- Mathematics and Science Performance in Grades 4, 8, and 12
- Algebra Performance of Ninth Graders in 2009
- International Comparisons of Mathematics and Science Performance

Increasing overall student achievement, especially lifting the performance of low achievers, is a central goal of education reform in the United States. This goal is reflected in the federal No Child Left Behind Act of 2001 (NCLB), which mandates that all students in each state reach the proficient level of achievement by 2014. This goal is also highlighted in the more recent federal Race to the Top program, which calls for states to design systemic and innovative educational reform strategies to improve student achievement and close performance gaps.^{[3]} The federal government also targets funds directly to low-performing schools through the School Improvement Grants program,^{[4]} for example, to support changes needed in the lowest achieving schools across the nation. These and other efforts to improve achievement are ongoing.

How has the performance of U.S. students changed over time? Are achievement gaps narrowing? How do U.S. students compare with their peers in other nations? This section addresses these questions by examining over time a series of indicators of student performance in mathematics and science in the United States. It begins with a review of recent results of mathematics and science assessments of U.S. students in grades 4, 8, and 12, followed by a review of the performance of ninth graders in algebra in 2009. The section ends by placing U.S. student performance in an international context, comparing the mathematics and science literacy of U.S. 15-year-olds with that of their peers in other countries.

The National Assessment of Educational Progress (NAEP), a congressionally mandated program, has monitored changes in U.S. students' academic performance in mathematics and science since 1969. NAEP has two assessment programs: main NAEP and NAEP Long-Term Trend (LTT).^{[5]} The main NAEP assesses national samples of 4th and 8th grade students at regular intervals and 12th grade students occasionally. These assessments are updated periodically to reflect contemporary curriculum standards in various subjects, including mathematics and science. (In 2014, NAEP will conduct its first nationwide assessment in technology and engineering literacy; see sidebar "Development and Content of NAEP Technology and Engineering Literacy Assessment.")

The NAEP LTT assesses the performance of students ages 9, 13, and 17. Its content framework has remained the same since it was first administered in 1969 in science and in 1973 in mathematics, permitting analyses of trends over more than 3 decades. This section examines recent performance results using main NAEP data only. Findings based on NAEP LTT data have been reported in previous editions of *Science and Engineering Indicators,* and no new data were available from the NAEP LTT for this volume.^{[6]}

The main NAEP reports student performance in two ways: scale scores and achievement levels. Scale scores place students along a continuous scale based on their overall performance on the assessment. For mathematics assessments, scales range from 0 to 500 for grades 4 and 8 and from 0 to 300 for grade 12. For science assessments, scales range from 0 to 300 for all grades.

NAEP also reports student results in terms of achievement levels. Developed by the National Assessment Governing Board (NAGB), achievement levels are intended to measure how well students' actual achievement matches the achievement expected of them in different subjects assessed by NAEP. Based on recommendations from educators, policymakers, and the general public, NAGB sets three achievement levels for all subjects assessed by NAEP (NCES 2010, 2011):

*Basic*denotes partial mastery of materials appropriate for the grade level.*Proficient*indicates solid academic performance.*Advanced*represents superior academic performance.

Based on their test scores, students' performance can be categorized as *below-basic, basic, proficient, *and *advanced.*^{[7]} Because achievement levels were developed independently at each grade level, they cannot be compared across grade levels.^{[8]} Although the NAEP achievement levels are useful in understanding student results and have been widely used by national and state officials, there is disagreement about whether these achievement levels are appropriately defined. A study commissioned by the National Academy of Sciences asserted that NAEP achievement levels were "fundamentally flawed" (Pellegrino, Jones, and Mitchell 1999). The National Mathematics Advisory Panel concluded in 2008 that NAEP scores for the two highest achievement categories (proficient and advanced) were set too high (NMAP 2008). Both NCES and NAGB acknowledged this controversy, and NCES, upon review of congressionally mandated evaluations of NAEP, has recommended that achievement levels be used on a trial basis and interpreted with caution (NCES 2011).

The following review of NAEP results reports both average scale scores and achievement levels, focusing on the percentage of students performing at or above the proficient level both overall and among various subgroups of students.

**Average Score.** For grade 4, the average mathematics score increased by 27 points from 1990 to 2007 and leveled off from 2007 to 2009 (figure ^{[9]}), and among students at both public and private schools (table

For grade 8, the average mathematics score increased steadily from 1990 to 2009 with a total gain of 20 points over the period, including a statistically significant 2-point gain from 2007 to 2009 (figure

For grade 12, only 2005 and 2009 results are examined here; substantial revisions of the mathematics framework for the 2005 assessment made comparison with earlier assessments impossible.^{[10]} Between 2005 and 2009, the average mathematics score for students in grade 12 increased by 3 points (appendix table ^{[11]} The gains in average scores were about 3–5 points for many subgroups, with the exception of Asian/Pacific Islander and American Indian/Alaska Native students, who posted gains of 12 and 10 points, respectively, from 2005 to 2009.

**Achievement Level.** Trends in the percentages of students in grades 4, 8, and 12 reaching the proficient level parallel the scale score trends. The percentage of fourth grade students performing at or above the proficient level increased steadily through 2007 but remained unchanged in 2009. Eighth grade students, on the other hand, showed continuous improvement from 1990 to 2009. Among 12th grade students, the percentage of proficient students increased from 2005 to 2009 (appendix table

Despite these gains, the percentage of students reaching the proficient level remains low. In 2009, the percentage of students performing at or above proficient was 39% for 4th graders, 34% for 8th graders, and 26% for 12th graders.

Although increasing student achievement is the central goal of educational reform in the United States, policies and reform efforts are aimed mainly at improving the achievement of low-achieving students (Hanushek, Peterson, and Woessmann 2010; Loveless 2008; NSB 2010a). Little nationally representative research has been conducted on high-achieving students.

Advances in STEM, however, often depend on originality and leadership from exceptionally capable individuals. Although such individuals are not easily identified, data on students who score unusually well on standardized assessments provide some indication of performance trends among highly capable students. The following analysis uses NAEP assessment data to focus on students who score in the top 1% of mathematics performance in grades 4 and 8.

In 2009, the 37,000–38,000 fourth and eighth grade students who performed at or above the 99th percentile on the NAEP mathematics assessment resembled higher performing students in the general population.^{[12]} However, compared with fourth and eighth graders nationwide, these top performers were more likely to be male, to be white or Asian/Pacific Islander, and to come from higher income families (table ^{[13]} Top performers in grade 8 were more likely than eighth graders overall to have parents with a college degree.^{[14]}

Average mathematics scores for fourth grade students in this top 1% were not only much higher than those for the average fourth grader (304 versus 240 in 2009), they also exceeded the eighth grade average (304 versus 283 in 2009)^{[15]} (table

Like fourth graders, the top 1% of eighth graders had much higher mathematics scores than average (e.g., 366 versus 283 in 2009). However, their trend pattern differed from that of their fourth grade counterparts: average mathematics scores for top eighth graders remained essentially unchanged between 2000 and 2003 and then increased steadily after 2003. The average scores for all eighth graders also increased (appendix table

Despite improvement in recent decades, gaps in mathematics performance persisted among many student subgroups (appendix table ^{[16]} Gaps between students of different racial/ethnic backgrounds or family income remained large, with white and Asian/Pacific Islander students and those from higher income families posting significantly higher scores than their counterparts who were black, Hispanic, or American Indian/Alaska Native students or who were from lower income families. Large gaps were also observed by school type, with private school students scoring significantly higher than their peers in public schools.^{[17]}

Some reductions in these gaps were observed among fourth grade students (table ^{[18]} (appendix table

The framework for the NAEP science assessment was updated in 2009 to reflect advances in science, curriculum standards, assessments, and research on science learning (NCES 2011). The new assessment placed a greater emphasis on what students can do with science knowledge. Because the framework changed significantly, the results from the 2009 assessment cannot be compared with earlier ones (NAGB 2008). This section, therefore, discusses only the 2009 assessment results, which will serve as a baseline for measuring students' progress on future science assessments. For earlier results on NAEP science assessments, see *Science and Engineering Indicators 2008,* pp. 1-13 and 1-14 (NSB 2008).

As in mathematics, science performance varies significantly by student demographics and by school type. At grade 4, the average score for boys was slightly higher than that for girls (151 versus 149) (figure

Most students failed to reach the proficient level on the science assessment. In 2009, 34% of 4th graders, 30% of 8th graders, and 21% of 12th graders performed at or above the proficient level in science (appendix table

The first year of algebra is a prerequisite for higher level mathematics courses in high school (NMAP 2008), opening doors to more advanced mathematics and a college preparatory curriculum. These, in turn, are associated with higher college attendance rates, higher college graduation rates, greater job readiness, and higher earnings once students have entered the workforce (Achieve, Inc. 2008; Adelman 2006; Allensworth and Nomi 2009; Bozick and Lauff 2007; Gamoran and Hannigan 2000; Ma and Wilkins 2007; Nord et al. 2011). The following section draws on the High School Longitudinal Study of 2009 (HSLS:09) to examine mathematics performance in algebra among a cohort of ninth graders in 2009.

HSLS:09, a nationally representative longitudinal study of more than 21,000 ninth graders in 944 schools, is following a sample of students who were ninth graders in 2009 through secondary and postsecondary education, providing insight into students' learning experiences from the beginning of high school into postsecondary education and work. The base year data collection of HSLS included an algebra assessment that provides indicators of ninth graders' proficiency in five specific algebraic skill areas (Ingels et al. 2011). These skill areas are arranged in a hierarchy such that proficiency at a higher level implies proficiency at all levels below it. In order of increasing difficulty, these five skill areas are as follows:

- Level 1, Algebraic expressions: Understands algebraic basics including evaluating simple algebraic expressions and translating between verbal and symbolic representations of expressions.
- Level 2, Multiplicative and proportional thinking: Under-stands proportions and multiplicative situations and can solve proportional situation word problems, find the percent of a number, and identify equivalent algebraic expressions for multiplicative situations.
- Level 3, Algebraic equivalents: Understands algebraic equivalents and can link equivalent tabular and symbolic representations of linear equations, identify equivalent lines, and find the sum of variable expressions.
- Level 4, Systems of equations: Understands systems of linear equations and can solve such systems algebraically and graphically and characterize the lines (parallel, intersecting, collinear) represented by a system of linear equations.
- Level 5, Linear functions: Understands linear functions and can find and use slopes and intercepts of lines and functional notation.

In 2009, a majority of ninth graders were proficient in lower level algebra skills such as algebraic expressions (86%) and multiplicative and proportional thinking (59%) (figure

Though there were no gender differences in algebra performance (appendix table

Differences by parents' education were also considerable (appendix table

This section examines the relative international standing of U.S. students in mathematics and science using assessment data from the Programme for International Student Assessment (PISA).^{[19]} Sponsored by the Organisation for Economic Co-operation and Development (OECD) and initially implemented in 2000,^{[20]} PISA assesses the performance of 15-year-olds in mathematics and science literacy every 3 years. Most countries participating in PISA are OECD members, although the number of participating non-OECD nations or regions has been increasing. Most OECD countries are economically advanced nations.

PISA is a literacy assessment, not a curriculum-based assessment; it measures how well students apply their knowledge and understanding to real-world situations.^{[21]} The term "literacy" indicates its focus on the application of knowledge learned in and out of school. In the PISA mathematics assessment, for example, students are asked to estimate an area, compare the best price for buying a product, or interpret the statistics in a news report or government document. In the PISA science assessment, students are asked to discuss acid rain, interpret erosion at the Grand Canyon, or predict the results of a controlled experiment (see sidebar "Sample Items from PISA").

Despite recent improvement, U.S. PISA scores in mathematics remain consistently below the OECD average and also below those of many non-OECD countries (figure

The top mathematics performers in the United States trailed behind their peers in many other nations as well. In 2009, the U.S. score at the 90th percentile in mathematics was 607, lower than the corresponding score in 12 of 33 other OECD nations (620–659) (OECD 2010b).

U.S. students performed relatively better in the PISA science assessment. The average science literacy score of U.S. 15-year-olds improved by 3 points from 2006 to 2009 (figure

Despite improvement, the 2009 U.S. score (502) was below that of 12 OECD nations (512–554) (appendix table