Learning Teaching (Math Teaching) Magazine (NCTM) for September 2022 Reviews

Author: Blake E. Peterson¹, Shari L. Stockero², Keith R. Leatham¹, and Laura R. Van ZOEST

THIS ARTICLE GIVES SOME ADVICE AND STRATEGIES FOR MANAGING MATHEMATICAL DISCOURSE IN THE CLASSROOM. This article gives some advice and strategies for manageing mathematical discourse in ther classroom. The fourth effective math teachiung practice from Principles to Actions (NCTM 2014) i: Facilitate meaningful mathematical discourse.  We all have mathematical discourse in our classrooms, but we all also need to have strategies to keep it focused and on track other than shouting “Stop Talking “  or “Put her down right now!) As we take on the challenge of facilitating a mathematical discussion, there are many strategies and methods to keep the conversation productive and on track.  This can be challenging for young children.  There are also many many ways the conversations could deviate, become unfocused and go off on a tangent.  This articles gives advice in stuations where a student makes a valuable contribution that is only loosely related to the topic at hand.  I recently was having a mathematical discussion with a 6^th grade class.  The conversation began on the topic of percents and ended up as a discussion of black holes, the nature of reality and how we see colors.  It was a lot of fun for me, but the original topic was quickly lost.  BTW they walked out thinking about the question “What if everyone had the same favorite color, but our eyes and brains perceive it differently?”  No one slept that night.

One example is when a student has offered a solution to a problem and the class is discussing the solution and then another student offers a different method.  Do you begin discussing the new solution or stay with the current idea?  If you divert, you could fragment the class and the discussion and also dishonor the original student’s contribution.

They introduce a couple different terms we can use to understand 

Focal Instance – a  student contribution that the teacher has chosen to pursue for some particular purpose.  It is that moment that a student offeres a contribution to the claa and you decide to focus the class on this contribution for a short discussion.  Choosing these focal instances is the foundation of mathematical classroom discourse.

Mathematical Opportunity in Student Thinking (MOST) – When a student offers their thinking and you recognize  a teachable moment in their contribution you have a MOST.

Facilitating mathematicsl discourse is often an exercise recognizing Mosts and turning them into focal instances.

CONERSATIONAL BUBBLE – When the teacher temporarily pauses the broader mathematically activity to explicitely take up the focal instance.

This article focuses on teacher moves that keep the whole-class discussion centered on a contribution that the teacher has decided is a focal instance. 

How to reduce the frequency of tangential student contributions

1.     Ask targeted questions – The questions should be explicitely about the the focal instance and not general questions like : “What do you think?” or questions aboiut the broader activity like: “Did anyone else use this solution?”

2.    Help keep students in the conversational bubble.  Calling on one of the random hands that go up is the “Box of Chocolates arpproach.  It leaves open the risk of a student sharing their own solution and taking the conversation away from the focal instance.  Instead the teacher can ask the spontaneous volunteer a targeted question like :  “DO you have something to share about Julie’s solution (the Focal instance).”  This can also be discussed in classroom norms.

 Responding to tangential Student contributions:  In this instanmce you can find yourself torn between remaining within the conversational bubble and honoring the new student’s contribution.  a teacher might say, “You’ve just shared another way you could think about this task, but remem- ber that right now we’re making sense of Sonia’s claim”—a statement that positions Edward as a legitimate mathematical thinker but keeps the discussion centered on the focal instance. You can use a combination of putting aside and recentering  moves like this to keep the discussion centered on the focal instance. The key is that it should be both putting aside and reentering.  If not, another tangential contribution will frequently surface.  Some teachers might go to another “box of Chocolates move” by calling for spontaneous contributions or returning to the general actifity by asking “What did others think?” which values more participation but not their thinking.

3.     You could stay with a tangential contribution:  The teacher may not want to immediately put aside a tangential contribution but rather allow a few conversational turns before deciding whether to put aside and recenter on thw focal instance.  Maybe the teacher askes for clarification in order to search for a better connection to the focal instance. Say .g., “Thank you for sharing” or “I understand what you’re saying”), and then recenter the focal instance (e.g., “So, where are we in making sense of [the focal instance]?”) It is helpful to try to connect the new contribution to the focal instance.  “How does this help us make sense of the focal instance?”

4.    In some cases, the teacher may need to stay with the tangential student contribution in order to address pre-requisite mathematics such as:  you cannot compare x and x+x or any variable equations unless you agree that the x values must be the same.  SO setting x to different values is not a valid solution.  Ask yourself  “Must the students make sense of this idea before they can make sense of the focal instance?” 

Conclusion:  One way to facilitate math discourse is to pause the broader activity and create a conversational bubble around a focal instance contributed by student thinking. It may take time and some classroom norming to become more effective at creating, facilitating and maintaining conversationbal bubbles to their effective conclusion.

There is a very good “DO AND DON’T” TABLE ON PAGE 624 THAT SUMMARRIZES THE TEACHER MOVES.