I joke with my students that I forget everything over the summer: not just the stuff I teach but even how to think about the stuff! We laugh at both aspects of it—that we all forget stuff during summer break but also the irony of thinking about thinking itself.

I remember the emphasis on reflection in George Pólya’s slim book How to Solve It (1945) back when I was in graduate school (and dinosaurs still roamed the earth). More recently, thinking and reflection are wound through the Common Core’s Standards for Mathematical Practice. This isn’t always what we’re used to in our math classes, so I focus on specific activities in class aimed directly at helping kids make their thinking visible.

My favorite activity for making thinking visible is using didactic triangles to help students draw out connections between ideas. Choose three ideas and put one at each vertex of a triangle. Use the sides to ask kids to list ideas and relationships that connect only those two vertices. You can always put a few ideas common to all three in the middle.  

For the triangle at left, kids are quick to point out that the Pythagorean theorem goes with right triangles, but they don’t always see that the distance formula simply captures the lengths of the sides of right triangles.

It’s amazing how differently kids see connections and how quickly this exercise can make their thinking visible. Some observations are superficial (“the bottom two are formulas”); others are deep (“the distance formula says that any distance between two points is the hypotenuse of a right triangle”).  

Didactic triangles are discussed in research papers, but they tend to highlight teacher, student, and concept relationships. See, for example, Reider Mosvold’s blog. Cool research, but not for class! Didactic triangles reimagined as a graphic thinking organizer was an idea I took away from an NCTM conference in Baltimore. Leigh Haltiwanger and Amber M. Simpson presented on writing as support for mathematical thinking, and I took notes as fast as I could! (My own focus is on high school mathematics, but you can find a middle school version of their presentation in our sister journal Mathematics Teaching in the Middle School [“Beyond the Write Answer: Mathematical Connections,” MTMS April 2013, vol. 18, no. 8, pp. 492–98].)

Now my students expect these triangles as warm-ups for a class activity and look for them on exams, as a way of wrapping up. I know the technique is successful when I can overhear the evolution of a concept as students discuss the connections they think they see.

We need powerful ways to get kids thinking and to make this thinking visible, but not just students. In our department meeting last spring, we had a lively discussion about the appropriate connections to make with the didactic triple “natural, rational, integer.” Try it with your own colleagues. After all, why should students have all the fun?

Regardless, best wishes for the new school year, and please send along your own ways to get the thinking going.

Greg Stephens, stephensg@hohschools.org, is a high school mathematics teacher, department chair, and instructional leader for the Hastings on Hudson School District in New York. He just rotated off the Mathematics Teacher Editorial Panel but is keeping busy in a doctoral program at Fordham University in New York City. At the moment, his thesis topic is the impact of digital literacy on the high school mathematics classroom, but the hardest thing of all is picking just one topic to focus on!