By Zachary Champagne, posted June 5, 2017 —

Recently, I’ve been thinking about and sharing some of my ideas on becoming a better listener. In a profession where we are envisioned as sharers of information, it was tough for me to swallow that I should be listening more and talking less. However, I’ve come to believe that it’s the single most important thing we can learn to do better as teachers. 

One of the ideas I’ve been exploring is how providing less structure can allow us to elicit more student thinking from our students’ written work. (Full disclosure: I was also recently inspired by my friend and colleague Brian Bushart and his NCTM Ignite talk “Do Less, Get More”). 

Consider this problem that Mike Flynn wrote for Teaching Children Mathematics (“Problem Solvers: Problem: The Cycling Shop,” vol. 23, no. 1, August 2016, pp. 10–13):

Imagine you work at a cycling shop building unicycles, bicycles, and tricycles for customers. One day, you receive a shipment of 8 wheels. Presuming that each cycle uses the same type and size of wheel, what are all the combinations of cycles you can make using all 8 wheels?

This problem is rich with various entry points for students. They can start in a variety of ways. Additionally, and perhaps more important, the problem is also rich in the variety of structures that different students may rely on to keep track of their work. I’ve seen students use pictures, charts, tables, tally marks, equations, and more when they begin this problem.

But it is only when we provide students with the problem as written above, and nothing else other than a pencil, that we can begin to see the power of the blank page. As teachers, we easily slip into trying to be too helpful. With good intentions, we take a problem like the Cycling Shop problem, and many times we give students something like this table:

In my opinion, providing this structure before students get started limits what we can find out about how our students are mathematizing this situation. We are stuck with learning if they can accurately fill out a table that we created. I think we are better suited to first let students be the problem solvers we know they can be, by letting them solve this problem using whatever structure (if any) makes sense to them! If students are struggling to get started or make sense of the problem, then we can help them think of ways that they might use structure to solve the problem. This “less is more” strategy provides us with much more information about students and does not require them to use a structure that we created.

I encourage you to respect the power of the blank page and give students more space to solve problems and less structure in the beginning.

Zachary Champagne is an assistant in research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) at Florida State University. He previously taught for thirteen years as an elementary school teacher with a specialization in math and science. During this time, he received many state and national awards for excellence in teaching, including the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST), Duval County Teacher of the Year, and Finalist for Macy’s Florida Teacher of the Year. Zak is the current president of the Florida Council of Teachers of Mathematics (FCTM) and is currently interested in learning how young students think about mathematics and how to help them understand that mathematics makes sense. He tweets at @zakchamp.