S&E Indicators 2006 - Chapter 1: Elementary and Secondary Education - Student Learning in Mathematics and Science

Student Learning in Mathematics and Science

Early Formal Learning: Kindergarten Through Third Grade
Performance of U.S. Students in Grades 4, 8, and 12
International Comparisons of Mathematics and Science Performance

The current performance of U.S. elementary and secondary students in mathematics and science is both encouraging and disappointing. Average mathematics scores on national assessments rose during the 1990s and early 2000s, and gains were widespread, with many demographic subgroups registering higher achievement. Performance in science has not improved recently, however. Substantial achievement gaps among some demographic subpopulations of students persist in both mathematics and science, and most 4th, 8th, and 12th grade students do not perform at levels considered proficient for their grade. On international assessments, recent data show that U.S. students performed above international averages that include scores from both developed and developing countries on tests closely aligned to the way mathematics and science are presented to them in the classroom. However, they performed below international averages for the 30 Organisation for Economic Co-operation and Development (OECD) nations in applying mathematical and scientific skills to situations they might encounter outside of a classroom.

This section presents information from recent national and international studies of U.S. student achievement in mathematics and science and compares them with earlier study results. It begins with a discussion of student performance during the primary grades, followed by a review of assessment results for students in grades 4, 8, and 12. The section ends by placing U.S. student achievement in a broader international context.

Early Formal Learning: Kindergarten Through Third Grade

The mathematics and science performance of U.S. students in upper-elementary and secondary grades has been reported since the late 1960s (Campbell, Hombo, and Mazzeo 2000). Much less has been known about student learning in these subjects during the first years of formal education, but this is changing with the release of initial findings from an ongoing study of students who began kindergarten in 1998 (Early Childhood Longitudinal Study, Kindergarten Class of 1998–99, ECLS–K).[1]

Kindergarten: Mathematics Skills and Knowledge

Children begin formal schooling with varying levels of mathematics skills, and over the course of the kindergarten year, the percentage of students proficient in specific skill areas increases (West, Denton, and Germino-Hausken 2000; West, Denton, and Reaney 2000).[2] In 1998, most beginning kindergartners (93%) could recognize single-digit numbers and basic shapes in the fall, and almost all (99%) demonstrated these skills in the spring (figure 1-1 ). In the fall, just more than half (57%) of the students could count beyond 10, recognize the sequence in basic patterns, and compare the relative size of objects, but by spring, 87% could do so. Increases occurred in other skill areas as well, although gains in more advanced skills such as addition, subtraction, multiplication, and division were relatively small (see sidebar "Mathematics Skills Areas for Primary Grade Students").

Disparities among subpopulations of students were evident when they started kindergarten. Mathematics performance was related to several student background factors, and the association between social disadvantages and performance was cumulative. Lower proportions of black and Hispanic students were proficient at each skill level compared with their white and Asian/Pacific Islander peers (appendix table 1-1 ).[3] Performance was also related to maternal education, with students whose mothers had less formal education demonstrating lower proficiency rates. For the kindergarten assessments, a family risk index was developed consisting of non-English primary home language, single-parent family, less than high school maternal education, and family receiving welfare assistance.[4] Students from families with no risk factors performed better than students from families with one risk factor, and students from families with one risk factor performed better than students from families with two or more risk factors.

As students progressed through kindergarten, gaps in basic mathematics skills decreased, but disparities in the more sophisticated skills increased. For example, by the end of kindergarten, blacks and Hispanics narrowed the proficiency gap with whites and Asians/Pacific Islanders in recognizing single-digit numbers and shapes and in comparing the relative size of objects (figure 1-2 ; appendix table 1-1 ). However, they did not acquire more advanced mathematics knowledge and skills, such as addition and subtraction, at the same rate as whites and Asians/Pacific Islanders. This resulted in even larger disparities in the more sophisticated skills by the end of kindergarten.

The First 4 Years of School

Mathematics. After 4 years of formal schooling, when most students were at the end of third grade, some performance gaps had widened (Rathbun and West 2004) (figure 1-3 ; appendix table 1-2 ).[5] Black students, who entered kindergarten with lower overall mathematics scores than white and Asian/Pacific Islander students, made smaller gains over the 4 years than did white, Asian/Pacific Islander, and Hispanic students, resulting in larger performance gaps. Students with one or more family risk factors started formal education with lower scores and made less progress than students with no family risk factors, also resulting in larger performance gaps.

Other research has shown that widening achievement gaps as students progress through school is, 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.

Studies of upper-elementary and secondary students dating back to the late 1960s have documented some sex differences in science and mathematics performance (e.g., Campbell, Hombo, and Mazzeo 2000; NCES 2003a and 2003b).[6] The ECLS–K study, the first national study of primary grade students, found no sex differences in average overall mathematics performance during the first 4 years of schooling (Rathbun and West 2004; West, Denton, and Germino-Hausken 2000; West, Denton, and Reaney 2000). However, at the end of third grade, boys were more likely than girls to demonstrate proficiency in the advanced mathematics skills of place value concepts and knowledge of rate and measurement to solve word problems (appendix table 1-3 ). These advanced math skills were first assessed in the third followup, when most students were in third grade.

The ECLS–K study examined associations between mathematics performance and two aspects of students' early school experiences: whether they attended public or private schools, and whether they attended full- or half-day kindergarten. Performance differences in mathematics by school type were evident as students started formal schooling (West, Denton, and Germino-Hausken 2000). Students beginning kindergarten in private schools had stronger mathematics skills than those at public schools. Although achievement differences persisted through the third grade, the growth rate in mathematics did not differ. Therefore, performance gaps between public and private school students did not increase (Rathbun and West 2004).[7] Students in full-day kindergartens experienced greater gains in mathematics compared with their peers in half-day classes (Watson and West 2004). At the end of third grade, however, the benefit of full-day kindergarten could no longer be detected (Rathbun, West, and Germino-Hausken 2004).

Science. The ECLS–K study began assessing students in science in spring 2002, when most were in third grade. The assessment placed equal emphasis on life science, earth and space science, and physical science and asked students to demonstrate understanding of the physical and natural world, make inferences, and understand relationships (Rathbun and West 2004). Students were also required to interpret scientific data, form hypotheses, and develop plans to investigate scientific questions.[8] Performance gaps observed in mathematics were also generally found in science (appendix table 1-4 ): white and Asian/Pacific Islander students had higher average science scores than blacks and Hispanics; Hispanic third graders outperformed their black peers; and students with no family risk factors scored higher, on average, than those with one or more risk factors. No sex differences were observed in third grade science performance.

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Performance of U.S. Students in Grades 4, 8, and 12

Many of the same performance gaps in mathematics and science achievement found among primary students also exist among upper-elementary and secondary students. Although mathematics performance in particular improved through the 1990s and early 2000s for many subgroups, substantial achievement gaps persist and, as will be detailed below, in some cases, have grown wider.

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.[9] This volume reports on recent trends, from 1990 to 2003 for mathematics and from 1996 to 2000 for science.[10] Previous Science and Engineering Indicators described long-term trends in mathematics and science results dating back to the first NAEP assessments.[11] Long-term trends in mathematics achievement from the 2004 administration were released too late for the text of this chapter but are reviewed briefly in the sidebar "Long-term Trends in Student Mathematics Achievement" at the conclusion of this section.

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 2003a). The mathematics assessment contains five broad content strands (number sense, properties, and operations; measurement; geometry and spatial sense; data analysis, statistics, and probability; and algebra and functions). It also assesses mathematical ability (conceptual understanding, procedural knowledge, and problem solving) and mathematical power (reasoning, connections, and communication). The science framework includes a content dimension divided into three major fields of science (earth, life, and physical), and a cognitive dimension covering conceptual understanding, scientific investigation, and practical reasoning (NCES 2001).

Student performance on the NAEP is measured with scale scores as well as achievement levels. The scale scores place students on a continuous ability scale based on their overall performance. For mathematics, the scale ranges from 0 to 500 across the three grades. For science, the scale ranges from 0 to 300 within each grade.

The 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 (NCES 2003a). The basic level represents partial mastery of the knowledge and skills needed to perform proficiently at each grade level. The proficient level represents solid academic performance and the advanced level represents superior performance. This review of NAEP results focuses on the proficient level (for definitions of the proficient level for grades 4, 8, and 12, see sidebars "Proficient Level in Mathematics in Grades 4, 8, and 12" and "Proficient Level in Science in Grades 4, 8, and 12").

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. Nevertheless, they warn readers to use and interpret the levels with caution (NCES 2003a).

In this section, the NAEP results are examined in a number of ways, including changes in average scores and the proportion of students reaching the proficient level, both overall and among 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 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 only high-achieving students or whether it reflects broader improvements.

Mathematics Performance

The average mathematics scores of fourth and eighth grade students increased from 1990 (the first year in which the current assessment was given) to 2003 (NCES 2001, 2003a) (figure 1-4 ; table 1-1 ).[12] The average performance of 12th graders also improved between 1990 and 2000, when they were last assessed. The pattern of increased average scores was widespread (table 1-1 ; appendix table 1-5 ). At each grade level, average mathematics scores improved for both male and female students, and for all students regardless of eligibility for free or reduced-price lunch (a commonly used indicator for poverty).[13] Generally, gains were observed for white, black, Hispanic, and Asian/Pacific Islander 4th and 8th grade students, although at grade 12, only the scores of white students improved.[14] Higher average scores for students at the 10th, 25th, 50th, 75th, and 90th percentiles in 2003, compared with 1990, provide further evidence that gains in mathematics were widespread. (Percentiles indicate the percentage of students whose scores fell below a particular score. For example, 75% of students had scores below the 75th percentile.)

Improvements in average mathematics scores were generally mirrored by increases in the percentage of students scoring at or above the proficient level for their grade (figure 1-5 ; table 1-1 ; appendix table 1-6 ). This growth was substantial at grades 4 and 8, with rates about doubling between 1990 and 2003.

Although gains in mathematics achievement are encouraging, despite the improvements, most students do not demonstrate solid mathematics skills and knowledge for their grade. In the latest NAEP mathematics assessments (2003 for grades 4 and 8, and 2000 for grade 12), only about one-third of 4th and 8th graders, and even fewer 12th graders (16%), reached the proficient level (figure 1-5 ; appendix table 1-6 ).

Science Performance

Recent trend lines for science are shorter than those for mathematics, and they suggest less improvement. Although average mathematics scores of fourth and eighth grade Students increased from 1996 to 2000 (appendix table 1-5 ), average science scores did not change (NCES 2003b) (table 1-1 ; appendix table 1-7 ). At grade 12, average science scores declined. The proportion of students reaching the proficient level in science did not change for any of the three grades. Subgroup results in science were also generally flat between 1996 and 2000, both in terms of average scores and in the percent at or above the proficient level.[15] (The current national NAEP science assessment was administered in 1996, 2000, and 2005. The 2005 data were not available in time to be included in this report.)

In results similar to the 2003 mathematics findings, only about one-third of fourth and eighth grade students reached the proficient level in science for their grade in 2000 (figure 1-5 ; appendix table 1-8 ). Rates were lower among 12th graders, with only 18% of these students scoring at or above the proficient level.

Achievement Gaps Between Demographic Subgroups

Gender Achievement Gaps. The most recent NAEP assessments report only small sex differences in mathematics and science performance at grades 4, 8, and 12, with boys performing slightly better than girls (appendix tables 1-5 , 1-6 , 1-7 , and 1-8 ).[16] For example, in 2003, 35% of fourth grade boys reached the proficient level in mathematics, compared with 30% of fourth grade girls (figure 1-6 ). The small gender gaps in mathematics have generally remained stable since 1990. However, the small gender gaps among fourth and eighth graders observed in science in 2000, for the most part, represent an increase from those observed in 1996 (table 1-1 ; appendix tables 1-5 , 1-6 , 1-7 , and 1-8 ).

Racial/ethnic Achievement Gaps. Substantial performance gaps exist between some racial/ethnic subgroups. At each grade level, white and Asian/Pacific Islander students performed better than black, Hispanic, and American Indian/Alaska Native students in both mathematics and science, both in terms of average scores and in percentage of students reaching the proficient level (figure 1-7 ; appendix tables 1-5 , 1-6 , 1-7 , and 1-8 ). These achievement differences were relatively large. For example, in 2003, between four and five times as many white and Asian/Pacific Islander fourth grade students reached the proficient level in mathematics as did black students (see sidebar "Tenth Graders' Proficiency in Specific Mathematics Skill and Knowledge Areas").

More subtle racial/ethnic differences in achievement were also observed.[17] For example, Asians/Pacific Islanders demonstrated slightly higher performance than whites in mathematics at each grade level, but the reverse was true for science at grades 4 and 8. In addition, in some instances, American Indian/Alaska Native and Hispanic students registered slightly higher performances than did black students (see sidebar "Projected School-Age Population of the United States").

Family Income Achievement Gaps. Mathematics and science performance also differed by family income (as measured by whether or not a student was eligible for the free or reduced-priced school lunch program) (figure 1-8 ; appendix tables 1-5 , 1-6 , 1-7 , and 1-8 ). At each grade level, in both mathematics and science, students eligible for the subsidized lunch program (i.e., students from low-income families) had lower average scores and were less likely to reach the proficient level than students who were not eligible. These gaps related to family income were substantial. For example, students eligible for free or reduced lunch were at least three times less likely to score at or above the proficient level for their grade in both mathematics and science.

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International Comparisons of Mathematics and Science Performance

Two mathematics and science assessments conducted in 2003 place U.S. student achievement in these subjects in an international context: the Trends in International Mathematics and Sciences Study (TIMSS) and the Programme for International Student Assessment (PISA). Results from the two assessment programs paint a complex picture. As detailed below, U.S. students scored above international averages on the TIMSS assessment and below international averages on the PISA assessment. The two programs are designed to serve different purposes, and each provides unique information about U.S. student performance relative to other countries in mathematics and science (Scott 2004). The differences in design and purpose of the assessments should be kept in mind when reviewing these divergent results.

One such difference is the grade/age of the students assessed. TIMSS provides data on mathematics and science achievement of students in primary and middle grades (grades 4 and 8 in the United States).[18] PISA reports the performance of students in secondary schools by sampling 15-year-olds, an age near the end of compulsory schooling in many countries.

Another difference between TIMSS and PISA is the relationship of the assessments to mathematics and science curriculum. TIMSS measures student mastery of curriculum-based knowledge and skills. Mathematics and science content experts and educators from many countries developed the framework behind the TIMSS assessment, and representatives from each participating country were asked to review and comment. The goal is to assess the mathematics and science content and skills that students are taught in school.[19] It is important to note that many of the participating countries have centralized, nationally mandated curriculums, whereas in the United States, curriculum, in the form of content standards, is developed at the state and local levels (Schmidt et al. 2001).

PISA , on the other hand, places more emphasis on Students' ability to apply scientific and mathematical concepts and thinking skills to problems they might encounter, particularly in situations outside of a classroom. To some degree, PISA mathematics questions tend to demand more complex reasoning and problem solving skills than those in TIMSS (Neidorf et al. forthcoming) (see sidebar "Sample Mathematics and Science Items From the Curriculum-Based TIMSS Assessment and the Literacy-Based PISA Assessment").

A third difference is the composition of the participating countries. The 46 countries participating in the 2003 TIMSS include 13 highly industrialized nations, as well as many industrializing and developing ones. TIMSS international averages are based on all of these participating countries. In contrast, the PISA results reviewed in this chapter are based on average scores from 30 OECD countries. Thus, although the TIMSS averages include scores from both developed and developing countries, the PISA averages reflect only the performance of industrialized countries.[20] In addition to comparing the performance of U.S. students to these two sets of international averages, the text and tables 1-5 and 1-6 compare the United States with other OECD and Group of 8 (G-8) nations. The G-8 are the eight most industrialized countries in the world that meet regularly to discuss economic and other policies issues: Canada, France, Germany, Italy, Japan, the Russian Federation, the United Kingdom, and the United States.

TIMSS 2003 Results for Students in Grades 4 and 8: Curriculum-Based Knowledge in Mathematics and Science

Curriculum-Based Mathematics Performance. In 2003, the average curriculum-based mathematics score of U.S. fourth and eighth grade students exceeded the TIMSS international averages for these two grades, which included scores from both developed and developing countries (Gonzales et al. 2004) (appendix tables 1-9 and 1-10 ). Compared with other participating G-8 nations, U.S. fourth graders were out-performed by their counterparts in England, Japan, and Russia but registered higher average scores than students in Italy (table 1-5 ). At grade 8, the average score of U.S. students was lower than the average score of students in Japan but higher than the average score of students in Italy. The average mathematics score of eighth grade U.S. students was approximately equivalent to the average scores of students in Russia.

TIMSS also was conducted in 1995, permitting an examination of changes in performance over time. The average mathematics score of U.S. fourth graders on this curriculum-based assessment did not change from 1995 to 2003, but eighth graders' scores improved (data not shown, see Gonzales et al. 2004). Based on these results and on changes in average performance in some of the other countries (both improvement and decline), the relative ranking of the United States in mathematics declined slightly at grade 4 but improved slightly at grade 8.[21]

Curriculum-Based Science Performance. Examination of science results shows that in 2003, the average science score of U.S. fourth and eighth grade students was higher than the TIMSS international averages, which were based on scores from both developed and developing countries (Gonzales et al. 2004) (appendix tables 1-11 and 1-12 ). Compared with the participating G-8 countries, the average score of U.S. students was higher than that of students in Italy in both grades 4 and 8 (table 1-6 ). In addition, U.S. eighth graders had higher average scores than their counterparts in Russia. However, Japan outperformed the United States at both grade levels and England outperformed the United States at grade 4.

Mirroring results for mathematics, average science scores of fourth graders did not change from 1995 to 2003, but science performance among eighth graders improved over this period (data not shown, see Gonzales et al. 2004). The relative ranking of U.S. students in science fell slightly between 1995 and 2003 for grade 4 but rose slightly for grade 8.[22]

PISA 2003 Assessments of Mathematics and Science Literacy of 15-Year-Olds

Although TIMSS measures how well students have mastered the mathematical and scientific content presented in school, PISA assesses students' literacy in these subjects (Lemke et al. 2004). PISA uses the term literacy to denote the program's goal of assessing how well students can apply their knowledge and skills to problems they might encounter, particularly in situations outside of a classroom.

In 2003, U.S. 15-year-olds performed below the OECD average in both mathematics and science literacy (appendix tables 1-13 and 1-14 ).[23] Among OECD nations, U.S. students were near the bottom in mathematics literacy, outperformed by students in Canada, France, Germany, the Netherlands, South Korea, Japan, and 14 other countries (table 1-5 ; appendix table1-13 ). The United States was at rough parity with Hungary, Poland, and Spain, and scored higher than Greece, Italy, Mexico, Portugal, and Turkey. In science, average literacy scores were higher in 15 other OECD countries compared with the United States and lower in 6 (table 1-6 ; appendix table1-14 ).

U.S. students' average science literacy scores did not change from 2000, the first year PISA was administered, to 2003 (data not shown, see Lemke et al. 2004). However, several other OECD countries registered improvements in science, and as a result, the relative position of the United States compared with the OECD average declined.[24] In 2000, the average score of U.S. 15-year-olds' science literacy did not differ from OECD averages, but in 2003, it was lower. U.S. performance in mathematics did not change from 2000 to 2003, and in both years, the U.S. average fell below the OECD average.[25]

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Mathematics Skills Areas for Primary Grade Students

The Early Childhood Longitudinal Study, Kindergarten Class of 1998–99 (ECLS–K) mathematics assessment measures core foundational mathematics skills, including conceptual understanding of numbers, shapes, mathematical operations, and processes for problem solving (West, Denton, and Germino-Hausken 2000). The assessment provides information on student performance in the form of an overall achievement score and proficiency in seven specific skill sets. The skill sets represent a progression of mathematics skills and knowledge. Levels 6 and 7 were first assessed in third grade. Each set of skills is labeled by the most sophisticated skill in the set.

Level 1: Number and shape: recognize single-digit numbers and shapes.

Level 2: Relative size: count beyond 10, recognize the sequence in basic patterns, and compare the relative size and dimensional relationship of objects.

Level 3: Ordinality and sequence: recognize two-digit numbers, identify the next number in a sequence, identify the ordinal position of an object, and solve simple word problems.

Level 4: Add and subtract: solve simple addition and subtraction items and identify relationships of numbers in sequence.

Level 5: Multiply and divide: perform basic multiplication and division and recognize more complex number patterns.

Level 6: Place value: demonstrate understanding of place value in integers to the hundredth place.

Level 7: Rate and measurement: use knowledge of measurement and rate to solve word problems.

SOURCE: West, Denton, and Reaney 2000.

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Proficient Level in Mathematics in Grades 4, 8, and 12

The National Assessment of Educational Progress (NAEP) ranks student performance according to three achievement levels: basic, proficient, and advanced. The levels are set by the National Assessment Governing Board (NAGB) based on recommendations from panels of educators and members of the public of what students should know and be able to do in the subject assessed. NAGB's definition of the proficient level for mathematics for grades 4, 8, and 12 is directly quoted below. Descriptions of the other achievement levels can be found in the report cited at the end of the sidebar.

Grade 4

Fourth grade students performing at the Proficient level should consistently apply integrated procedural knowledge and conceptual understanding to problem solving in the five NAEP content strands.

Fourth graders performing at the Proficient level should be able to use whole numbers to estimate, compute, and determine whether results are reasonable. They should have a conceptual understanding of fractions and decimals; be able to solve real-world problems in all NAEP content areas; and use four-function calculators, rulers, and geometric shapes appropriately. Students performing at the Proficient level should employ problem-solving strategies such as identifying and using appropriate information. Their written solutions should be organized and presented both with supporting information and with explanations of how they were achieved.

Grade 8

Eighth grade students performing at the Proficient level should apply mathematical concepts and procedures consistently to complex problems in the five NAEP content strands.

Eighth graders performing at the Proficient level should be able to conjecture, defend their ideas, and give supporting examples. They should understand the connections among fractions, percents, decimals, and other mathematical topics such as algebra and functions. Students at this level are expected to have a thorough understanding of basic-level arithmetic operations—an understanding sufficient for problem solving in practical situations. Quantity and spatial relationships in problem solving and reasoning should be familiar to them, and they should be able to convey underlying reasoning skills beyond the level of arithmetic. They should be able to compare and contrast mathematical ideas and generate their own examples. These students should make inferences from data and graphs, apply properties of informal geometry, and accurately use the tools of technology. Students at this level should understand the process of gathering and organizing data and be able to calculate, evaluate, and communicate results within the domain of statistics and probability.

Grade 12

Twelfth grade students performing at the Proficient level should consistently integrate mathematical concepts and procedures into the solutions of more complex problems in the five NAEP content strands.

Twelfth graders performing at the Proficient level should demonstrate an understanding of algebraic, statistical, geometric, and spatial reasoning. They should be able to perform algebraic operations involving polynomials, justify geometric relationships, and judge and defend the reasonableness of answers as applied to real-world situations. These students should be able to analyze and interpret data in tabular and graphical form; understand and use elements of the function concept in symbolic, graphical, and tabular form; and make conjectures, defend ideas, and give supporting examples.

Source: NAGB 2002.

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Proficient Level in Science in Grades 4, 8, and 12

The National Assessment of Educational Progress (NAEP) ranks student performance according to three achievement levels for their grade: basic, proficient, and advanced. The levels are set by the National Assessment Governing Board (NAGB) based on recommendations from panels of educators and members of the public of what students should know and be able to do in the subject assessed. NAGB's definition of the proficient level in science for grades 4, 8, and 12 is directly quoted below. Descriptions of the other achievement levels can be found in the report cited at the end of the sidebar.

Grade 4

Students performing at the Proficient level demonstrate the knowledge and reasoning required for understanding of Earth, physical, and life sciences at a level appropriate to grade 4. For example, they understand concepts relating to the Earth's features, physical properties, structure, and function. In addition, students can formulate solutions to familiar problems as well as show a beginning awareness of issues associated with technology.

Fourth grade students performing at the Proficient level are able to provide an explanation of day and night when given a diagram. They can recognize major features of the Earth's surface and the impact of natural forces. They are also able to recognize water in its various forms in the water cycle and can suggest ways to conserve it. These students recognize that various materials possess different properties that make them useful. Students at this level are able to explain how structure and function help living things survive. They have a beginning awareness of the benefits and challenges associated with technology and recognize some human effects on the environment. They can also make straightforward predictions and justify their position.

Grade 8

Students performing at the Proficient level demonstrate much of the knowledge and many of the reasoning abilities essential for understanding of Earth, physical, and life sciences at a level appropriate to grade 8. For example, students can interpret graphic information, design simple investigations, and explain such scientific concepts as energy transfer. Students at this level also show an awareness of environmental issues, especially those addressing energy and pollution.

Eighth grade students performing at the Proficient level are able to create, interpret, and make predictions from charts, diagrams, and graphs based on information provided to them or from their own investigations. They have the ability to design an experiment and have an emerging understanding of variables and controls. These students are able to read and interpret geographic and topographic maps. In addition, they have an emerging ability to use and understand models, can partially formulate explanations of their understanding of scientific phenomena, and can design plans to solve problems. Students at this level can begin to identify forms of energy and describe the role of energy transformations in living and nonliving systems. They have knowledge of organization, gravity, and motion within the solar system and can identify some factors that shape the surface of the Earth. These students have some understanding of properties of materials and have an emerging understanding of the particulate nature of matter, especially the effect of temperature on states of matter. They also know that light and sound travel at different speeds and can apply their knowledge of force, speed, and motion. These students demonstrate a developmental understanding of the flow of energy from the sun through living systems, especially plants. They know that organisms reproduce and that characteristics are inherited from previous generations. These students also understand that organisms are made up of cells and that cells have subcomponents with different functions. In addition, they are able to develop their own classification system based on physical characteristics. These students can list some effects of air and water pollution as well as demonstrate knowledge of the advantages and disadvantages of different energy sources in terms of how they affect the environment and the economy.

Grade 12

Students performing at the Proficient level demonstrate the knowledge and reasoning abilities required for understanding of the Earth, physical, and life sciences at a level appropriate to grade 12. In addition, they demonstrate knowledge of the themes of science (models, systems, and patterns of change) required for understanding how these themes illustrate essential relationships among the Earth, physical, and life sciences. They are able to analyze data and apply scientific principles to everyday situations.

Twelfth grade students performing at the Proficient level are able to demonstrate a working ability to design and conduct scientific investigations. They are able to analyze data in various forms and utilize information to provide explanations and to draw reasonable conclusions. Students at this level have a developmental understanding of both physical and conceptual models and are able to compare various models. They recognize some inputs and outputs, causes and effects, and interactions of a system. In addition, they can correlate structure to function for the parts of a system that they can identify. These students also recognize that rate of change depends on initial conditions and other factors. They are able to apply scientific concepts and principles to practical applications and solutions for problems in the real world and show a developmental understanding of technology, its uses, and its applications.

Source: NAGB 2000.

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Tenth Graders' Proficiency in Specific Mathematics Skill and Knowledge Areas

Achievement disparities by student and family backgrounds are observed in other national studies, such as the Education Longitudinal Study of 2002 (ELS: 2002). This base-year study assessed mathematics achievement of 10th grade students and placed their performance in one of five proficiency levels: simple arithmetical operations with whole numbers; simple operations with decimals, fractions, powers, and roots; simple problem solving requiring the understanding of low-level mathematical concepts; understanding of intermediate-level mathematical concepts and multistep solutions to word problems; and complex multistep word problems and advanced mathematics material (Ingels and Scott 2004). The skill levels represent a progression of mathematics skills and knowledge.

In 2002, a vast majority of 10th grade students (92%) were proficient in simple arithmetical operations with whole numbers, and 67% were also proficient in simple operations with decimals, fractions, roots, and powers (table 1-2 ). However, the proportions demonstrating proficiency in more advanced mathematics skills were lower and decreased with the progression of skill levels. The differences in proficiency in each skill area for male and female students were small, but they were larger for racial/ethnic and family socioeconomic subgroups. White and Asian/Pacific Islander students were more likely than black and Hispanic students to demonstrate proficiency in each level of mathematics skills, as were students from high-socioeconomic families compared with those from low-socioeconomic families. Followup data collection is under way. When these longitudinal data are available and can be used with other longitudinal studies such as High School and Beyond (HS&B) and the National Education Longitudinal Study (NELS), they will provide more valuable information about growth in student achievement and factors related to this growth.

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Long-Term Trends in Student Mathematics Achievement

This chapter presents indicators of student achievement in mathematics and science based on the national NAEP assessments. This sidebar briefly introduces indicators of mathematics learning based on the NAEP 30-year long-term trend assessment of 2004 that became available in July 2005, too late for incorporation into the text of this volume.[9]

Major differences between these two NAEP programs include:

Content in the long-term trend assessments has remained the same across administrations, whereas the national assessments have been updated periodically as the world and curricula have changed.
The long-term trend assessment is administered to 9-, 13-, and 17-year-olds, whereas the national assessments are given to students in the 4th, 8th, and 12th grades.
The long-term trend assessment reports achievement at the national level, whereas the national assessment reports achievement at the national and state levels and produces some district-level data.

This sidebar discusses scores on mathematics performance of representative samples of more than 11,000 students at each of the three ages assessed. More detailed data, as well as scores on reading, are available in the full report (Perie, Moran, and Lutkus 2005).

Overall Trend in Mathematics

Average scores on the long-term trend assessment in mathematics increased for 9- and 13-year olds in 2004 over the last assessment in 1999. The average score of 9-year-olds, after remaining flat throughout the 1990s, increased 9 points in 2004; the 2004 scores were 22 points higher than 30 years earlier. Thirteen-year-olds' average scale score increased 5 points in 2004 over 1999 and 15 points over 1973.

However, mathematics scores of 17-year-olds did not change from 1999 to 2004. The average score of 17-year-olds has increased 9 points since the lowest score in 1982, but has remained flat for more than a decade and is not significantly different from the average score for the first long-term trend mathematics assessment in 1973.

Trends in Mathematics Score Gaps

Samples of students for the NAEP long-term trend assessments are sufficiently large to allow reporting of scores separately for whites, blacks, and Hispanics. As table 1-3 shows, whites have, on average, scored higher than blacks and Hispanics throughout the 30-year assessment period. Although the gaps in achievement have decreased over the 30-year period, few of these declines occurred in the past 20 years.

Across the 30 years of the testing program, the gap in scores between whites and blacks decreased by 12, 19, and 12 points for 9-, 13-, and 17-year-olds, respectively. However, for each age group, the gap has remained significantly unchanged for at least the past decade.

The gap in average scores between white and Hispanic 9-year-olds was lower in 2004 than 1999 but did not differ from the 1973 gap. The gap in scores between white and Hispanic 13- and 17-year-olds decreased 12 and 9 points, respectively, between 1973 and 2004. However, this improvement was registered early in the assessment program; no statistically significant improvement has been measured since the 1970s.

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Projected School-Age Population of the United States

The No Child Left Behind Act of 2001 grew out of concerns about disparities in performance among subpopulations of students. Current population projections indicate increasing student population in coming decades, particularly among several racial/ethnic subgroups currently underperforming in mathematics and science. The number of children ages 5 to 17 is expected to increase by 33% between 2000 and 2050. Population growth is estimated to occur among each group shown in table 1-4 with the exception of non-Hispanic whites, whose population is projected to decline by 6% between 2000 and 2050.

Differential growth rates across these groups are expected to change the racial/ethnic distribution of the U.S. schoolage population. In 2000, Hispanic children made up 16% of the population ages 5 to 17 years, but by 2050, this percentage will almost double to 29%. The proportion of the schoolage population that is white, non-Hispanic will decrease from 62% in 2000 to 44% in 2050. The percentage of the population that is Asian/Pacific Islander is expected to almost double, from 4% to 7%. The proportion of children in the "all other races" category is also expected to grow substantially from 4% to 8%. The percentage of the schoolage population that is black is not forecast to change from 2000 to 2050.

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Sample Mathematics and Science Items From the Curriculum-Based TIMSS
Assessment and the Literacy-Based PISA Assessment

Example items from the two international assessments are provided below. Trends in International Mathematics and Sciences Study (TIMSS) assesses mathematics and science skills of fourth and eighth graders in a manner closely aligned with the way these subjects are typically presented in school. The Programme for International Student Assessment (PISA) measures 15-year-olds' abilities to apply mathematics skills and knowledge.

TIMSS Eighth Grade Mathematics Item

If n is a negative integer, which of these is the largest number?

(A) 3 + n
(B) 3 x n
(C) 3 – n
(D) 3 ÷ n

Correct Answer: C

Percent correct:
United States 48
International average 40

TIMSS Eighth Grade Science Item

The burning of fossil fuels has increased the carbon dioxide content of the atmosphere. What is a possible effect that the increased amount of carbon dioxide is likely to have on our planet?

(A) A warmer climate
(B) A cooler climate
(C) Lower relative humidity
(D) More ozone in the atmosphere

Correct Answer: A

Percent correct:
United States 56
International average 44

PISA 15-Year-Old's Mathematics Item

(See illustration below)

A carpenter has 32 meters of timber and wants to make a border around a garden bed. The carpenter is considering several designs for the garden bed.

Circle either "Yes" or "No" for each design to indicate whether the garden bed can be made with 32 meters of timber.

Correct Answers: Design A, Yes; Design B, No; Design C, Yes; Design D, Yes

Percent full credit:
United States 15
International average 20

PISA 15-Year-Old's Science Item

Drivers are advised to leave more space between their vehicles and the ones in front when they are traveling more quickly than when they are traveling more slowly because faster cars take longer to stop.

Explain why a faster car can take more distance to stop than a slower one.

Reasons: ___________________________________

Full credit: Answers that mention that:

The greater momentum of a vehicle when it is moving more quickly means that it will move further while slowing down than a slower vehicle, given the same force;

AND

It takes longer to reduce speed to zero from a greater speed, so the car will travel further in this time.

Partial credit: Answers that mention only one of the points above.

Results for this item not published.

SOURCES: Gonzales et al. 2004; OECD 2003b; and http://nces.ed.gov/surveys/pisa/Items.asp?SectionID=2&CatID=4.

Illustration for PISA 15-year-old's mathematics item: Garden bed design options.

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Footnotes

[1] A series of reports based on data from the ECLS-K study and released by the National Center for Education Statistics (NCES) can be found at: http://nces.ed.gov/ecls.

[2] The ECLS-K assessment measures students' overall mathematics achievement through both scale scores and their specific mathematics skills and knowledge as measured through a set of proficiency scores. The scale scores place students on a continuous ability scale based on their overall performance on the assessment, whereas the proficiency scores are based on clusters of items assessing particular skills and report whether students mastered those skills. When describing gains over the kindergarten year, this review focuses on proficiency in specific areas. When reporting on growth in achievement from kindergarten to third grade, scale scores are discussed. For more information on the ECLS assessment battery and scoring, including the Item Response Theory (IRT) methodology used, see Rathbun and West (2004) and West, Denton, and Reaney (2000).

[3] The studies reviewed in this chapter report combined results for Asians and Pacific Islanders. It is important to note that this category combines groups that have very different cultural and historical backgrounds, and whose achievement varies widely.

[4] In later years of the ECLS-K study, family income below the federal poverty level was substituted for the welfare assistance risk factor. Students were classified as having no family risk factors, one risk factor, or two or more risk factors.

[5] About 10% of the cohort was in second grade, and another 1% was in another grade. For the sake of simplicity, the students in the 2002 followup are referred to as third graders.

[6] Trends in mathematics and science performance by gender are not easily summarized, with girls outperforming boys in some age groups and boys outperforming girls in other cases. See Science and Engineering Indicators – 2004, page 1-7, for more details on long-term trends in mathematics and science performance of males and females. See sidebar in this issue "Long-Term Trends in Student Mathematics Achievement."

[7] Students were identified as attending private schools continuously, attending public schools continuously, or attending a combination of private and public schools between the beginning of kindergarten and the end of third grade. There were no statistically significant differences in gains in average mathematics scores across these three groups.

[8] Because students have been assessed in science only once in the ECLS, the study has thus far produced less information on science learning. As of yet, only science scale scores have been reported. As the study continues to follow these students, future reports will likely provide more detail on science achievement.

[9] 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.

[10] These recent trends are based on data from the national NAEP program. The current national mathematics assessment was first administered in 1990 and was given again in 1992, 1996, 2000, and 2003. In 2003, only fourth and eighth grade students were assessed. The current national science assessment was first administered in 1996 and was given again in 2000 and 2005. The 2005 results were not available in time for inclusion.

[11] The 2002 and 2004 volumes reviewed trends in science from 1969 to 1999 and in mathematics from 1973 to 1999. The long-term trend assessment in mathematics was administered again in 2004, but those data were not released in time to be included in the text of this chapter (see sidebar "Long-Term Trends in Student Mathematics Achievement"). The long-term trend assessment in science has not been given since 1999.

[12] NAEP is in the process of changing the way it includes students with disabilities and limited English proficiency in assessments. Before 1996, these students were not allowed to use testing accommodations (e.g., extended time, one-on-one testing, bilingual dictionary); as a result, many did not participate. In 1996 and 2000, the assessment was administered to split samples of "accommodations not permitted" and "accommodations permitted." In 2003, the NAEP mathematics assessment completed the transition to an "accommodations permitted" test.

[13] Using eligibility for the free or reduced-price lunch program as a proxy for family poverty is not as reliable in the higher grades because older students may attach stigma to receiving a school lunch subsidy.

[14] Sample size was insufficient to permit reliable mathematics estimates for American Indian/Alaska Natives prior to 1996 for grades 4 and 12 and prior to 2000 for grade 8.

[15] NCES did not publish 2000 science scores for fourth grade Asian/Pacific Islander students because of accuracy and precision concerns; therefore, those scores are not included.

[16] In science, the apparent difference at grade 12 in average scale scores by gender was not statistically significant. However, a greater proportion of 12th grade boys reached the proficient level in science than did girls.

[17] For detailed racial/ethnic group comparisons see NCES (2001, 2003a, 2003b).

[18] The primary grade assessed in each country was "the upper of the two adjacent grades with the most 9-year-olds" (Mullis et al. 2005). In the United States, and most other countries, this was the fourth grade. The middle grade assessed was defined as the "upper of the two adjacent grades with the most 13-year-olds." In the United States and most countries, this was the eighth grade. Students in their final year of secondary school (12th grade in the United States) were assessed with TIMSS in 1995. For a review of those results, see page 1-14 in Science and Engineering Indicators – 2004 or Takahira et al. (1998). Subsequent TIMSS administrations have focused on the middle grades.

[19] To be assessed in TIMSS, the specific content domains and topics had to be included in the curricula of "a significant number of participating countries" (Mullis et al. 2005). It is important to note that whereas the TIMSS program identified common mathematics and science curriculum across participating countries, there are many differences in the way countries delivered that curriculum and in their breadth of coverage (Sherman, Honegeger, and McGivern 2003).

[20] More information about TIMSS and PISA assessments can be found at http://nces.ed.gov/TIMSS/ and http://nces.ed.gov/Surveys/PISA/.

[21] Of the 14 other countries that participated in both the 1995 and 2003 grade 4 TIMSS mathematics assessments, the United States was outperformed by four countries in 1995 and by seven countries in 2003. Of the 21 other countries that participated in both the 1995 and 2003 grade 8 mathematics assessments, 12 had average scores higher than the U.S. average score in 1995 and 7 had higher scores in 2003.

[22] Of the 14 other countries that participated in both the 1995 and 2003 grade 4 TIMSS mathematics assessments, only 1 had a higher average score than the United States in 1995, but 2 did in 2003. At grade 8, of the 21 countries that participated in both years, 9 had higher average scores than the United States in 1995, whereas 5 did in 2003.

[23] Forty-one countries participated in the 2003 PISA assessment—30 OECD member countries and 11 non-OECD countries. This section summarizes a report released by NCES (2004c) that presents PISA results from a U.S. perspective. That report omitted data from the United Kingdom because of low response rates and from Brazil because these data were not yet available. That report and this section compare U.S. averages first to OECD averages (i.e., average of national averages from the 29 OECD countries for which data were available, including the United States) and, second, to individual country averages (both OECD and non-OECD countries).

[24] Data for both 2000 and 2003 are available for 26 OECD countries, including the United States. Of these countries, nine improved their science scores and five registered declines.

[25] Comparing change in mathematics performance is complicated by the fact that the 2003 PISA assessment was more extensive than the 2000 assessment. In 2000, two content areas were assessed: space and shape and change and relationship. In 2003, those two areas, along with two additional content areas (quantity and uncertainty) were tested. Thus, change in mathematics performance can be examined only for the two content areas assessed in both years. The average scores for U.S. students did not change from 2000 to 2003 on either the space and shape or the change and relationship content areas. Of the 25 other countries that participated in both assessment years, 18 outperformed the United States in the space and shape area in 2003 compared with 19 in 2000. In the change and relationship area, 17 countries outperformed the United States in 2003, and 14 did in 2000.