Mathematical discussions are a vital part of my daily work as a research mathematician. It’s true that when you walk around the building in which I work, people are often sitting in deep thought behind closed doors. But, when needed—which can be often—these same people also participate in animated mathematical conversations with each other: to help untangle difficult concepts, to see if a line of reasoning stands up, to sharpen arguments before presenting the work to the wider mathematical community, and to forge new ideas together.

Participating in mathematical discussions also has many, many benefits for our students. Mathematical communication:

- Facilitates learning: not only are students’ abilities to articulate and justify their own thinking improved, but as students learn to listen to and make sense of other’s ideas, they make deeper connections between mathematical concepts and develop the metacognitive skills needed to be more effective learners.
- Is a problem-solving tool: it helps to get started, it crystallises what we don’t understand, and it opens a way to get feedback from others.
- Shapes mathematical identity: students begin to appreciate that there is more than one ‘right way’ to think about maths, which increases confidence, encourages participation, and expands perceptions of maths.
- Activates peer-to-peer learning: it fosters learning from and with all members of the community as students realise that they can learn from one another, decreasing reliance on the teacher as the fount of all knowledge.

Yet many of my students report learning in classrooms where ‘we were always told to sit quietly and work by ourselves’ with the teacher doing all the talking. And, truth be told, my classrooms have often been one-sided talkfests. It seems that even when the benefits of having students discuss maths are clear, the path to making that happen in a meaningful way can seem too difficult.

**Enter Chris Luzniak, and his use of debate structures in mathematics.**

I first ‘met’ Chris as a teacher profiled in Tracy Zager’s book ‘Becoming the Math Teacher You Wish You’d Had’. (My review of Tracy’s book is here.) I was immediately enticed by Chris’ routines for teaching argumentation—underpinned by his hefty experiences in Speech and Debate Teams as both a student and a coach—and his structures for asking debatable questions. When the opportunity in 2018 came up to spend three mornings with Chris and Mattie Baker at Twitter Math Camp, digging deeper into getting students to discuss and debate math(s), I jumped at the chance.

When I returned home, I immediately put some of the ideas into practice, and shared them on Twitter. (Click through for tweets with more explanation.) For example:

- students practiced giving claims and warrants with Skyscraper puzzles
- I facilitated an impromptu dispute about whether or not a quadratic must always have a y-intercept, using the point/counterpoint structure I learned from Chris
- students contrasted different representations for linear relationships using Chris’ debate cards (modified for three person groups)
- pairs debated which compound areas are easiest/hardest to calculate

Every time I make questions debatable, I’m delighted by the vibrant mathematical conversations and arguments that students are engaging in, and yet I’ve still struggled to make maths debates a mainstay of my teaching—perhaps due to the perceived time required to devise debatable questions, and having to get to grips with how to teach students to have rich mathematical discussions.

**Re-enter Chris, and his succinct new book ‘Up for Debate!’.**

In Chapter 1 Chris states three goals for his book:

- To bring debates to life for the reader — the ‘what’
^{1} - To provide concrete structures and routines to help get students talking and debating — the ‘how’ and the ‘why’
- To provide guidance in how to transform existing maths questions into more debatable ones — the ‘nitty gritty’

and he amply meets all three goals across six captivating chapters. Chapter 1 sets the scene with rich classroom vignettes that demonstrate how maths debates enliven and humanise maths classrooms.

The next two chapters form the foundation for debating maths. Chapter 2 introduces us to the tools of debate—the talking and listening routines. This is where we encounter the two key parts of any argument: the claim (the controversial statement) and the warrant (the justification for the claim.) One big takeaway for me was the importance of the physical aspects of debate: a student stands and everyone else sits (including the teacher), turning their eyes and knees to the speaker to show they are listening, and because the behaviour follows from the physical action.

Chapter 3 is a goldmine of ways to transform, adapt and create debatable questions, and is crucial for getting started with and sustaining maths debate. Not only does Chris give us a ready supply of debatable words (e.g. best/worst, hardest/easiest, weirdest/coolest, most/least), questions, and other routines, he reminds us that student learning should be at the forefront of designing debatable maths questions and activities.

Chapters 4 and 5 extend the principles of short maths debates to more in-depth routines, where students summarise previous claims, argue multiple sides of an issue, and have full-scale multi-session debates. There are also useful tips for supporting written arguments and teaching proofs.

Chapter 6 closes the book by sharing experiences of three teachers (Karla, Patricia and Claire) who have implemented maths debates in their classrooms. It’s helpful to have these as a reference when taking your own first steps. Indeed, the entire book is infused with Chris’ experimentation with debating maths, and his readiness to share his own nervous moments around introducing debates in his classes makes it all the more accessible.

This book has the potential to be a game changer for many maths classrooms. I wholeheartedly recommend that you check it out. You can preview the book at Stenhouse Publishers, access some resources at Chris’ website, and share with the twitter hashtags #DebateMath and #Up4DebateBook.

[1] To be crystal clear, the ‘what’, ‘how’, ‘why’ and ‘nitty gritty’ descriptors are mine.