I have been thinking a lot about mathematical surprise these days. Specifically, I have been thinking about all the different ways we can be surprised in our math classrooms.

We have worked a lot at creating mathematical thinking spaces for our students. The lovely monograph by Dr. Chris Suurtamm Making Space for Students to Think Mathematically nicely formulates how our classrooms can be mathematical thinking spaces.

The above image is some student work on a proportional reasoning problem. (I was going to say unit rates, but leading in that direction might already be taking some of the thinking away, don't you think?)

The video clip linked here shows 3 of us analyzing the chart paper you see in the picture: analyzing the thinking. Our students used many surprising strategies. Some were radically different, using completely different thinking tools, or perspectives on the problem. Some were subtly different. There were huge mathematical implications on the rates they chose to use, for just one example.

I had a chat with a brilliant teacher the other day, and he told me he identified 26 distinct solution paths for one problem (the classic "tug of war") problem. It may be this Marilyn Burns one, or another- it's a classic context for a problem. One this one you may rightly note: we as experienced thinkers might choose to model it with algebra. This is only one type of solution! Don't underestimate the flexible thinking of the novice, with less experience and context to hem in their thinking! (Dr. Brent Davis out of Calgary calls this the mistake of the experienced thinker)

Once we remove our preconceived notions of what the problem "should" be, we can focus on what sort of thinking our students are doing. They will surprise us! I haven't been involved in one collaborative inquiry where we haven't been surprised by at least some element of one solution! We must be open to surprise, in our classrooms.

One minor caveat: I am not saying here we are "surprised" because we don't know the math at hand. We must know the fundamental big idea in the math, and the connections to curriculum (both content and process). Many of our teachers and coaches are working from the "Five Practices for Orchestrating Productive Mathematical Discussions" book. The key practice for me is anticipation: we must know the math, do the math, and anticipate student responses to the best of our ability.

But, I maintain: the element of surprise will still remain. Sometimes it's a powerful conjecture from a usually quiet student. Other times it's a subtle variation on the math you expected. It could be a question that arises from a student, and spurs them to create more math. Sometimes it's a lightning flash of insight- an amazing and new solution path that you have never seen before.

Last, what surprises will emerge from the murk and (seeming) mess, when students are allowed open spaces for mathematical thinking?