Grade 4
Grade 8
Grade 12
Grade 12 (1990—2003)
Specific definitions of the Basic, Proficient, and Advanced achievement levels for grades 4, 8, and 12 are presented in the tables that follow. Because of changes made to the NAEP mathematics framework in 2005, the achievement-level descriptions and cut points indicated below for grade 12 have been updated. To maintain trend, results for grades 4 and 8 are reported on a 0–500 scale while results for grade 12, beginning with the 2005 assessment, are reported on a 0–300 scale. View the grade 12 achievement-level descriptions and cut scores used prior to 2005.
The achievement levels are cumulative; therefore, students performing at the Proficient level also display the competencies associated with the Basic level, and students at the Advanced level also demonstrate the skills and knowledge associated with both the Basic and the Proficient levels. The cut score indicating the lower end of the score range for each level is noted in parentheses.
Basic (214) |
Fourth-grade students performing at the Basic level should show some evidence of understanding the mathematical concepts and procedures in the five NAEP content areas. Fourth-graders performing at the Basic level should be able to estimate and use basic facts to perform simple computations with whole numbers, show some understanding of fractions and decimals, and solve some simple real-world problems in all NAEP content areas. Students at this level should be able to use—though not always accurately—four-function calculators, rulers, and geometric shapes. Their written responses will often be minimal and presented without supporting information. |
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Proficient (249) |
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 areas. 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 explanations of how they were achieved. |
Advanced (282) |
Fourth-grade students performing at the Advanced level should apply integrated procedural knowledge and conceptual understanding to complex and nonroutine real-world problem solving in the five NAEP content areas. Fourth-graders performing at the Advanced level should be able to solve complex and nonroutine real-world problems in all NAEP content areas. They should display mastery in the use of four-function calculators, rulers, and geometric shapes. The students are expected to draw logical conclusions and justify answers and solution processes by explaining why, as well as how, they were achieved. They should go beyond the obvious in their interpretations and be able to communicate their thoughts clearly and concisely. |
Basic (262) |
Eighth-grade students performing at the Basic level should exhibit evidence of conceptual and procedural understanding in the five NAEP content areas. This level of performance signifies an understanding of arithmetic operations—including estimation—on whole numbers, decimals, fractions, and percents. Eighth-graders performing at the Basic level should complete problems correctly with the help of structural prompts such as diagrams, charts, and graphs. They should be able to solve problems in all NAEP content areas through the appropriate selection and use of strategies and technological tools—including calculators, computers, and geometric shapes. Students at this level also should be able to use fundamental algebraic and informal geometric concepts in problem solving. As they approach the Proficient level, students at the Basic level should be able to determine which of the available data are necessary and sufficient for correct solutions and use them in problem solving. However, these eighth-graders show limited skill in communicating mathematically. |
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Proficient (299) |
Eighth-grade students performing at the Proficient level should apply mathematical concepts and procedures consistently to complex problems in the five NAEP content areas. Eighth-graders performing at the Proficient level should be able to conjecture, defend their ideas, and give supporting examples. They should understand the connections between 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. |
Advanced (333) |
Eighth-grade students performing at the Advanced level should be able to reach beyond the recognition, identification, and application of mathematical rules in order to generalize and synthesize concepts and principles in the five NAEP content areas. Eighth-graders performing at the Advanced level should be able to probe examples and counterexamples in order to shape generalizations from which they can develop models. Eighth-graders performing at the Advanced level should use number sense and geometric awareness to consider the reasonableness of an answer. They are expected to use abstract thinking to create unique problem-solving techniques and explain the reasoning processes underlying their conclusions. |
Basic (141) |
Twelfth-grade students performing at the Basic level should be able to solve mathematical problems that require the direct application of concepts and procedures in familiar mathematical and real-world settings. Students performing at the Basic level should be able to compute, approximate, and estimate with real numbers, including common irrational numbers. They should be able to order and compare real numbers and be able to perform routine arithmetic calculations with and without a scientific calculator or spreadsheet. They should be able to use rates and proportions to solve numeric and geometric problems. At this level, students should be able to interpret information about functions presented in various forms, including verbal, graphical, tabular, and symbolic. They should be able to evaluate polynomial functions and recognize the graphs of linear functions. Twelfth-grade students should also understand key aspects of linear functions, such as slope and intercepts. These students should be able to extrapolate from sample results; calculate, interpret, and use measures of center; and compute simple probabilities. Students at this level should be able to solve problems involving area and perimeter of plane figures, including regular and irregular polygons, and involving surface area and volume of solid figures. They should also be able to solve problems using the Pythagorean theorem and using scale drawings. Twelfth-graders performing at the Basic level should be able to estimate, calculate, and compare measures, as well as to identify and compare properties of two- and three-dimensional figures. They should be able to solve routine problems using two-dimensional coordinate geometry, including calculating slope, distance, and midpoint. They should also be able to perform single translations or reflections of geometric figures in a plane. |
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Proficient (176) |
Twelfth-grade students performing at the Proficient level should be able to recognize when particular concepts, procedures, and strategies are appropriate, and to select, integrate, and apply them to solve problems. They should also be able to test and validate geometric and algebraic conjectures using a variety of methods, including deductive reasoning and counter examples. Twelfth-grade students performing at the Proficient level should be able to compute, approximate, and estimate the values of numeric expressions using exponents (including fractional exponents), absolute value, order of magnitude, and ratios. They should be able to apply proportional reasoning, when necessary, to solve problems in nonroutine settings, and to understand the effects of changes in scale. They should be able to predict how transformations, including changes in scale, of one quantity affect related quantities. These students should be able to write equivalent forms of algebraic expressions, including rational expressions, and use those forms to solve equations and systems of equations. They should be able to use graphing tools and to construct formulas for spreadsheets; to use function notation; and to evaluate quadratic, rational, piecewise defined, power, and exponential functions. At this level, students should be able to recognize the graphs and families of graphs of these functions and to recognize and perform transformations on the graphs of these functions. They should be able to use properties of these functions to model and solve problems in mathematical and real-world contexts, and they should understand the benefits and limits of mathematical modeling. Twelfth graders performing at the Proficient level should also be able to translate between representations of functions, including verbal, graphical, tabular, and symbolic representations; to use appropriate representations to solve problems; and to use graphing tools and to construct formulas for spreadsheets. Students performing at this level should be able to use technology to calculate summary statistics for distributions of data. They should be able to recognize and determine a method to select a simple random sample, identify a source of bias in a sample, use measures of center and spread of distributions to make decisions and predictions, describe the impact of linear transformations and outliers on measures of center, calculate combinations and permutations to solve problems, and understand the use of the normal distribution to describe real-world situations. Twelfth-grade students should be able to use theoretical probability to predict experimental outcomes involving multiple events. These students should be able to solve problems involving right triangle trigonometry, use visualization in three dimensions, and perform successive transformations of a geometric figure in a plane. They should be able to understand the effects of transformations, including changes in scale, on corresponding measures and to apply slope, distance, and midpoint formulas to solve problems. |
Advanced (216) |
Twelfth-grade students performing at the Advanced level should demonstrate in-depth knowledge of and be able to reason about mathematical concepts and procedures. They should be able to integrate this knowledge to solve nonroutine and challenging problems, provide mathematical justifications for their solutions, and make generalizations and provide mathematical justifications for those generalizations. These students should reflect on their reasoning, and they should understand the role of hypotheses, deductive reasoning, and conclusions in geometric proofs and algebraic arguments made by themselves and others. Students should also demonstrate this deep knowledge and level of awareness in solving problems, using appropriate mathematical language and notation. Students at this level should be able to reason about functions as mathematical objects. They should be able to evaluate logarithmic and trigonometric functions and recognize the properties and graphs of these functions. They should be able to use properties of functions to analyze relationships and to determine and construct appropriate representations for solving problems, including the use of advanced features of graphing calculators and spreadsheets. These students should be able to describe the impact of linear transformations and outliers on measures of spread (including standard deviation), analyze predictions based on multiple data sets, and apply probability and statistical reasoning to solve problems involving conditional probability and compound probability. Twelfth-grade students performing at the Advanced level should be able to solve problems and analyze properties of three-dimensional figures. They should be able to describe the effects of transformations of geometric figures in a plane or in three dimensions, to reason about geometric properties using coordinate geometry, and to do computations with vectors and to use vectors to represent magnitude and direction. |
Basic (288) |
Twelfth-grade students performing at the Basic level should demonstrate procedural and conceptual knowledge in solving problems in the five NAEP content areas. Twelfth-grade students performing at the Basic level should be able to use estimation to verify solutions and determine the reasonableness of results as applied to real-world problems. Twelfth-graders performing at the Basic level should recognize relationships presented in verbal, algebraic, tabular, and graphical forms; and demonstrate knowledge of geometric relationships and corresponding measurement skills. They should be able to apply statistical reasoning in the organization and display of data and in reading tables and graphs. They also should be able to generalize from patterns and examples in the areas of algebra, geometry, and statistics. At this level, they should use correct mathematical language and symbols to communicate mathematical relationships and reasoning processes, and use calculators appropriately to solve problems. |
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Proficient (336) |
Twelfth-grade students performing at the Proficient level should consistently integrate mathematical concepts and procedures to the solution of more complex problems in the five NAEP content areas. Twelfth-graders performing at the Proficient level should demonstrate an understanding of algebraic, statistical, and 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. |
Advanced (367) |
Twelfth-grade students performing at the Advanced level should consistently demonstrate the integration of procedural and conceptual knowledge and the synthesis of ideas in the five NAEP content areas. Twelfth-grade students performing at the Advanced level should understand the function concept, and be able to compare and apply the numeric, algebraic, and graphical properties of functions. They should apply their knowledge of algebra, geometry, and statistics to solve problems in more advanced areas of continuous and discrete mathematics. They should be able to formulate generalizations and create models through probing examples and counterexamples. They should be able to communicate their mathematical reasoning through the clear, concise, and correct use of mathematical symbolism and logical thinking. |