 |
|
|
|
|
|
|
|
|
|
|
|
|
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
|
|
||||||||||||||||||||||||||
|
|
Find more Kappan articles in the
Practicing Representation Learning with and About Representational Forms Every student's educational activities should include the rich variety of experience and learning made possible through participation in multiple practices of representation, Messrs. Greeno and Hall suggest. Our title is deliberately ambiguous. In one interpretation, "practicing representation" refers to a kind of exercise that is required of students, especially in mathematics and science. They have to practice using the standard forms of representation, such as arithmetic expressions, tables, graphs, and equations. In another interpretation, "practicing representation" must always be part of a social practice. Viewed in this way, learning to construct and interpret representations involves learning to participate in the complex practices of communication and reasoning in which the representations are used. This learning involves much more than simply learning to read and write symbols in arrangements corresponding to the accepted forms. A theoretical perspective -- sometimes called the situative perspective -- that is currently developing in research emphasizes those practices in which students participate as they learn. Teachers know the importance of student participation in the activities of learning, and they commit much effort to designing activities that will engage the participation of students. What is often overlooked is that, in their participation, students learn the prevailing practices of learning and knowing that occur in their school setting. For example, if students' learning activities include formulating questions and proposing and explaining alternative solutions, they can learn how to participate in those activities of inquiry. However, if they learn to give only the answers and explanations that are specified by teachers and textbooks, they are likely to learn the practices of memorizing. To be still more specific, if students need to construct tables and graphs to complete a project report in mathematics or science, they can learn how to consider whether and how these forms are effective in communicating the information that they think is important. Or if students draw graphs corresponding to equations for, say, linear versus exponential growth in order to explore the effects of changes in their parameters, they can learn how to use graphs to consider differences in meaning between the concepts of additive and multiplicative change. But if students simply complete assignments of constructing representations in forms that are already specified, they do not have opportunities to learn how to weigh the advantages and disadvantages of different forms of representation or how to use those representations as tools with which to build their conceptual understanding. We discuss here the process of learning to construct and interpret representations from the perspective of social practice. We and others have been conducting research on practices of representation in schools and in other work settings, and we believe that this research has implications for curriculum and instruction in school mathematics and science. Our discussion makes three main points.
Representations Are Important for Their Uses Many forms of representation are taught explicitly in schools, especially in mathematics and science. Students learn how to construct and interpret tables, graphs, equations, and other forms of technical representation because they are essential tools for communicating and reasoning about concepts and information in mathematics, science, and other domains. Unfortunately, technical representations are often taught as though they were ends in themselves. Learning to make graphs and to write equations and interpret them correctly becomes important not for understanding and communicating concepts but for getting high scores on tests. Under the pressure to cover the prescribed curriculum, teachers often feel that there is not enough time to teach students what representations are for and why the forms are useful and effective. The changes that many reformers are advocating in mathematics and science education include more emphasis on students' learning to apply what they learn to their activities of working, thinking, and socializing, both within and outside of school.1 In this way of thinking, forms of representation are tools that students can learn to use as resources in thinking and communicating.2 A classroom example. New mathematics curricula provide experiences in which students learn to use forms of representation for purposes other than encoding information in correct form. In one example, groups of middle school students work on a project in which they build models of a biological system with two populations, wolves and caribou. Their job is to develop recommendations that could be sent to the state of Alaska about alternative policies for controlling the populations of wolves on public lands. The students are to develop one or more simulation models that support their recommendations. We have observed middle school students working on such projects in classes where their teachers use a curriculum unit called "LifeLines," developed at the Institute for Research on Learning.3 This design-based unit includes a computer modeling environment called "HabiTech." LifeLines is one of several curriculum units that have been developed recently in line with the standards issued by the National Council of Teachers of Mathematics. We discuss it here as an example of a general trend in mathematics and science education. In their activity with the LifeLines unit, students use and construct representations involving tables, graphs, and equations. Students read information contained in a database. The different forms of representation in the database illustrate the different ways in which representations are useful. For example, the database includes tables of census data and text that present quantitative information about animal populations in different locales. These provide estimates of the sizes of wolf and caribou populations in different years. The tables and accompanying text are particularly useful if students want to find out the size of a population in any given year. But it is not easy for students to use the tables to see general trends in the population sizes. It is easier to judge from a graph of the data whether a population grew or declined in a regular way or stayed relatively the same over a period of years, whether the rate of increase or decrease was more or less constant over the period, whether there was a more rapid change during the early or later years, or whether there was a sudden change that occurred at one particular time. One student attempted to build an explicit link between the number of wolf packs that hunt caribou and the number of caribou that die every year. The form of this representation provided by the computer program was a network, with quantities in the model represented by boxes and with relations between the quantities shown by arrows between the boxes. (The arrows indicated that changes in one quantity depended on the values of other quantities in the model.) This network allowed the student to integrate two quantities: the number of wolf packs in Alaska and the number of caribou eaten over the course of a year. As the student put it, the model should show that, as more wolves are killed, fewer caribou will be eaten. He used this model to explore a policy recommendation by his group, advising the state to allow hunters to kill wolves in order to stabilize the caribou population. While working with the model, the student could move between modifying the functions in the network, changing the equations typed into the boxes, and evaluating the results in tabular and graphical form. For example, after running this model and watching a graph of the caribou population plummet below zero, the student completely revised the function that linked wolf and caribou population sizes and produced a model in which caribou and wolf populations both grew slowly over a five-year period. Teachers who engage their students in these project-based activities usually have groups of students present their work to the rest of the class. In some projects, presentations by students are videotaped and submitted to a panel of reviewers, as well as being seen by students in other classes who are working on projects involving the same problems.4 These presentations are a major source of information for assessment and a valuable learning activity. Teachers in the Middle-School Mathematics Through Applications Project have found it essential to have presentations midway through students' work. The presentations are reviewed by teachers and by other groups of students. The students preparing presentations learn to evaluate alternative ways of representing their ideas and findings. Those reviewing other students' presentations learn ways of judging the effectiveness of representations for communicating understanding. An important educational goal is for students to learn to use multiple forms of representation in communicating with one another. In classrooms that use the LifeLines curriculum, students use a variety of representational forms to communicate their understanding in several ways: 1) presentations using networks, tables, and graphs in complex narrative descriptions of the models they have constructed; 2) qualitative expectations about relations between quantities in their models, usually described with reference to the node/link representation of the network and the resulting graph; and 3) explicit descriptions of the relation between the model results and the group's policy recommendations. As in any classroom setting, the extent to which students give fully articulated versions of their work varies, but the practices found in these classrooms often involve extensive use of interrelated representational forms to communicate understanding. A heterogeneous mix of representational forms becomes a valuable resource for students when they communicate the sense of their work to teammates, to other students in the class, and to their teachers. An example of professional work. When students learn to use representations as tools, they are preparing for the kinds of activities that are common among mathematicians, scientists, engineers, and others who use mathematics in their professional work. Research examining these work practices has shown that people working on a problem use diagrams, equations, and other notations to keep track of partial results and to derive implications of results they have already developed. They also choose representations to communicate their conclusions effectively to other people. In one example, Rogers Hall and Reed Stevens studied the activity of civil engineers working on design projects and compared it with that of middle-schoolers working on projects in their mathematics class.5 In a case drawn from the adult workplace, two civil engineers were reviewing the design of a "collector" roadway that would cut through a hilly region in a large housing development. Their physical design documents consisted of "plan," "profile," and "section" views of the proposed development and associated roadways. These representations were printed out separately on large sheets of paper by using CAD (computer-aided design) software. Each view provided a different perspective on the design, but they all used a conventional set of measures and labels (e.g., feet of elevation above sea level, numbered surveying stations, and linear offsets from these stations). The use of standard representational forms in this roadway design provides an illustration of two points about representations as "boundary objects" -- that is, representations that can be interpreted by people from different communities in ways that allow them to share information.6 Our first point is that standard representational forms help communicate the results of analytical work to other people. This is one function of the representations that students can learn in such classroom activities as the LifeLines curriculum when they present their results to an audience. In the engineering firm studied, plan, profile, and section views were expected by clients and, in some cases (e.g., in the design of a clinic for a large health maintenance organization), were even mandated by law. The representations were constructed in working design sessions within the firm and were used in other settings, including project reviews with clients and senior management of the firm, design reviews by municipal boards, and eventual work with the developer's construction crews. For example, just before the design meetings that Hall and Stevens observed, preliminary design documents had been sent out for review by the city, a new soils report had been completed by a different group of engineers (using a plan view of the housing site with topographic contour lines), and an environmental impact report had been requested. Our second point about this example is that standard representational forms, when effectively combined, allow people to think about situations that they are not in or that may not yet exist. In a curriculum such as LifeLines, representations provide students with information about populations of animals in Alaska, and the students construct hypothetical models about those populations. In the engineering task, representations let many people (the civil engineers, their clients, municipal review boards, or construction workers) think about the complex system of roadways and residential sites being proposed. In a design meeting that Hall and Stevens analyzed closely, engineers focused on the various "paper views" of their design in progress. For example, when the grade (or slope) of a proposed collector roadway was found to exceed the city guideline maximum of 15%, the engineers began searching through a sizable collection of paper views, putting some views aside and shifting or rolling back others to construct a layered, coordinated display. With this display in hand, they could talk about the design tradeoff that was "driving" the unusually steep roadway. When the junior engineer in this pair cautioned that they would need a "brutal" amount of fill dirt under the proposed roadway, the more senior engineer (also the project manager) used a pencil to sketch a less desirable design. The alternative he sketched would save fill dirt in a relatively steep canyon, but it would rip down an entire forested hillside at the top of the proposed roadway. Given standardized forms and a shared understanding of how to assemble them to make a design proposal, these engineers were able to make present many objects and activities that could never literally have been brought into the office: the topography of a hilly development site, the existing distribution of trees and other foliage on the site, alternative roadways that construction crews would need to "snake through" the site, cubic yards of dirt they would "cut" and "fill" in the process, and the nonoverlapping concerns of both their client and a host of regulatory agencies.7 When we observe students working with LifeLines and other design-based curricula, we see that they, too, use representations to reason about situations that they are unable to experience directly. They rely on a kind of "hypothetical reality" that anchors their mathematical reasoning. As they watch a graph of increasing population, someone may say, "They're growing fast -- too fast." Or as they watch a graph decreasing to zero, someone might say, "They're dead -- goodbye." This kind of involvement with the meanings of representations is often missing when students work on traditional word problems. Constructing Understanding People use representations to aid understanding when they are reflecting on an activity or working on a problem. As individuals or groups work on problems, they may make drawings, write notes, or construct tables or equations. These representations help them keep track of ideas and inferences they have made and also serve to organize their continuing work. Sometimes people construct an analysis to be sure that a conclusion is valid or correct, and then the representation can also be used as an argument that supports the conclusion. The representational work that people do often uses nonstandard forms, which are constructed for the immediate purpose of developing their understanding. In most practices, people generate representational forms in ways that serve immediate local purposes. In addition to being representations of something, they are for something.8 This contrasts with a common practice in school, wherein students learn to construct representations of information without having a real purpose.9 Recent research on problem solving in mathematics and design provides examples of this distinction that suggest ways of rethinking how we teach mathematics in school. It might be thought that flexible construction of understanding using representations occurs only with problems that arise in such complex practices as engineering. But people working on school-like tasks also construct understanding in flexible, adaptive ways. Several findings in this line of research are important for our argument about how representational forms might be made and used in new kinds of classroom practices.
Rogers Hall conducted a study comparing the practices of students, teachers, and advanced engineering undergraduates in solving algebra "story" problems.14 The standard representational forms that students are taught to use for these problems are algebraic equations. Hall found that approximately half of the representational forms produced during the participants' solution attempts involved systems of representation other than algebraic expressions. Diverse representational forms were used by all three groups of participants in the study. Figure 1 shows two of these nonstandard forms: a drawing of successive "states" in motion and a table that organizes intermediate quantities during these states. Problem solvers also gave brief oral narratives that coordinated time and distance across related rates. For example, an algebra teacher, trying to explain an inference about times while working on a motion problem, produced the following narrative: "If you put two people on the trains and started their watches at the same time the train left, then their watches would say the same thing when you took the measurement between the trains."15
These nonstandard representational forms were frequently combined to construct models of a problem's structure. When problem solvers introduced correct inferences about the problem, they were more likely to use nonstandard forms (e.g., narratives or drawings) than their standard counterparts (e.g., algebraic expressions or formula "charts"). Nonstandard forms were also used to repair errors made while participants were using more standard forms. Hall characterized the use of such multiple representations as the construction of "material designs," arguing that their written features were important for inference and calculation and that combined representations were actively designed for specific purposes. The results of this study and others16 show that people do not use only the forms of representation that they have been taught to use, but they also construct novel forms. These constructions help them to identify or repair inferences about problem structure, as well as providing resources that help to manage calculation. We miss many of these nonstandard competencies when we restrict our attention (or that of our students) exclusively to standard representational forms. Using and Constructing Conventions of Interpretation There is a common (and unfortunate) assumption that physical notations (texts, numerals, pictures, equations, and so on) by themselves constitute representations -- that is, that information is somehow contained in the physical patterns that are written or drawn on paper or stored in computer files. This notion neglects the very basic principle, stated many decades ago by Charles Sanders Peirce, that for a notation to function as a representation, someone has to interpret it and thereby give it meaning.17 (In Peirce's view, with which we agree, there are three things involved whenever there is a representation: something that is represented, the referent; the referring expression that represents the referent; and the interpretation that links the referring expression to the referent.) According to this principle, such physical notations as tables, equations, and graphs are potential representations. They can become representations when someone gives them meaning by interpreting them. Of course, the person who produced a representation in the first place had an interpretation in mind, and therefore the notation was a representation for that person. But if it is to be a representation for other people, they have to do the interpreting. Standard forms -- such as equations, Cartesian graphs, and tables -- have widely shared conventions of interpretation, and it is important for students to learn these so that they can understand these forms when they encounter them and construct them to communicate their ideas. Standard instructional practices in mathematics provide students with opportunities to learn the conventions of interpretation of standard representational forms at an operational level. Teachers explain how to construct and interpret tables, graphs, and equations, and students are asked to construct representations of given information in these forms and to interpret representations that they are given. In these activities students can learn to follow the standard conventions of interpretation for the forms, and with this learning the forms function as representations for the students. These school practices are valuable, as far as they go, but they have an unfortunate limitation. They often treat written and drawn notations as though they are representations, rather than potential representations that depend on interpretation. School practice is now moving toward better recognition that interpretation is an essential part of representations in mathematics and science. In these practices, students not only learn to follow standard conventions of interpretation, but they also can come to understand how representations work. Understanding representations includes knowing that there can be different interpretations of the same notation. It is important to know the conventional interpretations of standard forms, but it is often productive to construct nonstandard representations with special interpretations in working on problems and communicating about ideas. Increasingly, classroom discussions are organized so that students participate in the construction and understanding of representations using mathematical and scientific notations.18 One example is the Algebra Project, led by Robert Moses, in which there is strong emphasis on involving students in processes of developing symbolic representations that express their understandings of mathematical concepts and principles in relation to their experience.19 The practices of learning in these classrooms include participation by students in discussions about conventions of interpretation used in their community. This participation establishes the role of students as contributing agents in the construction of the meanings of representations. It also can help students appreciate the fact that forms of representation are significant in the construction and communication of understanding. An example from records of teaching by Magdalene Lampert has been analyzed by Rogers Hall and Andee Rubin. Lampert's fifth-grade mathematics class spent part of a class period working in groups on the problem: "A car is traveling 40 miles per hour. How far will it go in 312 hours? How long will it take to go 70 miles?" At the end of the class period, Lampert led a discussion of the answer to the first part of the problem, and the students agreed that it should be 140 miles. Lampert then continued:
Karim said, "Well, like, every hour, you're going, every hour you're going 40 miles, and so I just added, um, I got three hours and then I added 40, three times, and then it gave me 120, then I had to, um, take a . . . I had to divide 40 in two, in half, and that gave me 20, so I added 20 'cause a half an hour is half of, like, an hour." Ellie gave her version using the same drawing. "Um, it's a good strategy because . . . it says miles per hour and [pointing to labels of "1 hr" and "40" below and above the line] one hour for each one and it's really, it's . . . when you get up to 120, which is four, which is three, fours, you, uh, put a [indicating a vertical mark between the "3 hr" mark and the mark beyond it on the line] half in there because it says three and a half hours, so you put a half right in there. And, uh, since half of 40 is 20, you add another 20 on and it's 140." By engaging students in these conversations, Lampert arranged for them to participate in the construction of representations that they used to construct and communicate the meanings of mathematical symbols and operations. They reached understandings of basic concepts and had opportunities to learn how to communicate their understandings through representations that they and their classmates had constructed. We have discussed research that emphasizes the expressive or inventive properties of representations in mathematics and science. Technical forms of representation, such as tables, graphs, and equations, are often contrasted with representations in fields such as painting, sculpture, and literature, which are adapted to particular uses of expression and communication, are flexibly constructed, and are open to multiple interpretations. But we argue that representations in mathematics and science also have these properties. We believe that educational purposes are better served if students are involved in activities in which they learn to construct versions of representations flexibly and to participate in discussions in which conventions of interpretation are developed. Such an approach enables them to understand and appreciate that mathematical and scientific representations, like those in other domains, are adapted for particular uses. Of course, there are fundamental differences in the forms and uses of representations in different domains. Numerical tables and algebraic equations are not literary texts, and Cartesian graphs are not abstract paintings. Different forms of representation are developed by different communities and focus on different aspects of experience. Forms of representation in science and mathematics are designed to represent highly selected properties of systems, often involving quantitative properties. Important differences exist between the kinds of learning afforded to students as they make and use representations in different disciplinary domains. Thus every student's educational activities should include the rich variety of experience and learning made possible through participation in multiple practices of representation. Across domains, these practices share important features of use, adaptability, and interpretation that should be included in classroom learning. 1. Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1989); and National Research Council, National Science Education Standards (Washington, D.C.: National Academy Press, 1996). JAMES G. GREENO is Margaret Jacks Professor of Education in the School of Education, Stanford University, Stanford, Calif., and a research fellow at the Institute for Research on Learning, Menlo Park, Calif. ROGERS P. HALL is an assistant professor in the Graduate School of Education, University of California, Berkeley, and a research affiliate at the Institute for Research on Learning. The research reported here was supported by grants from the National Science Foundation and the Spencer Foundation. |
|
|
