A Focus on Dialogue and the Construction of Knowledge

Mrs. Carmel's students enter her math class and she announces, "I have often heard people say that multiplication makes numbers bigger and division makes numbers smaller. Is this statement ever true? Is it ever false? Can you give examples? Be prepared to either defend or revise this statement." The buzz of conversations unfolds within her classroom. When planning to begin a lesson this way, Mrs. Carmel thought deeply about the development of conceptual knowledge, giving consideration to common misconceptions. The cognitive demands of her task - the skills and knowledge required for success - are high. Here, the responsibility to solving this dilemma lies squarely on her students. We could make the case that in an exercise such as this, all 8 of the mathematical practice standards are employed .(Core Standards)   Furthermore, as an outcome of the discussion that ensues, each child has an opportunity to expand his or her understanding of the concept of multiplication and division across various types of numbers.

It is not just the practice standards that reflect the need for such rich conversations, but the grade level content standards themselves. The verbs that are employed in the standards are reflective of a revision of bloom's taxonomy (RBT) where we think about the creation of knowledge as requiring a hierarchy of actions: remember, analyze, apply, evaluate, and create.( School Wires)  To inspire these actions, it is our job to pose both good tasks and questions. In a classroom that values dialogue and the construction of knowledge, the responsibility for learning lies with the students, and selection of tasks and questions lies with the teacher. There are many resources from books of prompts, to websites with video clips that can help teachers shift the responsibility of learning in their classrooms. Cathy Seely, past president of NCTM, asks us to consider our teaching with the following questions - "Who is doing the math? Who is doing the talking?" When we make this shift, we can confidently say it is the students, who through their struggles and collective discourse, are doing the talking and the math!

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