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.
As a middle school mathematics teacher who has stepped into a world here at the AIMS Center that is bigger than my experience, I am often overwhelmed by the scope of all I need to keep in mind. Still, many of the ideas are accessible, and allow me to reflect on my practice as an… Continue Reading
If you closed your eyes for just a moment, what childhood memory would come to mind? It might be tied to a person or maybe to an event. Researchers describe a memory as our brain’s ability to reconstruct a whole from fragments. Memories are much more powerful than we realize. Researchers studying the brain are… Continue Reading
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
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
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
When kids memorize times tables, we blithely believe that they understand multiplication. Too often, that is not the case. In multiplying, we are looking at coordinating units. What does that mean? I will use an example to introduce the idea: If I go on a trip for 5 days, but then I have a 2-day… Continue Reading
Are you, like me, an accidental teacher? Did the career of educator just sort of creep into your life? When I entered college I initially wanted to become a medical examiner, you know, like Quincy (circa 1976 – 1983). Well, in the process, I discovered I didn’t deal well with blood and gore. Hence, still… Continue Reading
Most recently I have taken on the task of creating an audio podcast for the AIMS Center. With this blog I would like to begin a series, of as yet undetermined length, regarding this project. Today I’d just like to explain the genesis and hopes for the podcast. I have, in recent years, been doing… Continue Reading
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