As Isaac Newton famously wrote in 1676, “If I have seen further, it is by standing on the shoulders of giants.” Many great, great bits of math are just making progress on problems, discovering new things about problems, even without coming to the actual solution. Perhaps most famous theorem in math, Fermat’s Last Theorem, was suggested in a margin of a book in 1637, and had bits and pieces proven by many great mathematicians (including the French mathematician Sophie Germain, my dog’s namesake). In the 1950’s, two Japanese mathematicians (Yutaka Taniyama and Gomo Shimura) introduced the Taniyama-Shimura conjecture (later the Taniyama-Shimura-Weil conjecture, and then the Modularity Theorem when it was finally proven a half century later). Sadly, Taniyama committed suicide, and in his suicide note said that he had “lost confidence in his future”.
It was discovered in the 1980’s that a proof of this conjecture would lead to the proof of Fermat’s Last Theorem, even though on the surface and to the layperson, they seem to be parts of completely unrelated fields of mathematics. However, when Andrew Wiles introduced the proof of Fermat’s Last Theorem, it was by working on, advancing, connecting, and completing, the mountains and mountains of work produced by giants (great and small) over a period of almost 350 years. I am sure that nobody studying this proof will see it as a straightforward proof of start at A and end at B, but rather as a meandering and complex journey from the margins of an amateur French mathematician’s text to a series of 3 lectures given by a British Princeton professor in June of 1993 at the Isaac Newton Institute at University of Cambridge.
So why this little history lesson? It’s a reminder that the process really is just as important as the result. Isn’t that part of why we ask students to show their work? What else can we do to encourage the process over the solution? In my Pre-Calculus class, we frequently encounter trig equations that don’t follow a cookie-cutter formula. One activity that I’ve been working on with students is to limit the amount of time they have on a problem to something very short. I explain that they will probably (hopefully) not have enough time to get the right answer, but that we would discuss different approaches taken, why they may or may not work, and what we can learn from them. One equation may lead a class of students in many different directions. Now, knowing that nobody will have the right answer at the end of 1 or 2 minutes, all students feel more comfortable writing something/anything down. It took time for most students to buy into this process, and some still want me to just give them an algorithm, a single method that will always work, convinced that I’m hiding away a magic formula or something. Still, the risk taking, the increased and productive conversations, have been well worthwhile.
A second activity that I’m going to try out next week is going to be even more collaborative. Each student will be given a different problem, and will be given a short period of time to write down one thought, question, or step in a solution. They then pass the paper on to a peer, who will also have a short time to write one thought, ask (or answer) a question, write a second step in a solution, or write a first step in a different solution. They then pass it on to a third student, or back to the first student. We will continue this process until significant progress has been made on several problems, before discussing. This is a silent activity, where students collaborate solely by writing, and giving each student in a group a different colored pen/pencil/marker can help to see who builds on whom, and how that happens.
In both of these activities, students get a chance to see several different approaches to a problem, and can discuss the merits of each one – some positives, some negatives, some suggestions, some ideas where an approach that doesn’t work in one case may work somewhere completely different. There is still a lot more work to do to adjust these activities, by starting them at the beginning of the year next year, and collecting some data to see just how well it seems to be working. Initially, though, many students who used to leave problems blank are starting to put some writing and math on paper, and taking more risks to develop their ideas, even if they don’t know for sure where their work will take them.
A Mathematician's Musings 