Recently I was speaking with a parent who was expressing frustration with their child’s work habits in their math class. It was a conversation I have experienced many times throughout my 20 years of teaching. It’s about that child who says they can do the math in their head and they do not need to show their work. Many of you have had experiences with situations like this and would probably agree that math work needs to be shown. In fact, many students have lost credit on assignments for not showing their work. The rationale for this seems solid; the teacher needs to see the work so they can advise the child of errors, it provides repetition for the child in the procedures and algorithms they are working with, it fosters a discipline needed for more complicated versions of the problem, and it provides some accountability that the child is actually working through the problem, not getting answers from another source. I have reiterated these ideas to parents during many parent-teacher conferences over the years while discussing their child.
As I begin to understand the research on how children come to know math, I realize we would be better served in asking students “to show their thinking” instead of “to show their work”. At the AIMS Center we are taking the time to understand the existing research on how children “come to know” their mathematical knowledge and how that understanding impacts their future academic performance. What we know is that for children to construct knowledge in mathematics they need to engage in meaningful thinking of their mathematical ideas. This means reflecting on their thinking and showing their reasoning through explanations, diagrams, and mathematical notations. This process develops the underlying mental operations in the child’s brain which are key to deep understanding and eventually success in mathematics.
I think you would agree that the common practice of children working 20 problems using a procedure they saw during math class is considered by many to be valuable to their formal education. The student’s work is then evaluated based on correct or incorrect final results. Errors may be explained to some students but we end up with little information about the cognitive processes and mental operations the child employed. If we are serious about teaching students from their errors, as we claim when asking for work, it is the errors in a student’s mathematical thinking that are critical for us to recognize. This is not quick or easy for the student or the teacher, but it is necessary if we are serious about the education of the child.
Teaching through “understanding the thinking of the student” is difficult. It calls for both knowledge and flexibility on the part of the teacher, who must provide support for students as they engage in mathematical sense making. This means knowing the “mathematics of children” as well as “mathematics for children”. It means having a sense of when to let students explore, when to tell them what they need to know, and knowing how to effectively nudge them in productive directions.