The same is true for rational numbers. Only 35% of 13-year-olds correctly ordered three fractions, all in reduced form,⁵⁶ and only 35%, asked for a number between .03 and .04, chose the correct response.⁵⁷ These findings suggest that students may be calculating with numbers that they do not really understand.

An overall picture of procedural fluency is provided by the NAEP long-term trend mathematics assessment,⁵⁸ which indicates that U.S. students’ performance has remained quite steady over the past 25 years (see Box 4–4). A closer look reveals that the picture of procedural fluency is one of high levels of proficiency in the easiest contexts. Questions in which students are asked to add or subtract two- and three-digit whole numbers presented numerically in the standard format are answered correctly by about 90% of 13-year-olds, with almost as good performance among 9-year-olds.⁵⁹ Performance is slightly lower among 13-year-olds for division.⁶⁰

Box 4–4 NAEP Scale Scores, Long-Term Trend Assessment, 1973–1999 SOURCE: Campbell, Hombo, and Mazzeo, 2000, p. 9. These scale scores include all content areas: number, geometry, algebra, and so on.