Tracy Zager’s new book ‘Becoming the Math Teacher You Wish You’d Had‘ is out, and it’s a treat. The central tenet of this important book is to ‘close the gap’ by making maths class more like mathematics, orienting our students towards the habits of mind of professional mathematicians. ‘Good teaching starts with us’ and Tracy companionably guides us through ten practices of mathematicians: taking risks, making mistakes, being precise, rising to a challenge, asking questions, connecting ideas, using intuition, reasoning, proving, working together and alone.

Tracy skillfully blends academic research, illuminating classroom dialogues, the thoughts of mathematicians and maths educators, and her own perceptive observations. This seamless mix is a real strength of the book; we not only see what habits are important and why, but how they can be enacted through specific teaching strategies, and the powerful effects they have on our students’ development as confident and capable mathematicians. The reader can’t help but be inspired by the teachers that Tracy holds up as exemplars of good practice. These teachers have so much respect for each of their students as serious mathematical thinkers. I was struck by the extent to which they would go to adapt instruction in response to student ideas and to support them in pursuing their own line of enquiry.

Tracy warns early on that the book is long—and it may be—but it is also captivating! The organisation is immensely practical; each chapter can be used as a self-contained guide for a particular mathematical habit. I can see myself repeatedly delving back into specific habits as the teaching year progresses. I read it cover-to-cover over a couple of days while curled up in a secluded cabin, pausing occasionally to stare out into the Australian bush and ponder what I can change in my own teaching. Some of my highlighted passages:

- From Chapter 3, Mathematicians Take Risks: ‘
*When we assign problems that have a single, closed path from start to finish, we’ve eliminated the possibility that students will take mathematical risks.*(pg 49)**There’s nothing to try if everything is prescribed.**‘*.*In my skills-based courses, I too infrequently give students opportunities to try and be successful with their own approaches. That’s something to work on. - From Chapter 4, Mathematicians Make Mistakes: ‘
*If we want students to learn from mistakes, we need to teach them how.’*(pg 57).*‘*How can I help students gain the skills to diagnose and learn from their mistakes, by themselves?**to teach students to make the most of the knowledge and experience they gained by figuring out their mistake**‘. - From Chapter 5, Mathematicians Are Precise: ‘
(pg 80). In my problem-solving course, I deliberately swung the pendulum from the typical procedure-based courses my students had mostly experienced towards creative, collaborative problem-solving. But I also need to find the middleground, where I place as much emphasis on rigour as I do on inquiry.**Math without inquiry is lifeless, but math without rigor is aimless.**There is no tension between teaching students how to solve problems accurately and efficiently and teaching students how to formulate conjectures, critique reasoning, develop mathematical arguments, use multiple representations, think flexibly, and focus on conceptual understanding.’ - From Chapter 12, Mathematicians Work Together and Alone: ‘
*If a major part of doing mathematics involves interacting with other mathematicians, then a major part of teaching students mathematics must be to teach students how, why, and whether to interact with one another mathematically.**Students need to learn how to ask for what they need from each other and to be what they need for each other**…***we need to teach students how to be good colleagues**…*it’s important we honor individual thinking and working time.**It’s not reasonable to expect students to collaborate at every moment, and that’s not how mathematicians work.’*(pg 312). This past semester, a few students in my problem-solving course commented that they needed more opportunities to work alone first, and more strategies to work effectively with group members. I’ll definitely be digging further into this chapter next year.

And, these phrases are going straight into my repertoire:

*‘Do you have more questions after doing this? What are you wondering about now?*(pg 149).*‘What does ______ have to do with _____?’*(Debbie Nicols, pg 191).*‘Remember that it’s hard to find mistakes when you assume that you’re right. So go back into it assuming something went wrong.’*(Jennifer Clerkin Muhammad, pg 284).*‘Would you recommend that strategy to someone you like?’*(pg 118). 😂

There is so much to love about this book. The writing is both encouraging and empowering. It’s labelled K-8 but Tracy offers important insights to help teachers across *all* year levels; I have been nodding furiously and making notes throughout. This particular passage had me shouting ‘yes!’:

*‘We need to give ourselves permission to say, publicly, and with delight, “I never thought about it that way before!” whether it refers to addition, fractions, or place value. It is long past time for us to respect the beauty, power, and importance of elementary mathematics, instead of having contempt for “the basics.”’ (pg 208)*

Listening carefully to student thinking,* especially about ideas I thought I understood, *always gives me new insight. It’s why I’ll never tire of teaching.

I can confidently say that, alongside ‘Thinking Mathematically‘ (Mason, Burton and Stacey, 1982; 2010), Tracy’s book will become a cornerstone for my teaching. It is a gift to all maths teachers. But don’t just take my word for it; you can preview the book in its entirety here. The companion website promises more, and I can’t wait to look around!

**Update (22 December 2016):** The companion website is now live, and it is packed full of goodies. Be sure to check out the free study guide under ‘Getting Started’, which works for either an individual or group book study.