Rethinking Math Fluency: Empowering Students Through Strategic Reasoning

When most educators hear the term math fluency, images of timed tests and rapid-fire fact recall often come to mind. But according to Savvas author and international leader in math education Dr. Jennifer Bay-Williams, this simplified definition doesn’t do justice to what fluency actually entails, or what students truly need to become confident, competent problem-solvers.

In her work with teachers, curriculum developers, and researchers, Bay-Williams has helped reshape the national conversation around fluency by emphasizing its three key components: efficiency, accuracy, and flexibility. She explains that true math fluency is not about memorizing procedures, but about equipping students with the tools and the confidence to make smart choices when solving problems.

Let’s explore what math fluency really means and how educators can create learning environments that nurture strategic, thoughtful, and empowered mathematicians.

What Is Fluency in Math? What Is Procedural Fluency? What’s the Difference?

Fluency in math, much like fluency in reading or language, goes beyond speed. In order to fully understand fluency in math, however, we must first clear up a common misconception: while math fact fluency and procedural fluency are related terms, they are distinctly different concepts within mathematics. In order to apply fluency to instruction effectively, it’s important to know the difference.

Math fact fluency refers to the ability to recall basic math facts, which are single digit operations (addition, subtraction, multiplication, and division) such as 7 + 8 accurately and without needing to consciously think through each step (automatic).

Procedural fluency is defined by the National Research Council as the ability to carry out procedures efficiently, accurately, and flexibly, and to select appropriate strategies for the task at hand. Procedural fluency includes basic facts, but also whole numbers, fractions, and decimal operations, as well as other procedures like solving for x in an algebraic equation.

Bay-Williams points out that efficiency is frequently equated with being fast, which can pressure students to prioritize speed over understanding. Instead, efficiency is about choosing a strategy for solving a problem that makes the most sense and is the most efficient for the numbers involved.

For example, if you need to solve 8 × 7, instead of counting by 7s eight times, you can break it into known facts. For example, you might know 8 × 5 = 40 and 8 × 2 = 16, then just add them: 40 + 16 = 56. This is more efficient than skip counting.

Flexibility, meanwhile, is at the heart of fluency. A student fluent in math knows how to assess a problem and select a strategy that works best, rather than blindly following one method. And accuracy isn’t just about getting the right answer; it’s also about executing the chosen strategy correctly, with attention to process and logic.

Another misconception Bay-Williams addresses is that fluency only applies to basic facts, such as single-digit addition, subtraction, multiplication, and division. While foundational fact fluency is important, computational fluency goes much further, extending into whole numbers, fractions, decimals, integers, and algebraic thinking.

For instance, students should be fluent in adding fractions not just through memorized procedures but by reasoning, such as moving fractional parts to make wholes or recognizing equivalent forms. Yet, according to Bay-Williams, we often see less reasoning with fractions and decimals, partly because current standards don’t explicitly call for it. Still, educators can create room for reasoning within their instruction, using real-world contexts like pizza slices or egg cartons to model and discuss strategies.

Math Fluency Practice

Bay-Williams emphasizes that students must build a repertoire of strategies and understand when and how to use them. Among the most critical strategies for addition and subtraction of whole numbers are:

• Counting on from a larger number
• Making tens or hundreds by adjusting numbers to easier benchmarks
• Compensation, or temporarily changing numbers to simplify computation
• Partial sums and differences to break numbers down by place value

These strategies should be more than isolated techniques, they should be accessible tools students can choose from. The more practice students have choosing the strategies that work best for them, the more confident and competent they become.

Bay-Williams also highlights the role of visual tools in supporting fluency. Open number lines, for example, help students estimate, visualize jumps, and make sense of subtraction as either removal or comparison. The use of a bottom-up 100 chart is another example of aligning visual tools with mathematical reasoning, helping students connect language (e.g., “going up by 10”) with visual representations that make sense.

Teaching for Strategic Choice

It’s not enough for students to learn strategies. They need experience choosing them. Bay-Williams advocates for skill-building lesson structures that provide opportunities for students to explore a problem using different approaches, have ample practice applying them, and receive opportunities to discuss and reflect on their decisions.

This process builds metacognition, which is the ability to think about one’s own thinking. Students become aware not only of what strategies are available but why they might be preferable in certain contexts. For example, in an addition problem where one number is near a multiple of 100 (e.g., 295 + 148), compensation might be more efficient than breaking the numbers into partial sums.

Compare the following:

• Compensation: Think 300 + 148. That equals 448. Take away the 5 you added. Answer is 443.
• Partial sums: (200 + 100) + (90 + 40) + (5 + 8) = 300 + 130 + 13 = 443

A component of fluency is being about to shift strategies mid-problem when a first approach isn’t working. For example, in the problem above, a student might start to do partial sums, notice that there are a lot of steps and regrouping, and then notice that 295 is close to 300 so they switch to use compensation. According to Bay-Williams, struggling students often overcommit to the one method they know, becoming stuck instead of trying a different strategy. Building flexibility means helping students recognize when to pivot and try something new.

Routines and Games That Build Fluency

Bay-Williams is also an advocate of using routines and games in building fluency. Effective routines offer repeated exposure to strategic thinking without becoming stale. By rotating routines throughout the year, for example, introducing one, retiring another, and revisiting familiar ones, teachers keep students engaged while reinforcing important skills.

Some examples include:

• Math Talks: Brief, focused discussions around solving problems using different strategies
• Number Strings: Sequences of related problems that help students notice patterns and build reasoning
• Strategize First Steps: Routines where students brainstorm possible first steps before solving a problem
• Fluency Games: Interactive experiences like “Show How You Know Bingo,” where students solve problems using chosen representations or strategies to claim spaces

These experiences aren’t just engaging, they’re essential for helping students internalize the strategies and reasoning they’ve learned and to build confidence in using them.

Teach for Thoughtfulness, Not Speed

According to Bay-Williams, math fluency is not about producing fast calculators. It’s about empowering students to approach problems thoughtfully, choose strategies wisely, and communicate their reasoning clearly. It’s about valuing how students solve problems as much as their answers.

Math educators have an opportunity to reframe fluency in their classrooms. By emphasizing strategic reasoning over rote memorization, by using routines that build flexibility and accuracy, and by fostering environments where students feel safe to try, reflect, and adjust, teachers can help every student become a confident and capable mathematician.