Position of the National Council of Teachers of Mathematics
What is procedural fluency, and how do we help
students develop it?
Procedural fluency is a
critical component of mathematical proficiency. Procedural fluency is the
ability to apply procedures accurately, efficiently, and flexibly; 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. To develop procedural fluency, students need
experience in integrating concepts and procedures and building on familiar
procedures as they create their own informal strategies and procedures. Students need opportunities to justify both informal
strategies and commonly used procedures mathematically, to support and
justify their choices of appropriate procedures, and to strengthen their
understanding and skill through distributed practice.
Procedural fluency is more than memorizing facts or
procedures, and it is more than understanding and being able to use one
procedure for a given situation. Procedural fluency builds on a foundation of
conceptual understanding, strategic reasoning, and problem solving (NGA Center
& CCSSO, 2010; NCTM, 2000, 2014). Research suggests that once students have
memorized and practiced procedures that they do not understand, they have less
motivation to understand their meaning or the reasoning behind them (Hiebert, 1999).
Therefore, the development of students’ conceptual understanding of procedures
should precede and coincide with instruction on procedures. Although conceptual
knowledge is an essential foundation, procedural knowledge is important in its
own right. All students need to have a deep and flexible knowledge of a variety
of procedures, along with an ability to make critical judgments about which
procedures or strategies are appropriate for use in particular situations (NRC,
2001, 2005, 2012; Star, 2005).
In computation, procedural fluency
supports students’ analysis of their own and others’ calculation methods, such
as written procedures and mental methods for the four arithmetic operations, as
well as their own and others’ use of tools like
calculators, computers, and manipulative materials (NRC, 2001).
Procedural fluency extends students’ computational fluency and applies in all
strands of mathematics. For example, in algebra, students develop general
equation-solving procedures that apply to classes of problems and select
efficient procedures to use in solving specific problems. In geometry,
procedural fluency might be evident in students’ ability to apply and analyze a
series of geometric transformations or in their ability to perform the steps in
the measurement process accurately and efficiently.
Procedural fluency builds from an initial exploration and
discussion of number concepts to using informal reasoning strategies and the
properties of operations to develop general methods for solving problems (NCTM,
2014). Effective teaching practices provide experiences that help students to connect
procedures with the underlying concepts and provide students with opportunities
to rehearse or practice strategies and to justify their procedures. Practice
should be brief, engaging, purposeful, and distributed (Rohrer, 2009). Too much
practice too soon can be ineffective or lead to math anxiety (Isaacs &
Carroll, 1999). Analyzing students’ procedures often reveals insights and
misunderstandings that help teachers in planning next steps in instruction. In the
same way, worked examples can serve as a valuable instructional tool,
permitting teachers to understand how students analyze why procedures work or
don’t work and consider what procedure might be most appropriate in a given
situation (Booth, Lange, Koedinger, & Newton, 2013).
Booth, J. L., Lange,
K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to
improve student learning in algebra: Differentiating between correct and
incorrect examples. Learning and Instruction, 25, 24–34.
Hiebert, J. (1999). Relationships between research and the
NCTM standards. Journal for Research in
Mathematics Education, 30(1), 3–19.
Isaacs, A. C., & Carroll, W.
M. (1999). Strategies for basic-facts instruction. Teaching Children Mathematics, 5(9), 508–515.
King, G., & Bay-Williams, J.
M. (2014). Assessing basic fact fluency. Teaching
Children Mathematics, 20(8), 488–497.
Lange, K. E., Booth, J. L.,
& Newton, K. J. (2014). Learning algebra from worked examples. Mathematics Teacher, 107(7), 535–540.
National Council of Teachers of
Mathematics. (2000). Principles and standards
for school mathematics. Reston VA: Author.
National Council of Teachers of
Mathematics. (2014). Principles to actions:
Ensuring mathematical success for all. Reston VA: Author.
National Governors Association
Center for Best Practices & Council of Chief State School Officers. (2010).
Common core state standards for mathematics.
Common core state standards (college- and career-readiness standards and K–12 standards
in English language arts and math). Washington, DC: Author. http://www.corestandards.org.
National Research Council
(2001). Adding it up: Helping children learn
mathematics. Washington, DC: National Academies Press.
National Research Council.
(2005). How students learn: History, mathematics,
and science in the classroom. Washington, DC: National Academies Press.
National Research Council.
(2012). Education for life and work:
Developing transferable knowledge and skills for the 21st century.
Washington, DC: National Academies Press.
Rohrer, D. (2009). The effects
of spacing and mixed practice problems. Journal
for Research in Mathematics Education,
Russell, S. J. (2000).
Developing computational fluency with whole numbers. Teaching Childdren Mathematics, 7(3), 154–158.
Star, J. R. (2005).
Reconceptualizing conceptual knowledge. Journal
for Research in Mathematics Education, 36(5), 404–411.