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.
I think solving this problem goes a long way to really helping students understand what learning is. Personally I try to encourage students to do fun hard things... They're the tasks that failing doesn't seem to dissuade as much...
ReplyDeleteIn my Graph Theory class this semester, my homeworks have a proofs component but also an exploration component. These can range from the simple "Find a graph that satisfies..." to playing games that highlight graph theoretical concepts (Hashi or planarity.net) to asking students to tinker with something unclear: "What is the maximal number of cycles in a graph that has n vertices and n edges?"
ReplyDeleteI totally agree! Tinkering is a form of mathematical play, which I spoke about in my Dean's List address last year.
ReplyDeleteGreat post from a much needed personal perspective. We all get discouraged sometimes, but if we tinker, try something, then we can start chipping away at our challenges. Thumbs up!
ReplyDeleteThanks for the comments.
ReplyDeleteVince, getting people to realize that "fun" and "hard" are compatible is important. I like to use sports analogies to support this idea.
Chris, thanks for the link to planarity.net.
Joel, I love your message about "playfulness" in your address. Thanks for sharing.
Stan, thanks!
You asked for obstacles. I think a large one is the way most everyone I know grades: 10/10 for getting it right, partial credit for getting parts of it right. No obvious reward for trying something interesting that doesn't work. When the teacher is faced with a stack of homework and not much time (or when the computer is grading for you!), it's easier to just check the answer and only look at the work if the answer is wrong.
ReplyDeleteAmy, you are absolutely correct. How we grade and the constraints on our time are serious obstacles. In Stan's Destigmatizing Mistakes post, he writes that Ed Burger implements the following:
ReplyDelete1. 5% of course grades are based on the quality of student failure. In order to get an A in the class students must fail productively.
2. Use specific activities to teach intentional mistake making as a positive strategy. An example of this strategy is to give a problem to students to work on. Their first task is to intentionally do the problem incorrectly and then share their mistake with a neighbor.
I think these sorts of ideas can go a long way to changing the culture of the classroom.
I encourage mistakes by making them myself! In class I present proofs and examples "off the cuff" - I explore and make mistakes on the board. I let students see that "real" math is about exploration and learning. I don¹t believe class should be a transcription exercise. When students see me making mistakes they don’t fear making their own. I also grade their homework on completion rather than “correctness". In exams I allow them make corrections later to improve their grade. So there is no fear about mistakes or homework or grades or anything. With fear removed the students can focus on learning rather than grades.
ReplyDelete