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Chapter 1. Elementary and Secondary Mathematics and Science Education

Student Learning in Mathematics and Science

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.

Mathematics and Science Performance in Grades 4, 8, and 12

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]

Reporting NAEP Results

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.

Trends in Mathematics Performance Through 2009

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 1-1). This overall trend was repeated in almost all demographic subgroups, across students at all performance levels (i.e., 10th to 90th percentiles[9]), and among students at both public and private schools (table 1-2).

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 1-1). Rising scores were widespread, occurring among both male and female students; almost all racial/ethnic groups; students from families that were financially disadvantaged and advantaged; students in the low-middle, middle, and high ranges of performance (i.e., 25th to 90th percentiles); and students attending public schools (table 1-2) (see sidebar "Mathematics and Science Achievement in Charter Schools"). The score at the 10th percentile, however, was unchanged from 2007 to 2009, indicating that mathematics performance did not improve significantly among very low-performing students during this period.

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 1-1). Improvement occurred across the board: for both sexes, across all racial/ethnic subgroups, for all performance levels, and among public school students (table 1-2).[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 1-2).

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.

Trends in Mathematics Performance of Top Students

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 1-3).[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 1-4). Average mathematics scores for this top group rose steadily from 2000 to 2005 and then remained flat after 2005. Between 2000 and 2009, the scores for the top 1% of fourth graders increased by 9 points, compared with a 14-point increase in scores for all fourth graders.

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 1-1) so that the improvements overall and among the top 1% were not measurably different.

Changes in Performance Gaps in Mathematics

Despite improvement in recent decades, gaps in mathematics performance persisted among many student subgroups (appendix table 1-1). In general, boys performed slightly better than girls.[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 1-5). For example, the white-black gap in mathematics performance among fourth grade students narrowed from 32 to 26 scale points between 1990 and 2009 because of larger gains by black students[18] (appendix table 1-1). The gap between public and private school fourth grade students also narrowed during the same period because of greater gains by public school students. Finally, fourth graders' score at the 10th percentile rose faster than that at the 90th percentile, reducing the gap between low- and high-performing students in grade 4. No similar gap reductions between 1990 and 2009 were observed at grades 8 or 12.

Science Performance in 2009

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 1-2). Differences by racial/ethnic background and family income were larger: scores for white and Asian/Pacific Islander students were at least 28 points higher than those for black, Hispanic, and American Indian/Alaska Native students, and the score for students from higher income families was 29 points higher than that for students from lower income families. Students from private schools outperformed their peers in public schools by 14 points. Similar performance gaps based on sex, race/ethnicity, and family income were observed among students in grades 8 and 12 (appendix table 1-3).

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 1-4). At grade 12, only 4% of black students, 8% of Hispanic students, and 8% of low-income students reached the proficient level.

Algebra Performance of Ninth Graders in 2009

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 1-3). Proportions demonstrating proficiency in more advanced algebra skills were lower and decreased as the difficulty level increased. Only 9% of ninth graders reached proficiency in linear functions, the highest algebra skill level assessed by HSLS.

Though there were no gender differences in algebra performance (appendix table 1-5), considerable differences were found among racial/ethnic subgroups (figure 1-3). In each skill area, Asian and white students demonstrated proficiency at higher rates than did black and Hispanic students. For example, 20% of Asians and 10% of whites were proficient in linear functions, compared with 6–7% of blacks and Hispanics.

Differences by parents' education were also considerable (appendix table 1-5). In every skill area assessed, proportionally more students whose parents had a bachelor's or advanced degree achieved proficiency than those whose parents had a high school education or less. For example, 35% of students whose parents had an advanced degree mastered systems of equations and 16% mastered linear functions; the corresponding percentages for students whose parents had not completed high school were 10% and 6%, respectively.

International Comparisons of Mathematics and Science Performance

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").

Mathematics Literacy Among U.S. 15-Year-Olds

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 1-4). On the most recent PISA test in 2009, the U.S. average score of 487 fell below the OECD average of 496 and was lower than the scores of 17 of 33 other OECD nations, including Republic of Korea (546), Finland (541), Switzerland (534), Japan (529), Canada (527), and the Netherlands (526) (appendix table 1-6). The U.S. score was also lower than scores in several non-OECD regions/countries/economies, such as Shanghai-China (600), Singapore (562), and Hong Kong (555). In 2009, U.S. students demonstrated higher mathematical literacy than students in only 5 out of 34 OECD countries (Greece, Israel, Turkey, Chile, and Mexico).

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).

Science Literacy Among U.S. 15-Year-Olds

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 1-4). Whereas U.S. students scored lower than the OECD average in 2006 (489 versus 498), this gap was not evident in 2009 (502 versus 501). The U.S. gains in science since 2006 were mainly driven by improvements at the bottom of the performance distribution; performance at the top remained unchanged (OECD 2010b).

Despite improvement, the 2009 U.S. score (502) was below that of 12 OECD nations (512–554) (appendix table 1-6). For example, U.S. students scored lower than students in 5 top-performing OECD nations (Finland, Japan, Republic of Korea, New Zealand, and Canada) by 27–52 points. U.S. students also lagged behind their peers in (non-OECD) Shanghai-China, Hong Kong, and Singapore (by 40–73 points), The U.S. 90th percentile score in scientific literacy was 629, below the corresponding scores in 7 of 33 other OECD nations (642–667) (OECD 2010b). Thus, U.S. top performers in science did better relative to other countries than did U.S. students on average.


[3] Race to the Top is a $4.35 billion competitive grant program funded by the U.S. Department of Education as part of the American Recovery and Reinvestment Act of 2009. This program is designed to encourage and reward states creating the conditions for education innovation and reform, achieving significant improvement in student outcomes, and implementing reform plans in four core areas: 1) adopting standards and assessments that prepare students to succeed in college and the workplace; 2) building data systems that measure student growth and success and inform teachers and principals how to improve instruction; 3) recruiting, developing, rewarding, and retaining effective teachers and principals; and 4) turning around the lowest performing schools. In March 2010, Delaware and Tennessee won grants in the first phase of the competition, receiving approximately $100 million and $500 million, respectively, to implement their comprehensive school reform plans. In August 2010, nine states (Florida, Georgia, Hawaii, Maryland, Massachusetts, New York, North Carolina, Ohio, and Rhode Island) and the District of Columbia won grants in the second phase of the competition. Grant levels depend on a state's student population: large states like New York and Florida receive up to $700 million and smaller states like Hawaii and Rhode Island receive up to $75 million. See the Race to the Top Fund website for more information:
[4] The U.S. Department of Education awarded School Improvement Grants to states under the Elementary and Secondary Education Act of 1965 (reauthorized in 2002 as the No Child Left Behind Act) to support focused school improvement efforts. In 2009, the department dramatically increased the funds that would be provided to states (from $491,265 in 2008 to $3.546 billion in 2009) and charged states with using the funds for leveraging changes needed to turn around persistently low-achieving schools.
[5] These two NAEP assessment programs differ in many aspects, including samples of students and assessment times, instruments, and contents. See
[6] The 2010 volume reviewed long-term trends in mathematics from 1973 to 2008, and the 2004 volume examined trends in science from 1969 to 1999. The long-term trend assessment in mathematics will be administered again in 2012; the long-term trend assessment in science has not been conducted since 1999.
[7] Students in the below-basic category have scores lower than the minimum score for the basic level. Students in the basic category have scores at or above the minimum score for the basic level, but lower than the minimum for the proficient level. Students in the proficient category have scores at or above the minimum score for the proficient level, but lower than the minimum score for the advanced level. Students in the advanced category have scores at or above the minimum score for the advanced level.
[8] See NAEP's mathematics and science achievement levels defined by grade at and
[9] Percentiles are scores below which a specified percentage of the population falls. For example, among fourth graders in 2009, the 10th percentile score for mathematics was 202. This means that 10% of fourth graders had mathematics scores at or below 202 and 90% scored above 202. The scores at various percentiles indicate students' performance levels.
[10] In 2005, NAGB adopted a new mathematics framework for the grade 12 assessment to reflect contemporary standards of high school curriculum and coursework. Based on this new framework, the 2005 assessment changed its content areas (e.g., increasing coverage on algebra, data analysis, and probability) and adopted a new reporting scale (i.e., 0–300 as opposed to 0–500 in earlier years). These changes made the 2005 assessment results not comparable to those in earlier years. Some changes were also made to the 2009 framework; the purpose was to enable NAEP to better measure how well prepared 12th grade students are for postsecondary education and training (e.g., adding content that is beyond what is typically taught in a standard 3-year course of study in high school mathematics). However, special analyses of 2005 and 2009 data determined that the 2009 grade 12 mathematics results could still be compared with results from the 2005 assessment despite the changes to the 2009 framework. More information about the mathematics frameworks for the 2005 and 2009 grade 12 assessments and how they differ from the previous framework is available at
[11] Results for private school students in 2009 could not be reported separately due to the low participation rate for private schools.
[12] Special NSF tabulations.
[13] Students' eligibility for free/reduced-price lunch is often used as a proxy measure of family poverty. Students who are eligible for free/reduced-price lunch are considered to come from low-income families, and those who are not eligible for free/reduced-price lunch are considered to come from relatively high-income families.
[14] Data on parental education for grade 4 were unreliable and therefore excluded from the analysis.
[15] Cross-grade comparisons are acceptable for mathematics scores of fourth and eighth graders because these scores were put on a common scale. However, mathematics scores for 4th and 8th graders cannot be compared to those of 12th graders because they used different score scales (0 to 500 for grades 4 and 8 and 0 to 300 for grade 12). Cross-grade comparisons are also not appropriate for other subjects because the scales were derived independently at each grade level. See
[16] Gender gaps are not consistent across racial/ethnic subgroups. For example, the results from the 2009 NAEP mathematics assessment show that, whereas white and Hispanic boys had higher scores than their girl counterparts at grade 4, the pattern was opposite among blacks—girls outperformed boys. Similar differences were also found among students in grade 8 (special NSF tabulations).
[17] Differences in performance between public and private school students reflect in part different types of students enrolled in public and private schools. Proportionally, private schools enroll more white students and students from advantaged socioeconomic backgrounds than public schools (Snyder and Dillow 2011).
[18] The reduction in the white-black gap at grade 4 is likely attributable to larger improvements made by black female students (Vanneman et al. 2009). From 1990 to 2007, the average mathematics score gains of black females at grade 4 were greater than those of their white peers, reducing the white-black gap. However, among male students at grade 4, no similar gap reductions were observed during this period.
[19] Previous volumes of Science and Engineering Indicators (e.g., NSB 2010b) also used data from the Trends in International Mathematics and Science Study (TIMSS) to examine the relative standing of U.S. students in mathematics and science achievement. No new data from TIMSS, however, were available when this chapter was prepared. The latest administration of TIMSS was in spring 2011, and international comparisons based on TIMSS data will be available in the 2014 volume of Science and Engineering Indicators.
[20] Information on OECD and its assessment programs is available at,2987,en_32252351_32235731_1_1_1_1_1,00.html.
[21] PISA differs from NAEP in several key aspects. NAEP assesses the knowledge and skills students need for an in-depth understanding of mathematics and science at various grade levels. PISA measures the "yield" of education systems, that is, the skills and competencies students have acquired and can apply in real-world contexts by age 15. NAEP emphasizes curriculum-based knowledge, whereas PISA focuses on literacy and applications, drawing on learning both in and outside of school. Although NAEP and PISA both are sample-based assessments, NAEP uses grade-based samples of students in grades 4, 8, and 12, and PISA uses an age-based sample of 15-year-old students nearing completion of compulsory schooling in many countries. Both assessments are developed from a framework specifying the content and skills to be measured, but the PISA framework is organized around overarching ideas (e.g., space and shape) with emphasis on the contexts in which concepts are applied (e.g., in school, in society), as opposed to curriculum-based topics, such as geometry and algebra.