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Stephanie Burroughs, Ed.D.

Curriculum Leader, K-12 Education

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Balancing Conceptual Understanding and Procedural Fluency

I did not become a Mathematics Teacher because I thought it would be exciting. In fact, I was pretty certain in signing up to teach Mathematics that I would be the resident dentist in my school, the class that every kid had to go to but considered to be a legal form of torture. As my teaching career began, I couldn't help but be jealous of my ELA and History counterparts who had kids at the edge of their seats, engaged in debate and discussion on a daily basis. Students would submit essays that were passionate reflections of their interpretation of political and literary figures alike, and I would pass out my standard 20 question quiz where my only chance at a chuckle was in how creative I could get with my projectile motion word problem on the back side of the paper. I even started giving out bonus points for Led Zeppelin Trivia, partly because I wanted my students to know that I was actually not an alien and partly because I knew there were those kids in the room that worked super hard only to not be able to perform well on their quiz. The moans and groans on quiz day were commonplace and the defeat on their faces was all too familiar. 

But how could I test them differently? Isn't this the way that Math is supposed to be taught?

Just as I was ready to accept my lot in life, out came the Common Core and a new mantra on how we should be teaching Mathematics. Reading, Writing, and Communicating in Mathematics class were finally defined as being critical components to teaching and learning. In fact, the Standards for Mathematical Practice highlighted the need for debate and discussion, something I had been craving since my first day on the job. The Common Core demanded that an effective Mathematics program place an equal emphasis on "conceptual understanding and procedural fluency." 

But what did this mean for me as a teacher? How can I create an equal balance?

Being that I am mathematical, I began creating graphic organizers that placed an equal emphasis on concepts and procedures. This literally meant that one side of the handout was about concepts and vocabulary and the other side of the handout was about processes and problems. This was a great baseline and allowed me to establish a 50/50 split in my instruction. But, this shift was only the tip of the iceberg. In order to place an equal emphasis on conceptual understanding and procedural fluency, I had to restructure everything that I do. My lessons had to create a mix of discussion and practice, my assignments had to support a mixture of making connections between concepts and practicing solution strategies, and my assessments had to test both student understanding of concepts and student ability to replicate the mechanics of mathematics. 

In order to place an equal emphasis on conceptual understanding and procedural fluency, we must treat mathematics as though it is it's own language. This means that we need to do ALL of the following consistently for the benefit of our students:

1. Emphasize Vocabulary

  • Define it, describe it, make connections and use it
  • Engage in discussions using vocabulary terms

2. Make Connections between concepts

  • read about the connections between concepts and the real world
  • write about the connections between concepts
  • communicate the connections between concepts through discourse

3. Study the procedures

  • practice the mechanics of solving mathematics problems
  • discuss solution strategies and evaluate their efficiency

4. Apply your understanding

  • find real world problems and solve them through mathematical reasoning
  • engage in global conversations on data and procedures and discuss their impact

 

Although my 50/50 split of concepts and procedures was a great way to get my feet wet and abandon this notion that the more practice problems students have, the better off they are in their understanding of Mathematics, I needed to dig much much deeper and think about how my Spanish class was structured in high school or how engaging my Literature class was. Practice problems in Mathematics all day is the equivalent of spending your entire English class going over the rules of grammar without ever applying it to what matters. We need to stop this nonsense and teach the language of Mathematics. 

 

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