Last summer I read a book by the NCTM that was all about how we should teach math. I wrote about HERE. One of the chapters was about how to orchestrate productive math discussions in class. Turns out the NCTM realized a lot of people struggle with this so they created a whole separate book that just focused on that: 5 Practices for Orchestrating Productive Mathematics Discussions. I read that this summer.
I’m really glad I spread these readings apart. I think if I read both books last summer, it all would have just been jumbled together in my mind. Having a year of really working at those math discussions in class and having super great days and days that flopped put me in a place where I had a bit more of a framework to digest the 5 practices. It’s sort of like when you let a student struggle a bit and then lead them to a fuller understanding of the concept. This past year was the struggle and this book providing feedback for me on what I was doing wrong and what I was doing right. I’ve written pretty extensively about the challenges of having a successful math classroom (here, here, and here); success being defined as seeing students struggle through a new concept and coming out with a firm understanding of it.
The text starts with a case study of a teacher named David Crane. He poses a problem to his students, let’s them work on it, has students share various methods to solve the problem, and told students they could choose the method that worked best for them. So many of my discussions last year looked like this, especially in my 6th grade math class. This case study served as an example of what not to do. It was pretty humbling because as I read it I thought, ‘hey that wasn’t too bad.’
So what are the 5 things that move a math discussion from being ineffective to enriching? I’ve listed them below and in parenthesis I’ve graded myself on a scale of 15 on how often I did that practice last year (1 being rarely, 5 being often)

Anticipating – how will students likely try to solve this problem (3)

Monitoring – see who’s doing what as they solve (2)

Selecting – thinking through what student work would be most meaningful to present (2)

Sequencing – being thoughtful of the order that you have students present (1)

Connecting – bringing it all together to the fundamental math concept you want them to understand (2)
Let’s break each down a little further shall we.
0. Setting Goals & Selecting Tasks
While it’s not one of the five practices, practice zero would be to develop a rich learning goal and task to go with it that will have a lot of potential to lead the students to that learning goal. I thought I did this well until I saw some excellent examples:
An OK learning goal – students will be able to use the Pythagorean theorem to solve a series of missing value problems.
A better learning goal – students will recognize that the area of the square built on the hypotenuse of a right angle is equal to the sum of the areas of the squares built on the legs and will conjecture that c^{2} = a^{2} + b^{2}.
I saw that and thought ‘jeez…every single one of my learning goals is going to need some work.’ I also thought how a learning goal like that makes it absolutely clear what mathematical concept you really want students to walk away with.
Selecting the tasks is really about finding quality questions. I was really hit & miss on this last year. Some of my questions and tasks really brought students to begin exploring the deeper mathematical concept I was wanting them to understand. Many of my questions did not. I’ve used 3Act problems but a lot of times they weren’t really hitting the exact learning goal I was aiming for. I’ll be using many problems from the Exeter collection of problems in Mathematics 1, at least for my PreAlgebra class this year. Still exploring for 6th grade math tasks.
Also, I’ve got to start using the word conjecture in class. Even thought it’s just a fancy word for guessing a mathematical relationship, it will make us all feel like the mathematicians that we are.
1. Anticipating
At this point in the text, they present various case studies to demonstrate what some teachers were doing right when trying to incorporate these 5 practices. While I saw many of the things I did in the ‘what not to do’ case studies, thankfully, I did see a few things that I did last year in the ‘what to do’ case studies as well.
Once you have your learning goals and task, the first step is to solve the problem yourself and try to anticipate what different avenues students may take when they solve it, as well as mistakes you think they will make. I’ve foolishly written a problem as an opening task, did not solve it, only to find out that it was a terrible problem as I began to watch students try to solve it and get frustrated. Thinking through these approaches and misconceptions on the forefront gives you time to think of the kinds of questions you can ask to lead students down the right track. You’ll also have an idea of which methods will lead themselves most to helping students grasp the concept at hand. Additionally, this will give you time to be more thoughtful to unanticipated methods that arise in class; you’ll have less things you’ll have to figure out on the spot.
2. Monitoring
This is where you pay attention to the mathematical thinking as they’re working on the task. Monitoring for me in the past has been walk around, making sure everyone’s on task, looking at what they’re doing but not commenting even if they’re completely off base (wouldn’t want to rob them of the opportunity to struggle through it), and then getting ready to ask for volunteers to share thinking.
I’m seeing that it’s OK to begin to steer students back in the right direction. If you can do so by asking questions instead of just saying ‘do this,’ you’re not robbing them of the struggle, but making their efforts more purposeful.
In the case study they provided, the teacher was recording data as he monitored. He listed all the strategies he anticipated and left room for other methods that he didn’t anticipate. He wrote down who did each strategy with a little note about how they went about it (ex. group 2 made a sketch of a graph but did not plot the points). He also had space to write which order he would want the groups to present in. I’ll get to that in practice 4.
I wonder how monitoring works in problembased learning classrooms where students do the problems as homework and then take part in the discussion in class. You don’t get to see the thinking as it happens.
In my lesson plans, in addition to the learning goal and task, I’d like to start noting my anticipations of the various ways students may approach the problems, some ideas of leading questions I can ask them to get them on the right track or to think more deeply, and to create a monitoring table where I list the methods with space to write who did what.
3. Selecting
My favorite line of this chapter was the following:
We have come to think of the question, “Who wants to present next?” as either the bravest or most naive invitation that can be issued in the classroom.
I laughed. I do that all the time. The text did say that asking for volunteers can still be effective if you choose a volunteer who will present a method that you feel is well timed for the way the discussion is going.
This was where most of my class discussions last year became most unproductive. I would ask if anyone did a different method than what had been shown and students would eagerly come up only to show the same method. I had students come up whose work I had not examined beforehand who were completely off base which confused many other students and wasted a lot of time. I’m totally down for examining mistakes but in the context of a class discussion, they must be productive mistakes that many students can learn from. During the monitoring stage, seeing those mistakes and meeting with those students privately is a better use of class time.
Selecting is the time to decide what mathematics you want highlighted and who you want to do it. You want all students to have opportunities to present their thinking; it communicates that everyone’s ideas are valued.
4. Sequencing
Once you know what you want presented and who will present it, the order in which you present is critical in giving students the most opportunity to make sense of the underlying concept. A good place to start is with the most commonly used method; it’s accessible to many of the students. Or perhaps a common misconception is a good starting point. Another option is to present methods that move from concrete to more abstract. Students who may struggle in class will often use more concrete methods when approaching a problem. This is where I try to bring those students into the conversation.
I was kind of surprised in the case study that the teacher basically talked about the answer to the problem first, then went through the various approaches. I remember reading somewhere last year that once you tell the students the answer, they stop thinking because they either are satisfied they got it right or just accept that they got it wrong. I guess that’s where culture of the classroom becomes pretty important; that the emphasis is on the mathematical thinking, not just getting the right answer.
5. Connecting
I struggled quite a bit with connecting the work to the underlying mathematical concept last year. Partly because I wasn’t exactly even clear on what concept I was aiming for (‘learn how to solve twostep equations’ isn’t very deep), and partly because the main question I asked was ‘how are these methods connected.’ Students would usually sort of stare blankly and slowly say ‘…because they’re all saying the same thing…???’ and wouldn’t be able to expound.
In the case study, students were finding an intersection point of linear equations in the context of two different phone plans (when would one phone plan become more expensive than the other). Several students drew graphs and a couple actually wrote out an algebraic equation. The teacher asked how various parts of the equation connected with the graphs that the other students made. Much more pointed than ‘how are all these methods connected.’ This practice is indeed the toughest, but most essential part of the discussion. Otherwise it’s just a bunch of students sharing disconnected ideas.
Yeah, but…
But what if the methods you hope to see don’t arise? I guess that would speak to the task; is it too advanced…are there other concepts that students are missing that are keeping them from being able to approach the task? Finding good tasks is the foundation for all of this. Is it a task that many students could approach, regardless of mathematical background? In addition to the Exeter problems, I’ve found some pretty quality problems at illuminations.nctm.org. Dan Meyer’s 3Act problems are always worth looking into as well.
But I’ve heard it said ‘never say anything that a kid can say in class.’ I’ve always interpreted that as ‘in an ideal classroom, the teacher wouldn’t really have to say anything.’ Reading the article again, I’m realizing that while much of the talking should be students sharing ideas, the teacher should be asking thoughtful questions that will bring students to a deeper mathematical understanding (rather than just lecturing and giving answers). There were times students were so lost in class last year and I just sort of sat there quietly and let them work through it. While that can sometimes be productive, it often wasn’t in my class.
If you’re looking to either begin having mathematical discussions or refine the ones you’re already having, check out Principles to Actions: Ensuring Mathematical Success For All, as well as the more specific 5 Practices for Orchestrating Productive Mathematics Discussions, both by the National Council of Teachers of Mathematics (NCTM).
Do you feel you’ve really refined one of the 5 practices? Share your secrets! Is there one of the practices you struggle with the most? Let’s start a dialogue about what’s worked and what hasn’t. We only become better by mutually refining each other. Thanks for reading.
[…] While not as inspiring as Rafe Esquith, the NCTM has really shaped my math pedagogy. The heart of it is having students struggle through the math, seek out patterns, and really teach to give them an opportunity to understand the concepts, not just memorize the algorithms. The two books that helped shape that for me were Principles To Actions and 5 Practices For Orchestrating Productive Math Conversation. […]