Do kids have an unheralded incentive to master math?
This week, I’m sharing responses to the question, “What insight or idea has thrilled or excited you?” This installment comes courtesy of John Allen Paulos, a Professor of Mathematics at Temple University who is here this week at The Aspen Ideas Festival. He adapted his answer from his forthcoming book, A Numerate Life: A Mathematician Explores the Vagaries of Life, His and Probably Yours.
I could mention my first introduction to Godel’s theorem about the essential incompleteness of mathematics; or my first encounter with the Banach-Tarski theorem in topology showing that a sphere the size of a pea can be decomposed into a finite number of pieces and put back together to get a sphere the size of a basketball; or Russell’s paradox about the set of all sets that do not belong to themselves; or any number of counterintuitive results in probability theory. All of these mathematical ideas excited me in high school and college, but I will concentrate instead on the thrill I felt in elementary school when I saw that the power of simple arithmetic was sufficient to vanquish a bully, my fifth-grade teacher. It still evokes the same emotions in me that it did decades ago.
I was about 10 years old and enthralled with baseball. I loved playing the game and aspired to be a major league shortstop. (My father played in college and professionally in the minor leagues.) I also became obsessed with baseball statistics and noted that a relief pitcher for the then Milwaukee Braves had an earned run average (ERA) of 135. (The arithmetic details are less important than the psychology of the story, but as I dimly recall, the pitcher had allowed the opposing team to score five runs and had got only one batter out. Getting one batter out is equivalent to pitching one-third of an inning, one-27th of a complete nine-inning game––and allowing five runs in one-27th of an inning translates into an ERA of 5/(1/27) or 135.)
Impressed by this extraordinarily bad ERA, I mentioned it diffidently to my teacher during a class discussion of sports. He looked pained and annoyed and sarcastically asked me to explain the fact to my class. Being quite shy, I did so with a quavering voice, a shaking hand, and a reddened face. (A strikeout in self-confidence.) When I finished, he almost bellowed that I was confused and wrong and that I should sit down.
An overweight coach and gym teacher with a bulbous nose, he asserted that ERA’s could never be higher than 27, the number of outs in a complete game.
For good measure he cackled derisively.
Later that season, The Milwaukee Journal published the averages of all the Braves players. Since this pitcher hadn't pitched again, his ERA was 135, as I had calculated. I remember thinking then of mathematics as a kind of omnipotent protector. I was small and quiet and he was large and loud. But I was right and I could show him. This thought and the sense of power it instilled in me was exciting. So, still smarting from my earlier humiliation, I brought in the newspaper and showed it to him. He gave me a threatening look and again told me to sit down. His idea of good education apparently was to make sure everyone remained seated. I did sit down but this time with a slight smile on my face.
We both knew I was right and he was wrong.
Oddly, this particular teacher did give me a potent reason to study mathematics that I think is underrated: show kids that with it and logic, a few facts, and a bit of psychology, you can prevail over blowhards no matter your age or size. Not only that, but you can sometimes expose nonsensical claims as well. For many students, this may be a much better selling point than being able to solve mixture problems or using trigonometry to estimate the height of a flagpole from across a river. (Incidentally, this mindset is not unrelated to some of my adult writings.)
As with mathematical ideas, so with scientific ones: There were many standard ones from plate tectonics to the double helix of DNA that gave me cerebral whiplash when I first heard them. Instead, however, let me focus briefly on a philosophically flavored idea, that of atomic materialism, which thrilled me as a 10-year-old. (My 11th year was a thrilling one.)
I'd read and had also been told by my grandfather, who seemed proud of the ancient Greek lineage of the idea, that everything was composed of atoms. It seemed obvious to me that an atom couldn't think, and so I "thought" that this proved that humans couldn't think either. I was so pleased with my ground-breaking idea about our essentially zombie natures that I explained it at length (all of two paragraphs) on a piece of paper, folded the paper carefully, put it inside a small metal box, taped it very securely, and buried it near the swing in our backyard where future generations of unthinking humans could appreciate my deep thoughts on this matter.
My attraction to the idea of atomic materialism wasn’t just intellectual, if that’s not too heavy a term to apply to a 10-year-old, but also visceral. Lying on the floor watching television or wrestling with my brother, I often had the inchoate idea that, in an important sense, there was no essential difference between me and not-me, that everything was composed of the same stuff and that the air above my forehead and the brain inside it were just patterned differently. The notion of emergent qualities, properties, and abilities didn't complicate my youthful certainty about these matters, and the dreary conclusion I came to that we couldn’t really think was one that I oddly found quite cheering.
Again, the excitement provided by my absurd interpretation of this fundamental idea preceded and in a way laid the groundwork for my appreciation of many other scientific ideas. Had I been told of emergent properties immediately, however, I doubt I would have been nearly as excited.
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