Procedural Fluency in Mathematics

• A Position of the National Council of Teachers of Mathematics

Reasoning and Decision-Making, Not Rote Application of Procedures Position

NCTM Position

Procedural fluency is an essential component of equitable teaching and is necessary to developing mathematical proficiency and mathematical agency. Each and every student must have access to teaching that connects concepts to procedures, explicitly develops a reasonable repertoire of strategies and algorithms, provides substantial opportunities for students to learn to choose from among the strategies and algorithms in their repertoire, and implements assessment practices that attend to all components of fluency.

## Introduction

Procedural fluency can be accomplished only when fluency is clearly defined and intentionally developed. Unfortunately, the term fluency continues to be (incorrectly) interpreted as remembering facts and applying standard algorithms or procedures. 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). For example, to add 98 + 35, a person might add 100 + 35 and subtract 2 or change the problem to 100 + 33. Procedural fluency applies to the four operations and other procedures in the K–12 curriculum, such as solving equations for an unknown. For example, to solve for x in the equation 4(x + 2) = 12, an efficient strategy is to use relational thinking, noticing that the quantity inside the parenthesis equals 3 and therefore x equals 1. As these examples illustrate, flexibility is a major goal of fluency, because a good strategy for one problem may or may not be as effective for another problem.

## Declarations

The following declarations describe necessary actions to ensure that every student has access to and develops procedural fluency. These declarations apply to computational fluency across the K–12 curriculum, including basic facts, multidigit whole numbers, and rational numbers, as well as to other procedures throughout the curriculum such as comparing fractions, solving proportions or equations, and analyzing geometric transformations.

1. Conceptual understanding must precede and coincide with instruction on procedures. Learning is supported when instruction on procedures and concepts is explicitly connected in ways that make sense to students (e.g., Fuson, Kalchman, and Bransford 2005; Hiebert and Grouws 2007; Osana and Pitsolantis 2013) and iterative (e.g., Canobi 2009; Rittle-Johnson, Schneider, and Star 2015). Conceptual foundations lead to opportunities to develop reasoning strategies, which in turn deepens conceptual understanding; memorizing an algorithm does not. When students use a procedure they do not understand, they are more likely to make errors and fail to notice when the answer does not make sense (Kamii and Dominick 1998; Narode, Board, and Davenport 1993). Examples of explicitly connecting procedures and concepts can be found in the Additional Resources section.
2. Procedural fluency requires having a repertoire of strategies. Before students can flexibly choose an appropriate strategy, they must have strategies from which to choose. Strategies are flexible ways to solve a problem (e.g., compensation); algorithms are step-by-step procedures. Although both are important in mathematics, strategies should not be presented as rigid, step-by-step processes. Students should be able to flexibly use and adapt strategies and switch to a different strategy when their first choice is not working well (NCTM 2020). Every student must have the opportunity to learn more than one method. Limiting students to only one method puts them at a disadvantage, denying them access to more intuitive methods and the opportunity to flexibly choose a method that fits the problem at hand.
3. Basic facts should be taught using number relationships and reasoning strategies, not memorization. Students who learn fact strategies outperform students who learn through other approaches (e.g., Baroody et al. 2016; Henry and Brown 2008; Brendefur et al. 2015). Basic fact strategies use number relationships and benchmarks and thus support students, emerging conceptual understanding and flexibility (Bay-Williams and Kling 2019; Davenport et al. 2019). Strategies such as Making 10 build a foundation for strategies beyond basic facts, such as Make-a-Whole with fractions and decimals (Bay-Williams and SanGiovanni 2021).
4. Assessing must attend to fluency components and the learner. Assessments often assess accuracy, neglecting efficiency and flexibility. Timed tests do not assess fluency and can negatively affect students, and thus should be avoided (Boaler 2014; Kling and Bay-Williams 2021; NCTM 2020; Ramirez, Shaw, and Maloney 2018). Alternatives include interviews, observations, and written prompts.

The way in which fluency is taught either supports equitable learning or prevents it. Effective teaching of procedural fluency positions students as capable, with reasoning and decision-making at the core of instruction. When such teaching is in place, students stop asking themselves, “How did my teacher show me how to do this?” and instead ask, “Which of the strategies that I know are a good fit for this problem?” The latter question is evidence of the student’s procedural fluency and mathematical agency, critical outcomes in K–12 mathematics.

# References

Baroody, Arthur J., David J. Purpura, Michael D. Eiland, Erin E. Reid, and Veena Paliwal. 2016. “Does Fostering Reasoning Strategies for Relatively Difficult Basic Combinations Promote Transfer by K–3 Students?” Journal of Educational Psychology 108, no. 4 (May): 576–91.

Bay-Williams, Jennifer M., and Gina Kling. 2019. Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention. Alexandria, VA: ASCD.

Bay-Williams, Jennifer M., and John J. SanGiovanni. 2021. Figuring Out Fluency in Mathematics Teaching and Learning, Grades K–8. Thousand Oaks, CA: Corwin.

Boaler, Jo. 2014. “Research Suggests That Timed Tests Cause Math Anxiety.” Teaching Children Mathematics 20, no. 8 (April): 469–74.

Brendefur, Jonathan, S. Strother, K. Thiede, and S. Appleton. 2015. “Developing Multiplication Fact Fluency.” Advances in Social Sciences Research Journal 2 (8): 142–54. https://doi.org/10.14738/assrj.28.1396.

Canobi, Katherine H. 2009. “Concept–Procedure Interactions in Children’s Addition and Subtraction.” Journal of Experimental Child Psychology 102, no. 2 (February): 131–49.

Davenport, Linda Ruiz, Connie S. Henry, Douglas H. Clements, and Julie Sarama. 2019. No More Fact Frenzy. Portsmouth, NH: Heinemann.

Fuson, Karen C., Mindy Kalchman, and John D. Bransford. 2005. “Mathematical Understanding: An Introduction.” In How Students Learn: History, Mathematics, and Science in the Classroom, edited by M. Suzanne Donovan and John D. Bransford, Committee on How People Learn: A Targeted Report for Teachers, National Research Council, pp. 217–56. Washington, DC: National Academies Press.

Henry, V., & Brown, R. 2008. “First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard.” Journal for Research in Mathematics Education, 39(2), 153-183.

Hiebert, James, and Douglas A. Grouws. 2007. “The Effects of Classroom Mathematics Teaching on Students’ Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 371–404. Charlotte, NC: Information Age.

Kamii, Constance, and Ann Dominick. 1998. “The Harmful Effects of Algorithms in Grades 1–4.” In The Teaching and Learning of Algorithms in School Mathematics, edited by L. Morrow, pp. 130–40. Reston, VA: National Council of Teachers of Mathematics.

Kling, Gina, and Jennifer M. Bay-Williams. 2021. “Eight Unproductive Practices in Developing Fact Fluency.” Mathematics Teacher: Learning and Teaching PK–12 114, no. 11 (November): 830–38.

National Council of Teachers of Mathematics (NCTM). 2014. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM.

National Council of Teachers of Mathematics (NCTM). 2020. Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.

National Research Council (NRC). 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press.

National Research Council (NRC). 2005. How Students Learn: History, Mathematics, and Science in the Classroom. Washington, DC: National Academies Press.

National Research Council (NRC). 2012. Education for Life and Work: Developing Transferable Knowledge and Skills for the 21st Century. Washington, DC: National Academies Press.

Narode, Ronald, Jill Board, and Linda Ruiz Davenport. 1993. “Algorithms Supplant Understanding: Case Studies of Primary Students’ Strategies for Double-Digit Addition and Subtraction.” Proceedings of the 15th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1), pp. 254–60. San Jose, CA: Center for Mathematics and Computer Science Education, San Jose State University.

Osana, Helen P., and Nicole Pitsolantis. 2013. “Addressing the Struggle to Link Form and Understanding in Fractions Instruction.” British Journal of Educational Psychology 83 (March): 29–56. https://doi.org/10.1111/j.2044-8279.2011.02053.x.

Ramirez, Gerardo, Stacy T. Shaw, and Erin A. Maloney. 2018. “Math Anxiety: Past Research, Promising Interventions, and a New Interpretation Framework.” Educational Psychologist 53, no. 3 (April): 145–64. https://doi.org/10.1080/00461520.2018.1447384.

Rittle-Johnson, Bethany, Michael Schneider, and Jon R. Star. 2015. “Not a One-Way Street: Bidirectional Relations between Procedural and Conceptual Knowledge of Mathematics.” Educational Psychology Review 27, no. 4 (March): 587–97. https://doi.org/10.1007/s10648-015–9302-x.

Star, Jon R. 2005. “Reconceptualizing Conceptual Knowledge.” Journal for Research in Mathematics Education 36, no. 5 (November): 404–11.

Bay-Williams, Jennifer M., John J. SanGiovanni, Sherri M. Martinie, and Jennifer Suh. 2022. Figuring Out Fluency: Addition and Subtraction with Fractions and Decimals. Thousand Oaks, CA: Corwin.

Bay-Williams, Jennifer M., John J. SanGiovanni, Sherri M. Martinie, and Jennifer Suh. 2022. Figuring Out Fluency: Multiplication and Division with Fractions and Decimals. Thousand Oaks, CA: Corwin.

Bay-Williams, Jennifer M., John J. SanGiovanni, C. D. Walters, and Sherri M. Martinie. 2023. Figuring Out Fluency: Operations with Rational Numbers and Algebraic Equations. Thousand Oaks, CA: Corwin.

Booth, Julie L., Karin E. Lange, Kenneth R. Koedinger, and Kristie J. Newton. 2013. “Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples.” Learning and Instruction 25 (June): 24–34. https://doi.org/10.1016/j.learninstruc.2012.11.002.

Cardon, Tina, and the MTBoS. 2015. Nix the Tricks: A Guide to Avoiding Shortcuts That Cut Out Math Concept Development. 2nd ed. https://nixthetricks.com/.

Renkl, A. 2014. “Learning from Worked Examples: How to Prepare Students for Meaningful Problem Solving.” In Applying Science of Learning in Education: Infusing Psychological Science into the Curriculum, edited by V. Benassi, C. E. Overson, and C. M. Hakala, pp. 118–30. http://teachpsych.org/ebooks/asle2014/index.php.

Schifter, Deborah, Virginia Bastable, and Susan Jo Russell. 2016a. Developing Mathematical Ideas Casebooks Facilitators Guides, and Video for Building a System of Tens in The Domains of Whole Numbers and Decimals. Reston, VA: National Council of Teachers of Mathematics.

Schifter, Deborah, Virginia Bastable, and Susan Jo Russell. 2016b. Developing Mathematical Ideas Casebooks, Facilitators Guides, and Video for Making Meaning for Operations in the Domains of Whole Numbers and Fractions. Reston, VA: National Council of Teachers of Mathematics.

Schifter, Deborah, Virginia Bastable, and Susan Jo Russell. 2018. Developing Mathematical Ideas Casebook, Facilitator’s Guide, and Video for Reasoning Algebraically about Operations. Reston, VA: National Council of Teachers of Mathematics.

Star, Jon R., and Lieven Verschaffel. 2016. “Providing Support for Student Sense Making: Recommendations from Cognitive Science for the Teaching of Mathematics.” In Compendium for Research in Mathematics Education, edited by Jinfa Cai. Reston, VA: National Council of Teachers of Mathematics.

January 2023