After reading the Five Practices for Orchestrating Mathematical Discussion, I began to analyze my own and other teachers’ teaching practices. The Five Practices are as follows:
- Anticipate—anticipating student thinking and response
- Monitor—monitoring student responses for ones that will direct discussion
- Select—selecting student work to create a mathematical discussion
- Order—place student work in logical order
- Connect—make connections throughout the discussion
One thing that I think my CT does really well is anticipating student thinking. She can always tell what questions her students will ask, what they will grasp easily, what concepts they will struggle with, and what strategies they will use to problem solve. On assessments, she is almost always spot on when describing which problems will be difficult for students and which problems they will make. I think a lot of this comes from knowing your students, individually, and as a group. For example, we just finished a unit on functions and on their test today, they had to use a table to create a linear equation, something they did in this unit and the one before it. Even before they began the test, I knew that few of my students would get a completely right equation. As a whole group, my students struggle with deciding which piece of information is the slope and which is the y-intercept. In this particular problem, the slope was negative. Even though a question asking whether it was positive or negative directly followed (and many got right), they still missed the negative part in their equation. From working closely with my students the past weeks, I have learned so much about their strengths and weaknesses to help me in anticipating their work in the future.
The second practice is to monitor. In my class, we are extremely lucky. My first two classes consist of special education students or low-medium performing math students. Though there are some negatives to this, the benefits are enormous. For one, we have four teachers in the classroom during these classes- my CT, the special education teacher, myself, and my partner. This means we have the ability to closely monitor each student. Between the four of us, we know the commonalities between students, and can help reach individual questions that would not help progress a mathematical discussion. In my class, my CT doesn’t often have students share their work, so this is where her involvement in the Five Practices ends. I begin teaching my unit on Monday and I’m planning on implementing more discussion to really help students build their understandings.
The third practice is to select and the fourth is to order. I think these go hand in hand. One of the best examples I’ve seen of this is in my middle school math education course with Lisa. Often, we were challenged to complete problems in more than one way. We used pictorial representations a lot and were often required to construct more than one on assignments. As we worked in class, Lisa would walk around, ask us about our thinking, and keep track of which methods were particularly of interest. She would then ask specific people to share their work, in a specific order. Some were incorrect, but developed our mathematical thinking. Others were correct and were usually set up in a way so that we saw the most common methods first and progressed to more unique ones. We would discuss the math behind each and work with each other to see why it worked. In this way, Lisa facilitated our discussion and ensured that we saw examples of many people’s work so we could develop our understanding. I like that many times she would show us examples that were very common. In this way, I could understand more fully and begin to feel comfortable with the material. As we moved into more eccentric ways of solving, I could use our prior discussion to assist me. I think this would work so well in the classroom. The common methods would validate thinking for most students and then expand their understanding as we moved into different methods.
The final practice is to make connections for your students. Often times, too often I think, teachers fail to explicitly state connections. We have already developed this network of math and don’t realize students need our help to do the same. So, math can start to seem like a bunch of unrelated, difficult topics for students, when, in fact, all math is built upon connections. Even in one discussion, multiple methods of solving one problem can seem disconnected to students. When my kids were learning about solving systems of linear equations, we learned the substitution and the elimination method. After learning both, my CT mentioned that they would both be on the assessment and a student asked why. They had no idea why they were connected and why they would be on the same test, or even why you could choose between the methods to solve. Some students had made the connection, but some hadn’t. Teachers need to be explicit about these connections, even if they seem obvious to us. The connections could be what solidifies a student’s understanding.
Overall, I think these Five Practices are brilliant for implementing productive and meaningful mathematical discussion. Too often, teachers forget that discussion is often the part of instruction in which students solidify their learning. With my upcoming unit, I want to ensure that students are seeing each other’s work, explaining their own work, making connections, and discussing mathematics.