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. The formal operational stage follows and extends into adulthood, with students able to understand and solve abstract problems without consciously depending on models or other prompts.
Interestingly, when we attempt to learn something new in an area where we have no knowledge background, we often regress to the concrete operational stage, regardless of our age. To confirm this, simply reflect on how helpful it is to have someone "walking you through" a new skill, such as developing your own Internet web page. The advantage of working at the concrete level is that it allows us to build a strong foundation for new understandings, thus increasing the probability that our problem solving will be accurate when we proceed to thinking at the formal level. Prompts offer the mental scaffolding that leads to deeper and accurate understanding. The implication for middle and high school mathematics teachers is that they should allow students to manipulate concrete materials or draw pictures when beginning new areas of study.
The body of research underlying the constructivist movement in education supports our argument that students at all grade levels should be actively engaged in learning activities. Using appropriate manipulative materials, questioning techniques, and discussions can accomplish this. According to the philosophy of constructivism (Good & Brophy, 1986; Wittrock, 1998), (a) students construct their own understandings when presented with new material, (b) new understandings depend upon prior knowledge, (c) authentic tasks make learning more meaningful, (d) learning is facilitated by active engagement by the learner, and (e) social interaction facilitates learning.
Research about the Concrete/Semi-concrete/Abstract (CSA) approach (Miller, Butler, & Lee, 1998) illustrates both Piaget's theory of cognitive development and the learning benefits of constructivist teaching for students with learning difficulties. In the concrete learning stage, students used manipulatives. When tasks were completed at an 80% success rate (and not before), teachers moved students on to the semi-concrete stage, when the same problems were presented using drawings on paper instead of concrete objects. Again, at the 80%-or-higher level of …