With the support of three NSF grants over eight years, the Boxer Project at U.C. Berkeley has developed and studied a very general purpose work environment, a computational medium, aimed at educational use. The system, called Boxer, incorporates facilities for text and hypertext production, for the production of interactive and dynamic graphics, for data management and networking activities, and for programming. The presumption is that computers can extend the basis for literacy in schools from written language to a more dynamic and interactive form, thus paving the way for very different and more effective modes of learning.

From this work, we have produced over 40 published papers and about 10 unpublished technical reports. Five of these papers appear in a special issue of the Journal of Mathematical Behavior devoted to Boxer. (See the second section of the bibliography for grant numbers and the list of publications, some of which are cited in this section.) Group members have given close to 100 public presentations on Boxer. Nine doctoral students and at least an equal number of undergraduates have been supported, in part, by funds from this project. Here we provide a brief summary of work completed in the two immediately prior grants, which included a small one-year "completion" supplement.

A Principled Design - We believe Boxer is the most thoroughly rationalized and documented design (from a human use point of view) of any large computer system. The primary theoretical contribution of this work has been the development of a mental models perspective on the learnability of complex computer environments. We developed a taxonomy of distinct types of models, which each have very different properties with respect to: (a) learning trajectory (e.g., some are easy to start with, but "run out of steam" for advanced users of the system); (b) differential use (e.g., some are better for construction tasks, others for comprehension or debugging). An early presentation of the theoretical framework appeared in Human-Computer Interaction (diSessa, 1985). Empirical work that largely confirmed predictions of the theory were documented in diSessa (1991a), Leonard (1991) and in two Ph.D. theses. Independent study of the learnability of Boxer (Schweiker & Muthig, 1986) showed a factor of three improvement of Boxer over the control, Logo, as measured by time to correctly code and debug.

Well-Studied Models of Learning - After creating a prototype system, we aimed to produce compelling models of new forms of instruction that are allowed by a computational medium like Boxer. Two substantial subprojects have designed a course for children as young as the sixth grade to learn about the physics of motion (diSessa, 1989; diSessa, et al., 1991; Sherin, diSessa & Hammer, 1993; diSessa & Minstrell, in press; diSessa, 1995 a and b) and a series of case studies of students learning modern biology through programming and developing computational representations (e.g., Ploger 1990 a and b; Ploger 1991 a and b; Ploger & Lay, 1992). This work generated not only instructional models that we believe can and should be emulated, but also cognitive results related to the learning of science. For example, we discovered and documented a surprising expertise in dynamic visual reasoning by children, which can be tapped in the instruction of relative motion (diSessa, 1989). Work related to meta-representational competence is reviewed in Section 3, below.

Other aspects of our work dealt with the nature of instructional activity and interaction. We developed a framework for understanding how activities work, both in their own terms ("sustaining goals"), and also to produce learning ("conceptual goals"). The framework was used to understand teacher strategies in the classroom activity (diSessa, et al., 1991; diSessa & Minstrell, in press). In diSessa (1992) we synthesized our analytic frame for understanding activities and used it and our empirical work to argue that the principles used to justify certain widely advocated instructional strategies, like cognitive apprenticeship, are not as generalizable as might be claimed.

PROPOSED WORK
1. Introduction

What do students know about the principles that make for good scientific representations? In asking this question, we are not proposing to determine what students know about the standard technical representations that are taught in mathematics and science classes, such as Cartesian graphing, tables, and algebraic notation. Instead, we are hoping to find out about student knowledge of some broader and deeper questions: How do we judge different representations-even ones we have never seen before-for their expressiveness, completeness, precision and aptness? Indeed, what are the appropriate criteria? Is expressiveness in communication to others more fundamental and important, for example, than the ability to compute with or reason with a representation? Or are these criteria redundant and tightly connected? These questions are about the matter of representation, per se, and we describe them as meta-representational.

There are a number of seemingly good reasons to believe students know very little about such questions. First, these questions seem to be deep and complicated. Furthermore, the answers to these questions are not part of the traditional subject matter of science and mathematics. As relative experts, what do we know about them? If scientists do know about these issues, it is very likely the knowledge is tacit. Mathematicians and scientists cannot name sanctioned principles, identify the history of these ideas, or point to texts on the subject. Instead, the study of representational systems themselves has been the purview of cognitive scientists and philosophers. Jerome Bruner and Nelson Goodman come to mind, and their ideas seem deep or speculative, or both. In short, this doesn't seem close to school kids' intellectual territory.

Worse, there is ample data that students seem barely to get the ideas we explicitly teach them about particular representations, much less transcendent principles. For example, there is a solid literature on graphing misconceptions (see, for example, references in the excellent review by Leinhardt, et al., 1990), which parallels the more extensive literature on misconceptions in science (Clement, 1982; Viennot, 1979; McCloskey, 1983; Eylon & Linn, 1988; Confrey, 1990). If they fail at the "concrete" level of particular representations, how could students turn out to be competent in lofty meta-representational abstractions?

Given this framing, it is surprising that recent data from several groups, our own included, have uncovered substantial meta-representational competence, even in elementary school students. This competence shows up in several contexts, including spontaneous representation to help problem solving, but, more dramatically, in contexts of design where students are explicitly given the problem of inventing a representation adequate to some class of phenomena. Students demonstrate a rich inventive capability to create cogent representations-sometimes apparently to the point of reinventing sanctioned standard representations, like graphing-and to critique and gradually improve these representations. Some of the data that supports the existence and importance of meta-representational competence will be reviewed in Section 3. First, in Section 2, we provide an overview of the research focus proposed.

2. Research Focus and Goals

The primary research focus of the proposed project is to document the breadth and limitations of student meta-representational capacities, and to investigate the role of these capacities in learning and instruction. More specifically, the goals of this research are threefold:

(1) Document student meta-representational competence. Our first goal is to investigate and document the nature of meta-representational competence (MC). (Although the origins of MC is a fascinating subject, it is beyond the scope of the current proposal.) Meta-representational competence may tentatively be defined by the following four interconnected components, which we intend to elaborate, independently and jointly.

(a) Invention. Meta-representational competence manifests itself with the ability to produce a wide range of new representational forms.

(b) Critique. MC involves the ability to critique given representations both for their cogency and effectiveness generally, but also for their adaptation to particular needs.

(c) Function. MC involves at least a tacit understanding of the many jobs representations do, and how particular representations manage to accomplish those jobs. Function is the "why" of representations.

(d) Operation. MC also involves the ability to learn to use diverse representations without extensive instruction in each case. Operation is the technical "how" to use representations.

There are two important caveats to this definition, both having to do with interpretations of the prefix "meta." First, meta sometimes connotes very general knowledge that applies to all contexts. We use meta in the more restricted sense of "about": meta-knowledge is knowledge about another knowledge domain, in this case, representation. We do not presume this knowledge is general or universal. It might be particular to both what is represented and to some selected forms of representation of that knowledge domain. Indeed, the generality of MC is one of the important sub-questions we intend to research. Part of our strategy is to investigate a spread of domains and representational forms in order to begin to map out generality and specificity. Our initial stance toward generality is cautious (diSessa, 1988).

The second caveat is that "meta" may call to mind some inappropriate associations from meta-cognition, for example, that meta-representational knowledge might be about a person's understanding of his/her own representational competence. In contrast, our initial hypothesis is that this component of MC exists, but is comparatively minimal; most MC will focus on representations, per se (although, clearly, this must be open to revision). Meta-cognition is also sometimes understood to be knowledge that stands separately from domain-specific knowledge and, as such, can be independently learned and instructed. In this proposal we jointly pursue MC and domain specific conceptual issues (discussed directly below) in order to explore this link, which we assume to be complex and variable.

In many respects, it would be appropriate to designate MC simply as representational competence. However, we use meta to emphasize that our focus is beyond the knowledge students need to operate standard, schooled representations.

(2) MC and conceptual change. The development of representational competence is, first, an important impetus for and, second, a component of conceptual change in science. In the first category, it makes sense that one learns scientific concepts in part by learning to represent them. In the latter case, one may tentatively hold that science entails, in part, a certain style of representing the world. Our research will include empirical and theoretical work on the role of MC in both these directions.

(3) Design and analysis of interventions. Finally, we intend to explore educational implications of MC in instruction. We believe that an understanding of meta-representational competence will allow us to tune the techniques that exist for teaching "standard representations" like graphing. More importantly, we also believe that this understanding will allow us to develop more substantially novel methods of teaching mathematics and science, which build on students' strengths in this area and compensate for weaknesses. This will involve designing, running and evaluating MC-based interventions in school settings. Our analyses of these will be two-fold, following the pattern of our prior work (described later). We will analyze these interventions as they reflect on MC, its instructability and impact; and we will also seek to document effective teacher strategies in promoting excellent MC-based activities.

The outcomes of our work will include:

(a) Synthetic writings about the nature, problems and opportunities associated with MC generally. MC is an unfamiliar area, and we will need to explicate it at a general level both for researchers and for interested practitioners. A chapter in a book designed to present general but state-of-the-art knowledge about instruction might be the ideal venue.

(b) Documentation for teachers that explains model MC activities, and what to look for and promote in student performance. An appropriate book chapter or an article in a journal read by teachers would be suitable.

(c) Cognitive scientific analyses of laboratory and classroom experiments dealing with the major issues, announced above. These analyses would be presented at scientific meetings and published in journals targeted at researchers.

3. Prior Work

As part of a prior project, we studied sixth grade and high school students engaged in learning to produce mathematical descriptions of physical motions, a curriculum that included such concepts as speed, acceleration, vectors, and the composition of motions. A major focus of this project was representational. This included graphing and functions represented as number lists, as well as students' spontaneous representations.

Our sixth grade and high school classes both began with a several week introduction to the computer system (Boxer), which was used extensively in these classes. Following this introduction, the first task given to students was to create simple computer programs to depict a few familiar motions. One of the motions was that of a dropped object, say, a ball dropped from your hand. Students of both age levels produced displays roughly like that shown below. This display is produced by a visible, graphical object (which looks like a ball) that moves along the path taken by the ball, leaving dots behind.

Figure 1. A student depiction of a drop.

To many, this display will probably appear wrong. A standard representation would show the object at equal time intervals, leading to increased dot spacing toward the bottom of the depiction. Indeed, since the computer takes approximately equal intervals of time between the creation of dots, the graphical object appears to slow down near the bottom of the motion, as it draws the display. Thus, one might presume that our students believed that a ball slows down as it falls. A little background and further inquiry, however, turns this vignette from a puzzling display of a misconception, into an illustration of the potential importance of the research we propose.

When students were queried about their depiction, it became clear not only that they had intended to show speeding up, but also that they felt they had achieved it. It seems "more dots" was intended to convey "more motion." When students were asked to comment on the fact that the graphical object slowed down, they indicated emphatically that they intended to show that the dropped object sped up, they had not intended to have the graphical object actually speed up as it drew the display!

There are several important points that are suggested here. First, it would be easy for an observer to misread the students' depiction simply as a conceptual error: Objects slow down when dropped. The data we intend to develop in this proposed work can help both teachers and scientists better understand student ideas by understanding the representations they make, and thus avoid such misreadings.

Second, what may appear to be a conceptual error is actually a display of meta-representational competence. It is clear that students would not have been taught this representational form. Instead, there is every reason to believe they constructed it on the basis of fairly general principles like "use more (of one quantity) to represent more (of another less visible quantity)." Indeed, our prior research and that of others has shown that close dot spacing is frequently interpreted as faster, even by non-technical university students. Furthermore, note that, although this representational form is unconventional, and possibly unscientific, there is little basis for classifying it as wrong or even ineffective. Such a judgment would be normative and depend very much on the context of use.

There is another more subtle display of competence in this little example. The fact that these students chose to represent that the object sped up and not to show speeding up suggests a surprising preference for (or at least a capability with) abstract depictions, rather than "iconic" ones. This appears the opposite of the kind of progression proposed by Piaget and Bruner, where abstract and symbolic depictions follow concrete and literal ones. Either students are more advanced than we might expect, or perhaps styles of representation are more mixed in development than some theorists may be interpreted to suggest.

The second example we discuss here occurred about 2 months later in our sixth grade class. We were about to begin a unit on graphing and decided to start by seeing what depictions of motion students could invent on their own. What followed were four intense days of collaborative design during which students developed, critiqued, and refined a wide range of representational forms. Figure 2 shows a sample. The student who proposed "Slants" (top right) explained that one could use the slant of a line to represent speed. This would free up the length of the line segment to represent another independent dimension of motion, either distance or time. The student went on to suggest connecting the line segments in sequence and then proceeded to produce a continuous graph-like depiction. Another student suggested overlaying the "graph" with a grid so one could easily read off numbers, and thus was born a rough approximation of traditional graphing. Over the final two days of the design exercise, students continued to refine their creations, and exercise each form on different motions. Eventually, the students selected a close approximation to the traditional Cartesian graphing of speed versus time as their favorite representation (diSessa, et al., 1991).

Figure 2. Representations of an object slowing to stop, then accelerating.

It would be easy to romanticize this design episode, so in our analyses we have been careful not to do so. Nonetheless, we believe it stands as a provocative piece of data that motivates not only more scientific research in the understudied area of students' meta-representational capabilities, but on instructional implications of what we are beginning to discover. Below we list a few generalizations and some details from this classroom work, and we outline some of the context it sets for proposed work. Where evident, we relate these to categories in the Section 3, on Research Focus.

1) Above all, this episode shows a rich creativity on the part of sixth grade students for representational forms [focus 1(a)]. Literally dozens of distinct forms and refinements were proposed. On the other hand, there are undoubtedly limits that have not been charted, and variation in individual capabilities is unexplored. Does representational capability follow along with general mathematical ability, or is it as much correlated with artistic or other creative capabilities? The latter might make this kind of activity especially attractive to engage different kinds of students in mathematical activities.

2) The students also showed a considerable capability to critique representational forms [focus 1(b)]. We documented over a dozen criteria students used to judge representation, including precision, spatial and symbolic economy, suitable abstractness, simplicity and completeness. Again, one wants to know: How far does this extend and what is the range of individual variation?

3) We also saw some limits of students' MC. For instance, creativity and flexibility are sometimes liabilities. In some discussions, students miscommunicated because each developed a different interpretation of what was depicted by a representation. Some students even changed interpretations, themselves, essentially mid-sentence.

4) Our students were enthusiastic about the activity, and we know they spent time out of class thinking about it. This is an extremely positive sign for general instructional application [focus 3].

5) As suggested in Figure 2, discrete or piecewise constant representations dominated at least early on; in other contexts, this has been labeled a "misconception" (McDermott, et al., 1987). Our preferred rendering is that this is more like a productive developmental stage (diSessa & Sherin, 1995). Note, for example, that discrete representations make certain attributes much easier to see. The slope of a piecewise linear presentation is clear and unambiguous compared to the known difficulties students have seeing slope in continuously changing graphs.

6) Although Figure 2 shows only surprisingly abstract representations, it took time for icons and the depiction of literal characteristics of motions to fade. We need to understand more about the meaning of this process.

7) Some historically prominent developmental patterns also appeared in our data. In particular, representing zero was a somewhat problematic accomplishment, and using absence to denote zero was a frequent detour from more adequate representations. Negative quantities also caused problems [focus 2].

8) Learning to use time as the base variable was a difficult accomplishment. The students often inadvertently replaced time by distance.

9) Transfer into competence with standard graphing tasks seemed more delicate than one might expect. However, our data and calibration need significantly more work to draw firm conclusions.

In addition to the above, our published studies looked at teacher strategies [focus 3] and some specific conceptual development [focus 2] (diSessa & Minstrell, in press).

4. Related Literature

Accounts of children's and students' meta-representational competence, especially spontaneously developed scientific representations, are quite rare in the literature. One notable recent exception is some work by Rogers Hall on the form and function of spontaneously produced representations in mathematical problem solving (Hall, 1989). Another notable exception is some recent work by Ricardo Nemirovsky and colleagues at TERC (e.g., Tierney & Nemirovsky, 1995). The core commonalty between their and our prior work is the documentation of early meta-representational competence, including the surprisingly early appearance of spontaneous representations similar to standard scientific ones.

We identify six other niches in the literature and calibrate, very briefly, their relevance.

1) General Cognition - Representation, of course, is a fundamental concern of cognitive science and psychology. Much of this is on the general nature and function of representations, and as such, is not strongly connected to the empirical and educational approach we propose here. Nelson Goodman's work on classifying representational forms may be prototypical (Goodman, 1968). Closer, especially because of its connection to instruction, is work by Jim Kaput (Kaput, 1987) and some others (Janvier, 1987) on the general nature of representation in mathematical cognition. Yet this work is very sparsely concerned with meta-representational skills, and does not discuss the use of, for example, representational design in instruction at all.

2) Technical Representation Design - There is a small and diverse, but interesting, literature on the design of technical representations. Tufte (1983) might be the classic exemplar. This literature deals generally with highly refined work of professionals, has no interest in the spontaneous development of these skills, and does not relate this to improving mathematics and science education.

3) Sociology of Science - Inscriptions and representations are a highly visible part of professional science and have not escaped the attention of those who study scientific practice (e.g., Latour, 1990). Again, this is likely to be peripheral to our main interests, except possibly in suggesting some interesting "novice/expert" comparisons.

4) Developmental Literature - Piaget and some others treated the development of representational thinking as an important general intellectual development. Relatively little of this work had to do with external representations of a scientific or quasi-scientific nature. Karmiloff-Smith came closer with her work on children's spontaneous depictions of "geography" for the purpose of navigation (Karmiloff-Smith, 1979). Jeanne Bamberger's (1991) work on spontaneous representations of music is close in spirit, but somewhat distanced in the target of representation. Sid Strauss has produced a study on very early (pre-instructional) competence in graphing, although invention and design are not probed (Strauss & Schneider, in press).

5) Schooled Representations - There is a large literature on learning schooled representations, especially graphing (e.g., Leinhardt, et al., 1990; Schoenfeld, Smith, & Arcavi, 1991; Confrey, 1990). A good proportion of this catalogs difficulties and misconceptions (e.g., Trowbridge & McDermott, 1987; McDermott, et al., 1987). This literature is important in that, in part, we share the same educational targets; improving school learning of representations is the practical goal for which we aim. On the other hand, our distinguishing characteristic of looking at meta-representational skills and especially student invention of representations is essentially absent from this literature.

6) Socio-cultural Approaches to Cognition - Vygotsky (1978), among others, emphasized representational systems, especially language, as a culturally created substrate of cognition. There are certainly interesting questions about how MC reflects on or plays into this story. We do not propose to make this a major focus or our work.

5. Focus and Goals, Elaborated

Given that this topic is relatively new and unresearched, we believe it is appropriate to cast a relatively wide net. In this section, we elaborate questions and hypotheses to be investigated, using the analytic framework introduced in Section 2. On some of these questions, we can guarantee results. For example, we will catalog, analyze and display a large number of student generated representations arising in contrasting contexts, and we will describe the contextual dynamic of their generation. We will also see how well and how quickly students learn to use new representations. On the other hand, because it requires theoretical development that will occur during the project, we cannot guarantee perspicuous, theoretically motivated and general descriptions of the range and limits of student MC. Similarly, development of an understanding of the relations between representation and conceptualization depends on theoretical progress driven by the empirical work proposed here.

Mapping MC. A central result of this work will be detailed, richly empirically and theoretically defended knowledge analyses of MC. This will entail description of the content, form, level of generality/specificity, limitations (e.g., instability), and so on, of MC, similar to (but with a different focus than) many conceptual change analyses of naive knowledge (e.g., Viennot, 1979; Clement, 1982; diSessa, 1993). Two orientations from our prior work particularly characterize what we hope to do here. First, many characterizations of naive knowledge do so exclusively as deficits or as opposed to expert knowledge. While comparisons to experts are useful, an exclusive focus on misconceptions and deficits will miss many productive aspects of naive knowledge and may not attend systematically to the nature of naive knowledge in its own right (Smith, diSessa & Roschelle, 1993). Our primary focus will be on the nature of naive knowledge, and not on deficits relative to a supposed standard. The examples in Section 3 on prior work and the form of proposed interventions, particularly design problems, ought to make this orientation clear.

Second, we, as others (e.g., Strike & Posner, 1992), have cultivated a view of knowledge as a complex, diverse and constantly evolving system. This means that we are unsatisfied with simple descriptions of knowledge as independent units and in commonsense terms-"ideas," "facts," "beliefs." In past work, this has been the core of our theoretical development, and we hope to maintain a similarly innovative theoretical stance in this proposed work. diSessa (1993) provides examples of appropriate theory building and empirical results illustrative of "richly theoretically and empirically defended knowledge analysis." See also diSessa (in press).

In mapping MC, we hope to answer several very general questions: How general or specific is the knowledge that constitutes MC, as judged by comparing different situations of elicitation? How stable is this knowledge; for example, do students lose track of their insights and have difficulty explaining them? Can we roughly describe the variation we find in the populations investigated? Does this correlate with other important capabilities, like "competence in mathematics"?

In addition, we intend to address issues that are specific to each of the subcomponents of MC:

Invention: Can we describe the knowledge and reasoning strategies used by students in developing new representations? These may range from general and heuristic (use more of X to represent more of Y) to quite specific (slope, length, and number can represent numerical quantities).

Critique: Here, we will start with and extend the list of criteria employed by students in our prior work. Furthermore, we want to know: Do criteria like "simple" and "understandable" mean the same thing to students from one context to another? How idiosyncratic or consensual are such judgments? When queried, how do students defend their judgments about better or worse representations? Are they capable of rational debate about these criteria? To what extent do students understand that any representation highlights some features and suppresses others?

Function: To what extent, and how articulately do students understand the various uses of representations? Some uses of representations that students could be aware of are: storing information for readout, providing a locus and support for computation, and making patterns and trends visible.

Operation: Can we identify any general knowledge and skills for tuning one's capability for using a representation quickly and effectively? How well and in what circumstances can students coordinate different representations to compensate for weaknesses in one representation?

Representations and Conceptual Change. How does weakness in conceptualization translate into weakness in representation? How, when and to what extent can representation be used as a lever to build conceptualization? For example, can negative numbers be motivated and explained as a "notational convenience"? Can some conceptual difficulties arise from differences between students' spontaneous representations and those used by teachers? To what extent is representation an intrinsic part of conceptualization, and to what extent is it separable? Many of the questions in this category will require theoretical innovation. A model being developed by Bruce Sherin (proposed post-doc for this work) explains how some knowledge elements may be inextricably both representational and conceptual (Sherin, 1994).

Design and Analysis of Interventions. We will provide motivation for, and analysis of the results of, classroom interventions in the categories of (1) conceptual content (mathematics of motion, etc.; see below) and (2) representational form (e.g., graphs, phase space, vector fields; again, see below). We will provide guidelines on what teacher strategies provide the best results-for example, the importance of naming and exercising students' own representations-and what are achievable results (for example, can one expect an episode like "inventing graphing," described briefly above, to transfer to good performance on standardized graphing tasks?). In what ways are students better prepared for standard instruction, having designed their own representations? How helpful is explicit instruction in representational criteria?

6. Experimental Design

Practical considerations prevent us from working with students of all age levels, while they employ all types of representations and for all purposes. We have therefore chosen a do-able slice that we hope will provide reasonable breadth for contrast, while allowing sufficient depth for scientific accountability. First, we will concentrate on two age ranges: a "pre-graphing, pre-algebra" range-sixth and seventh graders-and a "post-graphing, post-algebra" range-sophomores and juniors in high school. In addition, three primary dimensions cut across our research program.

Range of representational forms and focus conceptual domains

We will engage students in the design and use of a range of representational forms. On this dimension of our study, balancing breadth and depth is particularly critical. We must go deeply into some particular domains in order to discover the details of MC within a given domain, but it is also crucial that we deal with issues of generality-we would like to discover the degree to which MC spans domains and varieties of representation. To this end, we propose to focus on two main areas of application:

(1) Time-parameterized, scalar data. Our earlier graphing and motion research falls mainly in this area. In that work, students represented one dimensional motions of individual objects, producing such representations as velocity versus time graphs. The work strongly suggests that this will provide a rich area of study, and there are many open questions. Students were able to produce a wide-range of interesting alternative representations, and there were interesting short-term developmental phenomena to observe.

A focus on time-sequenced, scalar data will allow our work to make significant contact with much of the recent research concerning "the mathematics of change." Researchers such as Nemirovsky (1994), Rubin & Nemirovsky (1991), Thompson (1994) and Kaput (1994) have been working to describe how students learn to give accounts of changing, scalar quantities. In addition, these researchers have been developing instructional techniques in this area, many of them involving (standard) representations. Thus, this is a good area, well grounded in prior work in which to study the joint development of conceptual and representational competence.

(2) Space-time distributions. Our second major area of focus will be the representation of data associated with points in space. An example is the representation of temperatures in a given geographic region. Recent research in "scientific visualization" has often focused on representations of this sort (Gordin, Polman, & Pea, 1994). In addition to weather data, areas of interest here include astronomical and geographic data. Research on conceptual development related to these areas is less well-developed than the mathematics of change. However, this is an active, current area.

If time permits, we would like to do formative studies of MC in some very different areas-for example, algebraic notation and simple musical pitch and duration patterns.

Functional context of the tasks

In order to research a reasonable range of MC, we will need to engage students in a variety of types of tasks. Roughly, we classify these tasks according to how the function of the representation is conceived of by students. There is a broad range of possibilities here, but we will start by considering two prototypical functions. First, in some of our tasks, students will be asked to design general purpose representations for use by a broad community. The graphing activities in our earlier work were of this sort, and we will call them general representational. We believe that these tasks draw on some particularly interesting aspects of MC, including the ability to critique the communicative properties of representational forms.

The second class of tasks will involve a more instrumental function, and we call them locally adapted. In these tasks, students will need to produce representations for some immediate purpose, such as to aid in the solution of a specific problem or to facilitate the understanding of a particular phenomenon. Unlike the above tasks, the focus here is on building a representation to negotiate some other need, not to design a representation that is generally useful and that can be understood and applied by other people. Thus, these tasks may well draw on substantially different aspects of MC, including the ability to invent representations to support problem-solving inferences.

Underlying media and the use of computer technology

Potentially, students could employ a variety of underlying media for their representations. We expect most of our work to be pencil-and-paper based, but some of it will be computer based for three reasons. First, new dynamic and interactive representational forms are, with computers, becoming commonplace in mathematical and scientific work. Second, we do not have any strong reason to expect that spontaneous and easily instructable MC should be restricted to hand-drawn representations. (If it turns out to be so, this will be a stunning and provocative result.) Third, using some representations-even some very simple ones-with realistic data sets (e.g., weather information) is burdensome without computers, sometimes to the point of being, for all practical purposes, impossible.

We will use the Boxer programming environment (diSessa, Abelson & Ploger, 1991), which is well-adapted to implementing new representations and has been used extensively for these purposes. For example, one of our sixth grade students implemented automatic generation of several forms depicted in Figure 2 as an independent project. In addition, Boxer is a computer requirement in our collaborating high school, so students will automatically have requisite competence with it.

Very crudely, one can conceive of our experimental plan as involving 16 cells in a 2 (pre- and post-algebra/graphing students) X 2 (different representational areas) X 2 (functional contexts) X 2 (media types). In practice, we will use this more as a heuristic guide to maintain diversity and the possibility of learning from contrast. There may be substantial collapsing of cells. For example, in most instances we do not expect age difference to play a substantial role, a conclusion suggested by our prior work on inventing graphing with sixth graders and high school students.

7. Example Tasks

In this section we present, in brief, a sample of the tasks around which we propose to base our exploration and instruction of MC. We believe that all of these tasks will be appropriate for both age ranges, though modifications may be required in moving between age ranges.

"Inventing Graphing"

Representational focus: Time-parameterized, scalar data.

Functional context: General representational.

Media: Paper and pencil.

In this task, which is closely modeled on our prior work, we will ask students to invent a representation of motion that is as clear, simple, and general as possible. As in our prior work, the Inventing Graphing activities will be loosely structured around a series of "motion stories." For example, our original study began with the following story:

A motorist is speeding across the desert, and he's very thirsty. When he sees a cactus, he stops short to get a drink from it. Then he gets back in his car and drives slowly away.

Work in the Inventing Graphing task proceeds through a series of such motion stories. The stories are used first as a basis for inventing the representations, then to select among alternative representations, and finally as an arena for perfecting and mastering the invented representations.

In our prior work, this task was structured as a group design activity that spanned approximately five class sessions. In the proposed study we expect that the task will keep a similar form, though it will be necessary to make modifications to meet the demands of the particular classroom contexts in which our study is based.

"The Chaotic Water Wheel"

Representational focus: Time-parameterized, scalar data.

Functional context: Locally adapted.

Media: Paper and pencil; computer.

Following Nemirovsky (1993) we will inquire about how students make sense of a "chaotic water wheel." A version of a "Lorenzian Water Wheel" has been developed by researchers at TERC, specifically to allow students to study a system that exhibits chaotic behavior. To begin, we will encourage students to make their own representations to explore its behavior. Later, we will suggest graphing and introduce phase space representations, if students do not spontaneously pursue these lines.

In the project described by Nemirovsky, students worked in pairs in a clinical setting. We will employ similar exploratory activities in our laboratory, but we also plan to adapt this task for use in a classroom setting.

"Image Processing in Astronomy"

Representational focus: Space-time distribution.

Functional context: General representational; locally adapted.

Media: Computer.

In this task, students will develop representations suited to a particular variety of data used by astronomers. We will present students with a two-dimensional array of data representing the amount of light falling on a CCD (charge coupled device) in a telescope and ask them how astronomers should present such data. A realistic proposal-and one that is employed by astronomers-is to use a picture, where each number maps to a brightness. However there are many interesting and useful alternatives: (1) Use false colors to emphasize interesting features; (2) just present the numbers; (3) use the numbers to specify "height" on a 3-D contour presentation; (4) use lines to show constant amplitude contours; (5) process the data in some way before presenting it, for example, compute differences with neighboring picture elements (gradients). If students are unable to invent a sufficient diversity of representational forms on their own, we will suggest some of these possibilities. This will allow us to see how quickly students can grasp these alternatives and how sensitive students are to the strengths and weaknesses of these representations. Later in the exploration (the locally adapted component) we will ask students to use their representations in various tasks, such as determining the diameter of Jupiter from a data set including it. We know this is an interesting and representationally rich problem; one of our group members, Jeff Friedman, has baseline data on students performing the task without meta-representational preparation or coaching.

"Mapping the Wind"

Representational focus: Space-time distribution.

Functional context: General representational.

Media: Paper and pencil.

In this task, students are presented with the following scenario: A group of meteorological scientists wants to study wind. What pictures might they draw to help themselves? A standard representation that students could choose to employ is to display wind as a vector field, using arrows on a grid. Other relevant representations would be of related causal factors, such as air pressure. As in our other tasks, we will begin by allowing students relatively free rein to invent their own representations, and then we will suggest some of our own alternatives, if necessary.

8. Empirical Methods

In this section, we provide a sketch of the methods of data collection and analysis involved in this work. Broadly, we intend to move from a more controlled, laboratory elicitation and description of MC (accompanied by theory development), to the design and testing of classroom interventions. This will require three different classes of methods.

(1) Fine-grained analysis of laboratory interviews: The groundwork for our account of MC will be laid in laboratory interviews. Here we discuss briefly the data collection and analysis methods proposed.

Data Collection: (See also Section 10 for some practical details.) Two related traditions motivate the type of data collection proposed. First, clinical interviewing, made famous by Jean Piaget in his studies of children, will define the stance and role of the experimenter in our laboratory sessions. In clinical interviewing (Piaget, 1926; Posner & Gertzog, 1982) the experimenter tries to set up a situation or task that will elicit relevant knowledge of the interviewees. Then the interviewer poses questions of clarification and follow-up to help get the interviewees to reveal how they think about the situation. A notable feature of clinical technique is ceding initiative, as much as possible, to interviewees, letting them pose terms of description and paths of inquiry. In the proposed work, we hope to set up situations where students can work on problems substantially on their own resources. So the "interviewer's" role will be more to manage some aspects of the inquiry (such as suggesting reflection or posing counter-hypotheses), asking for clarifications and justifications, and possibly hinting at alternate possibilities and formulations (e.g., suggesting another representational form, if need be). This is in contrast to ascertaining knowledge by direct questioning.

The second related tradition of data collection relevant to proposed work is the teaching/learning experiment (Steffe, 1991) or "model elicitation activity" (Lesh, Hole, & Post, 1995). The most important feature we share with these traditions is the belief that the character of competence of individuals frequently can be revealed only by time-extended activities (typically a minimum of one hour) that are fostered in a supportive environment, including a careful but helpful experimenter. Learning and display of competence may require iterated trials and refinements on the part of the subject (in contrast to almost all traditional school testing of competence).

Analysis: Because all of our sessions will be videotaped, we can employ a version of microgenetic analysis, wherein interviewees' performance can be examined over and over and in great detail for hints and corroborating evidence concerning interviewees' competence. For example, in retrospect it can sometimes be determined that an interviewer accidentally provoked a line of thinking without knowing it, or a subtle shift of attention of an interviewee may suggest a reconceptualization. Interviews will be transcribed. We use electronic indexing (C-Video by Envisionology) to catalog and gradually build a database of on-line interpretations.

Because we have a focused set of initial questions, we have reasonable expectations about where relevant data will arise and what form it will take. For example, whenever students propose a new representation we can analyze the representational resources involved (e.g., what representational form is used, such as slope, to represent what perceived-as-relevant quality or quantity, such as speed). Similarly, whenever a critique is made, we need to characterize the basis of that critique, as the student perceives it. To be sure, we are less sure where and how data will arise for other questions, such as how students understand the functions of representations.

Our set of questions and theoretical orientation also make certain aspects of students' performance less relevant. We are not interested in everything that goes on in these sessions; we are not interested in collaborative or interpersonal strategies, general problem solving strategies, inquiry management skills or even knowledge of specific schooled representations, except as these bear directly on the issue of MC or the interpretations of the experimental sessions relevant to MC.

Knowledge analysis is not nearly as codified in the literature as other methodologies. A fair proportion of what has been written is unfortunately less than ideally matched to our purposes. For example, a landmark work on protocol analysis, Ericsson and Simon (1984), explicitly excludes clinical interviewing from its purview. Similarly, knowledge analysis for the purposes of producing expert systems has aimed at robust, procedural knowledge rather than possibly unstable conceptual knowledge of non-experts. While some recent expositions can be helpful (e.g., Chi, in press) our work will also entail an obligation to elaborate its analysis procedures, as well as its results. Here, we display four central heuristic principles (adapted from a more detailed and extensive list in diSessa, 1993) that help define the approach to be used.

Principle of Invariance: If one gets the description of a knowledge entity right, it will apply in all implicated contexts. So if a knowledge entity appears to be used in a situation where it is not evidently applicable for us as theorists, redescription is in order. Similarly, if an element is not used in a situation where it "should be," problems are suggested. Strong evidence is provided if one can invent a situation in which a predicted application of conjectured knowledge leads to surprising behavior. For example, we hope to predict constructions similar to the falling ball, Figure 1.

Principle of Functionality: Knowledge attribution to a subject is more plausible if we can discover how that knowledge might be useful in the subject's everyday experience.

Principle of Discrepancy: When people give unusual interpretations or use very non-standard representations, distinctive knowledge is implicated. This is the principle of misconceptions research. However, it does not override the principle of functionality.

Principle of Redescription: Basic descriptive terms are fundamental. Commonsense vocabulary and intuitively ready characterizations seldom suffice. Instead constant revision and tuning of descriptions and competitive argumentation against alternatives is important.

On the basis of well-rationalized knowledge analysis, it should be relatively straightforward to develop questions and response categories, for example, to measure some components of MC (Hunt & Minstrell, 1994).

(2) Interpretive analysis of classroom sessions. In addition to the above laboratory work, our trials will include in-school work in experimental but overtly instructional formats. We will begin by designing and testing preliminary versions of our classroom interventions, including variations on the tasks in Section 7. These classroom sessions will be videotaped and the videotapes will be analyzed in part using results and procedures of the fine-grained laboratory studies. These sessions and their analysis will undoubtedly help refine laboratory results, but they will also have a different audience and goal. By explaining workable classroom meta-representational activities, how to run them and what results may be achieved, we hope to help teachers understand and engage in an exciting new style of learning experience. For these purposes, analyses and write-up will be less formal than for the lab studies. (In prior work, we have published several such interpretive analyses of classroom sessions. See, for example, diSessa, et al., 1991; diSessa & Minstrell, in press; and Sherin et al., 1993).

One of the sites for these classroom studies will be a high school in San Francisco, California-the Urban School-with whom we have a long-term, open and productive relationship. We also plan to solicit subjects and give courses in an on-going summer program of instruction run by the U.C. School of Education. The program involves students from both proposed age groups. This will provide an opportunity for us to design short courses explicitly around the topic of representation and representational innovation.

(3) Formal assessment of designed interventions. In addition to the interpretive analysis of classroom sessions, we propose to perform more formal assessments of the interventions we design. These will take the form of written exams given to all participating students. We will be especially interested in determining if our interventions help to alleviate some of the known difficulties that students have with particular representations, such as specific graphing-related misconceptions. Work by one member of our group on image processing in astronomy (see Section 7) will provide calibration in the case of space-time distributed representations. In addition, since one of our main interests is in motivating students who tend to find science and mathematics uninteresting, we will use questionnaires and observations of participation to take a reading on how engaging students find our tasks.

9. Educational Relevance and Relation to RTL Priorities

The study of a form of spontaneous competence that, to this point, has been under-appreciated will clearly contribute both to cognitive science and to developmental research. However, the educational and practical relevance of this program may be less obvious. In this section, we will argue that the proposed work relates strongly to RTL's matrix of priorities. This is especially true of the student row of the matrix, with which we begin this discussion.

Preparation for Success in Post-Industrial Societies. Two goals stand out in revising instruction to prepare students for the rapidly changing present and future: flexibility and the capability to deal with complex systems. With regard to flexibility, students entering the modern work force will need to be prepared to adapt to changing requirements throughout their careers. We can thus no longer count on a small list of standard representational forms to assure coverage of new problems, new contexts and new ways of working. MC, by definition, includes the competence to develop and use new representations adapted to new tasks. Our proposed work will help define a more open, forward-looking mathematics and science curriculum in this respect. In addition, MC is especially critical in coping with complex, ill-understood systems. When closed-form solution fails, the perspicuous use of representation so as to reveal underlying patterns becomes progressively more important. In general, when the concepts necessary to understand a system or situation are not given in advance, representational explorations assume a greater importance.

Equal Access to Powerful Ideas. We maintain that some of the most powerful ideas in mathematics and science are representational-the ability to select, develop, or apply the appropriate representation may be the key to completing a task. In this regard, our research will help define MC as a goal for instruction, and we will develop additional data in support of its importance during the project. Furthermore, the educational activities we will investigate have several properties that can help ensure access by diverse populations, especially those under-represented in science. (1) Our prior work showed that meta-representational activities can be engaged in straight-away, without long lists of technical prerequisites. (2) Representational design activities involve skills and sensibilities (e.g., artistic presentation, creativity) not typical of standard mathematics and science instruction. Hence, they will help engage students with different profiles. (3) Activities structured as group design tasks, in particular, have niches for student participation (critical audience, inventor, articulate advocate, artist, conciliator) that can help insure a creative role for all students. (4) The activities we propose constitute accessible elements of authentic mathematical and scientific practice, rather than rehearsal or reproduction. Such activities are critical, but often relegated to "enrichment" and withheld from many students.

Standards and Assessments. MC may constitute an important new goal for instruction and a student capability that can be assessed. In addition, developing and using new representations is an excellent area in which to display multi-faceted, creative competence, for example, in a portfolio.

Influence of Technologies. Computers are the most protean representational medium ever created. New standard and non-standard representations, from spreadsheets to tools for exploratory statistical analysis and data visualization, appear almost daily. In addition to learning new forms quickly, many workers, especially (but not exclusively) in scientific and technical careers will have responsibility to develop new ones. Part of our work, notably that dealing with areas like spatially distributed data where paper-and-pencil presentation is impossible, will involve students in creating computer representations.

Connections between our work and the teacher row of the RTL matrix are also strong. We will provide information that can help teachers diagnose and address student difficulties. For example, as we suggested in our first example of a falling object, an understanding of students' spontaneous representational forms could help teachers interpret student ideas. As we move toward more open, student-centered classrooms, it will be of increasing importance to provide teachers with information of this sort. In addition, in diSessa, et al. (1991), we conjecture that students that have difficulty learning standard representations may need more coaching at higher levels of purpose and function, and less practice of low-level representational operation skills. Such a claim, if borne out, has serious implications for the manner in which teachers must address student difficulties.

Our work will provide an augmented research base for such curricular ideas as "progressive formalization," where students are brought slowly through intuitive and self-generated descriptions and representations of problems toward more standardized and formal versions of these representations and descriptions.

Learning activities are the life-blood of instruction. New priorities for extended, authentic and creative mathematical and scientific activities (e.g., in the NCTM Standards and the California State Frameworks) mean teachers need new models and ideas. Designing representations may be an important class of such activities. This is especially so if the level of enthusiasm shown by our sixth graders (and, to a lesser extent, by high school students) holds broadly. The parallel between this project and the well-developed work on invented algorithms (Kamii, 1985; Lampert, 1986) may be a powerful one. Note that developing such activities and describing teacher strategies for conducting them constitute one of three top-level goals of the proposed work. In addition, we will be opening up new areas of computer-based instruction, which can be helpful in extending teachers' repertoire. It may be that a progressive formalization approach to computer tools and their representations-where students first explore their own possibilities-makes for excellent instruction.

It is not incidental that the proposed work connects strongly to current and past RTL supported projects and closely related work. For example, it meshes well with work on the mathematics of change (Nemirovsky, Kaput and Roschelle, Thompson), with work on modeling and model-eliciting instruction and assessment (Hestenes, Lesh), and work on learning standard representations like algebra and graphing (Thornton, Clement, Confrey, Schoenfeld). On the other hand, our emphasis is distinctive, novel and independently important.

In closing this discussion of relevance and importance, we repeat the observation that the development of new representations is an important part of the practice of mathematicians and scientists. Journals and textbooks are full of variations on standard representations and even many new ones crafted for the needs of the moment. Thus, we maintain that meta-representational skills are legitimate and important targets for instruction, just as the teaching of standard representations, doing experiments and solving problems are important. If early work done in this area holds and it turns out students possess significant resources that can be capitalized on, the neglect of this area in current curricula will be particularly poignant.

10. Project Organization and Timeline

In year one, we will begin formative studies, especially in laboratory contexts. With our cooperating teachers, we will also begin to explore the mathematical and scientific contexts in which designing representations makes sense, either as full-class designs and inquiries or as individual or small group independent activities. In this regard, we will explore collaboration with Rogers Hall in his recently RTL funded work on problem solving in real contexts.

In year one we will also design and implement a computer tool for space-time distributed representations (first six months), then conduct formative trials and revision (second six months). (See also the description under programmer's duties, below.)

Because of the relative newness of this topic of research, we want also to engage in two other activities during year one. First, we would like to study in more detail and synthesize some of the broad and disparate literature tangentially related to our concerns such as some listed above. A graduate reading group would be an appropriate forum. Second, we would like to invite an advisory board of scientists to participate in the planning of the project. This would involve at least one two-day meeting.

Year two will involve a full deployment of both laboratory and classroom studies. We hope to have developed a systematic regimen of data cataloging, analysis and summarization that will make new studies easier to run and more cumulative. We expect to run about 4-6 studies during the academic year and 2 more during the summer.

Year three will involve: completion of the scope of planned experiments; replication with variation, if warranted (for a total of about one-half the number of experiments of year two); and intensive synthesis and writing.