In our February newsletter, we discussed the influential work of two British researchers, Eddie Gray and David Tall1.  They found that students who are low achievers are often learning a more difficult form of mathematics than their high achieving classmates. Gray and Tall found that the high-achieving students used derived facts (decomposing and recomposing numbers to make easier number combinations). For example, 4 + 13 can be thought of as 4 + [10 +3] = 10 + [4 + 3].

Jo Boaler2 points out that under-achieving students typically are put into classes where they are drilled on computational procedures. "This is the last thing these students need..." (Boaler, 2008, p. 152) Instead, low achieving students need practice in using numbers flexibly, which requires more of them than memorizing procedures. 

Both Principles and Standards for School Mathematics (NCTM 2000)3 and the NCTM Curriculum Focal Points4 emphasize the importance of computational fluency for all students. Computational fluency involves:

• Having efficient, flexible and accurate methods for computing
• Having a large repertoire of number relationships, including the addition and multiplication "facts" as well as understanding place value system relationships, such as how 4 x 5 is related to 4 x 50
• Understanding the meaning of operations and their relationships to each other—for example, the inverse relationship between multiplication and division
• Being fluent in mental math, paper and pencil methods and using technology such as a calculator
• Knowing which of those three methods is best for a given task
• Being able to determine if an exact answer or a close approximation (estimate) is sufficient

At the K-1 level, John Van de Walle and LouAnn Lovin5 encourage beginning to develop computational fluency with basic concepts such as:

• Relationships among numbers 1-10 and later among numbers 10-20
• Anchoring numbers to 5 and 10 by using the ten-frame
• Understanding numbers in terms of parts (e.g., “8” can be separated into 2 and 6 or 3 and 5, etc.)
• Developing a rudimentary understanding of counting patterns to 100 by using a hundreds chart.

From second grade on, Van de Walle and Lovin highlight the importance of base-10 concepts and place value.  They encourage flexible methods of computation, frequent use of a variety of base-10 models, ten-frame cards, and invention of computational methods to strengthen the understanding of place value and base-10 concepts.

To encourage computational fluency in the older grades, Jo Boaler6 engaged middle school students with "number talks". The teacher displays a calculation (usually an addition or multiplication problem) and asks students to work it out in their heads without using paper or pencil.  When students are finished, they share their methods.  Boaler reports the methods that four students used to solve 18 x 5:

Each of the methods involves decomposing and recomposing numbers.  Original calculations are changed into easier equivalent calculations.

Making Sense of Problem Solving offers many instructional suggestions and "possible solutions" that foster or illustrate computational fluency. A bibliography of some outstanding Computational Fluency resources follows this newsletter.  The Science Education Partnerships at Oregon State University provides a variety of free "number strategy" worksheets that teachers can use to help students (and parents) develop a repertoire of fluency tactics.

Please note that helping students learn to use numbers flexibly does not preclude helping them create useful algorithms or emphasizing the connection between physical models of mathematical processes and standard algorithms.  We invite you to join our discussion on computational fluency in our blog.

1 Gray, E. & Tall, D. (1994), Duality, Ambiguity and Flexibility: A "Proceptual" View of Simple Arithmetic. Journal for             Research in Mathematics Education, 25(2), pp. 116-140.
2 Boaler, Jo. (2008). What’s Math Got to Do With It? Viking Press, NY, NY.
3 NCTM (2000). Principles and Standards for School Mathematics. Reston, VA.
4 NCTM (2006). NCTM Curriculum Focal Points: A Quest for Coherence. Reston, VA.
5 Van de Walle, John & Lovin, LouAnn. (2006).Teaching Student Centered Mathematics, Grades K-3. Pearson/Allyn
and Bacon. NY, NY.
6 Boaler. op. cit., p. 155-157.

When the final version of the Common Core State Standards is published, Teacher to Teacher Publications will post a "crosswalk" from the NCTM, Washington and New Jersey editions of Making Sense of Problem Solving to those new standards on-line at:

www.teachertoteacher.com