A few days ago I was in my office working on a research problem related to the combinatorics of Coxeter groups. I’ve been thinking about this problem off and on for a few years and haven’t made any real progress in quite some time. The last time I worked on the problem, I was feeling pretty discouraged. On this particular day, however, I was just enjoying the process and feeling blessed that part of my career includes hunting for and occasionally discovering new mathematics. Someone actually pays me to put my head in the clouds and do mathematics.
It had been a while since I worked on the problem, so I started by reviewing all the things I had tried previously. I thought, “now what?” I couldn’t think of anything new to try that I thought had any chance of actually working. At this point, I was reminded of a recent post by Moebius Noodles, titled "Make Mistakes on Purpose," that contains a wonderful quote by the author Neil Gaiman.
Make interesting mistakes, make amazing mistakes, make glorious and fantastic mistakes.
This quote comes at the very end of Gaiman’s excellent keynote address from the 2012 commencement at the University of the Arts in Philadelphia. (I’ve included the whole address below.)
So, I set myself the task I trying to come up with clever mistakes. I intentionally followed what I expected to be dead ends. An hour later, I had several new insights. I still haven’t cracked the problem, but for the first time in a while I felt like I had made some headway.
This experience reminded me of something I’ve been pondering for a while in regards to teaching. How do we encourage students to tinker with mathematics? As a culture, it seems we are afraid of making mistakes. This seems especially bad when it comes to how most students approach mathematics. But making and then reflecting on mistakes is a huge part of learning. Just think about learning to walk or riding a bike. Babies are brave enough to take a first step even though they have no idea what will happen. My kids fell down a lot while learning to walk. But they kept trying.
I want my students to approach mathematics in the same way. Try stuff, see what happens, and if necessary, try again. But many of them resist tinkering. Too many students have been programmed to think that all problems are solvable, that there is exactly one way to approach each problem, and that if they can’t solve a problem in five minutes or less, they must be doing something wrong. But these are myths, and we must find ways to remove the misconceptions. The first step is to encourage risk taking.
A few months ago, Stan Yoshinobu addressed this topic over on The IBL Blog in a post titled "Destigmatizing Mistakes." I encourage you to read his whole post, but here is a highlight:
Productive mistakes and experimentation are necessary ingredients of curiosity and creativity. A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment. Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.
Here are a few related questions I have:
- How do we encourage students to tinker with mathematics?
- How do we destigmatize mistakes in the mathematics classroom?
- How do we encourage and/or reward risk taking?
- What are the obstacles to addressing the items above and how do we remove these obstacles?
I have some ideas about how to tackle these issues, but I’m curious what ideas you might have. I’m hoping for a fruitful dialogue in the comments.
Here is Gaiman’s keynote address in its entirety. Trust me, it’s worth 20 minutes of your time.
You can find the transcript for his speech here.