Chapter Three: 2005 NAEP Mathematics Objectives

In order to describe the specific mathematics that should be assessed at each grade level, it is necessary to organize the domain of mathematics into component parts. This is accomplished by utilizing the five content areas, as described in chapter 2. Though such an organization brings with it the danger of fragmentation, the hope is that the objectives and the test items built on them will, in many cases, cross some of the boundaries of these content areas.

One of the goals of this framework is to provide more clarity and specificity in the objectives for each grade level. To accomplish this, a matrix was created that depicts the particular objectives that are appropriate for assessment under each subtopic. Within Number, for example, and the subtopic of Number Sense, specific objectives are listed for assessment at grade 4, grade 8, and grade 12. The same objective at different grade levels depicts a developmental sequence for that concept or skill. An empty cell in the matrix is used to convey the fact that a particular objective is not appropriate for assessment at that grade level.

Mathematical Content Areas

NUMBER PROPERTIES AND OPERATIONS

Numbers are our main tools for describing the world quantitatively. As such, they deserve a privileged place in the 2005 NAEP framework. With whole numbers, we can count collections of discrete objects of any type. We can also use numbers to describe fractional parts and even to describe continuous quantities such as length, area, volume, weight, and time, and more complicated derived quantities such as rates, speed, density, inflation, interest, and so forth. Thanks to Cartesian coordinates, we can use pairs of numbers to describe points in a plane or triples of numbers to label points in space. Numbers let us talk in a precise way about anything that can be counted, measured, or located in space.

Numbers are not simply labels for quantities. They form systems with their own internal structure. The arithmetic operations (addition and subtraction, multiplication and division) help us model basic real-world operations. For example, joining two collections, or laying two lengths end to end, can be described by addition, while the concept of rate depends on division. Multiplication and division of whole numbers lead to the beginnings of number theory, including concepts of factorization, remainder, and prime number. Besides the arithmetic operations, the other basic structure of the real numbers is ordering, as in which is greater and lesser. These reflect our intuitions about the relative size of quantities, and provide a basis for making sensible estimates.

The accessibility and usefulness of arithmetic are greatly enhanced by our efficient means for representing numbers: the Hindu-Arabic decimal place value system. In its full development, this remarkable system includes decimal fractions, which let us approximate any real number as closely as we wish. Decimal notation allows us to do arithmetic by means of simple, routine algorithms, and it also makes size comparisons and estimation easy. The decimal system achieves its efficiency through sophistication, as all the basic algebraic operations are implicitly used in writing decimal numbers. To represent ratios of two whole numbers exactly, we supplement decimal notation with fractions.

Comfort in dealing with numbers effectively is called number sense. It includes firm intuitions about what numbers tell us; an understanding of the ways to represent them symbolically (including facility with converting between different representations); the ability to calculate, either exactly or approximately, and by several means (mentally, with paper and pencil, or with calculator, as appropriate); and skill in estimation. The ability to deal with proportion, including percents, is another important part of number sense.

Number sense is a major expectation of the 2005 NAEP. At fourth grade, students are expected to have a solid grasp of whole numbers, as represented by the decimal system, and to have the beginnings of understanding fractions. By eighth grade, they should be comfortable with rational numbers, represented either as decimal fractions (including percents) or as common fractions. They should be able to use them to solve problems involving proportionality and rates. In middle school also, number should begin to coalesce with geometry via the idea of the number line. This should be connected with ideas of approximation and the use of scientific notation. Eighth graders should also have some acquaintance with naturally occurring irrational numbers, such as square roots and pi. By 12th grade, students should be comfortable dealing with all types of real numbers.

Number Properties and Operations

MEASUREMENT

Measuring is the process by which numbers are assigned in order to describe the world quantitatively. This process involves selecting the attribute of the object or event to be measured, comparing this attribute to a unit, and reporting the number of units. For example, in measuring a child, we may select the attribute of height and the inch as the unit for the comparison. In comparing the height to the inch, we may find that the child is about 42 inches. If considering only the domain of whole numbers, we would report that the child is 42 inches tall. However, since height is a continuous attribute, we may consider the domain of rational numbers and report that the child is 4¹³/16 inches tall (to the nearest 16th of an inch). Measurement also allows us to model positive and negative numbers as well as irrational numbers.

This connection between measuring and number makes measuring a vital part of the school curriculum. Measurement models are often used when students are learning about number and operations. For example, area and volume models can help students understand multiplication and the properties of multiplication. Length models, especially the number line, can help students understand ordering and rounding numbers. Measurement also has a strong connection to other areas of school mathematics and to the other subjects in the school curriculum. Problems in algebra are often drawn from measurement situations. One can also consider measurement to be a function or a mapping of the attribute to a set of numbers. Much of school geometry focuses on the measurement aspect of geometric figures. Statistics also provides ways to measure and to compare sets of data. These are some of the ways in which measurement is intertwined with the other four content areas.

In this NAEP Mathematics Framework, attributes such as capacity, weight/mass, time, and temperature are included, as well as the geometric attributes of length, area, and volume. Although many of these attributes are included in the grade 4 framework, the emphasis is on length, including perimeter, distance, and height. More emphasis is placed on area and angle in grade 8. By grade 12, volumes and rates constructed from other attributes, such as speed, are emphasized.

Units involved in items on the NAEP assessment include nonstandard, customary, and metric units. At grade 4, common customary units such as inch, quart, pound, and hour and the common metric units such as centimeter, liter, and gram are emphasized. Grades 8 and 12 include the use of both square and cubic units for measuring area, surface area, and volume; degrees for measuring angles; and constructed units such as miles per hour. Converting from one unit in a system to another (such as from minutes to hours) is an important aspect of measurement included in problem situations. Understanding and using the many conversions available is an important skill. There are a limited number of common, everyday equivalencies that students are expected to know (see the Assessment and Item Specifications document for more detail).

Items classified in this content area depend on some knowledge of measurement. For example, an item that asks the difference between a 3-inch and a 1³/4-inch line segment is a number item, while an item comparing a 2-foot segment with an 8-inch line segment is a measurement item. In many secondary schools, measurement becomes an integral part of geometry; this is reflected in the proportion of items recommended for these two areas.

Measurement

GEOMETRY

Geometry began as a practical collection of rules for calculating lengths, areas, and volumes of common shapes. In classical times, the Greeks turned it into a subject for reasoning and proof, and Euclid organized their discoveries into a coherent collection of results, all deduced using logic from a small number of special assumptions called postulates. Euclid’s Elements stood as a pinnacle of human intellectual achievement for over 2000 years.

The 19th century saw a new flowering of geometric thought, going beyond Euclid, and leading to the idea that geometry is the study of the possible structures of space. This had its most striking application in Einstein’s theories of relativity, which describes the behavior of light, and also of gravity, in terms of a four-dimensional geometry, which combines the usual three dimensions of space with time as an additional dimension.

A major insight of the 19th century is that geometry is intimately related to ideas of symmetry and transformation. The symmetry of familiar shapes under simple transformations (that our bodies look more or less the same if reflected across the middle, or that a square looks the same if rotated by 90 degrees) is a matter of everyday experience. Many of the standard terms for triangles (scalene, isosceles, equilateral) and quadrilaterals (parallelogram, rectangle, rhombus, square) refer to symmetry properties. Also, the behavior of figures under changes of scale is an aspect of symmetry with myriad practical consequences. At a deeper level, the fundamental ideas of geometry itself (for example, congruence) depend on transformation and invariance. In the 20th century, symmetry ideas were seen to also underlie much of physics, not only Einstein’s relativity theories, but atomic physics and solid-state physics (the field that produced computer chips).

School geometry roughly mirrors the historical development through Greek times with some modern additions, most notably symmetry and transformations. By grade 4, students are expected to be familiar with a library of simple figures and their attributes, both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, spheres, and cylinders). In middle school, understanding of these shapes deepens, with the study of cross-sections of solids and the beginnings of an analytical understanding of properties of plane figures, especially parallelism, perpendicularity, and angle relations in polygons. Right angles and the Pythagorean theorem are introduced, and geometry becomes more and more mixed with measurement. The basis for analytic geometry is laid by study of the number line. In high school, attention is given to Euclid’s legacy and the power of rigorous thinking. Students are expected to make, test, and validate conjectures. Via analytic geometry, the key areas of geometry and algebra are merged into a powerful tool that provides a basis for calculus and the applications of mathematics that helped create the modern technological world in which we live.

Symmetry is an increasingly important component of geometry. Elementary students are expected to be familiar with the basic types of symmetry transformations of plane figures, including flips (reflection across lines), turns (rotations around points), and slides (translations). In middle school, this knowledge becomes more systematic and analytical, with each type of transformation being distinguished from other types by their qualitative effects. For example, a rigid motion of the plane that leaves at least two points fixed (but not all points) must be a reflection in a line. In high school, students are expected to be able to represent transformations algebraically. Some may also gain insight into systematic structure, such as the classification of rigid motions of the plane as reflections, rotations, translations, or glide reflections, and what happens when two or more isometries are performed in succession (composition).

Geometry

DATA ANALYSIS AND PROBABILITY

Data analysis covers the entire process of collecting, organizing, summarizing, and interpreting data. This is the heart of the discipline called statistics; it is in evidence whenever quantitative information is used to determine a course of action. To emphasize the spirit of statistical thinking, data analysis should begin with a question to be answered, not with the data. Data should be collected only with a specific question (or questions) in mind and only after a plan (usually called a design) for collecting data relevant to the question is thought out. Beginning at an early age, students should grasp the fundamental principle that looking for questions in an existing data set is far different from the scientific method of collecting data to verify or refute a well-posed question. A pattern can be found in almost any data set if one looks hard enough, but a pattern discovered in this way is often meaningless, especially from the point of view of statistical inference.

In the context of data analysis, or statistics, probability can be thought of as the study of potential patterns in outcomes that have not yet been observed. We say that the probability of a balanced coin coming up heads when flipped is one-half because we believe that about half of the flips would turn out to be heads if we flipped the coin many times. Under random sampling, patterns for outcomes of designed studies can be anticipated and used as the basis for making decisions. If the coin actually turned up heads 80 percent of the time, we would suspect that it was not balanced. The whole probability distribution of all possible outcomes is important in most statistics problems because the key to decisionmaking is to decide whether a particular observed outcome is unusual (located in a tail of the probability distribution). For example, 4 as a grade point average is unusually high among most groups of students, 4 as the pound weight of a baby is unusually low, and 4 as the number of runs scored in a baseball game is not unusual in either direction.

By grade 4, students should be expected to apply their understanding of number and quantity to pose questions that can be answered by collecting appropriate data. They should be expected to organize data in a table or a plot and summarize the essential features of center, spread, and shape both verbally and with simple summary statistics. Simple comparisons can be made between two related data sets, but more formal inference based on randomness should come later. The basic concept of chance and statistical reasoning can be built into meaningful contexts, though, such as, “If I draw two names from among those of the students in the room, am I likely to get two girls?” Such problems can be addressed through simulation.

Building on the same definition of data analysis and the same principles of describing distributions of data through center, spread, and shape, grade 8 students will be expected to use a wider variety of organizing and summarizing techniques. They can also begin to analyze statistical claims through designed surveys and experiments that involve randomization, with simulation being the main tool for making simple statistical inferences. They will begin to use more formal terminology related to probability and data analysis.

Students in grade 12 will be expected to use a wide variety of statistical techniques for all phases of the data analysis process, including a more formal understanding of statistical inference (but still with simulation as the main inferential analysis tool). In addition to comparing univariate data sets, students at this level should be able to recognize and describe possible associations between two variables by looking at two-way tables for categorical variables or scatterplots for measurement variables. Association between variables is related to the concepts of independence and dependence, and an understanding of these ideas requires knowledge of conditional probability. These students should be able to use statistical models (linear and nonlinear equations) to describe possible associations between measurement variables and should be familiar with techniques for fitting models to data.

Data Analysis and Probability

ALGEBRA

Algebra was pioneered in the Middle Ages by mathematicians in the Middle East and Asia as a method of solving equations easily and efficiently by manipulation of symbols, rather than by the earlier geometric methods of the Greeks. The two approaches were eventually united in the analytic geometry of René Descartes. Modern symbolic notation, developed in the Renaissance, greatly enhanced the power of the algebraic method; from the 17th century forward, algebra in turn promoted advances in all branches of mathematics and science.

The widening use of algebra led to the study of its formal structure. Out of this were gradually distilled the “rules of algebra,” a compact summary of the principles behind algebraic manipulation. A parallel line of thought produced a simple but flexible concept of function and also led to the development of set theory as a comprehensive background for mathematics. When it is taken liberally to include these ideas, algebra reaches from the foundations of mathematics to the frontiers of current research.

These two aspects of algebra, a powerful representational tool and a vehicle for comprehensive concepts such as function, form the basis for the expectations throughout the grades. By grade 4, students are expected to be able to recognize and extend simple numeric patterns as one foundation for a later understanding of function. They can begin to understand the meaning of equality and some of its properties, as well as the idea of an unknown quantity as a precursor to the concept of variable.

As students move into middle school, the ideas of function and variable become more important. Representation of functions as patterns, via tables, verbal descriptions, symbolic descriptions, and graphs, can combine to promote a flexible grasp of the idea of function. Linear functions receive special attention. They connect to the ideas of proportionality and rate, forming a bridge that will eventually link arithmetic to calculus. Symbolic manipulation in the relatively simple context of linear equations is reinforced by other means of finding solutions, including graphing by hand or with calculators.

In high school, students should become comfortable in manipulating and interpreting more complex expressions. The rules of algebra should come to be appreciated as a basis for reasoning.

Nonlinear functions, especially quadratic functions, and also power and exponential functions, are introduced to solve real-world problems. Students should become accomplished at translating verbal descriptions of problem situations into symbolic form. Expressions involving several variables, systems of linear equations, and the solutions to inequalities are encountered by grade 12.

Algebra