Show Your Thinking

davidpRecently 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.

The Why Before the How

I have a deep passion for mathematics education. More specifically, elementary math education is where I have spent most of my career. I began my career as an elementary teacher for ten years, and am now a mathematics coach and consultant with the Fresno County Office of Education (FCOE). In addition this year, I am… Continue Reading

Workshops at SER-Ninos Charter School, Houston, TX

I have the pleasure of working with workshop facilitators who live in different cities throughout the United States.  Care Butler, who has facilitated countless workshops for AIMS, resides in Arkansas. She works hard to design teacher days that will empower educators to be more effective in their classrooms and ultimately have positive effects on student… Continue Reading

What’s Wrong Here?

Another late night reading… Another night of pulling away from my family to try to make sense of new information, in order to make my lessons meaningful and productive for my students. I love teaching and had been teaching for 12 years before coming to the AIMS Center. I had taught Kindergarten, 1st, 3rd, and… Continue Reading

Algebraic Thinking with Rural Teachers in Florida

This past year, the AIMS Center had the privilege of working with hundreds of rural teachers in the state of Florida.  Our host was actually the Florida and the Islands Comprehensive Center (FLICC), operated by the Educational Testing Service.  We worked with three different consortia, the Heartland Educational Consortium (HEC) in Lake Placid, the North… Continue Reading