Understanding Deep Changes in Representational Infrastructures...

James J. Kaput, Professor of Mathematics, University of Massachusetts - Dartmouth

On the assumption that at least some part of meetings such as this should engage us in conversations of wider scope than those of the usual professional meeting, I'll try to offer a view on technology in education from a deliberately high perch-where the twigs are thin and the footing precarious. But if I lose my footing, I'll fly off with my good colleague, Frank Wattenberg. More detail and examples can be found in the companion paper pointed to below.

The Co-Evolution of Representational Infrastructures & Social Systems.

Adventuresome readers are invited to download and read a more detailed story of the learnability and availability of important representational infrastructures: How the transitions from ideographic writing to phonetic writing (which tapped into the pre-existing acoustic and spoken system) greatly expanded expressiveness of writing, and then the transition to alphabetic writing dramatically increased learnability and the associated literacy communities. I also look at the evolution of algebra (including its transition from rhetorical/phonetic to ideographic) and calculus. I draw an analogy between how writing became more expressive and learnable by tapping into the neurophysiologically well-evolved pre-existing system of spoken language and how the representation of quantitative relationships is making another huge transition from that of ideographic algebra to another system by tapping into the neurophysiologically well-evolved visual system, etc. I also draw a second analogy in the availability dimension between how the printing press changed the availability of written systems (during an extended period of 250+ years) and how the transition from rare & expensive computers of the recent past to the more diverse, ubiquitous and connected computing devices of the near future is changing the availability of the new representational infrastructures. That paper also includes concrete illustrations of the points made towards the end of this one.

Fundamental representational infrastructures such as writing systems, number systems, and algebra play a deep role in determining what and how people think, and what they are capable of doing (Cole, 1997). Hence their associated literacy communities and communities of practice, and their related social institutions, are very sensitive to their learnability and how they are physically implemented via such devices as the printing press and computers. Today's communities of practice, and social systems such as education, are greatly dependent upon, and in great measure defined by, the historically received representational infrastructures with which they co-evolved. Nowadays, you don't get a high school diploma without surviving algebra courses (which is at least indexically connected to learning the representational infrastructure of algebra), and you don't become an engineer without surviving the calculus sequence (which, again, perhaps remotely, suggests that you've learned something about that system of knowledge and representation). Importantly, these infrastructures evolved within the constraints of static, inert media, and, with the partial exception of the now standard number system, they evolved to serve the needs of a small intellectual elite.

The close relationships between major representational infrastructures and social institutions were historically quite stable across the millennia because they co-evolved, just as do the constituents of any mature ecosystem. However, the dynamic and interactive computational medium enables the rapid evolution as well as deliberate design of new representational infrastructures, such as spreadsheets, dynamic geometry, interactive diagrams, visually editable graphical systems, state-space visualizers of dynamical systems (Stewart, 1990), as well as by adding new interactivity and hot linkages to traditional representational infrastructures as is reflected in any Computer Algebra System (CAS). Certain of such changes are especially destabilizing because they change the means by which new systems of knowledge can be built. This was the case of writing, which provided the first extracortical means for creating and storing the products of human cognition, and which therefore made possible new kinds of feedback loops between knower and known (Donald, 1991). Another surge forward occurred with the development of operative algebra in the 16th and 17th centuries (Bochner, 1966) as western civilization attempted to create models of the physical world. This operative system and the hierarchical, placeholder system for numbers supported an entirely new level of computational and reasoning capability at the hands of a suitably trained human partner.

We are in the early stages of yet another surge, rooted as were the others in mathematical activity - in this case the development of the idea of formalism and operations on formalisms (programs, as defined by von Neumann & Turing) coupled with new physical devices that could instantiate these (Shaffer & Kaput, 1999; Kaput & Shaffer, in press; Riorden & Hoddeson 1997). This has provided the ability to run extracortical processes (not merely records) autonomously of a human partner which in turn has led to a new medium, the dynamic, interactive computational medium, within which one can create new representational infrastructures and new means for creating and sharing knowledge, embodying the combination of rapid computation, visualization and communication. Feedback loops are again changed, whereby we no longer need to interact with static, inert knowledge records, but can interact with dynamic and responsive representations and tools - so the central role of the Book is replaced by the Simulation. This marks yet another breakthrough in human representational capability. The impacts of this change will unfold unpredictably in the coming generations (because the means of building and sharing things have changed, as reflected, for example, in the Human Genome Project or in a field such as Artificial Life), but we have every reason to believe that their impact will be on the order of the invention of writing in the way humans shape their worlds, define themselves and their places in them, and live in them (plurals here are deliberate).

Historically, changes in representational infrastructures and changes in associated communities of practice (and literacy) as well as social institutions were slow, measured in centuries or longer, and, more importantly, on the same time-scale. But this consonance in time-scales no longer prevails, which creates new tensions and new opportunities, as changes in representations and their learnability are occurring on much shorter time-scales than the ambient social systems. However, discussions of technology in education largely reflect expectations defined by pre-existing social systems and the representational infrastructures with which they co-evolved. Hence we tend (with honorable exceptions!) to discuss doing old stuff better within slightly altered educational circumstances rather than doing better stuff within radically changed circumstances. At the worst, we worry about how to use technology to remediate K-12 educational failures (which is where the majority of post-secondary mathematics education resources are spent). Or we try to fix university calculus without changing anything else - surely some of us remember the Calculus Reform Movement.

Two Kinds of Uses of Electronic Technologies in Education.

I see two broadly drawn types of impacts of technology on education:

  1. Those that involve connectivity and all that changes in connectivity entail (e.g., in the forms of education, relations between education and the wider society or between constituents and levels of the education system), and

  2. those that involve representational infrastructures and hence the nature of the knowledge and its learnability, including forms of interaction. (A & B obviously interact-as seen in any good applet.)

My work involves both, although on the connectivity side, with Jeremy Roschelle at SRI International, I'm working on classroom connectivity - using wirelessly linked devices - which may have at least as large as impact as wider connectivity via the WWW (see the companion downloadable paper for a few examples).

Three Levels of Representational Infrastructure Change.

Here's a rough taxonomy of changes in knowledge based on changes in representational infrastructure that may help to situate the various kinds of innovations examined in our discussions. Change in representational infrastructures seems to have impact at three levels:

  • Level 1.
    The knowledge produced in static, inert media can become knowable and learnable in new ways by changing the medium in which the traditional notation systems in which it is carried are instantiated - for example, creating hot-links among dynamically changeable graphs equations and tables in mathematics. Most traditional uses of technology in mathematics education, especially graphing calculators and computers using Computer Algebra Systems, are of Level 1.

  • Level 2.
    New representational infrastructures become possible that enable the epistemic reconstitution of previously constructed knowledge through, for example, the new types of visually editable graphs and immediate connections between functions and simulations and/or physical data of the type developed and studied in the SimCalc Project - to be described briefly below.

  • Level 3.
    The construction of new systems of knowledge employing new representational infrastructures - for example, dynamical systems modeling of nonlinear phenomena, or multi-agent modeling of Complex Systems with emergent behavior, each of which has multiple forms of notations and relationships with phenomena. This is a shift in the nature of mathematics and science towards the use of computationally intensive iterative and visual methods that enable entirely new forms of dynamical modeling of nonlinear and complex systems previously beyond the reach of classical analytic methods - a dramatic enlargement of the Mathematics of Change & Variation that will continue in the new century (Kaput & Roschelle, 1998; Stewart, 1990).

An exercise for the reader: Where do the innovations discussed in our meeting fall relative to the two taxonomies offered above?

My own recent work in the SimCalc Project involves, as already noted, a reformulation of the core content of the Mathematics of Change & Variation (MCV), of which a subset concerns the ideas underlying Calculus is a Level 2 change.

Summary of SimCalc Representational Infrastructure Changes.
I summarize the core web of five representational innovations employed by the SimCalc Project, all of which require a computational medium for their realization. The fifth actually falls into Level 3. Space limitations force the examples into the downloadable companion paper. Cross-platform software, Java MathWorlds for desktop computers can be viewed and downloaded at http://www.simcalc.umassd.edu and parallel software for hand-helds can be examined and downloaded from http://www.simcalc.com.

  1. Definition and direct manipulation of graphically defined functions, especially piecewise-defined functions, with or without algebraic descriptions. Included is "Snap-to-Grid" control, whereby the allowed values can be constrained as needed-to integers, for example, allowing a new balance between complexity and computational tractability whereby key relationships traditionally requiring difficult prerequisites can be explored using whole number arithmetic and simple geometry. This allows sufficient variation to model interesting situations, avoid the degeneracy of constant rates of change, while postponing (but not ignoring!) the messiness and conceptual challenges of continuous change.

  2. Direct connections between the above representational innovations and simulations - especially motion simulations - to allow immediate construction and execution of a wide variety of variation phenomena, which puts phenomena at the center of the representation experience, reflecting the purposes for which traditional representations were designed initially, and enabling orders of magnitude tightening of the feedback loop between model and phenomenon.

  3. Direct, hot-linked connections between graphically editable functions and their derivatives or integrals. Traditionally, connections between descriptions of rates of change (e.g., velocities) and accumulations (positions) are usually mediated through the algebraic symbol system as sequential procedures employing derivative and integral formulas-but need not be. In this way, the fundamental idea, expressed in the Fundamental Theorem of Calculus, is built into the representational infrastructure from the start, in a way analogous to how, for example, an immensely powerful hierarchical system for structuring numbers is built into the placeholder system and which democratized access to numerical computation.

  4. Importing physical motion-data via MBL/CBL and re-enacting it in simulations, and exporting function-generated data to drive physical phenomena LBM (Line Becomes Motion), which involves driving physical phenomena, including cars on tracks, using functions defined via the above methods as well as algebraically. Hence there is a two-way connection between phenomena and mathematical notations. This work is being pursued in a new Project with Ricardo Nemirovsky at TERC.

  5. Use of hybrid physical/cybernetic devices embodying dynamical systems, whose inner workings are visible and open to examination and control with rich feedback, and whose quantitative behavior is symbolized with real-time graphs generated on a computer screen.

The result of using this array of functionality, particularly in combination and over an extended period of time, is a qualitative transformation in the mathematical experience of change and variation. However, short term, in less than a minute, using either rate or totals descriptions of the quantities involved, or even a mix of them, a student as early as 6th-8th grade can construct and examine a variety of interesting change phenomena that relate to direct experience of daily phenomena. And in more extended investigations, newly intimate connections among physical, linguistic, kinesthetic, cognitive, and symbolic experience become possible.

Importantly, taken together, these are not merely a series of software functionalities and curriculum activities, but amount to a reconstitution of the key ideas. Hence we are not merely treating the underlying ideas of Calculus in a new way, treating them as the focus of school mathematics beginning in the early grades and rooting them in children's everyday experience, especially their kinesthetic experience, but we are reformulating them in an epistemic way. We continue to address such familiar fundamentals as variable rates of changing quantities, the accumulation of those quantities, the connections between rates and accumulations, and approximations, but they are experienced in profoundly different ways, and are related to each other in new ways.

These approaches are not intended to eliminate the need for eventual use of formal notations for some students, and perhaps some formal notations for all students. Rather, they are intended to provide a substantial mathematical experience for the 90% of students in the U.S. who do not have access to the Mathematics of Change & Variation (MCV), including the ideas underlying Calculus, and provide a conceptual foundation for the 5-10% of the population who need to learn more formal Calculus. Finally, these strategies are intended to lead into the mathematics of dynamical systems and its use in the nonlinear science that is now growing so dramatically.

In terms of our historical perspective, we see this current work as part of a large transition towards a much more broadly learnable mathematics of quantitative reasoning, where both the representational infrastructure is changing as well as the material means by which those more learnable infrastructures can be made widely available. Taken together, could it be that the kinds of representational innovations outlined above and illustrated in the companion paper constitute steps toward the development of a new "alphabet" for quantitative mathematics which might do for mathematical representation what the phonetic alphabet did for writing?


Bochner, S. (1966). The Role of Mathematics in the Rise of Science. Princeton, NJ: Princeton University Press.

Donald, L. (1991). The Origins of the Modern Mind. Cambridge, MA: Harvard University Press.

Kaput, J. (2000) Implications of the shift from isolated, expensive technology to connected, inexpensive, ubiquitous, and diverse technologies. In M. O. Thomas (Ed.) TIME 2000: An International Conference in Mathematics Education. pp. 1-25. University of Aukland, New Zealand.

Kaput, J., & Roschelle, J. (1998). The mathematics of change and variation from a millennial perspective: New content, new context. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.), Mathematics for a New Millennium (pp. 155-170). London: Springer-Verlag.

Kaput, J., & Shaffer, D. (in press). On the development of human representational competence from an evolutionary point of view: From episodic to virtual culture. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, Modeling and Tool Use in Mathematics Education. London: Kluwer.

Riordan M., & Hoddeson, L. (1997). Crystal Fire: The Birth of the Information Age. New York: W. W. Norton.

Shaffer, D., & Kaput, J. (1999). Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education. Educational Studies in Mathematics, 37, 97-119.

Stewart, I. (1990). Change. In L. Steen (Ed.), On the Shoulders of Giants: New Approaches to Numeracy (pp. 183-219). Washington, DC: National Academy Press.