To conduct a meaningful assessment of mathematics proficiency, it is necessary to measure students’ proficiencies in various content strands. As in the 1990 and 1992 assessments, five content strands will be used to categorize content for the mathematics assessments. The strands are illustrated later in this chapter. Classification of topics into these strands cannot be exact and inevitably will involve some overlap. For example, some topics appearing under Data Analysis, Statistics, and Probability may be closely related to others that appear under Algebra and Functions. As assessment programs continue to be refined, it becomes less desirable to force every item into only one content strand. Students are expected to solve problems that naturally involve more than one specific mathematical topic. Consequently, the assessment as a whole will address the topics and subtopics identified in this chapter, and every item will be categorized under primarily one topic and subtopic so that analysis of results may be somewhat specific. Ideally, however, the items will require students to synthesize knowledge across topics and subtopics, and occasionally it may be difficult to identify a unique topic for each item. In fact, at least half of the new items for the assessment should involve content from more than one topic or even from more than one strand.

The following sections of this chapter provide a brief description of each content strand with a list of topics and subtopics to be included in the assessment. This level of specificity is needed to guide item writers and ensure adequate coverage of the content areas and abilities to be assessed. The five content strands for 1996, 2000, and 2003 are largely consistent with the strands used in the 1990 and 1992 assessments.

For each grade (4, 8, and 12), the following symbols are used: a “” indicates that the subtopic can be assessed at that grade level, a “” indicates that the subtopic should not be assessed at that grade level, and a “” indicates that the subtopic may be introduced at a simple level, probably using a manipulative or pictorial model. The test specifications include additional detail and descriptions of how item types, families, calculators, manipulatives, and special studies fit within and across topics and subtopics.

Number Sense, Properties, and Operations

This strand focuses on students’ understanding of numbers (whole numbers, fractions, decimals, integers, real numbers, and complex numbers), operations, and estimation, and their application to real-world situations. Students are expected to demonstrate an understanding of numerical relationships expressed in ratios, proportions, and percentages. Students are also expected to understand properties of numbers and operations, generalize from numerical patterns, and verify results.

Number sense includes items that address a student’s understanding of relative size, equivalent forms of numbers, and his or her use of numbers to represent attributes of real-world objects and quantities. Items that call for students to complete open sentences involving basic number facts are considered part of this content strand. Items that require some application of the definition of operations and related procedures are classified under Algebra and Functions.

The emphasis in computation is on understanding when to use an operation, knowing what the operation means, and being able to estimate and use mental techniques in addition to performing calculations using computational algorithms. In terms of actual computation, students are expected to demonstrate that they know how to perform basic algorithms and, in more complex situations, use calculators appropriately. Although some isolated computation items are included, a priority is placed on developing items in which mathematical operations are used in problem-solving situations.

The grade 4 assessment emphasizes the development of number sense through the connection of various models to their numerical representations, as well as an understanding of the meaning of addition, subtraction, multiplication, and division. These concepts are addressed for whole numbers, simple fractions, and decimals at this grade level, emphasizing the use of models and their connection to the use of symbols.

The grade 8 assessment extends number sense to include both positive and negative numbers and addresses properties and operations involving whole numbers, fractions, decimals, integers, and rational numbers. The use of ratios and proportional thinking to represent situations involving quantity is a major focus at this grade level, and students are expected to read, use, and apply scientific notation to represent large and small numbers.

At grade 12, the assessment includes both real and complex numbers and tests students’ competency in topics up to and including precalculus. Operations with powers and roots, as well as various real and complex numbers, may be assessed. Including a broad range of items at this level ensures that students who have had different types of high school mathematics courses will be able to demonstrate proficiency in some parts of this content area.

NAEP Mathematics Content Strand 1

Measurement

This strand focuses on an understanding of measurement and the use of numbers and measures to describe and compare mathematical and real-world objects. Students are asked to identify attributes, select appropriate units and tools, apply measurement concepts, and communicate measurement-related ideas.

Students should understand and be able to use the measurement attributes of length, mass/weight, capacity, time, money, and temperature. Students demonstrate their ability to extend basic concepts in applications involving, for example, perimeter, area, surface area, volume, and angle measure.

Students use measuring instruments and apply measurement concepts to solve problems. Due to the inherent imprecision of measurement tools, it is important for students to recognize that measurement is an approximation.

When students use technology for calculations with imprecise measurements, errors are often carried or increased. Students should be assessed on their judgments about such answers.

The assessment focus at grade 4 is on time, money, temperature, length, perimeter, area, capacity, weight/mass, and angle measure. Although assessment at grades 8 and 12 continues to include these measurement concepts, the focus shifts to more complex measurement problems that involve volume or surface area or that require students to combine shapes, translate, and apply measures. Students in grades 8 and 12 should also solve problems involving proportional thinking (such as scale drawing or map reading) and do applications that involve the use of complex measurement formulas. When appropriate and possible, measurement is assessed with real measuring devices.

Items requiring straightforward computation with measures, especially those involving time and money, are included not in this content strand but in Number Sense, Properties, and Operations.

Applications involving measurement are a rich source of questions that assess the connections among number sense and operations, algebra, and geometry.

NAEP Mathematics Content Strand 2

Geometry and Spatial Sense

Spatial sense must be an integral component of the study and assessment of geometry. Understanding spatial relationships allows students to use the dynamic nature of geometry to connect mathematics to their world.

This content strand extends well beyond low-level identification of geometric shapes into transformations and combinations of those shapes. Informal constructions and demonstrations (including drawing representations), along with their justifications, take precedence over more traditional types of compass-and-straightedge constructions and proofs. Although reasoning is addressed throughout the content areas, this strand addresses reasoning in formal and informal settings. The extension of proportional thinking to similar figures and indirect measurement is an important aspect of this strand.

In grade 4, students are expected to model properties of shapes under simple combinations and transformations and use mathematical communication skills to draw figures given a verbal description. In grade 8, students are expected to understand properties of angles and polygons and apply reasoning skills to make and validate conjectures about transformations and combinations of shapes. In grade 12, students are expected to demonstrate proficiency with transformational geometry and to apply concepts of proportional thinking to various geometric situations. They also have opportunities to demonstrate more sophisticated reasoning processes, and they are also expected to demonstrate various algebraic and geometric connections. The importance of these connections and their use in solving problems is indicated by the shifting emphasis in geometry to coordinate geometry, as described in chapter four.

NAEP Mathematics Content Strand 3

Data Analysis, Statistics, and Probability

Because of its fundamental role in making sense of the world, this content strand receives increased emphasis. The important skills of collecting, organizing, reading, representing, and interpreting data are assessed in various contexts to reflect the pervasive use of these skills in dealing with information. Statistics and statistical concepts extend these basic skills to include analyzing and interpreting increasingly sophisticated data. Dealing with uncertainty and making predictions about outcomes require an understanding of not only the meaning of basic probability concepts but also the application of those concepts in problem-solving and decisionmaking situations.

Questions emphasize appropriate methods of gathering data, the visual exploration of data, ways to represent data, and the development and evaluation of arguments based on data analysis. Students are expected to apply these ideas in increasingly sophisticated situations that require increasingly comprehensive analysis and decisionmaking.

In grade 4, students are expected to apply their understanding of number and quantity by solving problems involving data and to use data analysis to broaden their number sense. They are expected to be familiar with various graphs. They are asked to make predictions from data and explain their reasoning and to deal informally with measures of central tendency. Grade 4 students also are asked to use the basic concept of chance in meaningful contexts not involving the computation of probabilities.

Probabilistic thinking and various specialized graphs become increasingly important in grades 8 and 12. Students in grade 8 are expected to analyze statistical claims and design experiments, and they may use simulations to model real-world situations. They should have some understanding of sampling, and they should be asked to make predictions based on experiments or data. They will begin to use some formal terminology related to probability, data analysis, and statistics. By grade 8, students should be comfortable using various graphs to represent different types of data in different situations.

Students in grade 12 are expected to use a variety of statistical techniques to model situations and solve problems. Students at this level should apply concepts of probability to explore dependent and independent events, and they should be somewhat knowledgeable about conditional probability. They should be able to use formulas and more formal terminology to describe various situations. At this level, students should have a basic understanding of the use of mathematical equations and graphs to interpret data, including the use of curve fitting to match a set of data with an appropriate mathematical model.

NAEP Mathematics Content Strand 4

Algebra and Functions

This strand extends from work with simple patterns at grade 4 to basic algebra concepts at grade 8 and sophisticated analysis at grade 12; it involves not only algebra but also precalculus and some topics from discrete mathematics. Algebraic concepts are developed throughout the grades, emphasizing informal modeling at the elementary level and functions at the secondary level. Students are expected to use algebraic notation and thinking in meaningful contexts to solve mathematical and real-world problems, specifically addressing an increasing understanding of the use of functions (including algebraic and geometric) as a representational tool.

The assessment at all levels includes the use of open sentences and equations as representational tools. Students are expected to use equivalent representations to transform and solve number sentences and equations of increasing levels of complexity.

The grade 4 assessment involves the informal demonstration of students’ abilities to generalize from patterns and justify their generalizations. Students are expected to translate mathematical representations, use simple equations, and demonstrate basic graphing.

The grade 8 assessment includes more algebraic notation, stressing the meaning of variables and an informal understanding of the use of symbolic representations in problem-solving contexts. Students at this level are asked to use variables to represent a rule underlying a pattern. They should have a beginning understanding of equations as a modeling tool, and they should solve simple equations and inequalities through various methods, including both graphical and basic algebraic methods. Students should begin to use basic concepts of functions to describe relationships.

In grade 12, students are expected to be adept at appropriately choosing and applying a rich set of representational tools in various problem-solving situations. They should have an understanding of basic algebraic notation and terminology as they relate to representations of mathematical and real-world problem situations. Students should be able to use functions to represent and describe relationships.

NAEP Mathematics Content Strand 5