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.

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.

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?"

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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….")

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.

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.