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11. Common Themes

  1. Systems
    1. Kindergarten through Grade 2
    2. Grades 3 through 5
    3. Grades 6 through 8
    4. Grades 9 through 12
  2. Models
    1. Kindergarten through Grade 2
    2. Grades 3 through 5
    3. Grades 6 through 8
    4. Grades 9 through 12
  3. Constancy and Change
    1. Kindergarten through Grade 2
    2. Grades 3 through 5
    3. Grades 6 through 8
    4. Grades 9 through 12
  4. Scale
    1. Kindergarten through Grade 2
    2. Grades 3 through 5
    3. Grades 6 through 8
    4. Grades 9 through 12

Some important themes pervade science, mathematics, and technology and appear over and over again, whether we are looking at an ancient civilization, the human body, or a comet. They are ideas that transcend disciplinary boundaries and prove fruitful in explanation, in theory, in observation, and in design.

Science for All Americans

Some powerful ideas often used by mathematicians, scientists, and engineers are not the intellectual property of any one field or discipline. Indeed, notions of system, scale, change and constancy, and models have important applications in business and finance, education, law, government and politics, and other domains, as well as in mathematics, science, and technology. These common themes are really ways of thinking rather than theories or discoveries. (Energy also represents a prominent tool for thinking in science and technology, but because it is part of the content of science, it is not included here as a theme.) Science for All Americans recommends what all students should know about those themes, and the benchmarks in the four sections below suggest how student understanding of them should grow over the school years. Although the context of both Science for All Americans and Benchmarks is mainly science, mathematics, and technology, other contexts are identified here to emphasize the general usefulness of these themes.


One of the essential components of higher-order thinking is the ability to think about a whole in terms of its parts and, alternatively, about parts in terms of how they relate to one another and to the whole. People are accustomed to speak of political systems, sewage systems, transportation systems, the respiratory system, the solar system, and so on. If pressed, most people would probably say that a system is a collection of things and processes (and often people) that interact to perform some function. The scientific idea of a system implies detailed attention to inputs and outputs and interactions among the system components. If these can be specified quantitatively, a computer simulation of the system might be run to study its theoretical behavior, and so provide a way to define problems and investigate complex phenomena. But a system need not have a "purpose" (e.g., an ecosystem or the solar system) and what a system includes can be imagined in any way that is interesting or useful. Students in the elementary grades study many different kinds of systems in the normal course of things, but they should not be rushed into explicit talk about systems. That can and should come in middle and high school.

Children tend to think of the properties of a system as belonging to individual parts of it rather than as arising from the interaction of the parts. A system property that arises from interaction of parts is therefore a difficult idea. Also, children often think of a system only as something that is made and therefore as obviously defined. This notion contrasts with the scientific view of systems as being defined with particular purposes in mind. The solar system, for example, can be defined in terms of the sun and planets only, or defined to include also the planetary moons and solar comets. Similarly, not only is an automobile a system, but one can think of an automotive system that includes service stations, oil wells, rubber plantations, insurance, traffic laws, junk yards, and so on.

The main goal of having students learn about systems is not to have them talk about systems in abstract terms, but to enhance their ability (and inclination) to attend to various aspects of particular systems in attempting to understand or deal with the whole system. Does the student troubleshoot a malfunctioning device by considering connections and switches—whether using the terms input, output, or controls or not? Does the student try to account for what becomes of all of the input to the water cycle—whether using the term conservation or not? The vocabulary will be helpful for students once they have had a wide variety of experiences with systems thinking, but otherwise it may mistakenly give the impression of understanding. Learning about systems in some situations may not transfer well to other situations, so systems should be encountered through a variety of approaches, including designing and troubleshooting. Simple systems (a pencil or mousetrap), of course, should be encountered before more complex ones (a stereo system, a plant, the continuous manufacture of goods, ecosystems, or school government).

A persistent student misconception is that the properties of an assembly are the same as the properties of its parts (for example, that soft materials are made of soft molecules). Sometimes it is true. For example, a politically conservative organization may be made up entirely of conservative individuals. But some features of systems are unlike any of their parts. Sugar is sweet, but its component atoms (carbon, oxygen, and hydrogen) are not. The system property may result from what its parts are like, but the parts themselves may not have that property. A grand example is life as an emergent property of the complex interaction of complex molecules.


Students in the elementary grades acquire the experiences that they will use in the middle grades and beyond to develop an understanding of systems concepts and their applications. They also can begin to attend to what affects what. Frequent discussion of how one thing affects another lays the ground for recognizing interactions. Another tack for focusing on interaction is to raise the question of when things work and when they do not—owing, say, to missing or broken parts or the absence of a source of power (batteries, gasoline).

Students should practice identifying the parts of things and how one part connects to and affects another. Classrooms can have available a variety of dissectable and rearrangeable objects, such as gear trains and toy vehicles and animals, as well as conventional blocks, dolls, and doll houses. Students should predict the effects of removing or changing parts.

Current Version of the Benchmarks Statements

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

  • Most things are made of parts. 11A/P1
  • Something may not work if some of its parts are missing. 11A/P2
  • When parts are put together, they can do things that they couldn't do by themselves. 11A/P3
1993 Version of the Benchmarks Statements

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

  • Most things are made of parts. 11A/P1
  • Something may not work if some of its parts are missing. 11A/P2
  • When parts are put together, they can do things that they couldn't do by themselves. 11A/P3

Hands-on experience with a variety of mechanical systems should increase. Classrooms can have "take-apart" stations where a variety of familiar hardware devices can be taken apart (and perhaps put back together) with hand tools. Devices that are commonly purchased disassembled can be provided, along with assembly instructions, to emphasize the importance of the proper arrangement of parts (and incidentally, the importance of language-arts skills, which are needed to read and follow instructions).

Current Version of the Benchmarks Statements

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

  • In something that consists of many parts, the parts usually influence one another. 11A/E1
  • Something may not work well (or at all) if a part of it is missing, broken, worn out, mismatched, or misconnected. 11A/E2
1993 Version of the Benchmarks Statements

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

  • In something that consists of many parts, the parts usually influence one another. 11A/E1
  • Something may not work as well (or at all) if a part of it is missing, broken, worn out, mismatched, or misconnected. 11A/E2

Systems thinking can now be made explicit—suggesting analysis of parts, subsystems, interactions, and matching. But descriptions of parts and their interaction are more important than just calling everything a system.

Student projects should now entail analyzing, designing, assembling, and troubleshooting systems—mechanical, electrical, and biological—with easily discernible components. Students can take apart and reassemble such things as bicycles, clocks, and mechanical toys and build battery-driven electrical circuits that actually operate something. They can assemble a sound system and then judge how changing different components affects the system's output, or observe aquariums and gardens while changing some parts of the system or adding new parts. The idea of system should be expanded to include connections among systems. For example, a can opener and a can may each be thought of as a system, but they both—together with the person using them—form a larger system without which neither can be put to its intended use.

Current Version of the Benchmarks Statements

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

  • A system can include processes as well as things. 11A/M1
  • Thinking about things as systems means looking for how every part relates to others. The output from one part of a system (which can include material, energy, or information) can become the input to other parts. Such feedback can serve to control what goes on in the system as a whole. 11A/M2
  • Any system is usually connected to other systems, both internally and externally. Thus a system may be thought of as containing subsystems and as being a sub-system of a larger system. 11A/M3
  • Some portion of the output of a system may be fed back to that system's input. 11A/M4** (SFAA)
  • Systems are defined by placing boundaries around collections of interrelated things to make them easier to study. Regardless of where the boundaries are placed, a system still interacts with its surrounding environment. Therefore, when studying a system, it is important to keep track of what enters or leaves the system. 11A/M5** (SFAA)
1993 Version of the Benchmarks Statements

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

  • A system can include processes as well as things. 11A/M1
  • Thinking about things as systems means looking for how every part relates to others. The output from one part of a system (which can include material, energy, or information) can become the input to other parts. Such feedback can serve to control what goes on in the system as a whole. 11A/M2
  • Any system is usually connected to other systems, both internally and externally. Thus a system may be thought of as containing subsystems and as being a subsystem of a larger system. 11A/M3

Students should have opportunities—in seminars, projects, readings, and experiments—to reflect on the value of thinking in terms of systems and to apply the concept in diverse situations. They should often discuss what properties of a system are the same as the properties of its parts and what properties arise from interactions of its parts or from the sheer number of parts. They should learn to see feedback as a standard aspect of systems. The definitions of negative and positive feedback may be too subtle, but students can understand that feedback may oppose changes that do occur (and lead to stability), or may encourage more change (and so drive the system toward one extreme or another). Eventually, they can see how some delay in feedback can produce cycles in a system's behavior.

Current Version of the Benchmarks Statements

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

  • A system usually has some properties that are different from those of its parts, but appear because of the interaction of those parts. 11A/H1
  • Understanding how things work and designing solutions to problems of almost any kind can be facilitated by systems analysis. In defining a system, it is important to specify its boundaries and subsystems, indicate its relation to other systems, and identify what its input and output are expected to be. 11A/H2
  • The successful operation of a designed system often involves feedback. Such feedback can be used to encourage what is going on in a system, discourage it, or reduce its discrepancy from some desired value. The stability of a system can be greater when it includes appropriate feedback mechanisms. 11A/H3*
  • Even in some very simple systems, it may not always be possible to predict accurately the result of changing some part or connection. 11A/H4
  • Systems may be so closely related that there is no way to draw boundaries that separate all parts of one from all parts of the other. 11A/H5** (SFAA)
1993 Version of the Benchmarks Statements

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

  • A system usually has some properties that are different from those of its parts, but appear because of the interaction of those parts. 11A/H1
  • Understanding how things work and designing solutions to problems of almost any kind can be facilitated by systems analysis. In defining a system, it is important to specify its boundaries and subsystems, indicate its relation to other systems, and identify what its input and its output are expected to be. 11A/H2
  • The successful operation of a designed system usually involves feedback. The feedback of output from some parts of a system to input for other parts can be used to encourage what is going on in a system, discourage it, or reduce its discrepancy from some desired value. The stability of a system can be greater when it includes appropriate feedback mechanisms. 11A/H3
  • Even in some very simple systems, it may not always be possible to predict accurately the result of changing some part or connection. 11A/H4

Physical, mathematical, and conceptual models are tools for learning about the things they are meant to resemble. Physical models are by far the most obvious to young children, so they should be used to introduce the idea of models. Dolls, stuffed animals, toy cars and airplanes, and other everyday objects can stimulate discussions about how those things are like and unlike the real things. The term model should probably be used to refer only to physical models in the early grades, but the notion of likeness will be the central issue in using any kind of model.

The usefulness of conceptual models depends on the ability of people to imagine that something they do not understand is in some way like something that they do understand. Imagery, metaphor, and analogy are every bit as much a part of science as deductive logic, and as much at home in science as in the arts and humanities. Students cannot be expected to become adept in the use of conceptual models, however, until they get to know quite a bit about materials, things, and processes in the accessible world around them through direct, hands-on experience. The curriculum emphasis, therefore, should be on a rich variety of experiences, not on generalizations about conceptual models. Moreover, students need to acquire images and understandings that come from drawing, painting, sculpting, playing music, acting in plays, listening to and telling stories, reading, participating in games and sports, doing work, and living life.

By their nature, mathematical models are usually more abstract than physical and conceptual models. The connection of mathematics to concrete matters, and hence its value for modeling, could be substantially stronger if mathematics were often taught as part of science, social studies, technology, health, gym, music, and other subjects, rather than only during "mathematics time." One of the drawbacks of teaching mathematics entirely as a separate subject is that mathematics is taught before real-world problems are identified, so the related exercises may have mostly to do with learning the procedures rather than with solving interesting problems.


Every opportunity should be taken to get students to talk about how the things they play with relate to real things in the world. The more imaginative the conversation the better, for insisting upon accuracy at this level may hinder other important developments.

Current Version of the Benchmarks Statements

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

  • Many toys are like real things in some ways but not others. They may not be the same size, are missing many details, or are not able to do all of the same things. 11B/P1
  • A model of something is different from the real thing but can be used to learn something about the real thing. 11B/P2
  • One way to describe something is to say how it is and isn't like something else. 11B/P3
1993 Version of the Benchmarks Statements

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

  • Many of the toys children play with are like real things only in some ways. They are not the same size, are missing many details, or are not able to do all of the same things. 11B/P1
  • A model of something is different from the real thing but can be used to learn something about the real thing. 11B/P2
  • One way to describe something is to say how it is like something else. 11B/P3

As students develop beyond their natural play with models, they should begin to modify them and discuss their limitations. What happens if wheels are taken off, or weight is added, if different materials are used, or if the model gets wet? Is that what would happen to the real things? Students also can begin to compare their objects, drawings, and constructions to the things they portray or resemble (real bears, houses, airplanes, etc.). Since students are being introduced to geometry, graphs, and other mathematical concepts, they should at the same time reflect on how these representations relate to nature. Similarly, what they are learning in the arts and humanities can supply analogies. Students can begin to formulate their own models to explain things they cannot observe directly. By testing their models and changing them as more information is acquired, they begin to understand how science works.

Current Version of the Benchmarks Statements

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

  • Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and oral and written descriptions can be used to represent objects, events, and processes in the real world. 11B/E2*
  • A model of something is similar to, but not exactly like, the thing being modeled. Some models are physically similar to what they are representing, but others are not. 11B/E3** (SFAA)
  • Models are very useful for communicating ideas about objects, events, and processes. When using a model to communicate about something, it is important to keep in mind how it is different from the thing being modeled. 11B/E4**
1993 Version of the Benchmarks Statements

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

  • Seeing how a model works after changes are made to it may suggest how the real thing would work if the same were done to it. 11B/E1
    In the current version of Benchmarks Online, this benchmark has been deleted.
  • Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail. 11B/E2

Now models and their use can be dealt with much more explicitly than before because students have a greater general knowledge of mathematics, literature, art, and the objects and processes around them. Also, student use of computers should have progressed beyond word processing to graphing and simulations that compute and display the results of changing factors in the model. All of these things can give students a grasp of what models are and how they can be compared by considering their consequences. Students should have many opportunities to learn how conceptual models can be used to suggest interesting questions, such as "What would the atmosphere be like if its molecules were to act like tiny, high-speed marshmallows instead of tiny, high-speed steel balls?"

The use of physical models also can increase in sophistication. Students should discover that physical models on a reduced scale may be inadequate because of scaling effects: With change in scale, some factors change more than others so things no longer work the same way. The drag effects of water flow past a model boat, for example, are very different from the effects on a full-sized boat.

Current Version of the Benchmarks Statements

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

  • Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly. They are also used for processes that are too vast, too complex, or too dangerous to study. 11B/M1*
  • Mathematical models can be displayed on a computer and then modified to see what happens. 11B/M2
  • Different models can be used to represent the same thing. What model to use depends on its purpose. 11B/M3*
  • Simulations are often useful in modeling events and processes. 11B/M4** (BSL)
  • The usefulness of a model depends on how closely its behavior matches key aspects of what is being modeled. The only way to determine the usefulness of a model is to compare its behavior to the behavior of the real-world object, event, or process being modeled. 11B/M5**
  • A model can sometimes be used to get ideas about how the thing being modeled actually works, but there is no guarantee that these ideas are correct if they are based on the model alone. 11B/M6** (SFAA)
1993 Version of the Benchmarks Statements

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

  • Models are often used to think about processes that happen too slowly, too quickly, or on too small a scale to observe directly, or that are too vast to be changed deliberately, or that are potentially dangerous. 11B/M1
  • Mathematical models can be displayed on a computer and then modified to see what happens. 11B/M2
  • Different models can be used to represent the same thing. What kind of a model to use and how complex it should be depends on its purpose. The usefulness of a model may be limited if it is too simple or if it is needlessly complicated. Choosing a useful model is one of the instances in which intuition and creativity come into play in science, mathematics, and engineering. 11B/M3

In the upper grades, considerable emphasis should be placed on mathematical modeling because it epitomizes the nature and power of models and provides a context for integrating knowledge from many different domains. The main goal should be getting students to learn how to create and use models in many different contexts, not simply to recite generalizations about models. They can acquire such generalizations too, but that will occur through discussions of models already studied. Research in developmental psychology implies that high-school students may understand that the best model isn't found yet, or that different people prefer different models while waiting for more evidence, but not that there may be no "true" model at all.

Current Version of the Benchmarks Statements

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

  • A mathematical model uses rules and relationships to describe and predict objects and events in the real world. 11B/H1a*
  • A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning. 11B/H1b
  • Computers have greatly improved the power and use of mathematical models by performing computations that are very long, very complicated, or repetitive. Therefore, computers can reveal the consequences of applying complex rules or of changing the rules. The graphic capabilities of computers make them useful in the design and simulated testing of devices and structures and in the simulation of complicated processes. 11B/H2*
  • The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that other models would not work equally well or better. 11B/H3*
  • Often, a mathematical model may fit a phenomenon over a small range of conditions (such as temperature or time), but it may not fit well over a wider range. 11B/H4** (SFAA)
  • The behavior of a physical model cannot ever be expected to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. The inappropriateness of a model may be related to differences between the model and what is being modeled. 11B/H5** (SFAA)
1993 Version of the Benchmarks Statements

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

  • The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same ways as the objects or processes under investigation. A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning. 11B/H1
  • Computers have greatly improved the power and use of mathematical models by performing computations that are very long, very complicated, or repetitive. Therefore computers can show the consequences of applying complex rules or of changing the rules. The graphic capabilities of computers make them useful in the design and testing of devices and structures and in the simulation of complicated processes. 11B/H2
  • The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that the model is the only "true" model or the only one that would work. 11B/H3

Much of science and mathematics has to do with understanding how change occurs in nature and in social and technological systems, and much of technology has to do with creating and controlling change. Constancy, often in the midst of change, is also the subject of intense study in science. The simplest account to be given of anything is that it does not change. Because scientists are always looking for the simplest possible accounts (that are true), they are always delighted by any aspect of anything that doesn't change even when many other aspects do. Indeed, many historians and philosophers regard conservation laws in physics (such as for mass, energy, or electric charge) to be among the greatest discoveries in science. Somewhat different aspects of constancy are described by the terms stability, conservation, equilibrium, steady state, and symmetry. These various ideas are interrelated in some subtle ways. But memorizing the distinct meanings for these terms is not a high priority. More important is being able to think about what is happening.

Symmetry is another kind of constancy—or more generally, invariance—in the midst of change. Equilibrium, steady states, and conservation might all be thought of as showing symmetry. But more typically, symmetry implies a pattern whose appearance stays the same when it undergoes a change such as rotation, reflection, stretching, or displacement. The symmetry can be geometrical or more general, as in a social order, set of computer operations, or classification of atomic particles.

When change occurs in a variable, a major issue is the rate at which change occurs. Clearly students have to make sense of a constant rate of change before they can consider increasing or decreasing rates. Yet understanding a constant rate of change is not as simple as it might seem, because of the difficulty of the idea of rate. Graphs would seem to be an immense help for semiquantitative descriptions of change—such as whether the rate is constant, increasing, saturating, etc. But the research results are that, unless the graph is of literal altitude, graph heights and slopes are puzzling to most children. The goal for all Americans should be modest: to understand a graph of any familiar variable against time in terms of reading it and interpreting its ups and downs in a story about what is going on. Eventually, steepness as well as direction of change can become part of the story.

Considering the pattern of change usually involves a scale of observations and a scale of analysis. The rock may appear to sit there on the ground unchanging, but at a distance scale 108 times smaller its atoms are chaotically restless, and at a scale of 108 times larger its planet is turning and orbiting. An ecological system may seem stable over a few centuries, but over days individuals come and go, and over millions of years it is greatly transformed.

Very, very small differences in what a system is like now may produce very large differences in what it is like later. That's not a difficult idea even in the middle school. What is harder to understand is that no matter how small the initial uncertainty may be, the behavior is eventually unpredictable. At the finest level, that of individual atoms, uncertainty is unavoidable. So the future is not determined by the present. For example, long-range weather forecasting now seems to be impossible—in principle, not just because of the limits of observation and analysis.

For the most part, change should not be taught as a separate subject. At every opportunity throughout the school years, the theme of change should be brought up in the context of the science, mathematics, or technology being studied. The first step is to encourage children to attend to change and describe it. Only after they have a storehouse of experience with change of different kinds are they ready to start thinking about patterns of change in the abstract. When students have such a background, a short capstone course on the subject of change could help them integrate their knowledge of patterns of change in physical, biological, social, and technological systems.


When collecting and observing the things around them, students can look for what changes and what does not and question where things come from and where things go. They may note, for instance, that most animals move from place to place but most plants stay in place, that water left in an open container gradually disappears but sand does not, and so forth. Such activities can sharpen students' observation and communication skills and instill in them a growing sense that many different kinds of change go on all the time. Students should be encouraged to take, record, and display counts and simple measurements of things over time. This activity can provide them with many opportunities to learn and use elementary mathematics. To begin to work toward ideas of conservation, mathematics exercises in which the sum stays the same may be helpful—e.g., "How many ways can you add whole numbers to get 13?"

Current Version of the Benchmarks Statements

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

  • Objects change in some ways and stay the same in some ways. 11C/P1*
  • People can keep track of some things, seeing where they come from and where they go. 11C/P2
  • An object can change in various ways, such as in size, weight, color, or temperature. 11C/P3a*
  • Small changes can sometimes be detected by comparing counts or measurements at different times. 11C/P3b*
  • Some things change so slowly or so quickly that the changes are hard to notice while they are taking place. 11C/P4*
1993 Version of the Benchmarks Statements

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

  • Things change in some ways and stay the same in some ways. 11C/P1
  • People can keep track of some things, seeing where they come from and where they go. 11C/P2
  • Things can change in different ways, such as in size, weight, color, and movement. Some small changes can be detected by taking measurements. 11C/P3
  • Some changes are so slow or so fast that they are hard to see. 11C/P4

With greater emphasis than before on measurement, graphing, and data analysis, students can make progress toward understanding some very important notions about change. At this stage, becoming familiar with a large and varied set of actual examples of change is more important than being able to recite the generalizations set out in the benchmarks.

Notions of symmetry can begin with identifying patterns whose appearance stays the same when they undergo some change (such as rotation, reflection, stretching, or displacement). Children generally are interested in exploring the shapes of things (plants and animals, themselves, buildings, vehicles, toys, etc.) and looking for regularities of shape. Students should have many experiences in discussing and depicting all sorts of change: continuing in the same direction, reaching a high or low value, repeatedly reversing direction, and so on.

Current Version of the Benchmarks Statements

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

  • Some features of things may stay the same even when other features change. 11C/E1*
  • Things change in steady, repetitive, or erratic ways—or sometimes in more than one way at the same time. 11C/E2a*
  • Often the best way to tell which kinds of change are happening is to make a table or graph of measurements. 11C/E2b
  • Some things, such as a person's age, change in only one direction. 11C/E3**
  • Some things in nature have a repeating pattern, such as the day-night cycle, the phases of the moon, and seasons. 11C/E4** (BSL)
  • The number of objects in a group can stay the same even as some enter or leave, as long as each one that leaves is replaced by another one that is entering. 11C/E5**
1993 Version of the Benchmarks Statements

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

  • Some features of things may stay the same even when other features change. Some patterns look the same when they are shifted over, or turned, or reflected, or seen from different directions. 11C/E1
    In the current version of Benchmarks Online, the last sentence of this benchmark has been moved to grades 3-5 in chapter 9, section C, and recoded as 9C/E9**.
  • Things change in steady, repetitive, or irregular ways—or sometimes in more than one way at the same time. Often the best way to tell which kinds of change are happening is to make a table or graph of measurements. 11C/E2

Constancy in a system can be represented in two ways: as a constant sum or as compensating changes. When the quantity being considered is a count (as of students or airplanes), then constancy of the total is obvious. When the quantity being considered is a measure on a continuous scale, rather than a packaged unit, then "it has to come from somewhere and go somewhere" may be a more directly appreciable principle. For example, it seems easier to see that heat lost from one part of a system has to show up somewhere else than to say that the total measure for the whole system has to stay the same. This may be particularly true when the quantity can take various, interconvertible forms—say, forms of energy or monetary value.

In these grades, students can look for more sophisticated patterns, including rates of change and cyclic patterns. Invariance may be found in change itself: The water in a river changes, but the rate of flow may be constant; or the rate of flow may change seasonally, but the cycle may have a constant cycle length.

The idea of a series of repeating events is not difficult for students—that is what their day-by-day and week-by-week lives are like. Cyclic variation in a magnitude is more difficult. The cycle length is its simplest feature, whereas the range of variation has little interest unless students are familiar with and care about the variable. (A variation of one degree in body temperature—because of its relevance to whether they have to stay home from school—may be more interesting to students than a tenfold variation in the number of cases of measles.)

Current Version of the Benchmarks Statements

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

  • A system may stay the same because nothing is influencing it or the influences on it are balanced. 11C/M2*
  • Many systems contain feedback mechanisms that serve to keep changes within certain limits. 11C/M3*
  • Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes. 11C/M4
  • Cycles, such as the seasons or body temperature, can be described by what their cycle length or frequency is, what their highest and lowest values are, and when these values occur. Different cycles range from many years down to a fraction of a second. 11C/M6*
  • Cyclic patterns evident in past events can be used to make predictions about future events. However, these predictions may not always match what actually happens. 11C/M7**
  • The way some systems behave is so erratic that patterns of change are not apparent. 11C/M8**
  • Small differences in how things start out can sometimes produce large differences in how they end up. Some events are so sensitive to small differences in initial conditions that their outcomes cannot be predicted. 11C/M9**
  • Trends based on what has happened in the past can be used to make predictions about what things will be like in the future. However, these predictions may not always match what actually happens. 11C/M10**
  • The amount of something in a system may stay the same because nothing is entering or leaving the system or because something is being added to the system at the same rate as it is leaving the system. 11C/M11**
1993 Version of the Benchmarks Statements

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

  • Physical and biological systems tend to change until they become stable and then remain that way unless their surroundings change. 11C/M1
    In the current version of Benchmarks Online, this benchmark has been deleted.
  • A system may stay the same because nothing is happening or because things are happening but exactly counterbalance one another. 11C/M2
  • Many systems contain feedback mechanisms that serve to keep changes within specified limits. 11C/M3
  • Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes. 11C/M4
  • Symmetry (or the lack of it) may determine properties of many objects, from molecules and crystals to organisms and designed structures. 11C/M5
    In the current version of Benchmarks Online, this benchmark has been moved to grades 9–12 and recoded as 11C/H13**.
  • Things that change in cycles, such as the seasons or body temperature, can be described by their cycle length or frequency, what the highest and lowest values are, and when they occur. Different cycles range from many thousands of years down to less than a billionth of a second. 11C/M6

Most of what is appropriate to study about constancy and change in the high-school years has at least been touched upon in the earlier years, though mostly in a qualitative or semiquantitative way. Although it is still not necessary to become intensely quantitative, many of the applications of the ideas take on more concrete meaning when calculations are made.

Stability, like many concepts in science, has to be considered in some context of scale. On a familiar scale of space and time, a mountain may appear stable for centuries. Yet on the atomic scale, the mountain is a continuous hubbub of restless motion and absorption and radiation of energy. On the scale of millions of years, mountains rise up from plains and erode away. In a practical sense, stability of some object or system means only that for present purposes one does not notice or have to worry about changes in it.

Perhaps the most important ideas to be dealt with are the conservation laws, rates of change, and the general notion of evolutionary change. The emphasis on conservation laws should probably be practical—that is, should show how those concepts led, and continue to lead, to advances in science. The historical cases studied can contribute to this understanding. Rates of change that are approximately constant (or averageable) make possible a variety of practical calculations. Changing rates need not be calculated but can be identified in graphs and sketched. Especially important is the case in which change rate is proportional to how much there already is (as in population growth or radioactive decay).

Evolutionary change is a general concept, of which biological evolution is only one instance. Another point is more philosophical: Although evolution is the kind of change that emerges from and is influenced by the past, the past appears not to completely determine the future.

Two major arguments for indeterminism are included in Benchmarks—the principle of uncertainty at the submicroscopic level and the additional uncertainty owing to the complexity of systems and their sensitivity to vanishingly small differences in conditions. These arguments are not easy to grasp but at least students should be given a chance to debate them. Many students may be reassured to learn that scientists do not claim to be able to predict the future in every detail, nor do they claim that nature is a mechanical system in which every occurrence is already determined.

Current Version of the Benchmarks Statements

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

  • If a system in equilibrium is disturbed, it may return to a very similar state of equilibrium, or it may undergo a radical change until the system achieves a new state of equilibrium with very different conditions, or it may fail to achieve any type of equilibrium. 11C/H1*
  • Things can change in detail but remain the same in general (the players change, but the team remains; cells are replaced, but the organism remains). Sometimes counterbalancing changes are necessary for a thing to retain its essential constancy in the presence of changing conditions. 11C/H3
  • Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change. 11C/H4
  • Cyclic change is commonly found when there are feedback effects in a system—as, for example, when a change in any direction gives rise to forces or influences that oppose the change. 11C/H5*
  • The present arises from the conditions of the past and, in turn, affects what is possible in the future. 11C/H6*
  • Most systems above the molecular level involve so many parts that it is not practical to determine the existing conditions, and thus the precise behavior of every part of the system cannot be predicted. 11C/H7a*
  • The precise future of a system is not completely determined by its present state and circumstances but also depends on the fundamentally uncertain outcomes of events on the atomic scale. 11C/H7b
  • Trends that follow a pattern that can be described mathematically can be used to estimate how long a process has been going on. 11C/H8** (SFAA)
  • It is not always easy to recognize meaningful patterns of change in a set of data. Data that appear to be completely irregular may be shown by statistical analysis to have underlying trends or cycles. On the other hand, trends or cycles that appear in data may sometimes be shown by statistical analysis to be easily explainable as being attributable only to randomness or coincidence. 11C/H9** (SFAA)
  • Whatever happens within a system, such as parts exploding, decaying, or reorganizing, some features, such as the total amount of matter and energy, remain precisely the same. 11C/H10** (SFAA)
  • The amount of something in a system may stay the same because nothing is happening to it or because it is being transformed into something else at the same rate as something else is being transformed into it. 11C/H11** (SFAA)
  • Even though a system may appear to be unchanging when viewed macroscopically, there is continual activity of the molecules in the system. 11C/H12** (SFAA)
  • Symmetry (or a lack of it) may determine properties of many objects, from molecules and crystals to organisms and designed structures. 11C/H13** (BSL)
1993 Version of the Benchmarks Statements

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

  • A system in equilibrium may return to the same state of equilibrium if the disturbances it experiences are small. But large disturbances may cause it to escape that equilibrium and eventually settle into some other state of equilibrium. 11C/H1
  • Along with the theory of atoms, the concept of the conservation of matter led to revolutionary advances in chemical science. The concept of conservation of energy is at the heart of advances in fields as diverse as the study of nuclear particles and the study of the origin of the universe. 11C/H2
    In the current version of Benchmarks Online, this benchmark has been deleted.
  • Things can change in detail but remain the same in general (the players change, but the team remains; cells are replaced, but the organism remains). Sometimes counterbalancing changes are necessary for a thing to retain its essential constancy in the presence of changing conditions. 11C/H3
  • Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change. 11C/H4
  • In many physical, biological, and social systems, changes in one direction tend to produce opposing (but somewhat delayed) influences, leading to repetitive cycles of behavior. 11C/H5
  • In evolutionary change, the present arises from the materials and forms of the past, more or less gradually, and in ways that can be explained. 11C/H6
  • Most systems above the molecular level involve so many parts and forces and are so sensitive to tiny differences in conditions that their precise behavior is unpredictable, even if all the rules for change are known. Predictable or not, the precise future of a system is not completely determined by its present state and circumstances but also depends on the fundamentally uncertain outcomes of events on the atomic scale. 11C/H7

Most variables in nature—size, distance, weight, temperature, and so on—show immense differences in magnitude. As their sophistication increases, students should encounter increasingly larger ratios of upper and lower limits of these variables. But that is only the starting point for the idea of changes of scale. The larger idea is that the way in which things work may change with scale. Different aspects of nature change at different rates with changes in scale, and so the relationships among them change, too. Probably the most easily demonstrated example is that as something changes size, its volume changes out of proportion to its area. So properties that depend on volume (such as mass and heat capacity) increase faster than properties that depend on area (such as bone strength and cooling surface). Therefore a large container of hot water cools off more slowly than a small container, and a large animal must have proportionally thicker legs than a small animal of otherwise similar shape.

As another consequence of disproportional change of properties, some "laws" of science (such as how friction depends on speed) are valid only within a certain range of circumstances. New and sometimes surprising kinds of phenomena can appear at extremely large or small values of a variable. For example, a star many times more massive than the sun can eventually collapse under its own gravity to become a black hole from which not even light can escape.

Looking at how things change with scale requires familiarity with the range of values and with how to express the range in numbers that make some sense. So children should start by noticing extremes of familiar variables and how things may be different at those extremes. There is no problem here, in that most children are entranced by "biggest," "littlest," "fastest," and "slowest"—giants and superlatives in general. In any case, scale should be introduced explicitly only when students already have a rich ground of experiences having to do with magnitudes and the effects of changing them.

The range of numbers that people can grasp increases with age. No benefit comes from trying to foist exponential notation on children who can't grasp its meaning at all. It has been argued that people really can't comprehend a range of more than about 1,000 to 1 at any one moment. One can think of a meter being a thousand millimeters (they are there to be seen in a quick look at a meter stick) and that a kilometer is a thousand meters (it can be run off in a few minutes)—but one may not be able to think of a kilometer as a million millimeters. A million becomes meaningful, however, as a thousand thousands, once a thousand becomes comprehensible. Particularly important senses of scale to develop for science literacy are the immense size of the cosmos, the minute size of molecules, and the enormous age of the earth (and the life on it).


Children at this level are not yet comfortable enough with numbers to succeed much in comparing magnitudes. Their attention should be drawn repeatedly to simple comparisons in observations: What is smaller or larger, what might be still smaller or larger, what is the smallest or largest they could imagine, and do such things exist? A sense of changes in scale can be encouraged by perspective-taking games that challenge imagination (for example, "What would other people look like to you if you were as tall as a house or as small as an ant?").

Current Version of the Benchmarks Statements

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

  • Things in nature and things people make have very different sizes, weights, ages, and speeds. 11D/P1
1993 Version of the Benchmarks Statements

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

  • Things in nature and things people make have very different sizes, weights, ages, and speeds. 11D/P1

Children at this level tend to be fascinated by extremes. That interest should be exploited to develop student math skills as well as a sense of scale. Students may not have the mathematical sophistication to deal confidently with ratios and with differences among ratios but the observational groundwork and familiarity with talking about them can begin. At the very least, students can compare speeds, sizes, distances, etc., as fractions and multiples of one another.

Students should now be building structures and other things in their technology projects. Through such experience, they can begin to understand both the mathematical and engineering relationships of length, area, and volume. They can be challenged to measure things that are hard to measure on account of being very small or very large, very light or very heavy.

Current Version of the Benchmarks Statements

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

  • Finding out what the largest and the smallest values of something are is often as informative as knowing what the usual value is. 11D/E2*
1993 Version of the Benchmarks Statements

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

  • Almost anything has limits on how big or small it can be. 11D/E1
    In the current version of Benchmarks Online, this benchmark has been deleted.
  • Finding out what the biggest and the smallest possible values of something are is often as revealing as knowing what the usual value is. 11D/E2

As students' familiarity with very large and very small numbers, ratios, and powers of ten improves, extremes of scale become more meaningful. The use of ratios can now be explicit and comparisons of extremes that exceed 1010 may make some sense to students. Alternative representations of great scale differences should be used—such as Charles Eames' Powers of Ten film and Haldane's classic essay, "On Being the Right Size." Indeed, this essay might very well serve as the centerpiece for a seminar or short course dealing with the importance of size in nature and in construction.

The topic of scale also lends itself to the use of computer simulation, in which the user can change scales at will, and to the use of elementary statistics—large collections of things may have to be represented by summaries such as averages or typical examples. Approximate powers of ten (orders of magnitude) can be learned if students have become comfortable with estimates and approximations. This use of exponents for comparisons does not justify teaching the full apparatus of exponential notation to all students.

Understanding the notion that things necessarily work differently on different scales is more difficult than recognizing extremes, hence students should study a variety of different examples (for instance, cooling rates of different-sized containers of water, strength of different-sized constructions from the same material, flight characteristics of different-sized model airplanes).

Current Version of the Benchmarks Statements

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

  • Some properties of an object depend on its length, some depend on its area, and some depend on its volume. 11D/M1*
  • As the complexity of any system increases, gaining an understanding of it depends increasingly on summaries, such as averages and ranges, and on descriptions of typical examples of that system. 11D/M2
  • Natural phenomena often involve sizes, durations, and speeds that are extremely small or extremely large. These phenomena may be difficult to appreciate because they involve magnitudes far outside human experience. 11D/M3**
1993 Version of the Benchmarks Statements

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

  • Properties of systems that depend on volume, such as capacity and weight, change out of proportion to properties that depend on area, such as strength or surface processes. 11D/M1
  • As the complexity of any system increases, gaining an understanding of it depends increasingly on summaries, such as averages and ranges, and on descriptions of typical examples of that system. 11D/M2

Facility with powers of ten can make it easier to describe great differences of scale, but not necessarily to make them comprehensible. Students can bootstrap their comprehension of magnitude only by a few factors of ten at a time, perhaps grasping each new level only in terms of the previous one. For instance, once students have come to terms with a million, then they may have a better sense of what it means to say there are over a billion galaxies, each with over a billion stars.

Mathematical sophistication can now also include abstract, algebraic representation of the effects of powers; properties that increase by the square of linear size or the cube; and the relation between those increases. Still, the most important point is not the precise ratio of x3 to x2, but the more approximate idea that one changes out of proportion to the other—therefore relationships change. Things, systems, and models that work well on one scale may work less well, or not at all, if greatly expanded or shrunk. A meter-wide amoeba, for example, would never be able to get enough nutrients and oxygen through its surface to survive; a meter-long bird built like a sparrow could not fly.

Current Version of the Benchmarks Statements

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

  • Representing very large or very small numbers in terms of powers of ten makes it easier to perform calculations using those numbers. 11D/H1*
  • Because different properties are not affected to the same degree by changes in size, large changes in size typically change the way that things work in physical, biological, or social systems. 11D/H2*
  • As the number of parts in a system grows in size, the number of possible internal interactions increases much more rapidly, roughly with the square of the number of parts. 11D/H3*
1993 Version of the Benchmarks Statements

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

  • Representing large numbers in terms of powers of ten makes it easier to think about them and to compare things that are greatly different. 11D/H1
  • Because different properties are not affected to the same degree by changes in scale, large changes in scale typically change the way that things work in physical, biological, or social systems. 11D/H2
  • As the number of parts of a system increases, the number of possible interactions between pairs of parts increases much more rapidly. 11D/H3

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