All students should be prepared to share their thinking and reasoning, whether correct or incorrect, at any point in a lesson. Holding groups and individuals accountable for their explanations and justifications will add further value to the learning activity. As you have surely discovered, when you truly understand a concept or idea yourself, then you are able to explain that concept or idea to someone else.
In the book, 5 Practices for Orchestrating Productive Mathematics Discussions, by Margaret S. Smith & Mary Kay Stein (2011), a framework is presented for facilitating mathematical discourse. In a collaborative classroom, discussions should stem from the work produced by students (and not from the teacher). A productive mathematics discussion will surface mathematical ideas and misconceptions and lead to deeper student understanding. The teacher can utilize the following instructional practices to design a string of classroom dialogue.
Anticipate likely student responses to challenging mathematical tasks
- How might your students approach this problem?
- What types of misconceptions do you predict?
Monitor students’ actual responses to tasks
- How will you track what students are doing and saying?
- What questions will you ask students as they work?
Select purposefully particular students to present their work during the share phase.
- What methods and reasoning do you want to emphasize in the discussion?
- How will you choose specific student work to drive the discussion?
Sequence purposefully student responses that will be displayed in a specific order.
- How will you order the presentation of student work to best meet the mathematical goals?
- How will students share their work? (e.g., chart paper, document camera, white boards, verbally, video, online discussion boards, etc.)
Connect different students’ responses and connecting the responses to key mathematical ideas
- What questions will you ask to make the key ideas visible to students?
- How will you help students make connections across multiple approaches?
In a collaborative classroom, the 5 Practices still allow for the teacher to maintain control over the content discussed by students. Teachers should be mindful of holding all students accountable for being actively involved in the discussion. Students should be expected to present their ideas, ask and answer questions, and comment on others’ ideas. These practices should be embedded within your lesson planning to generate thoughtful discourse on a daily basis.
A Note About Students’ Written Responses
When working in the Student Text, students should always try to answer questions with complete sentences. Full sentences should also be encouraged in written answers to Assignment or Assessment questions. Complete sentences help students reflect on how they arrived at a solution, make connections between topics, and consider what a solution means both mathematically as well as in context. In the Teacher’s Implementation Guide, answers are provided in the form of complete sentences whenever appropriate.
Note that many answers in the Teacher’s Implementation Guide are samples of correct answers. Students may present correct responses that do not exactly match the sample answers provided in the teacher’s materials. Actual students' responses may include alternate solution paths or interpretations of a given question. This variety provides teachers with a rich opportunity for classroom exploration and discussion. Because correct student responses may vary, we recommend showing only student responses to the class and not the sample answers from the Teacher’s Implementation Guide.