Recently I’ve been working on a chapter for a book on creativity in mathematics. I am, to say the least, an unlikely choice to be an author in such a book—a fact of which I‘ve reminded the editors more than once. I’m not a mathematician. Sadly, I am from a generation in which, when I completed the required high school math courses several years early, I was told, “No problem, you don’t need the advanced courses” whereas my math-loathing future spouse was brow-beaten into continuing math as “essential to his career.” Such were gender expectations of the time. But regardless of the cause, my math experiences were quite limited. Perhaps because of that, my work on the chapter—and for the math section of Creativity in the Classroom–has been particularly interesting.
Did you know that some of the earliest modern analyses of creativity were by mathematician Poincaré? His story of dozing before the fire and envisioning a snake is often used in describing illumination, but he also wrote about intuition in mathematics and argued for attention to invention and intuition in math (1908). Somewhat later, Halmos (1968), describing mathematics as a creative art, clarified what real mathematics is, by describing what it is not.
Mathematics – this may surprise you or shock you some – is never deductive in its creation. The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof. The conviction is not likely to come early – it usually comes after many attempts, many failures, many discouragements, and many false starts (p 380-81).
As one who had far too many experiences as both a student and a teacher in which mathematics was defined as rows of calculations to be completed at optimum speed, the idea of failures and false starts being part of the process seemed incongruous. In many school-worlds of mathematics, incorrect calculations are still simply errors, noted with colored pen and destined to be corrected. The problem is, as I now understand, math taught the way I experienced was not only not creative math, but it also wasn’t mathematics at all. Oh my.
Mathematical creativity, like creativity in its other forms, entails new ideas, new applications, new discoveries of beauty. Just as science requires questioning and data, so mathematics requires exploration, problems, proofs and generalizations. It is a far cry from the textbook-driven rows of problems many of us experienced. Real mathematics involves uncertainty and open-ended questions. There is no single answer key for genuine mathematics, particularly of the creative variety.
Of course, students in P-12 schools are unlikely to solve genuinely new mathematical dilemmas. But just as students can experience asking questions and gathering data for science projects in which the principles are generally already known, they can experience mathematics as a place for many answers, or many approaches to discovering the same answer. If mathematical problem solving is taught only as a step-by-step algorithm, it is not surprising that students believe all math has one path and one solution—and those are in the back of the teacher’s book!
What to do? At the heart of the dilemma is teaching students to ask and answer mathematical questions, to speak the language of mathematics. They can investigate multiple paths to the same answer. For example, “How many ways can you get to 4?” or “How can we figure out the approximate number of sequins on this pillow?” They can explain their reasoning. They can work on problems with many possible answers, such as “What is the most important statistic when calculating COVID risk?” There are many other options, and I’ll share others in coming posts, but for today, we focus on understanding math as a creative activity, complete with the failures, discouragements, and false start Halmos described, but also with a spirit of adventure. Yes it is. Yes we can.
Halmos, P. R. (1968). Mathematics as creative art. American Scientist, 4, 380–381. https://www.jstor.org/stable/27828328
Poincaré, H. (1908). Science et méthode. Paris: Flammarion.