January 2025
As math teachers, we strive to help our students develop procedural fluency, a skill often associated with accuracy. When concerns arise about students’ mathematical fluency, they are often centered on automaticity—“the ability to complete a task with little or no attention to process” (Bay-Williams & SanGiovanni, 2021, p. 108). This emphasis stems largely from the historical value placed on these skills in standardized testing. Fluency has often been measured through timed assessments that prioritize speed, computation, and quick recall. However, these measures fail to fully reflect a student’s depth of understanding and can negatively affect their mathematical identity. Real fluency is more than just students being able to correctly answer questions; a student’s conceptual understanding is evidenced through their reasoning and the justification of their strategy and answer. Effective teaching of procedural fluency positions students as capable, with reasoning and decision-making at the core of instruction.
What is Procedural Fluency?
Before moving forward, let’s take a moment to refresh our understanding of what procedural fluency means and why it is important in the math classroom. One of NCTM’s 8 Effective Mathematics Teaching Practices includes “Building Procedural Fluency From Conceptual Understanding.” NCTM defines Procedural Fluency as “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).
So how do you see this in the classroom? In the book Figuring Out Fluency in Mathematics Teaching and Learning, authors Jennifer Bay Williams and John J. SanGiovanni break down three fluency components (efficiency, flexibility, and accuracy) into six observable student actions:
- Selects an appropriate strategy.
- Solves in a reasonable amount of time.
- Trades out or adapts strategies.
- Applies a strategy to a new problem type.
- Completes steps accurately.
- Gets correct answer.
Follow this link to see an image from Bay-Williams and SanGiovanni that shows how efficiency, flexibility, and accuracy are interconnected, and how attending to all three is crucial for achieving true fluency. The six observable student actions highlight the ways students can actively engage in these fluency components. Together, these actions position students as independent learners who are capable of checking for reasonableness—a skill that goes far beyond the classroom. So how do you attend to these in your student-centered CPM classroom? Let’s break them down one at a time.
Procedural Fluency and Flexibility
To support flexibility, we must provide students with opportunities to use their own reasoning strategies and methods for solving problems. This begins with strong conceptual understanding, which is built into CPM’s purposeful design with mixed, spaced practice.
The problems at the beginning of each chapter in each CPM course provide opportunities for students to explore concepts collaboratively, giving them the space to develop informal strategies while justifying their reasoning. Over time, these informal strategies are refined into formal strategies that can be applied to new types of problems or situations. (Be mindful of their progress so that we do not assess skills too early!) As you plan, look for opportunities where students can explain the thinking behind their strategies. Pocket questions such as, “Can you solve the problem in a different way?”, “Why does this strategy work?”, and “Does this method always work?” can give students an extra nudge to think more deeply about their approaches.
Study Team and Teaching Strategies (STTS) structure time for students to listen to each others’ thinking, justify their reasoning, and think flexibly. Using a Pairs Check allows students to reflect on and adapt their strategies as they share their thinking with a partner. Another way to do this is through Math Chats. These include routines such as Number Talks, Dot Talks, Data Chats, and Which One Is Unique? Consider providing opportunities for students to reflect on their strategies by asking questions such as, “Is the answer I got reasonable?”, “Should I trade out my strategy?”, and “How might I adapt my strategy?” This reflection can take place as students are working with their teams or during lesson Closure. (Follow this link to hear from a classroom teacher on her experience with Math Chats.)
Procedural Fluency and Efficiency
Efficiency is closely tied to flexibility. When students practice flexibility and get opportunities to solve problems in multiple ways (by trading out and adapting strategies), they develop the understanding and confidence needed to determine which strategies are appropriate for a given situation. This process takes time, and it can be tempting to rush students through their decision-making process or push them toward the standard algorithm. However, the standard algorithm is not always the most efficient method and might not make sense to a student—leading them to mimic and not think critically.
After students have had time to approach a situation in multiple ways, we want to narrow in on their decision-making when it comes to navigating multiple strategies. Ask students to explain why they chose a specific strategy. Have students consider: “Is it reasonable to use the standard algorithm or is there a more efficient method?” To support efficiency, utilize Study Team and Teaching Strategies like Reciprocal Teaching or Proximity Partner to help students to compare their strategies. During the lesson Closure, select and sequence different strategies, allowing students to connect and compare them. Ask questions such as, “Why did you choose that strategy?”, “How are these strategies similar/different?”, and “Is one strategy more efficient than the other?” Reflection journals are also a great place for students to process which strategies make the most sense to them and why.
Avoid funneling the conversation toward what works best for you personally. Instead, give students space to navigate their own preferences.
Procedural Fluency and Accuracy
Accuracy is important, and it is essential to support students in completing steps correctly and reaching right answers. However, be cautious about emphasizing accuracy over flexibility and efficiency, as this can lead to negative dispositions toward math. It is not uncommon for students’ anxiety to increase when teachers prioritize standard algorithms and memorization techniques. When students struggle to memorize someone else’s method, they may disengage and lose confidence. Lean into the moments when students confidently use strategies other than the standard algorithm. When discussing answers, focus on understanding the process rather than memorizing steps, and reassure students that different paths to the same solution can be equally valid.
How you teach procedural fluency either supports equitable learning or prevents it (See NCTM’s position paper). When we balance our focus across all three components of fluency, we position students as capable decision makers. They are able to stop asking, “What did my teacher do to solve a problem like this?” and instead ask themselves, “Which of the strategies that I know is a good fit?” (NCTM, 2023). NCTM reminds us that “Procedural fluency is an attainable goal for each and every student. All students are capable of developing a repertoire of strategies and learning skills to apply those strategies flexibly, efficiently, and accurately.”
Well-balanced fluency instruction, addressing all six fluency actions, supports equitable learning by countering negative dispositions towards math and helping students confidently approach problems in ways that make sense to them. It is important to reflect on the following: Which fluency actions are emphasized in your classroom? Which fluency actions tend to be overlooked?
As you plan your next lesson, ask yourself whether there are moments for students to practice fluency actions. Which fluency action is this lesson supporting students on?
Note: This is not a comprehensive overview of fluency. For a deeper understanding, refer to Figuring Out Fluency in Mathematics Teaching and Learning (Bay-Williams & SanGiovanni, 2021). To explore fluency in a CPM classroom, reach out about our one-day learning event, Developing Procedural Fluency from Conceptual Understanding.
