Show and tell activity (aka lecture method) may work for some but never in a mathematics class. Getting students to explain and ask questions are nice but only when the explanation and the questions are mathematical. Reasoning and justifying are good habits of mind but they are only productive if they are based on mathematical principles. Explaining, asking questions, and substantiating one’s conjecture or generalization make a productive class discussion but they are only productive for learning mathematics if the mathematics is kept in focus. Orchestrating a productive class discussion is by far the most challenging work of mathematics teaching. Stein, Engle, Smith, and Hughes* proposed five practices for moving beyond show and tell in teaching mathematics. I have always practiced them in my own teaching whether with students or with teachers and I find them effective especially when the lesson involve cognitively demanding tasks and with multiple solutions.
1. Anticipating Students’ Mathematical Responses
The first practice is for teachers to make an effort to actively envision how students might mathematically approach the instructional tasks(s) that they will be asked to work on. Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect—they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn.
2. Monitoring Student Responses
Monitoring student responses involves paying close attention to the mathematical thinking in which students engage as they work on a problem during the explore phase . This is commonly done by circulating around the classroom while students work. The goal of monitoring is to identify the mathematical learning potential of particular strategies or representations used by the students, thereby honing in on which student responses would be important to share with the class as a whole during the discussion phase.
3. Purposefully Selecting Student Responses for Public Display
Having monitored the available student responses in the class, the teacher can then select particular students to share their work with the rest of the class in order to get “particular piece[s] of mathematics on the table”. A typical way to do this is to call on specific students (or groups of students) to present their work as the discussion proceeds. Alternatively, a teacher might ask for volunteers but then select a particular student that he or she knows is one of several who has a particularly useful idea to share with the class.
4. Purposefully Sequencing Student Responses
Having selected particular students to present, the teacher can then make decisions about how to sequence the students’ presentations with respect to each other. By making purposeful choices about the order in which students’ work is shared, teachers can maximize the chances that their mathematical goals for the discussion will be achieved.
5. Connecting Student Responses
Teachers can help students draw connections between the mathematical ideas that are reflected in the strategies and representations that they use. They can help students make judgments about the consequences of different approaches for the range of problems that can be solved, one’s likely accuracy and efficiency in solving them, and the kinds of mathematical patterns that can be most easily discerned.