Jeremy Roschelle
University of California, Berkeley
These curricular, economic, and coordination difficulties have interacted to maintain the status quo for the great majority of schools, students, and teachers, despite a consensus that deep reform is urgently needed and will require a substantial cadre of energetic reformers. We will argue below that a deliberately orchestrated combination of the component and connectivity revolutions will enable many of the needed changes to occur.
We will now outline the kinds of changes needed in curriculum (including classroom practice), in the economics of technology-related materials, and in coordination of technologists and educators.
This paper is organized as follows. First we will offer a scenario involving serious mathematical activity and learning that embodies many of the features advocated in the outline above--a scenario that is not possible today, despite its surface plausibility. Second, we will examine this scenario from several perspectives to determine what needs to be done to render it not only possible, but likely. Third, based on recent experience with the ongoing SimCalc Project, we will suggest some policies and specific actions, relating to the development of standards for easily manipulable and interchangeable components in mathematics and science education, that will enable the necessary changes to occur.
The paper focuses on the content/curriculum side of reform more than on the teacher education/support side, partly due to limitations of space and time and partly due to a conviction that although much reform discussion gives content and curriculum a secondary or superficial role, these aspects of reform are utterly essential to all others. Approximately 4 million students at each grade-level cohort experience the core mathematics and science curriculum; to the extent that it remains the same, all the technological and other innovations advocated will continue to be marginalized--because the core is already filled.
The challenges and the changes identified above are not equally susceptible to impact by the connectivity and component revolutions. Nonetheless, we will attempt to identify how each is affected by the changes we foresee and advocate.
Parade is a team game played within a vehicle-simulation system. In the simulation, multiple vehicles follow behind a lead vehicle in a parade configuration. The motion of the lead vehicle can be generated or accessed either locally or elsewhere on the World Wide Web. The driver of each vehicle has access only to limited information about the vehicle immediately in front of it. The specific type of information available determines the kind and level of kinematics and calculus knowledge necessary to play. Below we illustrate some Parade activities that involve heavy use of several software components that readily exist in some form in 1996--indeed, each has been produced at least once by NSF-funded projects--a car simulation, real-time data acquisition, video-measurement, mouse-defined motion, and an algebraic/graphical function manipulator, as well as e-mail and browsable network documents. In this scenario, unlike with today's software, components coexist in a multimedia student notebook, and pages of the notebook can be shared easily.
Soon the eighth graders will be sent a challenge trip from another team from California that determines the motion of the lead parade car that they will try to follow in their own simulated cars.
Each car behind the lead car has two graphs on its dashboard: its own velocity-versus-time graph that is generated as it moves along, and the position-versus-time graph of the car immediately in front. Hence, the act of staying in the parade requires the driver to coordinate velocity and position data in real time. (For example, if the position graph of the car in front of you is getting steeper, you must immediately speed up.) The team from California has successfully completed the trip with a parade of four cars behind the lead car using position-velocity information (they had previously done it with position-position information, which is much easier). The Boston students are rehearsing with their own body movements to get a feel for the activity before ultimately trying for a parade in the simulation with five cars.
Another student, upon seeing her classmate's irregular velocity graph on the computer screen, decides to try to outdo him with her own wild function--but hers is algebraically defined using a mix of trig functions and polynomials. It's her algebra against his basketball moves! Which can be trickier? They test it out by having another student try to follow each of their graphs in the car simulation. Her algebra wins.
The question arises: why does this happen? Does it have something to do with reaction times, or is there something else involved? The students decide on a simplified experiment involving the Mouse-Based Lab (MBL). Now the lead student drags a car back and forth on the screen to create a fairly simple position-versus-time graph with a wiggly, up-and-down progression. Then the second student attempts to match that position graph by dragging the car on the screen with real-time feedback on only the velocity of the dragged car. The third student does likewise, but based on the position graph of the second student's car. The fourth student does likewise, but based on the position graph of the third student's car, and so on. Five students do this in succession, then they plot all the velocity graphs on the same axes. Lo and behold, besides being delayed in time, as was expected, the velocity swings get either bigger or smaller. There seems to be some kind of "exaggeration effect" going on: whatever trend started in the second student seems to be exaggerated in subsequent students. They record and describe this in their notebooks.
Next they decide to ask others across the country whom they know from previous interchanges to try the same experiment. Accordingly, they create another wiggly position graph by deforming a sine function and, using voice annotation, describe the experiment they want others to try. They then send their notebook to their colleagues with a request to send back the series of their velocity graphs in their experiment reports. They also post their experiment on their WWW server in the expectation of getting more data.
The students notice that a slight slowdown in the lead car can result in the more distant successive cars stopping. Their teacher says that they have "reinvented the traffic jam." When they post their result on the network, a group from the MIT Media Lab offers a new way to model such phenomena that does not even require a lead car (Resnick 1994). The new model even applies to geese flying in formation, where there is no lead goose but only individual geese, each following simple energy conservation rules. Indeed, the new way of modeling applies to a whole range of situations in biology and even sociology. Students in schools affiliated with this research center are anxious to share their insights and models with inquiring students involved in the Parade investigations.
A group of curriculum authors notices these Parade phenomena and decides to organize a middle school math curriculum that will study simple difference equations. Their "Finite Differences Workbook" is published to the Math Resource Server maintained by the National Council of Teachers of Mathematics (NCTM) as a linked set of network documents. (Users' schools pay royalties based on a prior registration with the NCTM.) Importantly, while the Workbook uses computer-based tables, graphs, and a simulation, it is entirely generic, not requiring any particular piece of software. The Workbook introductory activities use microcomputer-based labs, but students who do not have motion probes can employ an alternate introduction that uses mouse-based motions to generate data. In either case, students sample and compare data to graphs generated with their own favorite analysis tools. Students use their favorite word processors and graphics programs in their notebook and view graphs using a variety of plug-and-play simulations.
When the assessment folks see what is going on, they decide to collect student notebooks as a form of structured portfolio assessment. They prepare a variety of "tests" using different Parade conditions, with a set of questions geared to each condition, and publish these on the NCTM/ETS Assessment Resource server. Students can mix and match whichever representations they need to answer the questions and turn in a complete Portfolio showing their work. Of course, the Finite Differences Workbook is linked to the Assessment Resource server. Its formal reviews are quite positive, and users' comments concur, so the workbook becomes quite popular in the next 12 months.
It is increasingly apparent that the mathematics and science of the 21st century will differ more from today's mathematics and science than today's differs from that of the 18th and 19th centuries (Casti 1992; Kauffman 1993; Steen 1988), especially since mathematics and science now grow in a new, dynamic interactive medium and are becoming ever more intimately tied to economic and social destiny. As currently constituted in curricula (mostly at the university level), this newly emerging mathematics and the associated science appear at the end of a long series of algebraically oriented prerequisites that the great majority of students do not complete. This mathematics is available in formal language to those who master the formalisms (Kaput 1994). But in the United States, we have done a relatively poor job of making those formalisms available to the great majority of students: algebra is introduced late, abruptly, and in decontextualized form (Kaput 1995a, 1995b). The net result is that relatively few students reach the branches of mathematics and science that currently depend on facility with algebra, and those who do, often do not learn the subsequent material especially well.
Looking at the mathematics curriculum at the high school level, we see very little room for change except at the margins. The Algebra I, Geometry, Algebra II, Precalculus sequence covers virtually all the available ground for college-bound students, and a somewhat weakened version controls the mathematical worlds for the non-college bound. This organization is deeply entrenched, not only in the curriculum, but in teacher education and assessment systems, and especially as part of the relationship between the secondary and tertiary layers of education. Virtually all recent curricular innovations, including those employing significant use of technology, have left the larger features of this structure intact. We need to find the means to break this stranglehold on the future.
We need to begin to develop students' algebraic reasoning in the early grades (NCTM Algebra Framework 1995). More generally, we need to replace the layer-cake structure of the curriculum with a more strands-oriented structure that starts to develop major ideas in the early grades, draws heavily on students' experience, and taps into their naturally occurring linguistic, cognitive, kinesthetic, and visualization powers (Kaput, in preparation). While the scenario is sketchy, it should be apparent that the students are using a variety of personal resources in their work, including kinesthetic resources associated with the MBL activity (e.g., the basketball moves). Such an approach to learning serious ideas does not necessarily need to be incremental--the Parade students occasionally plunge into ideas that have aspects that they will not be able to appreciate until much further in their schooling. Nonetheless, thanks to the technology, the space of options for student activity and exp loration is richly endowed and accessible. Similar approaches should make sense for the mathematics of uncertainty, data, space, and dimension, for example (Steen 1990), and most certainly for science (NRC Standards 1988; AAAS 1989, 1993).
The traditional answers to the need for integrated software were the "X-Works" systems, which combine a word processor, spreadsheet, drawing, database, and, more recently, a communications program. They solve the file and screen management problems and also provide a common interface on a "per vendor" basis. While versions of these generic programs exist for education, they do not come anywhere near the capability needed for even a subset of the kinds of activities depicted in the scenario. Nor do sophisticated, extensible systems such as Mathematica, or computer algebra systems, which typically contain integrated graphics components, numerical functions, and algebra symbol manipulators. And none of these comes equipped with flexible pedagogical supports.
All of these approaches constitute stand-alone systems. They are produced by organizations that are well capitalized and can devote millions of dollars and years of work to the development, marketing, and maintenance of a product. These are quite unlike the organizations that produce educational software, which are financially fragile, orders of magnitude smaller, and narrowly focused. Moreover, no educational publisher would create and market the systems used in the scenario that are not education-specific, e.g., the communications programs. Hence compatibility is a matter of adhering to an independently created standard, which, at least until recently, involved significant risk.
The cross-teaching by students was intended, as were the mutual teacher support and connection with higher education. The expectation that royalty-generating products could or should be produced by a wide range of individuals or small groups of developers is central, as is the expectation that they be readily available electronically. Today, except for some supplementary written materials, textbooks are produced by large, centralized teams. The same is true for software. Little remains of that small cottage software industry of the early-mid 1980s, as the exponentially growing complexity-costs of contemporary software have risen out of reach. The materials development part of the scenario is utterly out of the question under today's monolithic software application technologies.
In addition, these components will be scriptable in the sense of being controllable by independently produced programs in high-level languages such as
The pull toward fragmentation runs counter to the need for richness, coherence, and pedagogical opportunism in classroom activity. It also tends to support artificial boundaries between instruction and assessment, and between inquiry and presentation. (One is reminded of the fraudulent relationship between inquiry and presentation in the classic physics laboratory and report.) And despite the evolution of common interfaces, the use of unrelated applications, often laden with features designed to anticipate large arrays of contingencies, tends to increase difficulty in learning to use software. Finally, the fact that much software is narrowly topic-specific, and often grade-level specific, further adds to the cost of software learning and training, as teachers and students repeatedly need to learn new applications. All these negative factors are mitigated by the use of interoperable components that are flexibly organizable into integrated suites that offer broad classes of functionality of the sort depicted in the scenario.
These design flexibility factors affect the use of software and its relation to activity structures. In particular, the core software need not embody a particular pedagogy. While this is not unusual in the sense that most software other than straight Computer-Aided Instruction (CAI) or Intelligent CAI (ICAI) systems can be subject to different uses, the new ingredient is that the scripted layer of a given core system software can deliberately embody different pedagogical strategies, depending on the intended use or audience. A given core piece can serve as the basis for a direct, heavy-handed chunk of direct instruction; or it may be the base for a game; or it might underlie a guided inquiry environment; or it might be connected to a lightweight tutor. This has important consequences for both development and use. Regarding development, the design task of anticipating in detail the myriad contingencies of end use is replaced by the task of anticipating broader categories of use--which translates into questions about which features to render scriptable, what the range of allowable values should be, how to ensure that the scripters' option spaces are coherent (so that a script, for example, does not configure the software into inconsistent or incoherent states, particularly as experienced by the user). While these are not trivial design issues, they are a step up in level of detail from the usual low-level design issues with which software developers must cope and which supply the complexity that is so costly and limiting. Scriptability is a major new tool in complexity and contingency management with potentially major consequences for educational software design, development, and deployment.
Another major impact of scriptability is on the critical relation between software producers and activity producers. As the management of complexity and contingency in software is redistributed away from the hard-coded level toward the scripting level, the authority for making decisions about what the software actually will do can be likewise redistributed, from the very narrow range of specialists who do core-level programming (typically C++) to a potentially much larger range of experts who are more intimately involved with educational and end-use issues. One might envision this broader group as including the kinds of people who once produced or designed HyperCard stacks in education. This redistribution helps respond to the third challenge mentioned at the outset, the coordination of technologists and education innovators and reformers. They can work much more closely together, since the act of scripting involves much less technical background than building core software systems in a low-level program ming language. In a sense, the features of rapid prototyping are now built into the design process as an intrinsic and continuing part of software design. This can have a major impact on educational R & D as well as on the creation of final products.
2Attached as an addendum to this paper is a concrete example of the kinds of alternatives that are under development. It is based on the SimCalc Project which James Kaput directs and for which Jeremy Roschelle is Co-Principal Investigator.
Our simulations, at various stages of development, include drivable vehicles (subways, cars), dynamic actors (people, animals, bugs) in various scenes, elevators, multilane swimming pools, dance halls, soda and candy factories, dripping faucets, systems of tanks, valves, and pipes, a schematic simulation of moving points, and so on. These are controllable in many different ways, including "direct" control (e.g., either driving or mouse-dragging a car), control via coordinate graphs of velocity, position, etc., rule-based state controls, or even algebraic controls. The simulations are linkable with MBL devices that allow a student's motion, for example, to be imported to a simulation and repeated or, more importantly, edited or compared with the motion of another object inside the simulation. We work mainly with minority students ranging in grades 3 through 13, with curricular materials and activities appropriate to the different students involved--answering student diversity with curricular diversity.
With SimCalc support, Ricardo Nemirovsky at TERC is studying student understanding of physical systems, while another team, including Jim Sandefur at GSU, biologists, and a USDA nutritionist, is studying students' understanding of nonlinear phenomena in the life sciences. These project reflect our effort to anticipate the kinds of extensions of the mathematics of change likely to become common in the next century.
By working simultaneously to develop new approaches to the key ideas underlying calculus and to develop new forms of software technologies, we hope to help set the stage for a new level of fundamental curriculum reform in mathematics and science education.
The Curriculum Vision
The central ideas of calculus--change and accumulation of quantity and the connections between them--are critically important tools for understanding science, engineering, and business and for producing informed citizens in a rapidly evolving democratic society. These ideas should be learnable by all, without a long series of prerequisites. Toward this end, we are creating and testing a series of engaging simulations and activities designed to link students' real experience with motion, fluid flow, and other continuously variable quantities to first graphical representations, then to more algebraic representations. Our ultimate goal is to help enable a fundamental reorganization of the core quantitative mathematics curriculum from the upper elementary school onward, one that treats calculus not as a capstone course for the intellectual and professional elite, but as a mainstream strand in the curriculum for all students. We create activities appropriate for students from grades 3 to 13, beginning with students from populations that are underrepresented in technical fields and are now alienated from mathematics in egregiously high numbers. Versions of our materials also serve as preparation for university calculus.
Scriptable Components: A New Level of Flexibility and Extensibility
If we are going to provide access for a very diverse group of students and teachers to the deep and subtle ideas of the mathematics of change, in a longitudinally coherent way across 10 or more years of education, and in many different kinds of learning situations in school and at home, then we must build software that embodies a new level of flexibility and extensibility. This software must be capable of supporting many different kinds of teaching and learning and must incorporate a steady flow of innovations from a wide range of activity producers. Therefore, we are building extensible simulation systems that will support maximal modularity and plasticity as the underlying development and system technologies evolve. The modularity takes the form of component structure, and the plasticity takes the form of scriptability. In this way, a given core system can support a full range of pedagogies: free exploration, guided inquiry, CAI, ICAI, and games.
Additionally, the same system can support ongoing assessment with the same flexibility, ranging from multiple choice quizzes, to interpreted feedback diagnostics, to structured portfolios.
Our software is designed in the form of "composable components" that can be put together in many ways to support a wide variety of activities. SimCalc MathWorlds fully supports scripting, to allow curriculum authors to customize toolbar palettes and menus, extend the interface, automate complex operations, and transmit data among a variety of educational software applications. We also extensively support drag and drop, to make it easier for teachers to assemble activities from preexisting parts. Furthermore, SimCalc interoperates with AppleGuide to provide active instructional assistance for each activity. Finally, we are actively exploring how OpenDoc can help us achieve maximally composable components that interoperate with software from other leading educational efforts.
The first SimCalc prototype system, MathWorlds, consists of a growing family of simulation-animations, accompanying graphs, visualization and analytic tools, and teacher and student support systems, including AppleGuide support. These can be linked directly to external supports such as mail and word processing systems. Most MathWorlds features, such as the available graphs, tools, constraints on these, user interface and input, and so on, are internally controllable via AppleScript (or other scripting languages). This scriptability provides enormous curricular and pedagogical flexibility without the complexity and expense that would be required if all these features had to be hard-coded in advance.
Some SimCalc MathWorlds involve motion simulations while others involve simulations of fluid flow; the motion is currently linear (one-dimensional, although occurring in two-dimensional space) and two-dimensional (occurring in three-dimensional space). We distinguish between "third person" and "first person" simulations. Third person simulations are controlled from the "outside" of the object(s) that move on the screen, by specifying the graphs of their motion (position, velocity, or acceleration versus time); by specifying position, velocity, or acceleration functions algebraically; by directly dragging the objects; by using a slider-controller (position, velocity, or acceleration); or by other means. First person simulations involve control from the "inside" of the simulation, as in "driving" a vehicle.
Existing, designed, and planned worlds include the following:
Linear motion-based (third person)--
Extensions to Three Dimensions and Nonlinear Modeling
In 1996 we will be extending SimCalc simulations to three-dimensional motion contexts, where the motion not only takes place in two and three dimensions, but also is controllable in multiple dimensions. Finally, to broaden the mathematical and scientific range of students' experience beyond traditional calculus and kinematics, we are developing additional modeling activities that involve fluid flow, money, and, especially, biological data, concentrating on dynamic, nonlinear biological and environmental systems with feedback and even chaotic behavior. These include studies of the human energy budget and nutrition, and animal foraging behavior.
Ricardo Nemirovsky and colleagues at TERC, under a SimCalc subcontract, are preparing for the latter work by examining in detail how students make sense of nonlinear phenomena in several contexts using physical devices such as bouncing carts and laser flashing devices. These devices are connected to data probes that allow the real-time recording of data and the analysis of that data. For example, a Macintosh software environment called the V-Scope Grapher has been developed for use with the V-Scope, a three-dimensional motion detector system. In the Bouncing Cart device, a motor moves a piston up and down along an inclined track on which a nearly frictionless cart rolls. The piston hits the cart at various points in the cart's travel, causing a range of different kinds of motion, some of which are chaotic. We are exploring the motion of the Bouncing Cart using the V-Scope and V-Scope Grapher software and will inform development of both simulations and modeling systems.
Work with Students and Teachers--Future WWW Access
We are actively engaged with students and teachers from an urban elementary school, several middle schools, and several high schools. Finally, we are testing software and curriculum materials in a class for academically disadvantaged college freshman who wish to major in technical fields and in a college calculus honors course. Thus, the range of activities is very large, and data gathering and analysis is quite complex. Our aim is to make MathWorlds and its scripting system available on the WWW in 1996, not only to engage a wider population in testing existing activities and worlds, but also to engage others in building, extending, and modifying activities using the scripting interface.
To write components, one must adhere to the additional standards of the component system one is using, such as OpenDoc; and to ensure scriptability, one must include appropriate "hooks" inside the components that can be accessed by the scripts. One way to think about scriptability is as a second "internal" interface--where the program can be controlled by other programs, the scripts. If others are to write scripts controlling the components, then the appropriate information needs to be made publicly available. In addition, in certain domains, additional data standards might be necessary to achieve additional functionality. For example, in the scenario, time-ordered motion data that involves functions play a critical role. In order to pass such data efficiently among components so as to achieve dynamic linking, additional standards need to be agreed upon and satisfied.Potential Impacts of Component Architectures
We will discuss three arenas in which interoperable components are likely to have impact: classroom practice, research and development work (R & D), and the commercial software market. In each case, the structural benefits of modularity pay off (Simon 1981). Classroom Practice
In the scenario, we examined how the natural boundaries of rich educational activities fail to fit with the artificial boundaries of stand-alone applications. Stand-alone applications encourage fragmentation of activity into relatively small pieces, despite the fact that applications already tend to be too large for educational software developers to produce efficiently. No one system that exists today could encompass more than a fraction of the activities depicted in the scenario, and even then the going would be rough, particularly if Internet communication were involved. R & D Work
Two problems plague R & D work: (1) high costs and resulting dilution of efforts due to redundant coding and (2) difficulty in bringing an R & D product to market. Most projects, by architectural necessity, tend to create most of the software code that they need, so the only level at which code is reused is at the programmer level. Not only does much of the same functionality--for example, a coordinate grapher--need to be coded across many different projects (redundancy across projects), each project has to code all the functionality it will need (dilution of effort within projects). Both of these factors cost funders dearly and limit the quality of the whole R & D effort, not just the quality of software products. The same dilution of effort leaves software products far from market quality and contributes to the difficulty of bridging from the R & D laboratories to the commercial market. By concentrating on its particular innovation and reusing components from an available pool of modifiable components, a project can lower costs and improve its focus. And, given a commercial market for interoperable components, the project may even be able to directly contribute to the market a high-quality, tested component that does not require deep recoding. Commercial Software Production and Marketing
Currently, only major, well-funded organizations can produce the large stand-alone applications that dominate the markets today, which limits competition and hence innovation. Similarly, schools must commit to large, proprietary systems, with all the concomitant risk and cost, for both the initial investment and the upgrading. A teacher or an administrator must compromise curricular needs to the constellation of givens in the respective available software systems. There is no possibility that the excellent grapher from publisher A might be matched with the very special tables system from publisher B. The availability of interoperable components changes both of these conditions fundamentally. Many more producers can participate in the market at a component level, and schools can purchase and assemble what they need from a wide variety of components, upgrading piecewise as needed over time. This fits the fluidity and adaptability depicted in the scenario, which contrasts starkly with the slow change of most schools and systems. It also fits the need to compose activity structures at a higherlevel of organization than was possible previously. Activity builders do not need to worry as much about what a particular application can do, and instead can draw upon what is available in the wider market to compose a variety of activities that use the mix-and-match, plug-and-play compatibility of widely accessible components.Potential Impacts of Scriptability and its Designed Flexibility
Changes in What Software "Is"
As described above, scriptability changes the meaning of software products, depending on the degree to which a particular component is "hard coded" into a given form in the low-level programming language part of the system. This in turn depends on how much scriptability it allows, i.e., how much of its internal structure and display characteristics are manipulable by scripts. As already noted, scriptability amounts to a second interface for the basic core program, one that is controllable by the scripter instead of the end user (unless the end user is a teacher who also happens to be a scripter, a very rare occurrence, of course). The scripts can be thought of as another layer of the software wrapped around the core system. It is quite possible for scriptable software to be entirely chameleonlike, where its interface is changeable by scripts. It is also possible to render scriptable virtually all the functional features of a piece of software. When exercised, these two design freedoms allow a scripter to tailor software to a particular activity or particular students. It allows the gradual unfolding of a system's attributes, the just-in-time appearance of tools or other components, or the preconfiguring of a system for particular purposes--either inquiry or assessment, for example.Pedagogical Agnosticism
Impacts on Relations between Software Producers and Activity Producers
Changes in Production, Marketing, and Distribution
While experience is needed to determine what is actually possible and what sorts of arrangements will eventually emerge, it seems apparent that the process of adding value to scriptable core software systems has been fundamentally altered. Given a scriptable system with published or licensable information on how to script it, the range of activity-producers who can create curriculum materials that use the core system has been greatly expanded and changed. One is no longer limited to writing off-line written materials to use a fixed (albeit flexible) system--as one does, to cite an extreme case, when writing for a graphing calculator. Nor does one need to be a major publisher with teams of programmers, curriculum authors, and artists using a large authoring system such as AuthorWare. When combined with the availability of components, a scriptable system enables much smaller groups, and possibly talented and energetic individuals, to produce activities, assessments, and the like, as depicted in the scenario. New means of marketing and distribution will have to be engineered to make available the products resulting from this new organization of technology. Undoubtedly, a period of adjustment will be necessary, since our current production, marketing, and distribution systems are organized around a very different technology and embedded in a very different culture. Exploiting Connectivity to Serve the Scriptable Component Revolution
The redistribution of complexity associated with components and scriptability seems to imply a reorganization of production, marketing, and distribution. Our next challenge is to redistribute and create incentive. Those who might create activities using technology need opportunities to be rewarded for doing so. If support for such work were widely available on the Internet--support for scripting and exploiting components--we would have one part of the challenge answered. The second part would be answered if marketing and royalty-based distribution were likewise possible over the Internet. This appears entirely possible to this author, particularly as it becomes in the economic self-interest of someone to make it happen. Perhaps the federal government can seed the market, possibly in conjunction with major professional organizations such as the National Council of Teachers of Mathematics or the National Science Teachers Association.Summary and Reflection
The admittedly rosy scenario and the optimism regarding components and scriptability must be set against the continuing failure of technology to affect current practice and the slow changes in curriculum to which we are accustomed. These failures are not accidents, and their causes are to be found in almost any direction we care to search. But we cannot overlook the factors that we have identified in this paper--factors that are fundamentally changeable. Changes in the directions offered may not be sufficient to accomplish the goals laid out at the beginning of this paper, but we suggest that they, or similar ones, are necessary.2
1The SimCalc Project is funded by the National Science Foundation Applications of Advanced Technology Program under grant RED 93-53507. The views expressed are those of the authors, however, and may not reflect those of the Foundation or the U.S. Department of Education. (Return to top of document)
References
Addendum
This addendum offers an example of an attempt at serious, technologically intensive mathematics innovation for mainstream students using scriptable components.Brief Overview of the SimCalc Project
SimCalc, a project which in 1996 is in its third year of funding from the NSF Applications of Advanced Technology Program, aims to democratize access to the mathematics of change, including calculus; historically, this area of study has been the province of a small intellectual elite who learn and use it after a long series of algebraic prerequisites. The specific goals of SimCalc are to:
The SimCalc Project--More Detail
SimCalc is building and testing innovations in two arenas: curriculum and software technology.
Planned two-dimensional motion-based (first person)--
Planned fluid-Based--
By using these graphical simulations, students can begin learning concepts of rate, mean value, accumulation, sampling, and approximation, even before they learn algebra. They begin with simple discrete step functions and work towards understanding continuously varying quantities. Students can input their own motion into the software (via a data probe) and compare empirical and simulated data.
[ Back to the Futures Page ]
Last modified September 19, 2001 (KJ).