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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.
- 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.
- 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.
- 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).
- 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 and
Additional Resources
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.
Additional Resources
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.