When you hear someone say memorization, knowing your facts, following the steps, and procedural fluency, do each of these terms have the same meaning?

When when it comes to describing procedural fluency, we often hear educators (and parents) using these terms interchangeably.

However, this is definitely NOT the case.

Being fluent in any area of mathematics means being:

• • efficient;
  • flexible; and,
  • accurate.

(Kilpatrick et al., 2001; National Council of Teachers of Mathematics (NCTM), 2014). 

Moreover, last week NCTM released a clarifying statement about their position on procedural fluency. 

“Procedural fluency is the ability to apply procedures efficiently, flexibly, and accurately; 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.”

(NCTM 2014, 2020; National Research Council 2001, 2005, 2012; Star 2005).

NCTM has included four (4) declarations that are necessary so that every student has access to developing procedural fluency: 

• Conceptual understanding must precede and coincide with instruction on procedures;
• Procedural fluency requires having a repertoire of strategies;
• Basic facts should be taught using number relationships and reasoning strategies, not memorization; and,
• Assessing must attend to fluency components and the learner. Assessments often assess accuracy, neglecting efficiency and flexibility.

Procedural fluency is much more than just knowing basic facts and algorithms and…

…teaching to develop true procedural fluency is so much harder to teach!

Here’s a question we’ve been asking ourselves more and more here at Make Math Moments:

“How can we help our students become more fluent and flexible in ______________________?”

At Make Math Moments, we believe that your mathematics classroom must be cultivated and fostered like a strong, healthy and balanced tree. To grow our own tree we must care and strengthen six (6) key areas:

• • The Trunk: Our Classroom Culture & Norms;
  • The Roots: Educator Mathematics Content Knowledge & Proficiency;
  • Soil, Water, Sunlight: Our Mindsets & Beliefs;
  • The Limbs: Educator (and District) Professional Learning Plans;
  • The Branches: Our Pedagogical Content Knowledge; and,
  • The Leaves: The Tools and Resources We Use as well as Our Classroom Environment.

Strengthening how we teach for the development of procedural fluency from conceptual understanding (as shared in NCTM’s guidelines above) means we will need to strengthen  all key areas of our tree, but in particular, the Roots, Branches and Leaves.

Why?

In order to develop the root system of our mathematics program, we must strengthen our own conceptual understanding in order to better scaffold our instruction to ensure we meet students where they are and push them along their developmental continuum.

As we continue to develop our own root system of mathematical content knowledge, we must also be strengthening the branches of our mathematics program. Only when we have the teacher moves and pedagogical strategies that promote learning will the work we do with students have the effect on fluency and flexibility we are striving for.

Of course, as our mathematical content knowledge and pedagogical practice evolve, we must also work towards growing the leaves of our tree by seeking out the lessons and resources that are most appropriate to bring out the big ideas, strategies and models we must highlight as we push each student further down the path towards procedural fluency.

Lessons With Fluency At The Core

Here are lesson examples that help you develop and strengthen your Roots, Branches, and Leaves. Each of these examples shows you how you can teach your standards while building procedural fluency as described by NCTM:

• • Teach The Standards To Build Fluency and Flexibility With Number and Operations (A Classroom Sneak Peek).
  • Building Multiplication Fluency Through Ratios, Rates, and Proportional Reasoning.
  • How To Teach Percent To Build Fluency.
  • 3 Models For Teaching 2-Step Equations.

“The way in which fluency is taught either supports equitable learning or prevents it.” ~ NCTM

If you’re interested in learning where your own mathematics program tree is flourishing and where you might focus your attention next, be sure to complete our Mathematics Program Assessment Screener.

With it, you’ll receive a customized report highlighting some immediate actions you can take to strengthen the area of your mathematics program you’ll see the most benefit from.

If you’re a mathematics leader in your district/organization, you can hop to the version designed specifically for your role as a mathematics decision-maker.

Let’s aim to put fluency at the core of our lessons each and every day.