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Nine Pivotal Understandings for Numerical Fluency


Adapted from Developing Numerical Fluency by Patsy Kanter and Steven Leinwand

The teaching practices delineated in Principles to Actions (NCTM 2014, 42) call on us to “build procedural fluency from conceptual understanding.” But the question teachers face is what constitutes this “foundation of conceptual understanding” that enables students to “become skillful in using procedures flexibly as they solve contextual and mathematical problems?”

Download a Sample Chapter of Developing Numerical Fluency

We believe that teachers must embrace nine pivotal understandings to support the development of numerical fluency in all students.

These nine pivotal understandings are:

  1. Everything begins with counting.
  2. All quantities are composed of parts and wholes, meaning that all whole numbers greater than 1 can be decomposed into smaller whole numbers.
  3. Acquisition of the language of the four operations (joining, separating, sharing, combining, comparing, each group, groups of, more, more than, less, less than, etc.) must precede the learning of facts.
  4. Powerful properties of operations reduce memory load and contribute to
    numerical fluency.
  5. Facility using 5 and 10 in computation underpins numerical fluency.
  6. Deeply understanding that 9 and 10 − 1 are the same number supports numerical fluency with a range of so-called “hard” facts.
  7. Multiplication by 2, 3, 5, and 10 are the foundation for fluency with all multiplication and division facts.
  8. The notion of balance or equivalence in equations and the meaning of the equal sign are fundamental to comprehending operations and algebraic thinking.
  9. Place value understanding is essential to build fluency with larger numbers.

These nine pivotal ideas provide a cognitive foundation upon which children become numerically fluent. It should be clear at this point that these understandings cannot be taught in only a lesson or two. Rather, they must be integrated into ongoing activities and conversations and infused throughout instruction.


Learn more about Developing Numerical Fluency at Heinemann.com

Download a Sample Chapter of Developing Numerical Fluency


Patsy Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana, for 13 years. Patsy is the co-author of Every Day Counts: Calendar Math and a consulting author for Math in Focus.

Follow Patsy on Twitter @patsykanter



stevenleinwandSteve Leinwand is the author of the bestselling Heinemann title Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement.He is Principal Research Analyst at the American Institutes for Research in Washington, D.C., where he supports a range of mathematics education initiatives and research. Steve served as Mathematics Supervisor in the Connecticut Department of Education for twenty-two years and is a former president of the National Council of Supervisors of Mathematics.

Follow Steve on Twitter @steve_leinwand


Topics: Steve Leinwand, Developing Numerical Fluency, Patsy Kanter

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