My wife works at Penguin Group, where naturally they have an incredible “take pile” of books they publish. That’s where I found Number: The Language of Science, a fascinating and completely accessible history of the human traditions that give us today’s number systems. Apart from the amazing stuff about numbers, my favorite thing so far is the variety of ways in which mathematicians have been wrong.
Too often our culture treats “real mathematicians” like infallable geniuses, but in fact, deep-seated misconceptions were central to the development of mathematics. This got me thinking about my students.
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My first year at Saint Ann’s, I had a rather heated exchange with a seventh grader over whether or not fractions fit infinitely in between each other. The class and I came up with all kinds of arguments to convince her otherwise, but she was sure that a stick could only be chopped so fine before it was just “in pieces.” At the time, this seemed like a major gap in her her understanding. I mean how were we supposed to get to _____ if she wasn’t “getting” this? I’m sad to say, though I never told her, I thought this was really bad.
Last year, my fifth graders were talking about stars and polygons, using protractors to make their own, when a student had a brilliant realization. After making a twelve pointed star from a dodecagon, she explained that a circle was just a polygon with 360 sides! This isn’t really true, so we had a conversation about what a 720-gon might be like. She got that, but “make enough sides,” she said, “and that’s a circle.” She was definitely onto something and really enjoying her work, so I let it slide.
I knew enough of Archimedes’ method of exhaustion to know that the fifth grader was thinking like a “real mathematician,” but even then I think I looked back in awe at the misconceptions of that seventh grader.
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Then I read the chapter, called “This Flowing World,” about the centuries of debate on these exact issues. The seventh graders concerns were legitimate issues articulated by Zeno and the greeks, and their resolution was of fundamental importance in the development of the differential calculus! Of course I want my students to see the poofs and results that will bring them “up to snuff,” but when they pause to wrestle with a classical paradox, why should I object?
When it comes to mathematics, this kind of wrongness is essential. As in the Monty Hall problem, our intuition often leads us astray, and it’s the act of proof that forms the heart of mathematical reasoning. Wrong notions represent a successful model of thought that reflects a current state of understanding. If a model is weak it will demand attention sooner or later. If not, then it’s not weak.
The genius view of mathematicians has meant, among other things, that most students leave school thinking they’re not one. Amazingly, mathematicians are people that have good ideas and bad ones – make mistakes and have breakthroughs. This year, I want to share in the growth and mathematical discovery of my students, without needing them to get it all right.
I can still be better about this.