# Our annual math burst gives students an extended experience with open-ended mathematical discovery.

During the mornings of the build week following block four, our students took part in an unprecedented activity—their first ever Proof School Math Burst. We divided the entire student body into seven groups, typically six students per group, and presented each with a different starting point. One group played a particular combinatorial game, another considered ways to optimally color the edges of a graph while satisfying a certain constraint, while a third wrote code to investigate an unexpectedly rich question involving iterating a process on piles of markers. My group began the week by counting how many ways there are to place up-arrows or right-arrows in some of the squares of a 2x2 grid so that no arrow points at another. (In case you’re wondering, the answer is 31.)

Each group was guided by a faculty advisor, but the teacher's role was much more about facilitating and encouraging progress rather than dictating the direction in which the exploration unfolded. At one point I gave all my students a particular question to answer as a group, suspecting that it would help to open the door to further headway. On another occasion I moderated a brainstorming session regarding what we might look at next, without offering any particular ideas myself. Towards the end of the week I joined one student who was interested in proving one case of a conjecture the group had formulated. In an effort to be supportive I followed this student’s lead, even though he set the computation up in a different manner than I would have, and to my surprise we proved the entire general theorem!

We had several purposes in mind by giving students eight hours of supervised but open-ended time to pursue a specific topic or problem. Certainly a primary goal was to allow students to participate in mathematical discovery: exploring new topics, asking questions, making conjectures, learning tools, and finding explanations. It’s an exciting process that I’ve been eager to share with our students ever since the very early stages of planning for Proof School.

But there are many other benefits to setting aside time for a math burst. For starters, students develop stamina—the ability to stick with a problem for an extended period of time. I’m also sure that these investigations will form the basis for future papers, science fair projects, and so forth. In the more immediate future, we will spend the week following Spring Break learning the art of presenting results to a general audience, whether via poster or oral presentation. Of all the skills we plan to impart to our students, the ability to effectively communicate complex ideas ranks up near the top.

At the end of that week, parents and classmates are invited to attend our first ever Math Mini-Symposium, to hear all about what our students discovered during this Math Burst. Regardless, it has been a remarkable week. Speaking for myself, I’m still playing around with some of the mathematics our group explored. I believe that it was an energizing and rewarding experience for all of our students.

-- Sam Vandervelde