Student Learning in Mathematics and Science

This section presents indicators of student performance in mathematics and science from two types of studies: longitudinal studies and repeating cross-sectional studies. Longitudinal studies follow the same group of students over time; for example, from kindergarten through fifth grade. These studies can show achievement gains in a particular subject from grade to grade. Repeating cross-sectional studies provide a snapshot of how certain students perform in a particular year and then take another snapshot of a similar group of students in a later year; for example, comparing fourth graders in 1990 to fourth graders in 2005.

Performance as Students Progress Through Elementary School

The Early Childhood Longitudinal Study (ECLS) followed a group of students who entered kindergarten in fall 1998 until spring 2004, when most were in fifth grade.[2] The 2006 volume of Science and Engineering Indicators provided data from ECLS through third grade (NSB 2006). Those indicators showed that mathematics achievement differences among subpopulations already existed when students entered kindergarten. Although all groups made gains by third grade, some gaps widened over the 4-year period (Rathbun, West, and Germino Hausken 2004). This volume updates those indicators of early mathematics learning to fifth grade. It also presents the first longitudinal data from ECLS on science learning, from third through fifth grade.

Mathematics: Fifth Grade Performance

The ECLS mathematics assessments provide indicators of student proficiency in nine specific skill areas that represent a progression of skills and knowledge (see sidebar "Mathematics Skills Areas for Elementary Grade Students"). This volume of Science and Engineering Indicators focuses on the skill areas assessed in fifth grade, whereas the 2006 volume focused on the lower-order skill areas assessed in kindergarten through third grade.

By the end of fifth grade, almost all students (92%) could solve simple multiplication and division problems, and about three-quarters demonstrated understanding of place value in integers to the hundreds place (figure 1-1figure. ; appendix table 1-1Excel.) (Princiotta, Flanagan, and Germino Hausken 2006). Other topics proved more challenging, with less than half of fifth graders (43%) able to solve word problems using knowledge of measurement and rate, 13% able to solve problems using fractions, and 2% able to solve problems using area and volume. However, in each of the mathematics skills areas assessed at both time points, the percentages of students demonstrating proficiency increased since the third grade (appendix table 1-1).

Mathematics: Achievement Gaps During Elementary School

Fifth grade mathematics performance was related to several student background factors (Princiotta, Flanagan, and Germino Hausken 2006). For each of the mathematics skill levels mentioned above, lower proportions of black and Hispanic students were proficient compared with their white and Asian peers (appendix table 1-1Excel.). Students whose mothers had less formal education and students who were living in poverty[3] also generally demonstrated lower proficiency rates than their peers.

Although many of these mathematics achievement differences were evident when these children started kindergarten, the ECLS data suggest that at least some gaps widened as students progressed through elementary school, and that other gaps, such as those between boys and girls, emerged that were not present when students started school (Princiotta, Flanagan, and Germino Hausken 2006; Rathbun, West, and Germino Hausken 2004). Changes in achievement gaps are most easily summarized by examining average scale scores, which place students on a continuous ability scale based on their overall performance. Results indicate that all demographic groups gain mathematical skills and knowledge during elementary school but the rate of progress varies.

Gender Gaps. Boys and girls started kindergarten at the same overall mathematics performance level (appendix table 1-2Excel.) but by the end of fifth grade, boys had made larger mathematics gains than girls, resulting in a small but observable gender gap of four points.

Race/Ethnicity Gaps. Gaps between white and black students and between white and Hispanic students existed when students started kindergarten and they widened over time. In mathematics, from kindergarten to fifth grade, white students posted a gain of 93 points; Hispanics, a gain of 89 points; and blacks, a gain of 80 points (table 1-1table.; appendix table 1-2Excel.). By fifth grade, the gap between white and black students in average mathematics scores was 19 points, and the average score of black fifth grade students was equivalent to the average third grade score of white students.

Mother's Education and Family Income Gaps. Students whose mothers had higher levels of education entered kindergarten with higher average mathematics scores than their peers whose mothers attained less formal education and these gaps increased as students progressed through elementary school (appendix table 1-2Excel.). By grade 5, the gaps in mathematics scores were substantial, with students whose mothers had dropped out of high school posting a lower average mathematics score than students whose mothers had graduated from college had posted at grade 3. Students living in families with incomes below the poverty threshold also entered school with lower mathematics skills than their peers from higher income families, and those discrepancies in scores grew by fifth grade.

Other research suggests that widening achievement gaps as students progress through school are, at least in part, a result of differential learning growth and loss during the summer (Alexander, Entwisle, and Olson 2001; Borman and Boulay 2004; Cooper et al. 1996). For example, although lower- and upper-income primary grade students made similar gains in mathematics during the school year, lower-income students experienced declines in mathematics skills during summer breaks, whereas higher-income students experienced gains (Alexander, Entwisle, and Olson 2001). These findings have been attributed to greater ability among higher-income parents to provide their children with mathematically stimulating materials and activities during the summer.

Science: Performance Gains and Gaps From Third to Fifth Grade

ECLS began assessing students in science in the third grade and tested those students' science knowledge again in fifth grade (Princiotta, Flanagan, and Germino Hausken 2006; Rathbun, West, and Germino Hausken 2004). The science assessments placed equal emphasis on life science, earth and space science, and physical science, asking students to demonstrate understanding of the physical and natural world, make inferences, and understand relationships. Assessments also required students to interpret scientific data, form hypotheses, and develop plans to investigate scientific questions. ECLS science assessments were not designed to measure proficiency in specific skill areas and therefore do not lend themselves to proficiency levels; results are instead summarized by average scale scores.

Gains in science skills and knowledge between third and fifth grade were seen across each demographic group, but performance gaps persisted (appendix table 1-3Excel.). Gaps were evident the first time students were assessed in science, in third grade. Boys had slightly higher average science scores than girls and they maintained this small difference in performance in fifth grade. In third grade, white and Asian students had higher average science scores than did blacks and Hispanics, and Hispanics outperformed their black peers. By fifth grade, none of these gaps had narrowed and the black-Hispanic gap had increased. The average score for black fifth graders was lower than the average score for white third graders.

Third graders whose mothers had more formal education performed better in science than did their peers with mothers who were less educated, and students who lived above the poverty threshold did better in science than those who lived below it (appendix table 1-3Excel.). By fifth grade these gaps in science performance by mothers' education and poverty status either remained constant or grew wider. Students from families below the poverty threshold had average fifth grade science scores equivalent to the third grade scores of students above the poverty threshold.

Mathematics Performance as Students Progress Through High School

Another longitudinal study, the Education Longitudinal Study (ELS), provides indicators of student learning during high school by following a nationally representative sample of students who were in 10th grade in 2002 (NCES 2007a). These students were assessed again in 2004 in 12th grade. ELS includes an assessment of student performance in mathematics, which provides information both on specific skills and on overall mathematics performance. The specific skills are divided into levels representing a progression of mathematics skills: (1) simple arithmetical operations with whole numbers; (2) simple operations with decimals, fractions, powers, and roots; (3) simple problem solving requiring the understanding of low-level mathematical concepts; (4) understanding of intermediate-level mathematical concepts and multistep solutions to word problems; and (5) complex multistep word problems and advanced mathematics material (NCES 2007a).

In 2004, almost all 12th grade students (96%) were proficient in simple arithmetical operations with whole numbers and 79% were also proficient in simple operations with decimals, fractions, roots, and powers (figure 1-2figure.; appendix table 1-4Excel.). However, the proportions demonstrating proficiency in more advanced mathematics skills were lower and decreased as more advanced skills were tested. Only 4% of 12th grade students reached proficiency at the highest level (solving complex multistep word problems). Nevertheless, at each level, the percentages of students demonstrating proficiency increased from the 10th to the 12th grade.

Each demographic subgroup examined improved in mathematics skills from 10th to 12th grade, but achievement disparities were evident. The ECLS data reviewed in the previous section found that boys and girls entered kindergarten with similar overall mathematics performance, but by the fifth grade, boys demonstrated slightly higher performance. This small gender gap favoring boys was also observed in the 10th and 12th grades in ELS, with the gap holding steady between those 2 years (appendix table 1-4Excel.).

Substantial differences among racial/ethnic groups were found in mathematics achievement at grade 10, with white and Asian/Pacific Islander students posting higher average scores than black and Hispanic students, and Hispanic students scoring slightly higher than black students (appendix table 1-4Excel.). After 2 additional years of schooling, white-Hispanic and Hispanic-black gaps held steady, and the white-black, Asian-black, and Asian-Hispanic gaps increased. By 12th grade, the average performance of black students was slightly lower than the average 10th grade performance of white and Asian students.

The mathematics skill gaps observed in kindergarten (and found to be greater in fifth grade) between students whose mothers had lower levels of education compared with students whose mothers were more educated were evident among ELS 10th graders (appendix table 1-4Excel.). These differences generally increased through the 12th grade. Students from low socioeconomic families[4] had lower average 10th grade mathematics scores than their peers in middle socioeconomic families, who in turn had lower scores than students in high socioeconomic families. By 12th grade these gaps had grown.

Performance of 4th, 8th, and 12th Grade Students Since the 1990s

The two longitudinal studies described above showed that students start school with different levels of knowledge and skills and that some of those differences grow as the same students move through the educational system. Notably, none of the achievement gaps reviewed above between historically privileged and underprivileged groups narrowed during elementary or high school.

Another type of assessment, a well-known repeating cross-sectional study, provided indicators that showed a somewhat more positive trend. As will be detailed below, fourth and eighth grade students in 2005 (including most subgroups) performed better on mathematics tests on average than fourth and eighth graders a decade and a half earlier. However, fewer gains were observed in science and substantial achievement gaps among subgroups of students in these grades persisted in both mathematics and science.

The National Assessment of Educational Progress (NAEP), also known as the "Nation's Report Card," has charted the academic performance of U.S. students in the upper elementary and secondary grades since 1969. Previous Science and Engineering Indicators reports described trends in mathematics and science results dating back to the first NAEP assessments.[5] This volume focuses on more recent trends, from 1990 to 2005 for mathematics (grades 4 and 8) and from 1996 to 2005 for science (grades 4, 8, and 12) (NCES 2006a, b). Twelfth graders were assessed in mathematics in 2005 but the assessment was not comparable with previous NAEP assessments, and therefore trend data are not available for grade 12 mathematics.[6]

The NAEP assessments are based on frameworks developed through a national consensus process that involves educators, policymakers, assessment and curriculum experts, and the public. The frameworks are then approved by the National Assessment Governing Board (NAGB) (NCES 2006a, 2007b). The mathematics grades 4 and 8 assessments contain five broad content strands (number sense, properties, and operations; measurement; geometry and spatial sense; data analysis, statistics and probability; and algebra and functions). The mathematics grade 12 assessment contains four content strands that are similar to the grade 4 and 8 strands, but with measurement and geometry collapsed together. The science framework includes a content dimension divided into three major fields of science: earth, life, and physical.

Student performance on the NAEP is measured with scale scores as well as achievement levels. Scale scores place students on a continuous ability scale based on their overall performance. For grades 4 and 8, the mathematics scales range from 0 to 500 across the two grades. For grade 12, the mathematics scale ranges from 0 to 300. For science, the scales range from 0 to 300 for each of the three grades.

Achievement levels are set by NAGB based on recommendations from panels of educators and members of the public, and describe what students should know and be able to do at the basic, proficient, and advanced levels for each grade (NCES 2006b and 2007b). The basic level represents partial mastery, proficient represents solid academic performance, and advanced represents superior performance on assessments measuring mastery of knowledge and skills for each grade level. This review of NAEP results focuses on the percentage of students deemed proficient (for more detailed definitions of the proficient levels, see Science and Engineering Indicators 2006, pp. 1–13 and 1–14 [NSB 2006 and NCES 2007b]).

Disagreement exists about whether NAEP has appropriately defined these levels. A study commissioned by the National Academy of Sciences judged the process used to set these levels "fundamentally flawed" (Pellegrino, Jones, and Mitchell 1998), and NAGB acknowledges that considerable controversy remains over setting achievement levels (Bourque and Byrd 2000). However, both the National Center for Education Statistics (NCES) and NAGB believe the levels are useful for understanding trends in achievement. They warn readers to use and interpret the levels with caution (NCES 2006b).

In this section, NAEP results are examined in various ways, including changes in average scale scores and in the proportion of students reaching the proficient level both overall and among various subgroups of students. In addition, achievement gaps between demographic subpopulations and changes in those gaps are reviewed.

Examining a set of measures reveals more about student performance than does examining just one measure (Barton 2004). For example, without examining changes in achievement for high-, middle-, and low-achieving students, it would be impossible to know whether a rise in average scores resulted from increased scores among one or a few groups of students, or whether it reflected broader improvements.

Mathematics Performance From 1990 to 2005

The average mathematics scores of fourth and eighth grade students have steadily increased since 1990 (the first year in which the current assessment was given), including small improvements during the more recent period 2003–05 (NCES 2006a) (figure 1-3figure.; table 1-2table.; appendix table 1-5Excel.). The pattern of higher average mathematics scores among fourth and eighth grade students was widespread (table 1-2; appendix table 1-5). At grades 4 and 8, average mathematics scores were higher for both male and female students in 2005 compared with both 1990 and 2003. This was also true for students regardless of eligibility for free or reduced-price lunch (a commonly used measure of poverty).[7] Generally, improvements were observed for white, black, Hispanic, and Asian/Pacific Islander populations.[8]

Examining trends for students at the lower, middle, and higher ranges of performance can uncover whether overall trends are driven by changes in only one or two of these groups. However, NAEP mathematics results indicate that the overall increase in mathematics performance was not driven by students at any one performance level (table 1-2table.; appendix table 1-5Excel.). Average scores for students in the 10th, 25th, 50th, 75th, and 90th percentiles in 2005 were all higher than those recorded in 1990 and 2003, providing evidence that gains in mathematics were widespread. (Percentiles are scores below which a specified percentage of the population falls. For example, among eighth graders in 2005, the 75th percentile score for mathematics was 304. This means that 75% of eighth graders had mathematics scores at or below 304, and 25% scored above 304).

The percentage of students reaching the proficient level for their grade also rose (figure 1-4figure.; appendix table 1-6Excel.). In 1990, 13% of fourth graders were deemed proficient in mathematics compared with 36% in 2005. Among eighth graders the percentage increased from 15% to 30%.

Mathematics Performance From 2005 to 2007

The NAEP 2007 fourth and eighth grade mathematics assessment results were released too late to incorporate more than a brief summary in this volume. Both fourth and eighth grade students registered continued improvements in mathematics achievement between 2005 and 2007 (Lee, Grigg, and Dion 2007). Improvements occurred across all performance percentiles and income levels in both grades. Among fourth graders, scores increased for whites, blacks, Hispanics, and Asians/Pacific Islanders but no significant increase could be reported for American Indians/Alaska Natives because of insufficient sample size. Among eighth graders, whites, blacks, and Hispanic students improved their scores but Asians/Pacific Islanders and American Indians/Alaska Natives registered no gain. The percentage of students scoring at or above proficient in both grades increased from 36% to 39% among fourth graders and 30% to 32% among eighth graders.

Although most groups showed improved performance from 2005 to 2007, performance gaps were resistant to improvement. In the fourth grade, the white-black and white-Hispanic gaps did not change between 2005 and 2007. In the eighth grade, the white-black gap decreased but the white-Hispanic gap remained about the same.

Science Performance From 1996 to 2005

Since 1996, the first year the current NAEP science assessment was given, average scores increased for 4th graders, held steady for 8th graders, and declined for 12th graders (table 1-3table., appendix table 1-7Excel.) (NCES 2006b). Trends in percentile scores suggest the increase in average scores at grade 4 was driven by lower- and middle-performing students: scores at the 10th, 25th, and 50th percentiles increased in 2005 compared with both 1996 and 2000, while scores at the 75th and 90th percentiles did not change over the same periods (appendix table 1-7).

The proportion of students reaching the proficient level for their grade in science did not change for grades 4 and 8, and declined slightly for grade 12 (figure 1-4figure.; appendix table 1-8Excel.). In 2005, 29% of fourth and eighth grade students reached the proficient level. Rates were lower among 12th graders (18% scored at or above the proficient level).

Changes in Achievement Gaps Since the 1990s

The longitudinal studies outlined in the beginning of this chapter reveal racial/ethnic gaps in mathematics and science performance as students start kindergarten, some of which grow as students progress through elementary and high school. NAEP, with snapshots of three grades over time, paints a slightly different picture. Since 1990, the white-black gap in mathematics achievement decreased among fourth graders and held steady for eighth graders (table 1-2table.; appendix table 1-5Excel.). The white-Hispanic mathematic gaps held steady over this time for students in grades 4 and 8. In science, fourth grade black students narrowed the achievement gap with white students from 1996 to 2005 (table 1-3table.; appendix table 1-7Excel.). Despite some narrowing, substantial racial/ethnic gaps in mathematics and science remained. For example, among 12th grade students in 2005, 24% of white students and 23% of Asian/Pacific Islander students were proficient in science compared with 13% of American Indian/Alaska Native students, 5% of Hispanic students, and 2% of black students (appendix table 1-8Excel.). Although grade 12 trends are not available for mathematics, the 2005 data reveal substantial racial/ethnic gaps in this subject as well: 36% of Asian/Pacific Islander 12th graders, 29% of white 12th graders, 8% of Hispanic 12th graders, and 6% each of black and American Indian/Alaska Native 12th graders reached the proficient level in mathematics (appendix table 1-6Excel.).

In 2005, boys in grades 4, 8, and 12 performed slightly better than girls in both mathematics and science (appendix tables 1-5Excel., 1-6Excel., 1-7Excel., and 1-8Excel.). These small gender gaps have remained stable since 1990 in mathematics (for grades 4 and 8) and 1996 in science (for grades 4, 8, and 12). In 2005, students in grades 4 and 8 who were eligible for the federal subsidized lunch program had lower average mathematics scores than their peers who were not eligible (appendix table 1-5). However, the grade 4 gap with regard to subsidized lunch was slightly less in 2005 than it had been in 1996 (table 1-2table.; appendix table 1-5). Achievement differences with regard to subsidized lunch eligibility were also found in science, with fourth and eighth grade students eligible for the lunch program performing below their ineligible peers (appendix table 1-7). Between 2000 and 2005, these science gaps by subsidized lunch eligibility in grades 4 and 8 decreased somewhat (table 1-3table.; appendix table 1-7).

International Comparisons of Mathematics and Science Performance

Two assessments help compare mathematics and science performance in the United States to other countries: the Trends in International Mathematics and Sciences Study (TIMSS) and the Program for International Student Assessment (PISA). Results from the most recent administration of these assessments are included in more detail in Science and Engineering Indicators 2006 and are only summarized here.

In 2003, U.S. students scored above international averages on the TIMSS assessment and below international averages on the PISA assessment, differences that may be explained, in part, by each test's focus and the set of countries participating in each assessment (Neidorf et al. forthcoming). TIMSS tests primary and middle grade students on curriculum-based knowledge and skills. PISA tests 15-year-olds on their ability to apply scientific and mathematical concepts and thinking skills to real-world problems. Although TIMSS includes results from 46 industrialized and developing countries, PISA results reported here include 30 countries, all of which are industrialized.

According to TIMSS data, U.S. fourth and eighth graders performed above the international average in mathematics and science in 2003 (Gonzales et al. 2004). However, because TIMSS includes many developing countries in its international average, it also can be helpful to compare U.S. performance to two similarly industrialized countries, the United Kingdom and Japan. Japan outperformed U.S. fourth and eighth graders in both mathematics and science. The United Kingdom outperformed U.S. fourth graders in both subjects, but had insufficient numbers participating in eighth grade to make a comparison. According to PISA results, U.S. 15-year-olds performed below the average for industrialized countries in both mathematics and science (Lemke et al. 2004). Of 30 participating industrialized nations, 20 outperformed the United States in mathematics and 15 outperformed it in science (see sidebar "Achievement Negatively Correlated With Confidence in Learning Across Countries/Economies").


Two national longitudinal studies found that students enter kindergarten with varied mathematics knowledge and skills, and all groups made gains during elementary and high school but at different rates. The result is that most mathematics achievement gaps remain, or have grown, by the time students graduate from high school. The national longitudinal data for science report achievement gaps in third grade (the first time students are assessed) and gains among all groups from third to fifth grade, but also no narrowing and even some widening of the achievement gaps over this 2-year period.

Repeating cross-sectional studies of mathematics and science performance provide different types of indicators. In 2005, students in grades 4 and 8 posted higher mathematics scores than students in those same grades in 1990. The pattern of higher scores was widespread, occurring among males and females, across racial/ethnic groups, for students from financially advantaged and disadvantaged families, and for students in the lower, middle, and higher ranges of performance. Additionally, some achievement gaps narrowed. In science, average scores increased for fourth grade students, held steady for eighth graders, and declined for 12th graders between 1996 (the first year the assessments were given) and 2005. Trends in percentile scores suggest the increase in overall science scores of fourth graders were driven by improved scores among lower- and middle-performing students.

Despite the gains made in mathematics (and to a lesser extent, science) from the 1990s to 2005, most 4th, 8th, and 12th graders do not perform at levels considered proficient for their grade. Just more than one-third of fourth graders reached the proficient level in mathematics in 2005, and the rates were lower for mathematics at grades 8 and 12, and at all three grades for science. International comparisons of student mathematics and science performance indicate U.S. students perform below average in mathematics and science for industrialized countries.


[2] In the 2004 followup for the ECLS kindergarten class of fall 1998, 86% of cohort members were in fifth grade, 14% were in a lower grade, and less than 1% were in a higher grade. For the sake of simplicity, students in the ECLS followups are referred to by the expected grade; that is, they are referred to as first graders in the spring 2000 assessment, as third graders in the spring 2003 assessment, and as fifth graders in the spring 2004 assessment.

[3] The poverty status variable in ECLS is based on information provided by the parent. The variable is derived from household income and total number of household members (Princiotta, Flanagan, and Germino Hausken 2006). Federal poverty thresholds are used to define households below the poverty level. For example, if a household contained two members, and the household income was lower than $12,015, the student was considered to be living below the poverty threshold.

[4] Socioeconomic status was based on five equally weighted components: father's education, mother's education, family income, father's occupational prestige score, and mother's occupational prestige score.

[5] NAEP consists of three assessment programs. The long-term trend assessment is based on nationally representative samples of 9-, 13-, and 17-year-olds. It has remained the same since it was first given in 1969 in science and 1973 in mathematics, permitting analyses of trends over three decades. A second testing program, the national or main NAEP, assesses national samples of 4th, 8th, and 12th grade students. The national assessments are updated periodically to reflect contemporary standards of what students should know and be able to do in a subject. The third program, the state NAEP, is similar to the national NAEP but involves representative samples of students from participating states.

[6] These recent trends are based on data from the national NAEP program. The current national mathematics assessment for grades 4 and 8 was first administered in 1990 and was given again in 1992, 1996, 2000, 2003, and 2005. In 2003, only fourth and eighth grade students were assessed. The current grade 12 mathematics assessment has only been administered once: in 2005. Trend analyses for grade 12 mathematics are therefore not available. The current national science assessment was first administered in 1996 and was given again in 2000 and 2005.

[7] Although the NAEP program collects information about eligibility for the free or reduced-price lunch program for grade 12 students, it does not report these data. Because other reasons for not applying for school lunch programs (including food preferences, ability to buy lunch outside school, and wanting to avoid embarrassment) generally increase with student age, program eligibility becomes an increasingly unreliable indicator of poverty at higher grade levels. For example, approximately 35%–45% of fourth grade and 30%–40% of eighth grade public school students have been eligible in recent years for the subsidized lunch program. In contrast, only about 15%–25% of 12th grade public school students have been eligible (determined using the online NAEP Data Explorer tool at The relatively low percentage of grade 12 students noted as eligible for the program raises concerns that it is not a reliable indicator of low family income for these students.

[8] Insufficient sample size in 1990 for Asian/Pacific Islanders and American Indians/Alaska Natives precluded calculation of reliable estimates for this group. Increases in average scores for Asian/Pacific Islanders in grades 4 and 8 were observed between 2003 and 2005. Scores increased for grade 4 American Indians/Alaska Natives between 2003 and 2005, but not for grade 8 American Indians/Alaska Natives.

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