Book Reading: 5 Practices for Orchestrating Productive Mathematical Discussions

Writing about the teaching books I’ve read is fast becoming a series, because this is the third post in a row about a teaching book I’ve read.  The book I finished earlier this week is “5 Practices for Orchestrating Productive Mathematical Discussions” by Margaret S. Smith and Mary Kay Stein and coming out of the National Council of Teachers of Mathematics (NCTM) in the USA.

I’ll get straight to the point: everyone in any sort of classroom where maths happens should read this book. It gives a simple and practical framework for using student work and class discussion to promote maths learning. The authors have a direct, clear style that make the nuances of the practices seem almost obvious, using careful studies of classroom scenarios to illustrate. Let me say again: read this book!

In a nutshell, the idea of the book is that you can help students learn mathematical content by giving them tasks rich enough to be worth talking about and connected to the mathematical goals you have in mind, and then orchestrating class discussion of the methods students use and their connections. They give five practices, and a smattering of other strategies and ideas to guide this.

I think this book should be required reading and/or the basis of training for staff who are teaching tutorials at university. University tutors are often given no training in teaching, and even then don’t get tools to help them choose what to do in their classrooms. In some schools here at the Uni of Adelaide, they are instructed to get students working in groups. This is great, but the part where the mathematical ideas of the week are brought out is not strong. I am hoping to take these practices to these schools, and to the ones where it’s more just another lecture, in the hope I can help to improve the learning happening in the tutes. I’ll be mentioning how I think it applies to tutorials as I go.

Here’s a summary in my own words:

“Practice 0”: Worthwhile tasks and mathematical goals

You’re not going to be able to have a class discussion about a task which is routine procedure-following, because everyone will do it the same way. You need something that has some level of challenge and has decisions to make about how you do could do it — something actually worth discussing! Also, you need to have a goal in mind for what you want to achieve so that you have a chance of achieving something. This goal needs to be about the mathematical ideas involved. For example, about the connection between the different types of equations for lines, or about the distributive law, or about the relationship between squares and rectangles.

This isn’t technically one of the five practices, since it happens “outside” the context of the discussion. Plus, you may not always have total control over the tasks that students have to do or the mathematical goals. (More likely a school teacher is in control of this, but a classroom tutor at university this will be less often true.) Even so, if you do have control, it’s very important, which is why the authors call this “Practice 0” a couple of times, because it’s needed before you even start.

As I already said, in classroom tutorials, someone else often chooses the tasks. But you can add your own question to the end to make it more open to discussion. Maybe something like “What would happen if…” are good to extend learning. Someone else may set the goal, but it’s more likely the people coordinating your course won’t tell you what the learning goal is. So you’ll have to choose for yourself. It’s so important to choose the goal so that the tutorial doesn’t end up feeling like a whole lot of activity and discussion, but with nothing of substance to take away.

Practice 1: Anticipate

When you have a goal and a task, the first thing to do is anticipate how the students will respond to the task. At the very least, you need to do the task yourself, but even better, imagine as many correct and incorrect, helpful and unhelpful approaches as you can.

One reason for this is so that you don’t have to make so many decisions on the fly during the class. You can figure out in advance some of the ways you will respond to these before you get there.

I see another advantage and it is about putting yourself in the mindset of your students. We university teachers are often so blind to how our students think, and tutors are often very focused on their own way of doing things. By explicitly trying to think of multiple approaches, it can help to break down this egocentric focus we fall into.

Practice 2: Monitor

Once you’re in class and the students are working on their task, the role of the teacher is to monitor the students’ work and thinking. The anticipating you did earlier helps you to respond appropriately to them, and sets you up into a mindset where you’re focused on their thoughts, so even unexpected methods are easier to process. It’s while monitoring their work that you will make the final decision of how you want to run the discussion, and who will be involved. It’s also while monitoring their work that you’ll ask the students questions to help them learn in-the-moment.

One thing I particularly like about this practice is how it gives us a focus while the students are working. Just the other day when talking to tutoring staff, they expressed a distaste for groupwork because it meant they, the teacher, weren’t “doing anything”. This practice says you’re not doing nothing — you’re monitoring.

The authors recommend asking students two types of questions during the working (and hence monitoring) phase:

  • Ask questions about student thinking
    Help students while they are working to express their thinking about the problem and the maths. Actually ask them to tell you how they are thinking. This gets them ready for the discussion to follow, and also helps them with the problem-solving too.
  • Ask questions about maths meaning and relationships
    Help students to express what the maths ideas mean and what they mean to them. In particular draw out relationships between concepts. This is what your goal is ultimately, and it front-loads this discussion so students are ready for it.

I see these two types of questions as really important for classroom tutors at university. Too often the questions we ask are about yes/no correct/incorrect answers, rather than about thinking and ideas. Encouraging tutors to focus on these types of questions makes thinking and meaning the focus of the learning activity.

Practice 3: Select

The last three practices are about making the discussion part of the class happen productively. They work together to help make sure that the discussion both uses student work, but also proceeds towards the mathematical goal. Also they prevent the random show-and-tell which often just ends up with students confused or with no particular idea of what they learned.

First, you want to select what student work you want to discuss as a whole class, and whose work it will be. The authors list a few considerations here, not least of which is choosing students who up to now haven’t participated much in class. It’s worth noting that in their examples, even though students worked in groups, specific single students are asked to talk about their work, which means people can’t hide from participating! It also means that people can’t monopolise the participation either! We all know that one person who seems to think the tutorial is just there for them to show how clever they are. By preselecting students to show their work, you’re making it less likely for this person to take over.

The thing I like most about the concept of selecting student work is that it has the potential to help students feel like their work is a valid and important contribution (which of course it is). By using student work and student generated ideas to forward the maths discussion, we can help them be more engaged in the learning and feel like we care about them. This is not a small thing to consider!

I am particularly interested in applying this idea to classes where students are expected to do preparation for the tutorial in advance and hand it in (like they do in several courses here at Uni of Adelaide). At the moment, what usually happens in these classes is that students do the homework, hand it in, and then the tutor presents their own preprepared solutions. But think what might happen if the students handed in the homework, and the tutor used the homework itself as a tool for class discussion. I think it might help the students feel like their homework was actually worth all the effort!

Practice 4: Sequence

After choosing which student work to present, you need to choose what order it will be presented in so that you can progress towards the mathematical goal. The authors give a number of things you might consider with your sequencing. For example, you might want to choose to start with a solution method that a lot of people have so that everyone can get buy-in to the discussion (I did this when I did Quarter the Cross in my daughter’s classroom). You might want to start with a solution containing a misconception to get it out of the way. You might want to avoid a specific solution because it will just send everyone off on a tangent (though you might also want to talk to that student one-on-one separately). You might want to have two particular solutions in quick succession in order to be able to compare them.

The important bit is to think about what order would be most helpful to get to where you hope to go. Importantly, the way you hope to make connections between ideas will dictate how you might sequence the students’ work.

Practice 5: Connecting

Now that you’ve chosen what student work to focus on in the discussion and in what order, it’s now time to actually have the discussion. It’s important here to remember there is a mathematical goal we’re working towards, which will often be about understanding a concept, and understanding is a sensation that happens when ideas are connected to other ideas. It’s our job to help students make these connections.

The authors suggest five “moves” you can make during your discussion to make sure it stays focused on the connections you want to draw out.

  • Revoicing
    This is when you repeat what a student says to make sure you and everyone else heard it and understands it. Importantly it’s not about you making what they said more correct, simply making it heard. A good phrase to end with when you revoice a students’ words is “Is that right?” This lets them know that the point is to make their thought heard (not yours) and they get to decide if it’s been voiced right.
  • Asking students to restate someone else’s reasoning
    Instead of you revoicing a students’ words, you can ask another student to explain the reasoning. This includes even more students into the discussion in a more active way.
  • Asking students to apply their own reasoning to someone else’s reasoning
    This time you’re not just asking the other student to explain the first student’s reasoning, you’re asking them to explicitly explain the connection between two different types of reasoning (one of which is their own reasoning). For example, suppose you’re doing Quarter the Cross and John proved the house-shape was a quarter by cutting and overlaying, whereas Jane proved the L-shape was a quarter by folding, you could ask Jane to prove the house-shape works by folding.
  • Prompting students for further participation
    There are times where a student will close off with a quick answer, and it might be more productive if they stayed in the discussion a little longer. The questions listed above of asking them to explain their thinking or focus on the meaning and relationships are useful now as well. In the Maths Learning Centre, I find “Tell me more about that” to be a good all-purpose request to participate further.
  • Waiting
    This may seem paradoxical, but leaving some silence can help to promote discussion. The authors say that whenever anyone asks someone else to say something, it’s appropriate to give them plenty of time to respond. Giving them this time helps to actually make the point that their answer is important to you. You giving yourself time to form your response to their question helps to make the point that their question is important to you. Waiting a bit after an explanation to let it sink in before asking people for any questions helps to make the point that it does in fact take time to process information. These last couple were new thoughts for me (though obvious in hindsight).

It’s this last practice that we often don’t do in tutorial discussions. I was talking to some tutors from the Faculty of Arts recently, whose tutorials are traditionally only discussion. They talked about how often the discussion just goes for a while and then stops at the end of the class, without coming to any conclusion the students can take away about the concepts or the process of learning them. They recognised a need to explicitly make connections during the discussion. Over in maths tutorials, I think we assume the connections are obvious, but I can attest that they are not, if all the students complaining that the tute doesn’t teach them anything are anything to go by.


It may seem that I’ve given you the content of the whole book, and indeed my aim was to present the ideas clearly, mostly for my own future reference! But I would still encourage teachers and tutors to actually read the book. The vignettes of actual classroom use are vitally important to come to an understanding of what the practices look like and where they are useful, plus there’s whole chapters about how to seek support for teaching and how to include it in formal lesson planning that I haven’t even mentioned (until just now).

I am excited to take the ideas here and use them to help support classroom tutors here at University. I think this book could really be a tool that people might actually get behind. Here’s hoping.

To wrap up, here’s the headings in dot point form for future reference:

  • Practice 0: Worthwhile tasks and mathematical goals
  • Practice 1: Anticipate
  • Practice 2: Monitor
    • Ask questions about student thinking
    • Ask questions about meaning and relationships
  • Practice 3: Select
  • Practice 4: Sequence
  • Practice 5: Connect
    • Revoicing
    • Asking students to restate someone else’s reasoning
    • Asking students to apply their own reasoning to someone else’s reasoning
    • Prompting students for further participation
    • Waiting
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