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May 9, 2013 / Jeff d.

Protocols for Mathematical Discussions

I’ve tried to make mathematical discussions a much larger part of my class.  Here’s what I’ve done.

The Protocols

To begin, my school is very protocol-driven.  We use many of the protocols from the National School Reform Faculty, among others, along with protocols we create, tune, and share amongst ourselves.  I use all kinds of protocols in class, including to run discussions.  So these are what I’ve used this year.

Text-Based Seminar

For discussions based on texts, nothing beats the good old NSRF Text-Based Seminar.  We use it often in Advisory (and for staff meetings).  I used it toward the beginning of the year to discuss an excerpt from What’s Math Got to Do With It? (around page 40-41 if you’re interested…I don’t want to break any copyright laws).  The trick as facilitator – as with any protocol – is to stick to the protocol, which in this case means holding learners accountable to referring to the text when speaking.

Pros: Keeps participants focused on the topic at hand without becoming a random opinion-spouting session.

Cons: This is a great protocol, but the opportunity to use it doesn’t come up all that much in my class.

Progress Fishbowl

This is a protocol I created based on a standard fishbowl when most of the class was struggling with a problem.  I mean really struggling.  A couple of groups in each class were finding some good solution paths and I wanted a way for them to share it productively.  This one’s a bit lengthy – it could probably be pared in half.

Pros: Allows those with good ideas for solutions paths to help others without resorting to a “tutoring” situation.  Those outside the fishbowl can come to their understanding through participation in the discussion rather than being told what to do.

Cons: It can get messy, particularly if some are totally lost.  One time the protocol broke down entirely because the discussion was good enough on its own merits that a protocol was no longer needed.  In that case, the class renamed it “the broken fishbowl”…we even drew a logo for it.

Linking Fishbowl

This protocol was my attempt to be sure all voices were heard.  I’ve used it for conjectures on problems that were complex, yet had a low entry point.  Most of the thinking is done beforehand, so it kind of amounts to a glorified share-out…until a learner pushes the previous speaker’s reasoning.  Then it becomes glorious.

Pros: All voices are definitely heard, the discussion moves quickly, and it can go fantastically well when learners push each others’ thinking.

Cons: Since it is more of a share-out, learners can become disengaged once they’ve shared.

Yes, And… Protocol

My most recent creation, I used this for a similar reason as the Progress Fishbowl – to gather all the good ideas in one place and help those that needed it come to a better understanding.  However, I wanted to avoid the breakdowns that can happen when disagreements occur, so learners had to build on what the previous speaker had said by prefacing with “Yes, and…” – which made for some great conversation.  This went so well that it sparked me to write this post.  I hardly had to say a word – my favorite kind of discussion!

Pros: All voices were heard, and great ideas were shared and improved upon.

Cons: It was actually fairly time-consuming.  If you’ve got short class periods, it may not work for you (I have 90 minute periods, so I can pretty much do whatever I want).

What makes these kinds of discussions work?  As a class and as a school, we focus on building a culture of trust, respect, and responsibility.  We spend the first week-and-a-half of the school year exclusively on building culture – we don’t even touch content.  Trust me, this pays off.  Among other things, it creates the environment where these kinds of discussions are possible.

Also, it isn’t as if we do nothing but protocols.  We have discussions everyday that are more typical of a classroom.  But these are a great way to come to a wider understanding as a group for an important topic.

I’m leaving so much out; these protocols in a vacuum are worthless.  But I thought they might spark some ideas for anyone looking for new ways to discuss mathematics in their classrooms.

Side note: I recently purchased Smith & Stein’s Five Practices for Orchestrating Productive Mathematics Discussions, but I also bought three other books at the same time (including the long version of A Mathematician’s Lament) so I haven’t even cracked it.  That is to say that this post is a baseline of sorts; I want to get my thoughts and current strategies for discussions in writing before it all (probably) changes after reading Five Practices.


Leave a Comment
  1. Judy / May 10 2013 9:31 am

    Peg Smith’s book on Orchestrating Productive Mathematics Discussions is fabulous. It provides a structure/protocol for thinking about how to use students’ ideas and where they are in their understanding. Based on your reflections in your post-it seems that you will find it really helpful. I’ve used it quite extensively over the past few years in various settings-and it’s an amazing to tool to help provoke and support thinking. I’m interested in your impressions!

    • Jeff d. / May 11 2013 10:13 am

      Thanks Judy, I’m looking forward to reading it…as soon as I finish Lockhart’s Measurement!


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