Exactly What Is Math Fluency?

-Rethinking Fluency-

I didn't mean to do it, but I did it! I was answering a colleague's question about comprehension and fluency and found myself speaking about it as if fluency and comprehension were two different things with no connection to one another – I caught myself mid-sentence and had to readdress what I believe to be true about mathematical fluency (and how it is tightly interwoven with math comprehension).

Fluency is often described in simple terms as "fast and accurate", but there really is so much more to it than that. There are volumes of research that help us to understand the importance of developing students' conceptual understanding before beginning instruction on procedures. As we begin to build procedural fluency with students, we should continue to embed that instruction within the conceptual knowledge that students hold. 

Procedural fluency is a critical component of mathematical proficiency. Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another. To develop procedural fluency, students need experience in integrating concepts and procedures and building on familiar procedures as they create their own informal strategies and procedures. Students need opportunities to justify both informal strategies and commonly used procedures mathematically, to support and justify their choices of appropriate procedures, and to strengthen their understanding and skill through distributed practice.

~The National Council of Teachers of Mathematics (NCTM), July 2014

I can't help but notice the use of the word efficiently rather than the word fast in NCTM's position statement on fluency. Years ago, I read an article written by Linda Gojak, NCTM President (2012), where she said, "​focusing on efficiency rather than speed means valuing students’ ability to use strategic thinking to carry out a computation without being hindered by many unnecessary or confusing steps in the solution process."

Speed. How would we go about determining if something is fast? Use a timer? But how do we determine if it is fast enough? On the other hand, the term efficient needs no specific context or timer to be clearly understood. To be efficient in math, we want our students to find ways of solving problems that waste the least amount of time and effort; after all, mathematics was designed to be used as a tool to help us in our daily life, not an end unto itself. One example of how we support the notion of efficient in our county's elementary math program is by considering some of the tasks contained within our Dreambox program. Unlike many math video games, students must choose good strategies for computing answers and often are challenged to find the most efficient way to solve the problem before the game allows them to move to the next level. In our classrooms, we should be looking at these same practices to help our students strengthen their fluency skills.

- Another Way of Looking at Fluency -

At a math conference I attended a few years back, the speaker asked the room of elementary teachers to consider how we assess reading in the elementary classroom:


Think about how you assess a student’s reading ability: 

1. Do you time students to see how many words they can read correctly in a specified amount of time? (yes)
2. Do you listen and observe as students read? (yes)
3. Do you ask questions to see if students understand what they’re reading? (yes)

​
Now imagine that you ONLY used timed tests to assess reading: 

1. Do you time students to see how many words they can read correctly in a specified amount of time?

-Assessing Fluency-

Attaining fluency and assessing fluency are related, but should never be confused as being synonymous. We cannot teach students to become mathematically fluent by testing them over and over again and expecting different results each time without incorporating fluency instruction and practice that is embedded in comprehension tasks. Assessing fluency allows us to collect data that sheds light on our students' progression toward greater fluency in the Standards. As we collect this data, we should be continually creating and modifying our instructional plans to help our students attain greater mathematical fluency by deepening their understandings of the math concepts and patterns.