Mathematical Content Areas and Assessment Strands

To conduct a meaningful assessment of mathematics proficiency, it is necessary to measure students' proficiencies in various content areas. As in the 1990 and 1992 assessments, five mathematical strands will be used to categorize mathematical content for the 1996 and 2000 mathematics assessments. The strands are illustrated later in this chapter. Classification of topics into these strands cannot be exact, however, 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 description category. 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 involve students in synthesizing knowledge across topics and subtopics, and occasionally it may be difficult to identify a unique topic for each item. In fact, it is desirable that 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 illustrative of those 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 are largely consistent with the strands used in the 1990 and 1992 assessments. The titles and emphases of the content areas have been modified to reflect more clearly the directions for curriculum and evaluation described in the NCTM Standards.

For each grade (4, 8, and 12), the following symbols are used: a "" indicates that the subtopic could 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 might be introduced in the assessment at a very 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 will be expected to demonstrate an understanding of numerical relationships as expressed in ratios, proportions, and percentages. Students also will be 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 area. Items that require some application of the definition of operations and related procedures are classified under the area of 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 will be expected to demonstrate that they know how to perform basic algorithms and use calculators in appropriate ways, given more complex situations. While a few isolated computation items may be included, a priority will be placed on including items in which operations are used in meaningful contexts.

The grade 4 assessment will emphasize the development of number sense through the connection of a variety of models to their numerical representations, as well as emphasizing an understanding of the meaning of addition, subtraction, multiplication, and division. These concepts will be addressed for whole numbers, simple fractions, and decimals at this grade level, with continual emphasis on the use of models and their connection to the use of symbols.

The grade 8 assessment will include number sense extended to include both positive and negative numbers and will address 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 will be expected to know how to read, use, and apply scientific notation to represent large and small numbers.

At grade 12, the assessment will include both real and complex numbers and will allow students to demonstrate competency through approximately the precalculus or calculus level. Operations with powers and roots, as well as a variety of real and complex numbers, may be assessed. Including a broad range of items at this level will ensure 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.

Measurement

The measurement strand focuses on an understanding of the process of measurement and on the use of numbers and measures to describe and compare mathematical and real-world objects. Students will be 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 will demonstrate their ability to extend basic concepts in applications involving, for example, perimeter, area, surface area, volume, and angle measure.

Students will 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 need to be assessed on their judgments about such answers.

Of these measurement concepts, the focus at grade 4 is on time, money, temperature, length, perimeter, area, capacity, weight/mass, and angle measure. While 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 at these grade levels also should 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 will be assessed with real measuring devices.

Items requiring straightforward computation with measures, especially those involving time and money, are included not as part of this content area but as a part of number sense, properties, and operations instead.

Applications involving measurement provide a rich source for families of questions that illustrate the connections among number sense and operations, algebra, and geometry.

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 area is designed to extend 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. While reasoning is addressed throughout all of the content areas, this strand continues to lend itself to the demonstration of reasoning within both formal and informal settings. The extension of proportional thinking to similar figures and indirect measurement is an important connection here.

In grade 4, students are expected to model properties of shapes under simple combinations and transformations, and they are expected to use mathematical communication skills to draw figures given a verbal description. For grade 8, students are expected to have extended their understanding to include properties of angles and polygons and to apply reasoning skills to make and validate conjectures about transformations and combinations of shapes. At grade 12, students are expected to demonstrate proficiency with transformational geometry and to apply concepts of proportional thinking to a variety of geometric situations. They will have opportunities to demonstrate more sophisticated reasoning processes than at earlier grade levels, and they will be expected to demonstrate a variety of algebraic and geometric connections. The importance of these connections and their use in solving problems is indicated by the shifting emphasis in geometry toward coordinate geometry, as described in Chapter Four.

Data Analysis, Statistics, and Probability

Because of its fundamental role in making sense of the world, this content area will receive increased emphasis. The important skills of collecting, organizing, reading, representing, and interpreting data will be assessed in a variety of 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 communicating increasingly sophisticated interpretations of data. Dealing with uncertainty and making predictions about outcomes require an understanding not only of the meaning of basic probability concepts but also the application of those concepts in problem-solving and decisionmaking situations.

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

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

Probabilistic thinking and a variety of specialized graphs become increasingly important in grades 8 and 12. Students in grade 8 will be 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 with a variety of graphs to represent different types of data in different situations.

Students in grade 12 will be expected to use a wide 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 a variety of 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.

Algebra and Functions

This strand extends from work with simple patterns at grade 4 to basic algebra concepts at grade 8 and to sophisticated analysis at grade 12, and involves not only algebra but also precalculus and some topics from discrete mathematics. These algebraic concepts are developed throughout the grades with informal modeling done at the elementary level and with increased emphasis on functions at the secondary level. Students will be 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 will include the use of open sentences and equations as representational tools. Students will use the notion of equivalent representations to transform and solve number sentences and equations of increasing levels of complexity.

The grade 4 assessment will involve informal demonstration of students' abilities to generalize from patterns, including the justification of their generalizations. Students will be expected to translate mathematical representations, to use simple equations, and to do basic graphing.

At grade 8, the assessment will include 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 will be 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 by a variety of methods, including both graphical and basic algebraic methods. Students should begin to use basic concepts of functions as a way of describing relationships.

By grade 12, students will be expected to be adept at appropriately choosing and applying a rich set of representational tools in a variety of 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 as a way of representing and describing relationships.

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