Have you ever seen students solve a problem confidently one day, only to stumble the next when the numbers looked different? I saw this often in my classroom. Students had memorized steps, but when the context changed, they could no longer apply what they knew. As a teacher and researcher, I found this was a common challenge in classrooms across the globe. Memorization without flexibility is fragile.
True fluency is not about recalling facts on demand; it’s an ongoing process of setting the right conditions for learning so that understanding grows. For students, procedural fluency is the ability to use procedures accurately, efficiently, and flexibly. Fluency isn’t about racing through drills; it’s about giving students a toolkit of strategies they can adapt with confidence.
Marcus’s Story
Marcus, a Grade 4 student, knew his multiplication tables by heart. But when asked to solve 15 × 12, he froze.
When his teacher introduced multiple strategies, Marcus realized he had choices:
- “15 × 10 = 150, plus 15 × 2 = 30. That makes 180.”
- Or: “Split 15 into 10 and 5. Then 10 × 12 = 120, and 5 × 12 = 60. Add them: 180.”
For Marcus, the breakthrough was knowing there wasn’t just one “right” way and having options built his confidence. That is procedural fluency in action: students solve problems by seeking patterns and using multiple strategies, not just memorized steps. Being good at math depends more on reasoning than on speed or working memory alone.
Global Perspectives
The National Council of Teachers of Mathematics (NCTM, 2024) defines procedural fluency as accuracy, efficiency, and flexibility, not speed alone. The Education Endowment Foundation (2025) indicates that fluency frees up working memory, enabling students to reason and solve problems more easily. Recent studies reaffirm the five strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Corrêa & Haslam, 2021; NRICH, 2025). Fluency is most powerful when developed alongside the other strands, helping students build both skill and confidence.
Across the world, research points to the same lesson: fluency must connect to understanding. Here in Australia, the Mathematics Association of Victoria (2025) has emphasized that fluency is not about speed, but about efficiency with understanding. They encourage teachers to balance fluency with reasoning and problem-solving, highlighting how students who engage with multiple strategies become more resilient and confident mathematicians. The OECD’s PISA 2022 report found that students who combined fluency with conceptual understanding scored far higher than those who relied only on memorization, and felt more confident tackling unfamiliar problems. Like other major global studies, the OECD’s PISA 2022 report shared that fluency grows best when paired with reasoning and understanding. Four principles for teachers that guide effective fluency:
- Start with understanding. Show why procedures work.
- Model multiple strategies. Normalize flexibility.
- Value efficiency, not just speed. Quick answers don’t always mean fluent thinking.
- Design purposeful practice. Use tasks that highlight connections and invite reasoning.
Fluency develops through the principles above, alongside frameworks such as the SOLO Taxonomy, Depth of Knowledge, or Bloom’s Taxonomy, can also help teachers to plan, design, and implement procedural fluency and provide a language of learning for both teachers and students.
Phase 1: Diagnose and Build Understanding
- Verbs: recall, identify, define (Bloom’s remember; DOK 1; SOLO unistructural).
- Use number talks and diagnostics to see who relies only on memorization.
- Link procedures to visuals and models to ground understanding.
Phase 2: Introduce and Compare Strategies
- Verbs: describe, explain, compare (Bloom’s understand/analyze; DOK 2–3; SOLO Multistructural).
- Model multiple methods, then ask: “Which is most efficient? Which is clearest?”
- Build norms where flexibility is valued more than speed.
Phase 3: Purposeful Practice
- Verbs: apply, organize, analyze (Bloom’s apply/analyse; DOK 2–3; SOLO relational).
- Provide varied tasks where students must choose and adapt strategies.
- Use routines like Which One Doesn’t Belong? to spark discussion.
Phase 4: Apply and Extend
- Verbs: justify, generalize, adapt (Bloom’s evaluate/create; DOK 3–4; SOLO extended abstract).
- Embed fluency in problem-solving and real-world tasks.
- Use AI tools for adaptive practice that strengthen reasoning.
Phase 5: Assess and Reflect
- Verbs: explain, critique, justify (DOK 3–4).
- Move beyond speed tests, then ask: “Can students explain choices? Switch strategies?” Apply fluency in new contexts?
- Gather reflections: Which strategies felt most efficient? How has your thinking changed?
Signs of Fluency
Students are becoming fluent when they:
- Choose strategies that fit the task.
- Switch methods when one doesn’t work.
- Explain their reasoning clearly.
- Apply procedures in new contexts.
- Self-correct through understanding.
- Approach unfamiliar problems with confidence.
- Try new approaches when they don’t know what to do.
Reflection for This Week
- Reflection: How have you approached fluency, through speed, understanding, or flexibility?
- Key Insight: Which idea about fluency resonated most with you?
- Commitment: What small change will you try in the next two weeks?
- Student Impact: How might this shift your students’ confidence?
Conclusion: Building Toward Problem Solving
Procedural fluency is more than quick answers. It equips students with strategies they can adapt and explain. When fluency develops alongside understanding, students are prepared for deeper reasoning and real-world application.
Join the Conversation
This blog is part of an ongoing conversation among educators. I’d love to hear from you:
- Share your reflections and stories using #MathSuccessJourney.
- Tag @CengageSchool, @BigIdeasLearning, and @SophieSpecjal
- Join the mid-series webinar for interactive Q&A and teacher spotlights.
- Contribute classroom examples for future spotlights by contacting me at [email protected]
Your experiences bring these strategies to life. Together, we can build a community where students develop fluency that goes beyond memorization, giving them the tools to reason and apply knowledge with confidence.
Looking Ahead
Problem-solving opens doors to curiosity and confidence. When we prioritize reasoning over steps, we prepare learners for challenges inside and outside the classroom. How are you making problem-solving central in your classroom?
Next blog: Strategy #5: Problem-Solving.
References
Bay-Williams, J. M., & SanGiovanni, J. J. (2021). Figuring out fluency in mathematics teaching and learning. NCTM.
Boylan, M., Zhu, H., Jaques, L., Birkhead, A., & Rempe-Gillen, E. (2024). Secondary maths: Practice review. Education Endowment Foundation. https://educationendowmentfoundation.org.uk
Corrêa, R., & Haslam, F. (2021). Mathematical proficiency as the basis for assessment: A literature review. Journal of Curriculum Studies, 53(6), 742–761.
Education Endowment Foundation. (2025). Mathematics evidence projects: Fluency and flexible learning with AI. https://www.hfleducation.org
Fisher, D., Frey, N., & Hattie, J. (2017). Visible learning for mathematics, grades K–12: What works best to optimize student learning. Corwin.
National Council of Teachers of Mathematics. (2024). Procedural fluency in mathematics: Position statement. https://www.nctm.org
NRICH. (2025). Five strands of mathematical proficiency. https://nrich.maths.org
OECD. (2023). PISA 2022 results (Volume 1): Learning during and from disruption. OECD Publishing. https://doi.org/10.1787/8f38d5c0-en