“Math discussions aren’t just about show-and-tell: stand up, sit down, clap, clap, clap.”
Designing and implementing quality mathematical discussions takes effort. It’s not as easy as just having students get up and share their answers to a problem. But don’t let that turn you away from working to improve your practice! This is just chapter 1, after all, and there’s still so much to read and learn. In this chapter the authors lay out four principles that should guide our classroom discussions:
- Each discussion should have a goal. This means thinking in advance what it is mathematically you want students to get out of the discussion.
- Be explicit! Students likely don’t come to you with the skills needed to participate in classroom discussions. The teacher’s role is to help the students learn what they should be sharing and how they should be sharing it.
- Students should be talking and responding to one another. It’s easy in a classroom “discussion” for all comments from students to be directed at you, the teacher. Instead the discussion should be a conversation amongst all the students around a particular mathematical idea.
- Students must believe that they can make sense of math, and their ideas are valuable, even when they aren’t fully correct. There’s a lot of learning to be found in mistakes, and we need to value those as much as correct answers. Getting students to share means they have to be willing to take risks, and as teachers our job is to make our students feel safe to do so.
In addition to presenting these guiding principles, this chapter also differentiates two types of classroom discussion: open strategy sharing and targeted discussion. Open strategy sharing is what many teachers already do to some degree in their classrooms. This is when you let students share their answers and solutions to a problem. The goal is to get a variety of responses out in the open.
However, sometimes you have a particular mathematical goal you want to focus on, such as having the students justify why a particular strategy works. That’s when you would use a targeted discussion instead. These types of classroom discussion are much more nuanced and planning for each one is different. This is why open strategy sharing only has one chapter in the book while 5 chapters are focused on the different types of targeted discussion:
- Compare and Connect – comparing similarities and differences among strategies
- Why? Let’s Justify – justifying why a certain strategy works
- What’s Best and Why? – determining the best (most efficient) strategy in a particular situation
- Define and Clarify – defining and discussing how to use models, tools, notation, etc. appropriately
- Troubleshoot and Revise – determining which strategy produces a correct solution or figuring out what went wrong with a particular strategy
The chapter includes three vignettes to help illustrate the differences between open strategy sharing and targeted discussion. I love how the authors insert comments about the intentional decision-making the teacher did throughout each conversation. It shows early on in the book that the teacher isn’t being herded into some lock-step approach. Rather, at every moment you have the power to guide and steer the conversation based on the needs of your students.
One thing that really stood out to me that I didn’t catch the first time I read this chapter is that it’s okay to stop a conversation and come back to it later. I know as a teacher I often let conversations run so long that I wouldn’t get to other things I had planned. In my mind the conversation was so great, it was okay that we were cutting into our next subject by 10-15 minutes. I think this speaks to how I wasn’t planning my discussions in advance. I just let them happen and let them run their course for as long as they were interesting. As the vignettes in this chapter show, however, important mathematical topics can be discussed over several class periods instead of trying to cram it all in to one sitting.
The other thing that stood out to me was how these discussions have the power to give a voice to all our students, not just the high achievers or the outspoken ones. Creating a sense of community where all ideas are valued and respected allows all children the opportunity to be heard and to demonstrate what they understand about math. As the authors say in the book, there are many different ways to be smart in mathematics:
- making connections across ideas
- representing problems
- working with models
- figuring out faulty solutions
- finding patterns
- making conjectures
- persisting with challenging problems
- working through errors
- searching for efficient solutions
How much more exciting to look for and honor these skills in our students rather than seeking out just correct answers! Just think of what valuing these skills tells students about what it means to learn and do mathematics.