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 the Partnership Liaison for the AIMS Center for Math and Science Education, through an agreement between the AIMS Center and FCOE.

I am on a steep learning curve as I begin to study radical constructivism and professional noticing with my new colleagues.  It is daunting to experience the incredible range of knowledge that the AIMS team has amassed in terms of the research on how children come to know and learn mathematics. My charge here at the Center is to work with our partnership schools, providing professional development that combines the tenets of our research with the goals of the school site.  I hope to take the fundamentals of this research and share some of it here in this blog as well.  

Recently, I was in a fifth grade classroom where a teacher had taught a beautifully constructed direct instruction lesson. The students had listened to the steps required to work the exercises, practiced the steps, and were then completing the exercises on their own and getting correct answers. The bulk of the class was happily working in class so that they wouldn’t have any homework that night. A number of the students were having to regularly refer to their steps in order to complete each new exercise, but others were able to finish each one quickly and correctly without referring to their notes.

Perhaps it was because I was in the back of the room that morning as a coach, but the teacher glanced at me and then said to the class, “Now I want to tell you why the steps work.” I perked up, hoping to hear about the conceptual development of the topic. A few students stopped what they were doing, but most continued working their homework problems. The teacher was about to launch into an explanation for the concept they had learned that day when a student raised her hand. The student asked, “Can’t we just use the steps that you taught us to get the right answer every time?” The teacher replied to the affirmative and the student then said, “Well then, does it really matter why? Because I already know I can get the answer.”

This is an attitude that I have run into repeatedly throughout my career and it brings me to my point. Students must construct their own understanding.  Memorizing steps does not equate to understanding.  Whenever possible, the conceptual or “why” the math works, should be explored by our students long before they practice solving exercises within a given area of mathematics. If not, then we may lose the chance to explore those ideas forever.  Students live in the moment.  When students already have a tool that they believe will always work, but no understanding of why that tool works, it is difficult to convince them of the need to “understand”. And when the tool no longer works, they are lost.

There are so many mathematical ideas that can and need to be explored in a conceptual manner. Why do we “put a three in the ones place and carry the one” when adding? Why do we cross-multiply, and when is it even appropriate to do so? Why, when we are dividing fractions, do we use the reciprocal of the second fraction and then multiply? When we teach conceptually and allow students the chance to feel the “disequilibrium” of not understanding something, we give them the chance to make sense of what they are experiencing. We allow them time to connect to other experiences, ideas, and concepts for which they already have an understanding. We allow them the opportunity to construct their own understanding.  In essence, we give them the chance to make connections and do real mathematics. Where in your practice have you seen the need for conceptual understanding before practicing problems?

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