Overview of the Assessment

Reporting the Assessment Results—Scale Scores and Achievement Levels

Description of Mathematics Performance by Item Maps for Each Grade

Results Are Estimates

NAEP Reporting Groups

Exclusion Rates

Statistical Significance

Cautions in Interpretations

NAEP assesses mathematics performance by administering assessments to samples of students who are representative of the nation's students. The content of the NAEP mathematics assessment is determined by a framework incorporating expert perspectives about mathematics knowledge and its measurement. Read more about what the assessment measures, how it was developed, who took the assessment, and how the assessment was administered.

Beginning in 2003, all of the states and jurisdictions have participated in the grade 4 and 8 NAEP mathematics assessments. For grades 4 and 8, the NAEP national sample was obtained by aggregating the samples of public school students from each state and jurisdiction, and then supplementing the aggregate sample with a nationally representative sample of students from nonpublic schools, rather than by obtaining an independently selected national sample. As a consequence, the national sample size increased, and smaller differences between years or between groups of students were found to be statistically significant than would have been detected in previous assessments. In keeping with past practice, all statistically significant differences are indicated in the current web results pages.

At grade 12, the national sample of schools and students is drawn from across the country, and results from the assessed students are combined to provide accurate estimates of the overall performance of twelfth-graders in the nation. There are no state results for grade 12 in 2015.

National results from the 2015 mathematics assessment are compared to 10 previous assessment years at grades 4 and 8 and 3 previous assessment years at grade 12. State results for 2015 are compared to 9 previous years at grade 4, and 10 previous years at grade 8.

From 1996 and later, national results are based on administration procedures in which testing accommodations were permitted for students with disabilities and for English language learners. Accommodations were not permitted in earlier assessments. Read more about NAEP's policy of inclusion. Comparisons between results from 2015 and those from assessment years in which both types of administration procedures were used (1996 for national and 2000 for state results) are discussed based on the results when accommodations were permitted, even though significant differences in results when accommodations were not permitted may be noted in figures and tables. Changes in student performance across years or differences between groups of students in 2015 are discussed only if they have been determined to be statistically significant.

The results of student performance on the NAEP mathematics assessment are presented in two ways: as average scores on the NAEP mathematics scale and as the percentages of students attaining NAEP mathematics achievement levels. The average scale scores represent how students performed on the assessment. The achievement levels represent how that performance measured up against set expectations for achievement. Thus, the average scale scores represent what students know and can do, while the achievement-level results indicate the degree to which student performance meets expectations of what they should know and be able to do.

Average mathematics scale score results are based on the NAEP mathematics scale, which ranges from 0 to 500 for grades 4 and 8 and 0 to 300 for grade 12. The 2005 mathematics framework initiated minor changes at grades 4 and 8 and more substantial changes at grade 12. This meant that the trend line could be maintained at the lower grades but a new trend line needed to be established in 2005 at grade 12. For 2009, the grade 12 mathematics framework was updated, adding objectives addressing mathematics content beyond that typically taught in a standard 3-year course of study in high school mathematics. Results of a 2009 mathematics trend study determined that the 2009 grade 12 results could be compared to the results from 2005 even with the changes made in 2009. Find out how the 2005 and 2009 mathematics frameworks compare to previous frameworks.

Results of the NAEP mathematics assessment are reported as a composite scale that combines the results of separately-estimated scales for each of the mathematics content areas: (1) number properties and operations, (2) measurement, (3) geometry, (4) data analysis, statistics, and probability, and (5) algebra. (Note that measurement and geometry make up one of four content areas at grade 12.) Average scale scores are computed for groups of students. NAEP does not compute individual student scores. In the base year of the trend line, the assessed grades were analyzed together to create a 0-500 cross-grade scale. In subsequent years, the data for grades 4 and 8 were analyzed separately and then linked to the original cross-grade scale established in the base year. Comparisons of overall national performance across grade levels on a cross-grade scale are acceptable; however, other types of comparisons or inferences may not be supported by the available information. Note that while the scale is cross-grade, the skills tested and the material on the test increase in complexity and difficulty at each higher grade level, so different things are measured at the different grades even though a progression is implied.

Achievement-level results are presented in terms of
mathematics achievement levels as adopted by the
National Assessment Governing Board, and are intended to measure how well students' actual achievement matches the achievement desired of them. For each grade tested, the Governing Board has adopted three achievement levels:
*Basic*,
*Proficient*, and
*Advanced*. For reporting purposes, the achievement-level cut scores are placed on the mathematics scales, resulting in four ranges: below
*Basic*,
*Basic*,
*Proficient*, and
*Advanced*.

The Governing Board established its achievement levels based upon the mathematics content framework and a standard-setting process involving a cross section of educators and interested citizens from across the nation who were asked to judge what students should know and be able to do relative to the content set out in the NAEP mathematics framework. At grade 12, new achievement levels were established in 2005 based on the revised content framework. As provided by law, NCES has determined that the achievement levels are to be considered on a trial basis and should be interpreted and used with caution. However, both NCES and the Governing Board believe these performance standards are useful for understanding trends in student achievement.

Item maps illustrate the knowledge and skills demonstrated by students performing at different points on the NAEP mathematics scale. In order to provide additional context, the cut points for the three NAEP achievement levels are marked on the item maps. The map location for each question represents the probability that, for a given score point, 65 percent of the students for a constructed-response question, 74 percent of the students for a four-option multiple-choice question, and 72 percent of the students for a five-option multiple-choice question answered that question successfully. For constructed-response questions, responses may be completely or partially correct; therefore, a question can map to several points on the scale.

Approximately 30 mathematics questions per grade have been selected and placed on each item map. Explore the mathematics item maps.

The average scores and percentages presented are estimates because they are based on representative samples of students rather than on the entire population of students. Moreover, the collection of subject-area questions used at each grade level is but a sample of the many questions that could have been asked. As such, NAEP results are subject to a measure of uncertainty, reflected in the standard error of the estimates. The standard errors for the estimated scale scores and percentages in the figures and tables presented are available through the NAEP Data Explorer.

Results are provided for groups of students defined by shared characteristics—gender, race or ethnicity, eligibility for free/reduced-price school lunch, students with disabilities, students identified as English language learners type of school and location, and highest level of parental education. Based on participation rate criteria, results are reported for subpopulations only when sufficient numbers of students and adequate school representation are present. The minimum requirement is at least 62 students in a particular group from at least five primary sampling unit (PSUs). However, the data for all students, regardless of whether their group was reported separately, were included in computing overall results. Explanations of the reporting groups are presented below.

Results are reported separately for male and female students.

Prior to 2011, student race/ethnicity was obtained from school records and reported for the six mutually exclusive categories shown below:

- White
- Black
- Hispanic
- Asian/Pacific Islander
- American Indian/Alaska Native
- Other or unclassified

Students who identified with more than one of the other five categories were classified as “other” and were included as part of the "unclassified" category along with students who had a background other than the ones listed or whose race/ethnicity could not be determined.

In compliance with new standards from the U.S. Office of Management and Budget for collecting and reporting data on race/ethnicity, additional information was collected beginning in 2011 so that results could be reported separately for Asian students, Native Hawaiian/Other Pacific Islander students, and students identifying with two or more races. Beginning in 2011, all of the students participating in NAEP were identified as one of the seven racial/ethnic categories listed below:

- White
- Black or African American
- Hispanic
- Asian
- Native Hawaiian or other Pacific Islander
- American Indian or Alaska Native
- Two or more races

Students identified as Hispanic were classified as Hispanic even if they were also identified with another racial/ethnic group. Students who identified with two or more of the other racial/ethnic groups (e.g., White and Black) would have been classified as “other” and reported as part of the "unclassified" category prior to 2011, but from 2011 on were classified as “two or more races”.

When comparing the results for racial/ethnic groups from 2011, 2013, and 2015 to earlier assessment years, the 2013 data for Asian and Native Hawaiian/Other Pacific Islander students were combined into a single Asian/Pacific Islander category. Information based on student self-reported race/ethnicity will continue to be reported in the NAEP Data Explorer.

As part of the Department of Agriculture's National School Lunch Program (NSLP), schools can receive cash subsidies and donated commodities in turn for offering free or reduced-price lunches to eligible children. NAEP first began collecting data in 1996 on student eligibility for NSLP as an indicator of poverty. Based on available school records, students were classified as either currently eligible for the free/reduced-price school lunch or not eligible. Eligibility for free and reduced-price lunches is determined by students' family income in relation to the federally established poverty level. Students whose family income is at or below 130 percent of the poverty level qualify to receive free lunch, and students whose family income is between 130 percent and 185 percent of the poverty level qualify to receive reduced-price lunch. For the period July 1, 2014 through June 30, 2015, for a family of four, 130 percent of the poverty level was $31,005 and 185 percent was $44,123. The classification applies only to the school year when the assessment was administered (i.e., the 2014–15 school year) and is not based on eligibility in previous years. If school records were not available, the student was classified as "Information not available." If the school did not participate in the program, all students in that school were classified as "Information not available." Because of the improved quality of the data on students' eligibility for NSLP, the percentage of students for whom information was not available has decreased compared to the percentages reported prior to the 2003 assessment. See the proportion of students in each category in the NAEP Data Explorer.

Results are reported for students who were identified by school records as having a disability. A student with a disability may need specially designed instruction to meet his or her learning goals. A student with a disability will usually have an Individualized Education Program (IEP), which guides his or her special education instruction. Students with disabilities are often referred to as special education students and may be classified by their school as learning disabled (LD) or emotionally disturbed (ED).The goal of NAEP is that students who are capable of participating meaningfully in the assessment are assessed, but some students with disabilities selected by NAEP may not be able to participate, even with the accommodations provided. Beginning in 2009, NAEP disaggregated students with disabilities from students who were identified under section 504 of the Rehabilitation Act of 1973.

Results are reported for students who were identified by school records as being English language learners. (Note that English language learners were previously referred to as limited English proficient (LEP).)

The national results are based on a representative sample of students in both public schools and nonpublic schools. Nonpublic schools include private schools, Bureau of Indian Affairs schools, and Department of Defense schools. Private schools include Catholic, Conservative Christian, Lutheran, and other private schools. Results are reported for private schools overall, as well as disaggregated by Catholic and other private schools. Sample size was insufficient to permit a reliable estimate for other private schools in 2005, 2007, 2009, 2011, 2013, and 2015. The state results are based on public school students only.

NAEP results are reported for four mutually exclusive categories of school location: city, suburb, town, and rural. The categories are based on standard definitions established by the Federal Office of Management and Budget using population and geographic information from the U.S. Census Bureau. Schools are assigned to these categories in the NCES Common Core of Data based on their physical address. In 2007, the classification system was revised; therefore, trend comparisons to previous years are not available. The new locale codes are based on an address's proximity to an urbanized area (a densely settled core with densely settled surrounding areas). This is a change from the original system based on metropolitan statistical areas. To distinguish the two systems, the new system is referred to as "urban-centric locale codes." The urban-centric locale code system classifies territory into four major types: city, suburban, town, and rural. Each type has three subcategories. For city and suburb, these are gradations of size—large, midsize, and small. Towns and rural areas are further distinguished by their distance from an urbanized area. They can be characterized as fringe, distant, or remote.

Parents' highest level of education is defined by the highest level reported by eighth-graders and twelfth-graders for either parent. Fourth-graders were not asked to indicate their parents' highest level of education because their responses in previous studies were highly variable, and a large percentage of them chose the "I don't know" option.

All 50 states and 2 other jurisdictions (District of Columbia and Department of Defense Education Activity) participated in the 2015 mathematics assessment at grades 4 and 8. To ensure that the samples in each state are representative, NAEP has established policies and procedures to maximize the inclusion of all students in the assessment. Every effort is made to ensure that all selected students who are capable of participating meaningfully in the assessment are assessed. While some students with disabilities (SD) and/or English language learners (ELL) students can be assessed without any special procedures, others require accommodations to participate in NAEP. Still other SD and/or ELL students selected by NAEP may not be able to participate. Local school authorities determine whether SD/ELL students require accommodations or shall be excluded because they cannot be assessed. The percentage of SD and/or ELL students who are excluded from NAEP assessments varies from one jurisdiction to another and within a jurisdiction over time. Read more about the potential effects of exclusion rates on assessment results.

See additional information about the percentages of students with disabilities and English language learners identified, excluded, and assessed at the national and state level in grades 4 and 8.See the types of accommodations permitted for students with disabilities and/or English language learners at the national level in grades 4 and 8.

Exclusion rates for other subjects, as well as rates of use of specific accommodations, are available.

The differences between scale scores and between percentages discussed in the results take into account the standard errors associated with the estimates. Comparisons are based on statistical tests that consider both the magnitude of the difference between the group average scores or percentages and the standard errors of those statistics. Throughout the results, differences between scores or between percentages are discussed only when they are significant from a statistical perspective.

All differences reported are significant at the 0.05 level with appropriate adjustments for multiple comparisons. The term "significant" is not intended to imply a judgment about the absolute magnitude or the educational relevance of the differences. It is intended to identify statistically dependable population differences to help inform dialogue among policymakers, educators, and the public.

Comparisons across states use a t-test (the method most commonly used to evaluate the differences in means between two groups) to detect whether a difference is statistically significant or not. There are four possible outcomes when comparing the average scores of jurisdictions A and B:

- Jurisdiction A has a higher average score than jurisdiction B,
- Jurisdiction A has a lower average score than jurisdiction B,
- No difference in scores is detected between jurisdiction A and B, or
- The sample does not permit a reliable statistical test. (This may occur when the sample size for a particular group is small.)

When comparing all jurisdictions to each other, the testing procedures are based on all pairwise combinations of the jurisdictions in a particular year or pair of years. It may be possible that a given state or jurisdiction has a numerically higher average scale score than the nation or another state but that the difference is not statistically significant, while another state with the same average score may show a statistical significance compared to the nation or the other state. These situations may arise due to the fact that standard errors vary across states/jurisdictions and estimates.

Users are cautioned against interpreting NAEP results as implying causal relations. Inferences related to group performance or to the effectiveness of public and nonpublic schools, for example, should take into consideration the many socioeconomic and educational factors that may also impact performance.

The NAEP mathematics scale makes it possible to examine relationships between students' performance and various factors measured by NAEP. However, a relationship that exists between achievement and another variable does not reveal its underlying cause, which may be influenced by a number of other variables. Similarly, the assessments do not reflect the influence of unmeasured variables. The results are most useful when they are considered in combination with other knowledge about the student population and the educational system, such as trends in instruction, changes in the school-age population, and societal demands and expectations.

Return to the mathematics subject information.

Last updated 21 October 2015 (FW)

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