5 Practices for Orchestrating Productive Mathematics Discussions

I teach three sections of College Math every semester - usually one right after the other on MWFs.  I usually have about 100 students in total.  While teaching the same class three times a day can sometimes be challenging (I always hope that my most talkative class is last), it has given me momentum to improve the class each semester.   Usually there are five sections of College Math each semester and I collaborate with the two adjunct professors who also teach the course.  Last year these colleagues and I started experimenting with the flipped class.  Our motivation was the desire to have more class time for discussion and application of the knowledge that students were learning.

One example is the lesson on measures of central tendency and measures of variation - mean, median, mode, range, five-number summaries, and standard deviation.  We usually had to spend a significant portion of time reviewing how to calculate these measures and ran out of time to analyze when to use them or what they reveal about sets of data.  So I created a video explaining how to calculate these measures and assigned watching it and filling out guided notes as homework due before class.  During class we spend time discussing when to use each measure and how to interpret what the measures are saying, especially in comparing sets of data.

I realized pretty early on that I needed to learn how to structure the flipped class time differently and that I easily slid back into a lecture mindset on those days.  I have not experienced a math class as a student where discussions are the central format, nor did I learn how to facilitate mathematical discussions in my teacher preparation program in college.  Since I was able to choose the topic for a project in my current grad. school class, learning about how to do this better seemed like the obvious choice.

I just finished reading a book by Margaret S. Smith and Mary Kay Stein called 5 Practices for Orchestrating Productive Mathematics Discussions, recommended to me by my professor.  I learned so much from this relatively short book.  Here are some quotes:

"In mathematics classrooms, high-quality discussions support student learning of mathematics by helping students learn how to communicate their ideas, making students' thinking public so it can be guided in mathematically sound directions, and encouraging students to evaluate their own and each others's mathematical ideas." (p. 1)

"Teachers often think of lessons in relation to the activities that they will ask students to do rather than what students will come to know and understand about mathematics as a result of having engaged in the lesson." (p. 15)

"I must consider the mathematics in relation to the children and the children in relation to the mathematics." (p. 50)

"What students learn is intertwined with how they learn it." (p. 61)

"Almost all good classroom discussions begin in the same way: by inviting a student to share how he or she solved a particular problem.  After the initial student response, however, classroom discussions diverge--separating into the relatively rare fruitful ones and the much more frequent unproductive show-and-tells.  It turns out that issuing an invitation to students to reveal their thinking is a relatively easy thing to learn how to do.  And most students will comply......Typically, however, teachers treat all presentations as equally good, they ask few questions of the students, and they do not connect different student presentations with one another or the disciplinary ideas under investigation." (p.69)

This book lays out five practices designed to remedy this situation:

1. Anticipating - considering ahead of time how students might approach a mathematical task, both correctly and incorrectly, and how those relate to the things that a teacher wants students to learn
2. Monitoring - paying close attention to students are they are working on a mathematical task and keeping track of strategies used as well as asking questions of students to help guide their thinking and keep them engaged
3. Selecting - picking particular students to share their work in order to get certain mathematics into the open for examination by the class
4. Sequencing - purposeful ordering of the student work being presented so as to maximize the chances of achieving the goals for the lesson
5. Connecting - drawing connections between the student work that is presented and the key ideas of the lesson

A necessary foundation  for these practices to be successful is setting clear goals and selecting tasks that are conducive to discussion.  The goal of the lesson needs to clearly identify what students should know and understand about mathematics and consequently will guide the selection of appropriate tasks.  The tasks need to be of sufficient depth to stimulate and provide opportunity for discussion.

I also appreciated the chapter on asking good questions and keeping students accountable for their understanding of the material.  Here is a list of questions asked by teachers in various classroom scenario presented in the book:
"What does this mean?"
"Why did you do it that way?"
"Can you explain more to me about that?"
"Does anybody want to add to that?"
"How do you know that?"
"Do those two things mean the same thing?"
"Does everyone agree with this?"
"Is there another way?"
"Can you repeat what [insert student's name] did to solve this?"
"How can we know for sure?"
"Can you always do that?"

The value in these practices is to provide a structure that will help transform class discussions from a "show-and-tell" format to an intentional interweaving of student work with the mathematical goals of the lesson.

The next step is to implement these five practices with a lesson in the College Math class that I teach, involving the adjuncts along the way.  Collaboration will improve the process.  Let the project begin!