The important role mathematical fluency, or automaticity, plays in daily life is perhaps best appreciated when the fluency is absent. Have you ever spent time in a checkout line as the shopper in front of you slowly counted out the correct payment for groceries? In the classroom, have you ever felt concern during a lesson as the momentum ground to a halt, while students looked up facts and formulas they should have memorized? Incidents like these remind teachers that comprehension is necessary but insufficient for mathematical proficiency. Automaticity, the ability to perform a skill fluently with minimal conscious effort, is also necessary (Bloom, 1986; Schneider & Shiffrin, 1977). According to Hasselbring, Goin, and Bransford (1988), "[t]he ability to succeed in higher-order skills appears to be directly related to the efficiency at which lower-order processes are executed" (p. 1).
The purposes of this article are to: (a) discuss a rationale for moving beyond mathematical accuracy to automaticity, (b) offer a model illustrating the fluid relationship between comprehension and fluency training in the promotion of mathematical proficiency, and (c) provide recommendations for increasing mathematical automaticity. The prominent role that mathematical proficiency plays in today's global, information-driven economy supports the need for addressing this issue. Efficiency reduces the costs of achieving results in terms of both time and effort. The development of automaticity enables standard mathematical processes, such as facts about families of functions and formulas, to become useful tools for facilitating higher-order thinking. Underlying our discussion of automaticity is the assumption that teachers interested in building their students' mathematical fluency have first taught and confirmed students' comprehension of the material.
Based on our experiences and discussions with students in middle and secondary school settings, many students are developing mathematical skills without concurrent development of automaticity. Teachers commonly relate that students do not know basic mathematical operations. Many educators discuss this problem when talking about elementary school students. We also observed similar deficiencies with older students. One high school teacher reported that her students do not know the basic arithmetic facts. A precalculus teacher said her students do not know how to graph the basic families of functions. Calculus students could not give the formulas for the derivatives of trigonometric functions without consulting a reference sheet.
Perhaps the availability of formula sheets and calculators in the classroom contributes to the lack of automatic recall by students. We observed students in higher-level high school mathematics classes (algebra and precalculus) using calculators to multiply single-digit numbers. There were even calculus students using graphing calculators to graph the function y = [x.sup.2].
Rationale for Building Automaticity
Although there is no universal agreement about the use of the term comprehension, we are using it to include the understanding of mathematical concepts, rules, principles, and generalizations. Comprehension is essential for the development of proficiency in mathematics (National Research Council, 2001) and should be developed along with fluency "in a coordinated, interactive fashion" (p. 11). There are various ways to teach students to understand and develop comprehension. According to Piaget (1977), logical thinking develops systematically through stages. During the concrete operational stage, typical from ages seven to eleven, students can think logically with the help of concrete materials. For instance, working with pie-shaped manipulatives can help students understand that one-fifth is equal to two-tenths. …