled me, however, is that they took the answer and the method for obtaining it as indicative of an esoteric nature of math, mentioning how bad they were at math. All I was doing was multiplying by two!
Then it hit me. My question was out of their comfort zone, yet they wanted to stay in their comfort zone. It was easier for them to reply based on their gut feelings, rather than thinking the problem through.
Many people get to the point where a problem like this is not really a problem. The others have to think about it, and serious thinking while eating breakfast in a diner surrounded by friends and relatives is not an easy task.
Does Whitehead's famous quote apply to individuals as well? "Civilization advances by extending the number of important operations which we can perform without thinking of them."
I wonder; is a school classroom the proper environment for serious thinking? It is a good environment for factual learning and algorithmic learning (how to solve quadratics, write a complete sentence, learn arithmetic, learn how to spell, for example.) But deep, serious, thinking? Especially on a timed test? My best mathematical thinking is done alone, with smooth jazz or light classical playing, with the opportunity to take my dog out when the thinking is in the mind and not yet ready for putting pencil-to-paper. In college I used to go into an empty classroom in the evening and make major use of the chalkboards when working on assignments. Class time was for acquiring the specifics which, presumably, would give a helpful knowledge base for the assignments. Deep thinking in a class full while at a small lecture desk? No way.
Thinking is what we have to do when we don't know what to do. If we know what to do, there's no problem. Yes, we would like future generations to be good problem solvers, but, more than that, we need to prepare them with a factual and knowledge base so that many potential problems will not be problems.
Stanford has a course called Introduction to Mathematical Thinking. The course description states
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
Why do they use the box analogy this way? Why don't they just say that they will be addressing mathematics as it sits in a larger box? They put down high school math while ignoring that high school students are not professional mathematicians, and they fail to mention that "professional mathematicians" is a gross generalization. Many mathematicians I know work in very small boxes anyway. One quick web click and I find a representative of this group listed with
Specialties: Abstract algebra, Galois theory, geometryInterests: Commutative rings and algebra
Most professional mathematicians are equally specialized. (Just as you would not go to a dermatologist when you need a pacemaker) Stanford also overlooks the fact that all those "professional mathematicians" were once high school students.
I would like to believe that Stanford's description is referring to mathematicians hired by business and industry to apply their special skills and talents in very specific arenas. If that is what they meant, it is not what they said.
Back to breakfast. My diner neighbors, at least one of them if not more, added that they "never saw any use in the math they were seeing in school". They missed the fact that they will never use what they do not know. Yes, I do not know the Greek language, and naturally I have never been able to use that knowledge. Would my life be different if I had learned Greek? Maybe yes, maybe no, but doors of opportunity are and were definitely closed to me. I cannot say what was on the other side of those doors since I could never go through them.