Build Procedural Fluency from Conceptual Understanding
A Professional Development Approach
Cluster One: Instructional Practices – Planning & Assessment (Interpreting, Responding)
Two of the legs of the three-legged stool that constitutes mathematical rigor is Procedural Skill & Fluency and Conceptual Understanding. It is these two together that are now explored here. Does conceptual understanding arise from learning and practicing procedures, or does procedural fluency arise from conceptual understanding? This is a debate that still rages among educators. The former is the argument for direct instruction. The latter is the argument for problem-solving based classroom instruction. Heibert et al. (1997) captures this spectrum elegantly in arguing for how these two elements, conceptual and procedural, need to arise together.
|
In traditional systems of instruction, teachers often describe their mathematical goals by listing the skills and concepts they plan to teach. These goals then push teachers toward activities like demonstrating, explaining, showing, and so on because these are thought to be the best and clearest ways of teaching the skills and concepts. The trouble is that these plans often are based on objectives of covering material and do not consider carefully the kinds of residues that might get left behind. The plans often are not sensitive to what we know about how students construct understanding, and do not allow for the kind of reflection and communication that is essential. In contrast, there is growing interest in nontraditional curricula that are filled with interesting problems, often large-scale real-life tasks. Teachers are to present the problems to students and allow them to work. Sometimes the work on a task will extend over a period of days or even weeks. The goals include engaging students in doing mathematics and solving the problems. The goals are not lists of skills and concepts. Although such programs place a positive emphasis on respecting students' autonomy and respecting their intellectual capabilities, the content goals often are unspecified. It can be difficult for teachers to identify the mathematical goals that can and should be planned for and worked toward during the year. The system of instruction we are recommending takes a different approach than either of these two. As in the nontraditional approach, mathematics begins with problems. But, the system encourages teachers to use their learning goals for students, and their vision of how these goals might be achieved over time to select sequences of problems. Simon (1995) describes this vision as a "hypothetical learning trajectory." The trajectory is the teacher's vision of the mathematical path that the class might take, and its hypothetical nature comes from the fact that it is based on the teacher's guess about how learning might proceed along the path. The trajectory guides the teacher's task selection, but feedback from students and the teacher's assessments of the residues that are being formed lead to revisions in the trajectory. Tasks are selected purposefully, but the sequence can be revised. (pp. 33-34) |
NCTM in Principles to Actions (2014) cites various research views that clearly identifies that procedural fluency arises from having a solid conceptual understanding of the mathematics involved. The document quotes Martin (2009, p. 165) for some of this reasoning.
|
To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. |
With these two elements arising together, further defining of terms is necessary.
Defining Fluency
Procedural fluency is defined by NCTM in a 2023 position paper as,
|
Procedural fluency is the ability to apply procedures efficiently, flexibly, and accurately; to transfer procedure 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, 2021; Star 2005). |
The position paper is organized into four declarations capturing the combination of instructional stances that need to be in place to meet the conditions of fluency.
- Conceptual understanding must precede and coincide with instruction on procedures.
- Procedural fluency requires having a repertoire of strategies
- Single-digit combinations (Basic facts) should be taught using number relationships and reasoning strategies, not memorization
- Assessing must attend to fluency components and the learner. Assessments often assess accuracy, neglecting efficiency and flexibility
Defining Efficiency
Efficiency is not a synonym for speed. To be efficient, one needs to be able to draw upon a variety of strategies. To be efficient, one must be able to make judgments, based on the given context and given the numbers or spatial information within that context, which strategy would serve as the least amount of work to determine the solution. One can be very fast with an inefficient strategy such as counting on one’s fingers or skip counting through multiples of a number. Such strategies, while allowing for access in solving a task (Low-Floor), they remain entry level strategies.
Defining Proficiency
Mathematical proficiency develops over time. It is the integration of several attributes that combine to make informed, efficient, and fluent decisions. The National Research Council in 2001 in its seminal publication Adding It Up: Helping Children to Learn Mathematics (2001) defined five strands that comprise mathematical proficiency.
 Intertwined Strands of Proficiency, (NRC 2001, p. 117) |
Conceptual Understanding – Conceptual understanding refers to “an integrated and functional grasp of mathematical ideas” (NRC, 2001, p. 118). This includes knowing facts and methods, but also why those facts and methods are important and how they are connected. Students sometimes develop conceptual understanding before they can adequately verbalize it, so a teacher needs to look at how students represent mathematical situations. The ways students use and connect multiple representations of mathematical ideas are an indication of their conceptual understanding.
Procedural Fluency – Procedural fluency refers to “knowledge of procedures, knowledge of when and how to use them appropriately, and skills in performing them flexibly, accurately, and efficiently” (NRC, 2001, p. 121). Conceptual understanding and procedural fluency are sometimes seen as competing for attention when they are really two closely related, interwoven strands. Conceptual understanding makes learning skills easier, less error-prone, and easier to remember, while procedural fluency can help develop and strengthen conceptual understanding. Procedural fluency includes basic computation and estimation strategies, such as computing with whole numbers, multiplying by multiples of ten or powers of ten, and using mental strategies (like knowing that 98+37 is the same as 100+35) without the use of additional tools or technology.
Strategic Competence – Strategic competence refers to “the ability to formulate mathematical problems, represent them, and solve them” (NRC, 2001, p. 124). In school settings, students are given problems to solve and, quite often, the exact procedure needed to solve them. This is rarely how mathematics is encountered in the world, where often the greatest challenge is understanding what the problem is and deciding on a mathematical approach to solving it. To develop strategic competence, students need experience and practice with unfamiliar problem situations so they can formulate strategies and try different problem-solving approaches. This is often described as mathematical modeling, which is a process that “uses mathematics to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena” (COMAP & SIAM, 2016, p. 8).
Adaptive Reasoning – Adaptive reasoning refers to “the capacity to think logically about the relationships among concepts and situations” (NRC, 2001, p. 129). This includes being able to justify valid approaches and conclusions, and it connects the many facts, procedures, concepts, and methods in ways that make sense. Even the youngest students are capable of some level of reasoning, provided that they have sufficient knowledge, the task is understandable, and the context is familiar and motivating. Justifications in mathematics can be very formal, such as in rigorous proofs, or less formal, where students provide sufficient reasons for their choices and conclusions. Adaptive reasoning takes time to develop. Students need regular opportunities to talk about the concepts and procedures they are using and the reasons for what they are doing.
Productive Disposition – Productive disposition refers to “the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics” (NRC, 2001, p. 131). For students to develop conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning, they need a productive disposition towards mathematics. Likewise, students’ productive disposition will improve as they develop conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning. Most children enter school with positive attitudes towards mathematics, but to sustain those beliefs, students need reinforcement that math ability is not a fixed trait, that math is about reasoning and sense making, and that everyone – regardless of background – can be a capable doer of mathematics.
Resources
General
Building Fluency
- Building (Powerful) Numeracy for Middle + High School Students, Harris, 2011
- Learning Single-digit Combinations: Developing Important Mathematical Ideas, Brickwedde, 2022
References
Heibert, J., Carpenter, T.P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H. Olivier, A., and Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. pp. 29-41. Heinemann Press: Portsmouth, NH.
Martin, W. G. (2009) The NCTM High School Curriculum Project: Why It Matters to You. Mathematics Teacher, 103, no 3: 164–66.
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.
Star, Jon R. 2005. “Reconceptualizing Conceptual Knowledge.” Journal for Research in Mathematics Education 36, no. 5 (November): 404–11.
Search for a Framework
Search the Mathematics and Science Frameworks.
Begin your Search »
Feedback
We want your feedback! Tell us what's working, what's not, and how you are using these Frameworks.
Give us your Feedback »