Students Thinking Like Mathematicians: Good or Bad?
Many mathematics teachers claim that they want their students to think like mathematicians. But, do they (or anyone...even mathematicians) really know how mathematicians think? Having been around mathematicians most of my life, I suggest that their thinking process is not easy to understand and certainly not predictable.
Do mathematicians sit around all day and pose problems and then solve them? Or, do they spend most of their waking hours doing mathematical proofs? Exactly what is it that mathematicians think and do? Two authors have written texts on their responses to these questions. Your task is to read the books....see if you understand their responses...and then decide if that is what you want students to do!
Michael Fitzgerald and Ioan James' The Mind of the Mathematician 2007) asks "What makes mathematicians tick?" To answer this question, Part I of the text explores topics such as behaviorial traits, personality traits, savants, gender issues, and creativity. Then, Part II is an interesting analysis of the thought-worlds of twenty significant mathematicians (e.g. Lagrange, gauss, Cauchy, Galois, Riemann, Cantor, ramanujan, and Godel), raising issues such as the impact of autism and mental illness. resources that enable students to explore mathematics online.
In contrast, David Ruelle's The Mathematician's Brain (2007) focuses on the question: Are the "world's most brilliant yet eccentric mathematical minds...brilliant because of their eccentricities or in spite of them?" His arguments are based on mathematicians he has known (e.g. Turing, Grothendieck, Thom, Riemann, and Klein), openly discussing "their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the sublime, inespressible beauty of their mopst breathtaking mathematical discoveries." Along the way, Ruelle raises philosophical issues, discusses the aesthetics and beauty of mathematical work, and illustrates strategy of mathematical invention.
The conclusions of the two text's overlap, and each offers its own insights. The end result, if you can read both books or even one of them, is that you perhaps will have a better understanding of how mathematicians think...and perhaps may not want your students to think like mathematicians!