Recently in schools I’ve noticed just how fleeting representations are in mathematics classrooms. I’ve seen numerous slides full of notes and worked examples that are here one minute and gone the next. I’ve seen students working on individual whiteboards who quickly erase their work. I’ve seen students working on computers, going from one screen to the next with no trace of thinking left behind. When did representations of our mathematical thinking and practice become so evanescent? (I had to look up synonyms for fleeting *wink*)

The first time this phenomenon really perturbed me (apparently I’m in a GRE vocab kind of mood) was a few months ago when I was working with a student in a classroom. In this class, it was common for students to work on individual whiteboards. I get it, students who don’t enjoy writing on paper will sometimes jump into action when they get to write on something novel like a whiteboard. However, the problem was that we were working on problem after problem and I couldn’t draw connections between them because there was no record of our work.

This was incredibly problematic. How can we look for patterns and make sense of structure without being able to look at multiple problems? How can my students go back and revise their thinking or strategies if there is no record of where they’ve been? How can we look back at how far we’ve gone without our historical records? How can students study their work to prepare for assessments? How do families discuss mathematical ideas from school if there are no artifacts to anchor these discussions? How can teachers connect one worked example to another without the ability to compare and contrast the images simultaneously? How can teachers really understand student thinking and writing (particularly in large classes) if there are no artifacts to consider? These questions have been troubling me for several months.

In light of my reflection on this (more and more common) approach in school mathematics classrooms, I encourage you to leave behind a trace. Have students use a notebook to document thinking as they work on whiteboards or screens. Embed questions that they should explore as they go. For example, have them stop after 5 minutes on a computer game and discuss the strategies they are using and how their thinking is progressing. Then, have them look back on prior updates to see how their strategies are evolving. If students are on whiteboards, have them note their thinking for problems every so often so you can see how they learn from their practice and they can revisit their thinking.

I’m not saying that there should never be activities for which there is no record. I am saying that we should consider what we lose and what we gain from the creation of records of practice so that the process for our and our students’ documentation (or lack thereof) of mathematical representations and/or thinking is purposeful. I’d love to hear your thoughts below!

I think that much of this comes from the feeling that I had growing up and early in my teaching that the only thing that mattered in math was the answer. I just followed the steps too so my work wasn’t all that interesting. Now I see some amazing strategies and I want to frame them and never forget. So perhaps the issue is helping people see the value and beauty of the representations?

I think that is part of it, but also the SmartBoards have limited space if you are writing over say a PDF and so I see teachers erasing constantly instead of leaving up ideas. Some of the doctoral students from Korea have mentioned that they had to plan for the use of board over a lesson, which I think is really interesting. To think about what should remain and what can go across a lesson is a powerful way to plan.