Procedural Fluency and Computational Fluency…are these "interchangeable"?

[bpesnell]

Some confusion has sprung over the terms “procedural fluency” and “computational fluency”. 

The CCSSM p.6 states “procedural fluency is skill in carrying out procedures flexibly, accurately, efficiently, and appropriately”.   The Principles and Standards for School Mathematics p.152 discuss computational fluency… “Computational fluency refers to having efficient and accurate methods for computing…computational fluency is exhibited in flexibility of computational methods, understanding of methods and ability to explain them, producing accurate answers efficiently…methods should be based on the mathematical ideas the student understands well; including structure of base ten system, properties of operations and number relationships.”

I was wondering if you could shed some light on these terms.  When I read the descriptions, they sound as if they are referring to the same attributes of mathematical practice.   Are these terms being used in an interchangeable manner or are they in fact two different pieces of fluency?

Thanks…Beth Pesnell

[Cathy Kessel]

Beth, I think there’s some overlap but also some difference.

In CCSS, a procedure isn’t necessarily a procedure for arithmetic computation. In PSSM, computational fluency is discussed in the Number and Operations strand and “method” refers to arithmetic computations (p. 144) and means “algorithm,” but I know there’s sometimes confusion about the meaning of “algorithm.”

What CCSS means by “algorithm” is discussed in the NBT progression. There “algorithm,” “method,” and “strategy” are each distinguished. The only mention of “procedural fluency” in CCSS occurs in discussion of Adding It Up. That report focused mainly on arithmetic, where “procedure” is likely always to be a reference to “algorithm.” (There are many references to “procedure” in Adding It Up, which can be read online here: http://www.nap.edu/catalog.php?record_id=9822.)

In CCSS, procedures occur in OA and NBT, but also in other areas:

Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. (Grade 8 Expressions & Equations, p. 52)

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. (high school Statistics & Probability, p. 81)

PSSM also discusses fluency with things other than arithmetic computations, e.g., “develop fluency in operations with real numbers, vectors, and matrices” (Number & Operations, p. 290); “write equivalent forms of equations . . . and solve them with fluency” (Algebra, p. 296).

PSSM “fluency” might not be identical to CCSS “procedural fluency” but might be in some cases, e.g., expanding (a + b)(x + y). There’s some subtlety about the meaning of “procedure” in this context. For example, FOIL seems to me to be an algorithm because it gives a specified (and unnecessary) order for performing the multiplications. (There’s no advantage to the order specified by FOIL, FILO would be just as good. I’m not claiming that either PSSM or CCSS advocates FOIL!) The main issue is a systematic use of the distributive property. A systematic use of the distributive property in expanding binomials might be considered a procedure that’s not an algorithm.

In summary: “computational fluency” and “procedural fluency” may not be the same because they may sometimes refer to fluency with different things, but “fluency” is likely to be the same. 

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