## 9. The Mathematical World

Mathematics is essentially a process of thinking that involves building and applying abstract, logically connected networks of ideas. These ideas often arise from the need to solve problems in science, technology, and everyday life—problems ranging from how to model certain aspects of a complex scientific problem to how to balance a checkbook.

Science for All Americans

Mathematical knowledge is interesting in its own right and because it can contribute to the understanding of nature and human inventions. The benchmarks in this chapter deal with basic mathematical ideas, especially those with practical application. Project 2061 takes the view that adult literacy consists primarily of enhanced perception and insight. It is this knowledge that provides the basis for solving problems, making decisions, understanding the world, and learning more. Separation of knowing and doing is unusual in mathematics education, but the distinction is as important for mathematics as it is for other sciences. Consistent with the format of Chapters 1 to 11, the benchmarks in this chapter are expressed as "students should know that." The related "students should be able to" benchmarks appear in Chapter 12: Habits of Mind.

However, claiming the primary value of knowledge does not address the question of how the knowledge is acquired. As with other sciences, understanding mathematics almost always requires extensive experience using it—solving problems, communicating ideas, connecting ideas to one another. The National Council of Teachers of Mathematics' report Curriculum and Evaluation Standards for School Mathematics (referred to hereafter as NCTM Standards) is an excellent source of inspiration and of suggestions for how instruction can promote both knowledge and practice. The mathematical components of benchmarks provide a level of specificity required for the Project 2061 strategy of designing rich, interconnected curriculum blocks with explicit goals for what students will learn.

Numbers are everywhere and enter people's lives in many guises. School experience with numbers should foster an appreciation of the beauty and versatility of numbers and contribute to the development of number sense. Although difficult to define in detail, number sense is what enables literate people to judge when mathematical thinking is making sense and whether the results are reasonable. Most important, it gives people confidence to use mathematics in solving problems and communicating ideas.

If students are to gain such confidence, curricula must be designed so that (1) the roles that numbers play in different activities—sports, history, music, election communications, lotteries, coding, etc.—are made explicit and discussed; (2) time is allotted to examining some of the more fascinating mathematical ideas, such as zero, negative numbers, pi, and primes; and (3) the numbers used in problem solving are frequently those that come from actual measurements-students' whenever possible and databases when not. Inquiry and design projects provide rich opportunities for students to work on problems that interest them and that engage them in making estimates, counting, taking measurements, graphing, and otherwise using numbers in ways that contribute to the growth of number sense.

Young children should have two kinds of experiences with numbers. One is simply to have fun with them. Counting and counting games in which students are challenged to count forward and backward, skip count, match numbers and things, guess how many things there are in a set and then count to see who is right, and so forth, are popular with students and help them become comfortable with numbers. These counting games should be extended to include having students compare, combine, equalize, and change numbers as well as "take away" and "add to." But counting and estimating—and of course doing sums and differences—are not the only use of numbers that students can learn in the early grades. The use of numbers for naming things, for instance, can be brought out by having students assemble a display or portfolio of all the different ways, such as car licenses and room numbers, they can discover in which numbers are for naming.

The other kind of number experience that is essential has to do with measurement (which is, after all, but a form of counting). Students should be doing things, especially in science and design projects, that require them to pose questions that can be answered only by numbers associated with things. In this way, they can begin to understand that answers to such questions as, say, "How big?" "How far?" or "How long?" can be, respectively, "9 pounds," "9 blocks," or "9 days"—but not "9." Although students should be encouraged to make relative physical comparisons directly whenever they can, concluding, say, that B is taller than A, C holds more than D, etc., they should also begin to develop a preference for numerical comparisons—B is 2 inches taller than A, box C holds 14 more marbles than box D. Graphing at this level should be mostly in the form of pictographs for the purpose of relative comparisons rather than the plotting of numbers.

##### Current Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Numbers can be used to count things, place them in order, measure them, or name them. 9A/P1*
• Sometimes in describing things there is a need to use numbers between whole numbers. 9A/P2*
• Simple graphs can help to tell about observations. 9A/P4
• An important kind of relationship between things is one thing being part of a whole. 9A/P5** (ASL)
• The first digit of a two-digit number describes how many sets of 10 there are in the number. 9A/P6**
• A quantity is stated as a number and a label, such as 4 inches or 7 blocks. 9A/P7**
##### 1993 Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Numbers can be used to count things, place them in order, or name them. 9A/P1
• Sometimes in sharing or measuring there is a need to use numbers between whole numbers. 9A/P2
• It is possible (and often useful) to estimate quantities without knowing them exactly. 9A/P3
In the current version of Benchmarks Online, this benchmark has been moved to grades 3-5 and recoded as 9A/E7**.
• Simple graphs can help to tell about observations. 9A/P4

At this level, students will be learning multiplication and division as necessary skills, using paper and pencil and calculators. Some of the practice needed to master these skills can be carried out using context-free numbers. But if students are to learn about the meaning of numbers and to use them properly, much of what they do must be based on solving problems in which the answers matter and the numbers used are measured quantities. A great source of number lore for students this age (and older) from which interesting problems can be crafted by the students themselves is The Guinness Book of World Records.

Students are now able to make more precise and varied measurements than in the earlier grades, and it is not too early to point out and discuss some of the realities of numbers based on measurement, especially that measurements are estimates that vary somewhat, that how a number is written says something about how precise the measurement was, and that specifying the unit of measurement is always necessary. These realities can be treated as general ideas and obvious examples can be given without requiring sophisticated rules.

As a practical matter, zero is important in measurement and graphing because it anchors scales, but students should have a chance now to explore it as an interesting mathematical concept. It can be made part of their introduction to the idea of a number system and place value.

##### Current Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• The meaning of a digit in a many-digit number depends on its position. 9A/E1*
• In some situations, "0" means none of something, but in others it may be just the label of some point on a scale, such as a number line. 9A/E2*
• Specifying a quantity requires both a number and a unit. 9A/E3*
• Measurements are always likely to give slightly different numbers, even if what is being measured stays the same. 9A/E4
• Fractions are numbers used to represent part of something. 9A/E5** (SFAA)
• Symbols are used to signify which operations to perform on numbers. The most common are +, -, x, and ÷. 9A/E6** (SFAA)
• It is possible (and often useful) to estimate quantities without determining them exactly. 9A/E7** (BSL)
##### 1993 Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• The meaning of numerals in many-digit numbers depends on their positions. 9A/E1
• In some situations, "0" means none of something, but in others it may be just the label of some point on a scale. 9A/E2
• When people care about what is being counted or measured, it is important for them to say what the units are (three degrees Fahrenheit is different from three centimeters, three miles from three miles per hour). 9A/E3
• Measurements are always likely to give slightly different numbers, even if what is being measured stays the same. 9A/E4

This may be the most important period of all in helping students develop an understanding of numbers. Up to now, they have dealt mostly with positive whole numbers and their manipulation, even though the numbers came from measurement. Negative numbers and fractions can now come into the picture because students will need to use them in carrying out the kinds of science and technology activities that should be on their agenda. This practical introduction to the value of fractions and negative numbers should be complemented by opportunities for students to reflect on those and other mathematical ideas, including relations of operations to one another, number systems, and abstract number patterns.

Except in instances in which the purpose is clearly practice of operations, teachers should insist that students think about the numbers they used in solving quantitative problems. Students should ask themselves and each other such questions as "What units should be associated with the measurements and the calculated answer?" "How many digits are enough in the answer, no matter what the calculator shows?" (Formal rules for significant figures are difficult, and most people tend to keep more digits than necessary "just to be safe," but at least they should realize that it matters and that there are ways to deal with it.) "Does the number make sense?"

##### Current Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• The system of using the Arabic numerals 0-9 is just one way of representing numbers. The very old Roman numerals are now used primarily for dates, clock faces, or ordering chapters in a book. 9A/M1*
• A number line can be extended on the other side of zero to represent negative numbers. Negative numbers allow subtraction of a bigger number from a smaller number to make sense, and are often used when something can be measured on either side of some reference point (time, ground level, temperature, budget). 9A/M2
• The same number can be written in different forms, depending on its intended use. 9A/M3a*
• How a quantity is expressed depends on how precise the measurement is and how precise an answer is needed. 9A/M3b*
• The operations + and - are inverses of each other—one undoes what the other does; likewise x and ÷. 9A/M4
• A number expressed in the form a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b. 9A/M5*
• Numbers can be represented by using sequences of only two symbols (such as 1 and 0, on and off); computers work this way. 9A/M6
• Computations (as on calculators) can give more digits than make sense or are useful. 9A/M7
• Some interesting relationships between two variables include the variables always having the same difference or the same ratio. 9A/M8**
• Exponents can be used to represent how many times a number is to be multiplied by itself. For example, 43 = 4 x 4 x 4. 9A/M9**
##### 1993 Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• There have been systems for writing numbers other than the Arabic system of place values based on tens. The very old Roman numerals are now used only for dates, clock faces, or ordering chapters in a book. Numbers based on 60 are still used for describing time and angles. 9A/M1
• A number line can be extended on the other side of zero to represent negative numbers. Negative numbers allow subtraction of a bigger number from a smaller number to make sense, and are often used when something can be measured on either side of some reference point (time, ground level, temperature, budget). 9A/M2
• Numbers can be written in different forms, depending on how they are being used. How fractions or decimals based on measured quantities should be written depends on how precise the measurements are and how precise an answer is needed. 9A/M3
• The operations + and - are inverses of each other—one undoes what the other does; likewise x and ÷. 9A/M4
• The expression a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b. 9A/M5
• Numbers can be represented by using sequences of only two symbols (such as 1 and 0, on and off); computers work this way. 9A/M6
• Computations (as on calculators) can give more digits than make sense or are useful. 9A/M7

With regard to numbers, the 9th through 12th grades are mostly an elaboration of the 6th through 8th grades. Students at this level encounter numbers and computations chiefly in the context of solving problems or learning more advanced mathematics. Through repeatedly encountering problem situations with numerical demands, students can deepen and refine their understanding of and facility with number relations, operations, ratios, estimation, measurement, graphs, and so on. Similarly, they should become more experienced in using calculators for a variety of computational tasks.

##### Current Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of ten. 9A/H1
• Numbers can be written with bases other than ten. The simplest base, 2, uses just two symbols (1 and 0, or on and off). 9A/H2*
• When calculations are made with measurements, a small error in the measurements may lead to a large error in the results. 9A/H3
• The effects of uncertainties in measurements on a computed result can be estimated. 9A/H4
##### 1993 Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of 10. 9A/H1
• Numbers can be written with bases different from ten (which people probably use because of their 10 fingers). The simplest base, 2, uses just two symbols (1 and 0, or on and off). 9A/H2
• When calculations are made with measurements, a small error in the measurements may lead to a large error in the results. 9A/H3
• The effects of uncertainties in measurements on a computed result can be estimated. 9A/H4

For all its popularity as a sign of academic respectability, its prominence as the gateway to advancement (college, if not in life more generally), and its undeniable importance in science, engineering, and many other fields, algebra remains a subject about which it is not altogether clear what the average adult needs to know. The current trend toward a year of algebra for all students is often justified by citing the strong correlation between taking algebra and vocational success in life, but of course correlations do not imply cause and effect. The question that Project 2061 asked in Science for All Americans was what should all students learn, not how much algebra they should take—just as it did not ask how much biology or chemistry they should have.

Certainly everyone should know what algebra is, for it stands as one of the great human inventions of all time. The first step toward this understanding is for students to learn about symbols, including that the use of symbols is widespread, takes many forms, and is not the exclusive property of algebra or mathematics. Algebra, however, represents numbers, sets of numbers, quantities, and relationships with letters and signs (for operations) in a systematic way that turns out to be useful for describing relationships between variables. The second step is for students to learn what it means to manipulate symbolic statements. Students can use algebraic symbols and even make up simple symbolic statements long before they know they are "doing algebra." Later, as students encounter examples of how algebra is used in various contexts (natural and social sciences, design), they will develop a sense of what it is.

The difficult practical questions have to do with how much students should learn about the nature and uses of algebraic equations and how adept they can be expected to become in manipulating equations. Developing a sense of what equations are and how they correspond to other ways of expressing relationships among things is more important for science literacy than being able to derive or use them. Students who take a year or more of algebra often learn to manipulate symbols and solve equations (at least until exam time) but come away with little grasp of what a solution means or why anyone would want it.

Over a period of years, therefore, students should have experiences leading to the realization that symbolic equations can be used—interchangeably with graphs, tables, and words-to summarize data and to model real-world relationships (as in physics, finance, engineering, etc.). Care must be taken, however, not to go over the heads of the students. Variables should be selected for study that are interesting and observable (or measurable), and the focus should be on the simple relationships between one variable and another outlined in Science for All Americans.

Science uses algebra in modeling how changes in one quantity affect changes in other quantities. Much of physics, chemistry, and engineering, and increasingly biology too, depends on algebraic representation. In Project 2061, we don't expect students to remember formulas for accelerations or parallel circuits or mass action; nor do we expect them to be able to perform algebraic manipulations or solve simultaneous equations. We do expect them to acquire an understanding of proportionality, the ability to read an algebraic formula, and to develop the ability to relate the shape of a graph to its implications for how some aspect of the world behaves.

The transformation of equations into graphs has been greatly simplified by calculators and computers. But before students begin to use that capability, they need to have considerable experience making data tables by numerical substitution in simple equations and then graphing the data. Then they can reverse the process by trying to find a curve, and hence a formula, that seems to fit the points on the graph. Perhaps the most practical way for them to learn about the transformation between data tables, graphs, and formulas is from using computer spreadsheet and graphics software. They can use data that interest them; make up mathematical formulas; use existing formulas (spreadsheet "functions"); carry out series calculations; and print out tables and line, bar, and circle graphs.

When algebraic equations and graphs of equations are used in studying science, the point should be made frequently that formulas and graphs are intended to describe phenomena but may not necessarily do so well. "Why doesn't it fit exactly?" is a question to which increasingly sophisticated answers should be given. Answers about errors of observation should come first, then answers about choosing the wrong formula to fit idealized data; later, answers should include uncontrolled influences and inappropriate ranges of application; finally should come the answer, "The world just doesn't seem to act as simply as the mathematics."

From the earliest grades, students should be asked to look for regularities in events, shapes, designs, and sets of numbers. Especially they should look for situations in which changes in one thing seem to be associated with changes in other things, but it would be a mistake to introduce dependence between two variables in all of its algebraic glory. A sense of function can start to be built both mathematically (as in trying the same calculator steps on different numbers) and physically (as by adjusting faucets, television-set controls, or thermostats, or observing the effects of exercise on heart and breathing rates).

##### Current Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Similar patterns may show up in many places in nature and in the things people make. 9B/P1
• Sometimes changing one thing causes changes in something else. In some situations, changing the same thing in the same way has the same result. 9B/P2
##### 1993 Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Similar patterns may show up in many places in nature and in the things people make. 9B/P1
• Sometimes changing one thing causes changes in something else. In some situations, changing the same thing in the same way usually has the same result. 9B/P2

Symbols are just things that stand for other things or sets of other things or kinds of other things. They can be objects or marks, even sounds. Perhaps it is not too soon to engage students in collecting or identifying symbols—state flags, the school mascot, "happy faces," candles on birthday cakes, etc.—and making up symbols to represent things and a combination of symbols to represent relationships (specified by other students) such as "this is bigger (or faster or more expensive) than that." In this activity, students should be helped to realize that the idea of symbols is not the sole property of mathematics, and letters are not the only kind of symbol used. They should gather and compare the uses of as many different kinds of symbols as they can find in mathematics and elsewhere—hieroglyphics, numbers, icons, musical notation, etc.

The dependence of one quantity on another can be appreciated first as simply "changing x causes a change in y." That need be no more than noticing the change in y and saying whether it gets bigger or smaller. Also feasible at this level is whether a noticeable change in y requires a lot of change in x or just a little. It is probably premature to introduce dependence between two variables in formal symbols. Some foreshadowing can take place, however. The unknown box in equations at this level typically stands for a single value that will make the equation a true statement. Two unknown boxes (or inputs and outputs of a "function machine") allow paired sets of values to satisfy the equation. It is possible for students to figure out what y is implied by a given x—and what x would be required to give a desired y. In any case, graphs and tables, rather than equations, should be used to explore relationships between two variables.

##### Current Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Mathematical statements using symbols may be true only when the symbols are replaced by certain numbers. 9B/E1
• Tables and graphs can show how values of one quantity are related to values of another. 9B/E2
##### 1993 Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Mathematical statements using symbols may be true only when the symbols are replaced by certain numbers. 9B/E1
• Tables and graphs can show how values of one quantity are related to values of another. 9B/E2

During these years, students can begin to see what it means to explore the relationships among different quantities by representing them as symbols and by manipulating statements that relate the symbols—and yet not themselves be ready to handle equations algebraically. That will happen if students are shown simple equations that represent some of the relationships they can extract from tables and graphs they have created and if they learn to use equations by numerical substitution. The use of substitution is suggested by the following stream-of-consciousness scenario of a student confronted with a problem that requires some algebra:

I need to know how long would it take a dropped object to fall 10 feet. And here is the equation relating fall distance to time: s=1/2at2. (I wonder why they use s—shouldn't the symbol be d for distance? Well, it doesn't matter as long as you know what it stands for.) I know s and a, but I want to find the right value for t. There is probably some way to rewrite this equation to get t by itself, but I don't have any confidence in my ability to do it right. So let's see if I can figure out what value of t would give 10 feet for s. How about 1 second? Nope, that results in a fall of 16 feet. How about 1/2 second? Nope, t is squared, so that gives just a quarter as much—4 feet. Well, try in between: 3/4 second. O.K., nine feet is pretty close. That's close enough for my purpose: a ten-foot fall takes a little over 3/4 second. (Or, if better precision were needed, "3/4 is .75, so let me try .80 ….")

In building on and drawing from students' experiences with patterns and regularity, emphasis shifts toward an exploration of functions—the basic notion that changes in one variable result in change in another. However, as stated in NCTM Standards, at this level "work with patterns should emphasize concrete situations and be informal and relatively unburdened by symbolism." More relevant than formal symbolic representation at this level is exploration of the notion of function, including maximum and minimum values, behavior at specially interesting values such as zero, approaches to limiting values, and so on.

The concept of a variable, pervasive as it is in mathematics, is difficult and often not understood. Even adult veterans of algebra may think of variables only by imagining particular numerical values for them. Letter names for variables may be taken to stand for single units (P to stand for a professor rather than some number of professors). Variables should not be approached through abstract definition but rather through real-world situations familiar to students in which they can understand, perhaps even be interested in, the multiple possibilities for value.

##### Current Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• An equation containing a variable may be true for just one value of the variable. 9B/M1
• Rates of change can be computed from differences in magnitudes and vice versa. 9B/M2*
• Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these. 9B/M3*
##### 1993 Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• An equation containing a variable may be true for just one value of the variable. 9B/M1
• Mathematical statements can be used to describe how one quantity changes when another changes. Rates of change can be computed from differences in magnitudes and vice versa. 9B/M2
• Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease indefinitely, increase or decrease in steps, or do something different from any of these. 9B/M3

Students should practice using tabular, graphical, and symbolic representations of functions and translating among them—and they should be called upon to describe their tables, graphs, and equations in clear English. With the help of calculators and computers, they should explore the effects of changing terms in an equation on the general behavior of its graph. Computing technology enables schools to provide a richer set of algebra experiences for all students than ever before. Students should spend less time plotting curves point by point but more time interpreting graphs, exploring the properties of graphs, and determining how these properties relate to the forms of the corresponding equations. Of course, students should continue to plot a few points to check the reasonableness of their graphs.

In modeling phenomena, students should encounter a variety of common kinds of relationships depicted in graphs (direct proportions, inverses, accelerating and saturating curves, and maximums and minimums) and therefore develop the habit of entertaining these possibilities when considering how two quantities might be related. None of these terms need be used at first, however. "It is biggest here and less on either side" or "It keeps getting bigger, but not as quickly as before" are perfectly acceptable—especially when phenomena that behave like this can be described.

In high school, students should encounter the idea that one quantity may relate, not to the amount of some other quantity, but to its rate of change—as force relates to the rate of change of velocity, or the induced electric "field" relates to the rate of change of the magnetic field. There are also many examples of the rate of change of a quantity being proportional to the quantity itself (for instance, radioactive decay, compound interest, or unhampered population growth). Prior to the availability of cheap calculators, such an ostensibly changing rate might have been treated, for purposes of general literacy, as an instance of successive multiplication.

##### Current Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• In some cases the more of something there is, the more rapidly it may change (as the number of births is proportional to the size of the population). 9B/H1a
• Sometimes the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting). 9B/H1b
• Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. 9B/H2a
• Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time. 9B/H2b
• Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation. 9B/H3
• Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another. 9B/H4
• When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all. 9B/H5
##### 1993 Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• In some cases, the more of something there is, the more rapidly it may change (as the number of births is proportional to the size of the population). In other cases, the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting). 9B/H1
• Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time. 9B/H2
• Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation. 9B/H3
• Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another. 9B/H4
• When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all. 9B/H5
• The reasonableness of the result of a computation can be estimated from what the inputs and operations are. 9B/H6
In the current version of Benchmarks Online, this benchmark has been deleted because the ideas in it are addressed in benchmark 9A/E7**.

Long before they can use the language of geometry, children become aware of shape. Before entering school, they have had lots of experiences with points, lines, planes, and spaces. In school, students need to extend that knowledge, developing spatial sense and learning to see the world through the eyes of geometry. That can come from activities that require them to use geometry in constructing, drawing, measuring, visualizing, comparing, describing and transforming things. The progression of experiences should take students from recognizing shapes as wholes to recognizing explicit properties of shapes, and only then to the analysis of relationships among shapes.

Because students in these grades are engaged in projects that have them collecting and building things, there are bound to be many opportunities to get them thinking about shapes. They should make drawings of the things they collect and of things they observe outside the classroom, and then discuss them from many perspectives such as color, size, and of course shape. At first, students tend to describe the shape of one thing by comparing it to something else—a marble is shaped like a basketball, a sheet of paper like a rug, a jump rope like a shoelace. As they organize different things that have sort of the same shape into groups, the need for names for the shared property will begin to become apparent to them.

Art is especially important in fostering spatial sense. Students should construct recognizable two-dimensional images (faces, people, buildings, beds, etc.) using only rectangles, triangles, and circles, and then do the reverse—that is, identify those same shapes in pictures of things. Also, a start can be made in laying the groundwork for the introduction, later, of the idea of symmetry by having students practice drawing pictures of a given object—geometrically simple ones—in which the position of the object is rotated or the observer changes position. And, in all of this, students should be given descriptive tasks that require using words such as above, below, behind, inside, outside, and upside-down.

##### Current Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Circles, squares, triangles, spheres, cubes, cylinders and other shapes can be observed in things found in nature and in things that people build. 9C/P1*
##### 1993 Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Shapes such as circles, squares, and triangles can be used to describe many things that can be seen. 9C/P1

The geometric description of objects includes size, orientation, symmetry, and proportions, as well as shape. Students should begin to use all these features in describing and designing things and increase substantially the number of geometric shapes and concepts they are familiar with. Concepts of area and volume should first be developed concretely, with procedures for computation following only when the concepts and some of their practical uses are well understood. Graphing can help students grasp some of the connections between quantity, shape, and position.

##### Current Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Length can be thought of as unit lengths joined together, area as a collection of unit squares, and volume as a set of unit cubes. 9C/E1
• If 0 and 1 are located on a line, any other number can be depicted as a position on the line. 9C/E2
• Graphical display of quantities may make it possible to spot patterns that are not otherwise obvious, such as cycles and trends. 9C/E3*
• Objects can be described in terms of their shape or the shapes of their parts. 9C/E4*
• Areas of irregular shapes can be found by dividing them into squares and triangles. 9C/E5
• Scale drawings show shapes and compare locations of things very different in size. 9C/E6
• Two shapes can match exactly or be identical except for their sizes. 9C/E7** (BSL)
• Two lines can be parallel, perpendicular, or slanted with respect to one another. 9C/E8** (BSL)
• Sometimes two shapes will match if one of them is rotated or flipped. 9C/E9** (BSL)
##### 1993 Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Length can be thought of as unit lengths joined together, area as a collection of unit squares, and volume as a set of unit cubes. 9C/E1
• If 0 and 1 are located on a line, any other number can be depicted as a position on the line. 9C/E2
• Graphical display of numbers may make it possible to spot patterns that are not otherwise obvious, such as comparative size and trends. 9C/E3
• Many objects can be described in terms of simple plane figures and solids. Shapes can be compared in terms of concepts such as parallel and perpendicular, congruence and similarity, and symmetry. Symmetry can be found by reflection, turns, or slides. 9C/E4
In the current version of Benchmarks Online, the last sentence of this benchmark has been moved to grades 6-8 and recoded as 9C/M8**.
• Areas of irregular shapes can be found by dividing them into squares and triangles. 9C/E5
• Scale drawings show shapes and compare locations of things very different in size. 9C/E6

The expanding logical capabilities of students at this level enable them to draw inferences and make logical deductions from geometric problems. Students should investigate and use geometric ideas rather than memorizing definitions and formulas. Similarity and congruence can be explored through transformations. Figures should be oriented in various positions to aid in forming generalizations that won't be bound to standard orientations. That is made particularly convenient by computer software that performs "flips" and "stretches." Photographs, overhead projectors, and photocopying machines are other common tools for shrinking and stretching shapes.

Exploration of how linear measures, areas, and volumes change with size will strengthen the concepts themselves and help, generally, in leading students toward the ideas of scale that appear in Chapter 11: Common Themes. (Most children in this grade range expect area and volume to change in direct proportion to linear size.)

Learning to find locations in reality and on maps using rectangular and polar coordinates can contribute to an understanding of scale and illustrate one of the important connections between numbers and geometry. Shape in these grades is strongly related to spatial measurements. Students should have extensive experience in measuring and estimating perimeter, area, volume, and angles, choosing appropriate measurement units and measuring tools. As much as possible, these activities should be carried out in the context of actual projects, that is, in order to design and build something.

##### Current Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• Some of the properties an object has depend on its shape: triangular shapes tend to make structures rigid, and spheres give the least possible boundary for a given amount of interior volume. 9C/M1*
• Shapes on a sphere like the earth cannot be depicted on a flat surface without some distortion. Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages. 9C/M3*
• The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps, or clusters that are useful when making predictions about the phenomena being graphed. 9C/M4*
• It takes two numbers to locate a point on a map or any other flat surface. The numbers may be two perpendicular distances from a point, or an angle and a distance from a point. 9C/M5
• The scale chosen for a graph or drawing makes a big difference in how useful it is. 9C/M6
• For regularly shaped objects, relationships exist between the linear dimensions, surface area, and volume. 9C/M7** (BSL)
• Shapes can be compared in terms of concepts such as parallel and perpendicular, congruence and similarity, and symmetry. 9C/M8** (BSL)
• Relationships exist among the angles between the sides of triangle and the lengths of those sides. For example, when two sides of a triangle are perpendicular, the sum of the squares of the lengths of those sides is equal to the square of the third side of the triangle. 9C/M9** (SFAA)
• Geometric relationships can be described using symbolic equations. 9C/M10** (SFAA)
##### 1993 Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• Some shapes have special properties: Triangular shapes tend to make structures rigid, and round shapes give the least possible boundary for a given amount of interior area. Shapes can match exactly or have the same shape in different sizes. 9C/M1
In the current version of Benchmarks Online, the last sentence of this benchmark has been moved to grades 3-5 and recoded as 9C/E7**.
• Lines can be parallel, perpendicular, or oblique. 9C/M2
In the current version of Benchmarks Online, this benchmark has been moved to grades 3-5 and recoded as 9C/E8**.
• Shapes on a sphere like the earth cannot be depicted on a flat surface without some distortion. 9C/M3
• The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps, or clusters. Such patterns sometimes can be used to make predictions about the phenomena being graphed. 9C/M4
• It takes two numbers to locate a point on a map or any other flat surface. The numbers may be two perpendicular distances from a point, or an angle and a distance from a point. 9C/M5
• The scale chosen for a graph or drawing makes a big difference in how useful it is. 9C/M6

Deductive proofs should be encountered in discussions of proof in a larger context than just geometry. How do people know when something has been "proven" to be true? Is it the same in astronomy as in biology, chemistry as in law, geometry as in algebra? The nature of logic and evidence are topics that should come up frequently in science, history, social studies, and mathematics. Although it is not worth trying to teach all students to become good at working out Euclidean proofs, they should learn something of what such proofs entail and why they are important in mathematics.

Computers can be enormously useful at this stage. Students should use them to explore complex shapes in three dimensions, to analyze the geometry of objects of interest to them, to work out scale problems, and to graph data from their science activities. But computers cannot substitute altogether for direct experience. Thus students should, for example, have opportunities to solve problems requiring triangulation—such as the classical experience of determining the distance across a river or to the moon, which can be done with scale drawings. They should also do some mechanical drawing the old-fashioned way before using the graphic capabilities of the computer.

##### Current Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas. 9C/H1
• When the linear dimensions of an object change by some factor, its area and volume change disproportionately: area in proportion to the square of the factor and volume in proportion to its cube. Properties of an object that depend on its area or volume also change disproportionately. 9C/H2*
• Geometric shapes and relationships can be described in terms of symbols and numbers—and vice versa. 9C/H3a
• The position of any point on a surface can be specified by two numbers. 9C/H3b
• A graph represents all the values that satisfy an equation, and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where the graphs intersect. 9C/H3c
• Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages. 9C/H4
• Although real objects never perfectly match a geometric figure, they more or less approximate them, so that what is known about geometric figures and relationships can be applied to objects. 9C/H5**
• Both shape and scale can have important consequences for the performance of systems. 9C/H6** (SFAA)
##### 1993 Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas. 9C/H1
• There are formulas for calculating the surface areas and volumes of regular shapes. When the linear size of a shape changes by some factor, its area and volume change disproportionately: area in proportion to the square of the factor, and volume in proportion to its cube. Properties of an object that depend on its area or volume also change disproportionately. 9C/H2
In the current version of Benchmarks Online, the first sentence of this benchmark has been moved to grades 6-8 and recoded as 9C/M7**.
• Geometric shapes and relationships can be described in terms of symbols and numbers—and vice versa. For example, the position of any point on a surface can be specified by two numbers; a graph represents all the values that satisfy an equation; and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where their graphs intersect. 9C/H3
• Different ways to map a curved surface (like the earth's) onto a flat surface have different advantages. 9C/H4

The sections on uncertainty, summarizing data, and sampling in Chapter 9 of Science for All Americans are important for learning to deal with evidence. An important distinction must be made for this goal, as for others, between what students are expected to understand—that is, notice, talk about, critique-and what they are expected to be able to do themselves—that is, plan and carry out. The principal intent is to make them informed consumers, not producers, of data. For example, students should know that people can be alert to possible bias in choosing samples that others take but may be unable to take adequate precautions against bias in designing a study of their own.

There are some very difficult ideas under this goal, ideas with which even most adults have trouble. One common misunderstanding is the belief that averages are always highly representative of a population; little or no attention is given to the range of variation around averages. This point is particularly important when comparing groups. For example, elaborating a minuscule (but believable) difference in average x for boys and girls into statements such as "Boys have high x, whereas girls have low x." So it is essential that talk about averages is always accompanied by some indication of the actual distribution of the data. And there is no point in introducing averages until some question arises for which the average supplies a useful answer. Some research studies suggest that learning the algorithm divorced from a meaningful context tends to block students from ever understanding what averages are for.

Another misunderstanding is the assumption that variables are always linked by cause and effect. When being told of a correlation between two variables, adults almost invariably leap quickly to imagine a cause or believe the cause that is offered to them. A correlation between A and B should always evoke five hypotheses for consideration: (1) A might cause B; (2) B might cause A; (3) A and B might have no cause between them at all, but both are caused by C; and (4) chance alone may have produced the apparent dependence. There may be no greater contribution of mathematics to science literacy than fostering an understanding of what a correlation is and what it is not.

One of the many misunderstandings of probability that teachers have to deal with is that a well-established probability will be changed by the most recent history: People tend to believe that a coin that has come up heads ten times in a row is more likely on the next flip to come up tails than heads or that the number that won the lottery last week is less likely to win this week. Those and other confusions about probability are purely mathematical and can be addressed as such, but it is also important to take up some of the questions related to how probabilities are established. Examples should come from medicine, natural catastrophes such as floods and earthquakes, weather patterns, sports events, stock-market events, elections, and other topical contexts.

In the very earliest grades, learning can begin that will eventually lead to students' having a good grasp of everyday statistics. Children at this level can array things they collect by size and weight and then ask questions about them, such as which one is in the middle, how many are the same, and so forth. From there they can go on to make simple pictographs that show how a familiar variable is distributed and again ask questions about the distribution. They can begin to find out about sampling in the context, say, of reporting on the kinds of stones found on the school playground.

Children will be keeping track of many different phenomena, some of which they will come to see have patterns of one kind or another. From time to time they should be asked, working in small groups, to review their records to see if they can figure out if they can predict some future events. The most important part of such exercises is that the students give reasons for their predictions and for not being able to make predictions. Of course they should follow up to see if they were right or not.

##### Current Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Some things are more likely to happen than others. 9D/P1a
• Some events can be predicted well and some cannot. 9D/P1b
• Sometimes people aren't sure what will happen because they don't know everything that might be having an effect. 9D/P1c
• Often a person can find out about a group of things by studying just a few of them. 9D/P2
##### 1993 Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• Some things are more likely to happen than others. Some events can be predicted well and some cannot. Sometimes people aren't sure what will happen because they don't know everything that might be having an effect. 9D/P1
• Often a person can find out about a group of things by studying just a few of them. 9D/P2

The questions about data only explored in the earliest grades can now be made into formal questions. Data distributions should be made of many familiar features and quantities: heights, weights, number of siblings, or kinds of pets. The important thing to emphasize at this level is the kind of questions that can be posed and answered by a data distribution: "Where is the middle?" is a useful question; "What is the average?" probably is not. Because there is a persistent misconception, even in adults, that means are good representations of whole groups, it is especially important to draw students' attention to the additional questions, "What are the largest and smallest values?" and "How much do the data spread on both sides of the middle?" Children also should be invited to suggest some circumstances in their studies that might bias the results—for example, making "random" measurements of student height just as a basketball team comes along or collecting only the insects that were easy to spot.

##### Current Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now. 9D/E1
• Statistical predictions (as for rainy days, accidents) are typically better for how many of a group will experience something than for which members of the group will experience it—better for how often something will happen than for exactly when. 9D/E2
• Summary predictions are usually more accurate for large collections of events than for just a few. 9D/E3a
• Even very unlikely events may occur fairly often in very large populations. 9D/E3b
• Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are. 9D/E4a
• A summary of data includes where the middle is and how much spread is around it. 9D/E4b
• A small part of something may be special in some way and not give an accurate picture of the whole. 9D/E5a
• There is a danger of choosing only the data that show what is expected by the person doing the choosing. 9D/E5c
• Events can be described in terms of being more or less likely, impossible, or certain. 9D/E6
##### 1993 Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now. 9D/E1
• Statistical predictions (as for rainy days, accidents) are typically better for how many of a group will experience something than for which members of the group will experience it—and better for how often something will happen than for exactly when. 9D/E2
• Summary predictions are usually more accurate for large collections of events than for just a few. Even very unlikely events may occur fairly often in very large populations. 9D/E3
• Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are. A summary of data includes where the middle is and how much spread is around it. 9D/E4
• A small part of something may be special in some way and not give an accurate picture of the whole. How much a portion of something can help to estimate what the whole is like depends on how the portion is chosen. There is a danger of choosing only the data that show what is expected by the person doing the choosing. 9D/E5
• Events can be described in terms of being more or less likely, impossible, or certain. 9D/E6

Building on previous experience, students can now delve into statistics in greater detail. The work should be directly related to student investigations and utilize computers. As stated in NCTM Standards:

Instruction in statistics should focus on the active involvement of students in the entire process: formulating key questions; collecting and organizing data; representing the data using graphs, tables, frequency distributions, and summary statistics; analyzing the data; making conjectures; and communicating information in a convincing way. Students' understanding of statistics will also be enhanced by evaluating others' arguments.

Database computer programs offer a means for students to structure, record, and investigate information; to sort it quickly by various categories; and to organize it in a variety of ways. Other computer programs can be used to construct plots and graphs to display data. Scale changes can be made to compare different views of the same information. These technological tools free students to spend more time exploring the essence of statistics: analyzing data from many viewpoints, drawing inferences, and constructing and evaluating arguments.

Students should make distributions for many data sets, their own and published sets, which have already inspired some meaningful questions. The idea of a middle to a data set should be well motivated—say, by asking for a simple way to compare two groups—and various kinds of middle should be considered. The algorithm for the mean can be learned but not without recurrent questions about what it conveys—and what it does not.

In studying data sets, questions like these should be raised: What appears most often in the data? Are there trends? Why are there outliers? How can we explain the data, and does our explanation allow a prediction of what further data would look like? What difficulties might arise when extending the explanation to similar problems? What additional data can we collect to try to verify the ideas developed from these data?

The distinction between ends and means should be kept in mind in all of this. The ultimate aim is not to turn all students into competent statisticians but to have them understand enough statistics to be able to respond intelligently to claims based on statistics; without the kind of intense effort called for here, that understanding will be elusive.

Probability, too, should be continued at this level through the use of tables of actual frequencies of events, begun in the 3rd through 5th grades. Every time, however, students should be asked to consider whether the data (necessarily collected in the past) are still applicable. How well, for example, would last year's daily temperatures apply to this year?

After they have had many occasions to count possible outcomes (such as the faces of a die) and discuss their equal probability (is each face as likely to come up as any other?), students can begin to move to generalizations about theoretical probabilities. Students' attention should consistently be drawn to the assumptions that all possible outcomes of a situation are accounted for and are all equally probable. Computers should be used to generate simulated probabilistic data for analysis, but only after students have worked on problems in which they use their own data.

##### Current Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• How probability is estimated depends on what is known about the situation. Estimates can be based on data from similar conditions in the past or on the assumption that all the possibilities are known. 9D/M1
• Probabilities are ratios and can be expressed as fractions, decimals, percentages, or odds. 9D/M2
• The mean, median, and mode tell different things about the middle of a data set. 9D/M3
• Comparison of data from two groups should involve comparing both their middles and the spreads around them. 9D/M4
• The larger a well-chosen sample is, the more accurately it is likely to represent the whole. But there are many ways of choosing a sample that can make it unrepresentative of the whole. 9D/M5
##### 1993 Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• How probability is estimated depends on what is known about the situation. Estimates can be based on data from similar conditions in the past or on the assumption that all the possibilities are known. 9D/M1
• Probabilities are ratios and can be expressed as fractions, percentages, or odds. 9D/M2
• The mean, median, and mode tell different things about the middle of a data set. 9D/M3
• Comparison of data from two groups should involve comparing both their middles and the spreads around them. 9D/M4
• The larger a well-chosen sample is, the more accurately it is likely to represent the whole. But there are many ways of choosing a sample that can make it unrepresentative of the whole. 9D/M5

As their mathematical sophistication grows during these grades, students are able to perform and make sense of more subtleties in collecting, describing, and interpreting data. They should have multiple opportunities to plan and carry out studies of their own observations and of large databases. Their written reports should include the reasoning that went into decisions about sampling method and size, about models chosen, about the display used, and about alternative interpretations. They should look for selection bias, measurement error, and display distortion in news reports as well as in their own studies.

Important, too, is frequent discussion of reports in the news media about scientific studies. Students should identify weaknesses in the studies and offer alternative interpretations of the results—perhaps writing alternative versions of the news stories or writing letters to the editor about what the stories may have been missing.

##### Current Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Even when there are plentiful data, it may not be obvious what mathematical model to use, or there may be insufficient computing power to use some models. 9D/H1*
• When people estimate a statistic, they may also be able to say how far off the estimate might be due to chance. 9D/H2
• The middle of a data distribution might be misleading when the data are not distributed symmetrically, when there are extreme high or low values, or when the distribution is not reasonably smooth. 9D/H3*
• The way data are displayed can make a big difference in how they are interpreted. 9D/H4
• Both percentages and actual counts have to be taken into account in comparing different groups; using either category by itself could be misleading. 9D/H5*
• Considering whether and how two variables are correlated requires inspecting their distributions, such as in two-way tables or scatterplots. 9D/H6a
• A correlation between two variables doesn't mean that one causes the other; perhaps some other variable causes them both or the correlation might be attributable to chance alone. A true correlation means that differences in one variable imply differences in the other when all other things are equal. 9D/H6bc
• The larger a well-chosen sample of a population is, the better it estimates population summary statistics. 9D/H7a
• For a well-chosen sample, the size of the sample is much more important than the size of the population. To avoid intentional or unintentional bias, samples are usually selected by some random system. 9D/H7bc
• A physical or mathematical model can be used to estimate the probability of real-world events. 9D/H8
##### 1993 Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• Even when there are plentiful data, it may not be obvious what mathematical model to use to make predictions from them or there may be insufficient computing power to use some models. 9D/H1
• When people estimate a statistic, they may also be able to say how far off the estimate might be. 9D/H2
• The middle of a data distribution may be misleading—when the data are not distributed symmetrically, or when there are extreme high or low values, or when the distribution is not reasonably smooth. 9D/H3
• The way data are displayed can make a big difference in how they are interpreted. 9D/H4
• Both percentages and actual numbers have to be taken into account in comparing different groups; using either category by itself could be misleading. 9D/H5
• Considering whether two variables are correlated requires inspecting their distributions, such as in two-way tables or scatterplots. A believable correlation between two variables doesn't mean that either one causes the other; perhaps some other variable causes them both or the correlation might be attributable to chance alone. A true correlation means that differences in one variable imply differences in the other when all other things are equal. 9D/H6
• For a well-chosen sample, the size of the sample is much more important than the size of the population. To avoid intentional or unintentional bias, samples are usually selected by some random system. 9D/H7
• A physical or mathematical model can be used to estimate the probability of real-world events. 9D/H8

The appearance of reasoning in this chapter in no way implies that it should be taught only in mathematics classes. Indeed, reasoning should be studied in all science courses, social-studies classes, and wherever critical thinking is being taught. Part of what is to be accomplished is for students to acquire the kind of understanding of deductive logic necessary for telling good logic from bad logic in the arguments people make. They should also become aware of why reasoning is so important in mathematics. Another part of the reasoning agenda should deal with inductive logic-making generalizations based on instances-and its uses in science and everyday life. It is important that students become clear on the limitations of such logic because of the widespread tendency of people to offer an example as a proof.

But as important as it is that students come to understand the nature of logic, it is even more important that they learn how to use logic and evidence in making valid, persuasive arguments and in judging the arguments of others. That will only happen if students have a lot of practice in formulating arguments, presenting them to classmates, responding to their criticisms, and critiquing the arguments of others. Furthermore, this experience should build over many years, becoming gradually more complex as students learn to organize evidence, and should take place in the context of interesting problems and issues raised in social-studies, history, science, and mathematics classes.

At the beginning level, the goal is more for students to develop expectations about reasoning than for them to acquire reasoning skills. The question "How do you know?" should become routine—children should come to expect it to be asked and should feel free to ask it of others. The quality of the answer is not yet important, although there should sometimes be discussion of what is most believable in other people's answers. Science activities provide daily opportunities for students to get practice in referring to evidence.

##### Current Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• People are more likely to believe your ideas if you can give reasons for them. 9E/P1
• Reasoning can be distorted by strong feelings. 9E/P2
##### 1993 Version of the Benchmarks Statements

By the end of the 2nd grade, students should know that

• People are more likely to believe your ideas if you can give good reasons for them. 9E/P1

The quality of the answer to "How do you know?" now starts to become more important. When asked for a reason for an assertion, children are likely at this age to just repeat the assertion, add emphasis ("Just because."), or appeal to authority ("My big brother said so."). Undermining authority is not a very good idea here, but the appeal to reason can be shifted to the authority ("What do you suppose his reasons might have been?"). Supporting claims with reasons should be modeled by the teacher. In science, questions can be raised suggesting that sometimes the trouble with an argument is that the evidence offered is weak. Teachers can set the tone by asking, "Do you think it would help to collect some more samples?" "If you did the investigation over again, do you think the same thing would happen?" "What evidence might change your mind?"

At this level, students are still very concrete in their thinking, but it is probably a good time to introduce reasoning by analogy. Analogies should be simple and obvious at first, and attention should focus on how the analog is like and unlike what is being studied. Reflection on analogies should not make the students so analytical that they back away from their poetic use. Analogies should be used freely in speculation and artistic expression. But when they are used as the basis of argument, they should be challenged. ("My love is like a red, red rose; therefore….")

##### Current Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• One way to think about something is to compare it to something more familiar. 9E/E1*
• The claims people make are sometimes based on how they feel about something rather than on what they observe. 9E/E2*
##### 1993 Version of the Benchmarks Statements

By the end of the 5th grade, students should know that

• One way to make sense of something is to think how it is like something more familiar. 9E/E1
• Reasoning can be distorted by strong feelings. 9E/E2

Many students are able to think more abstractly in the middle grades than in the prior years. Hence they can now consider the principles of reasoning in more detail and begin to appreciate the critical part that logic plays in clear thinking, whereas up to now more emphasis had been placed on the quality of the evidence being offered in support of a claim. This shift entails insisting on the careful use of particular words and phrases, such as If…, then…, and, or, not, all, and some.

Science and mathematics are obvious places for paying attention to logic, but they are not the only ones. Designing projects and troubleshooting mechanical objects and systems provide excellent opportunities for students to apply logic, and such activities have the virtue of providing concrete feedback on how good the logic was. In social studies, students should examine the use of logic in retrieving data from databases and in political and social controversies.

##### Current Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• If people have generalizations that always hold, and good information about a particular situation, then logic can help them to figure out what is true about it. This kind of formal logic requires care in the use of key words such as if, then, and, not, or, all, and some. 9E/M1*
• Sometimes people invent a generalization to summarize a set of observations. But sometimes people overgeneralize, imagining generalizations on the basis of too few observations. 9E/M3*
• People are using incorrect logic when they assume that a statement such as "If A is true, then B is true" implies that "If A isn't true, then B must not be true either." 9E/M4*
• In formal logic, a single example can never prove that a generalization is always true, but sometimes a single example can prove that a generalization is not always true. Proving a generalization to be false is easier than proving it to be true. 9E/M5*
• An analogy has some likenesses to but also some differences from the real thing. 9E/M6
• Reasoning by similarities can suggest ideas to consider but can't prove them one way or the other. 9E/M7** (BSL)
##### 1993 Version of the Benchmarks Statements

By the end of the 8th grade, students should know that

• Some aspects of reasoning have fairly rigid rules for what makes sense; other aspects don't. If people have rules that always hold, and good information about a particular situation, then logic can help them to figure out what is true about it. This kind of reasoning requires care in the use of key words such as if, and, not, or, all, and some. Reasoning by similarities can suggest ideas but can't prove them one way or the other. 9E/M1
• Practical reasoning, such as diagnosing or troubleshooting almost anything, may require many-step, branching logic. Because computers can keep track of complicated logic, as well as a lot of information, they are useful in a lot of problem-solving situations. 9E/M2
In the current version of Benchmarks Online, this benchmark has been deleted because the ideas in it have been incorporated into benchmark 9E/H5**.
• Sometimes people invent a general rule to explain how something works by summarizing observations. But people tend to overgeneralize, imagining general rules on the basis of only a few observations. 9E/M3
• People are using incorrect logic when they make a statement such as "If A is true, then B is true; but A isn't true, therefore B isn't true either." 9E/M4
• A single example can never prove that something is always true, but sometimes a single example can prove that something is not always true. 9E/M5
• An analogy has some likenesses to but also some differences from the real thing. 9E/M6

Transfer of formal logic skills to real-world situations requires a great deal of practice. Claims made in print, radio, and television (including news items, editorials, letters to the editor, and advertisements) should regularly be critiqued by students for the quality of the arguments they make. Students should be able to identify the premises (whether explicit or not), logic, and evidence used, and then evaluate the claim. They should also be able to point out where something other than a sound argument is being used to convince the reader, listener, or watcher. History can provide documented cases of the uses of good and bad logic on a grand scale.

##### Current Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• A sound argument should have both true statements and valid connections among them. Formal logic is mostly about connections among statements, not about whether they are true. People sometimes use logic that begins with untrue statements, and they sometimes use poor logic even if they begin with true statements. 9E/H1*
• Logic requires a clear distinction between those conditions that are necessary to get a result and those that are sufficient to get the result. Some conditions may be both necessary and sufficient. 9E/H2*
• In using logic in real-world situations, one often has to deal with probabilities rather than certainties. 9E/H3*
• Once a person believes a generalization, he or she may be more likely to notice cases that agree with it and to overlook cases that don't. 9E/H4*
• Because computers can store, retrieve, and process large amounts of data, they can rapidly perform a long series of logic steps. They are therefore being used increasingly to help experts solve complex problems that would otherwise be very difficult or impossible to solve. Not all logic problems, however, can be solved by computers. 9E/H5*
• A failure to find an exception to a generalization after reviewing a large number of instances increases the confidence in the accuracy of the generalization. 9E/H6**
##### 1993 Version of the Benchmarks Statements

By the end of the 12th grade, students should know that

• To be convincing, an argument needs to have both true statements and valid connections among them. Formal logic is mostly about connections among statements, not about whether they are true. People sometimes use poor logic even if they begin with true statements, and sometimes they use logic that begins with untrue statements. 9E/H1
• Logic requires a clear distinction among reasons: A reason may be sufficient to get a result, but perhaps is not the only way to get there; or, a reason may be necessary to get the result, but it may not be enough by itself; some reasons may be both sufficient and necessary. 9E/H2
• Wherever a general rule comes from, logic can be used in testing how well it works. Proving a generalization to be false (just one exception will do) is easier than proving it to be true (for all possible cases). Logic may be of limited help in finding solutions to problems if one isn't sure that general rules always hold or that particular information is correct; most often, one has to deal with probabilities rather than certainties. 9E/H3
• Once a person believes in a general rule, he or she may be more likely to notice cases that agree with it and to ignore cases that don't. To avoid biased observations, scientific studies sometimes use observers who don't know what the results are "supposed" to be. 9E/H4
In the current version of Benchmarks Online, the second sentence of this benchmark has been moved to chapter 1, section B, and recoded as 1B/H11**.
• Very complex logical arguments can be made from a lot of small logical steps. Computers are particularly good at working with complex logic but not all logical problems can be solved by computers. High-speed computers can examine the validity of some logical propositions for a very large number of cases, although that may not be a perfect proof. 9E/H5

VERSION EXPLANATION

During the development of Atlas of Science Literacy, Volume 2, Project 2061 revised the wording of some benchmarks in order to update the science, improve the logical progression of ideas, and reflect the current research on student learning. New benchmarks were also created as necessary to accommodate related ideas in other learning goals documents such as Science for All Americans (SFAA), the National Science Education Standards (NSES), and the essays or other elements in Benchmarks for Science Literacy (BSL). We are providing access to both the current and the 1993 versions of the benchmarks as a service to our end-users.

The text of each learning goal is followed by its code, consisting of the chapter, section, grade range, and the number of the goal. Lowercase letters at the end of the code indicate which part of the 1993 version it comes from (e.g., “a” indicates the first sentence in the 1993 version, “b” indicates the second sentence, and so on). A single asterisk at the end of the code means that the learning goal has been edited from the original, whereas two asterisks mean that the idea is a new learning goal.

Copyright © 1993,2009 by American Association for the Advancement of Science