Technology as a Transformative Force in Math Education: Transforming Notations, Curriculum Structures, Content and Technologies

 

James J. Kaput

Department of Mathematics, University of Massachusetts-Dartmouth

 

Introduction

Looking Beyond the Rear-View Mirror

Computational technology is a profoundly transformative force across our society. In importance, it is analogous to technological inventions such as the printing press or the internal combustion engine, and such representational inventions such as alphabetic writing. Impacts of such enormous inventions unfold across generations in utterly unpredictable ways, because they alter the conditions under which and means by which inventions are made; and, by changing the conditions under which people live, they spawn needs for other inventions. Marshall McLuhan (1967), drawing on his historical and cultural studies of the printing press, coined the term "rear-view mirror syndrome" to describe the understandable dual tendency to view new technologies within the conceptual and cultural framework of prior technologies and to superimpose the new technologies on the world as it existed before the new technologies appeared. Depending on the technology used as a frame, different rear-view mirrors are used to understand computational technologies in education. Particularly quaint examples were offered by Cuban (1986), who saw the computer in the tradition of slide projectors and filmstrips. More subtle rear-views occur when early forms of a technology are used to think about its future. These abound in educational technology discussions and share the feature that they accept the world as it existed prior to the technology as the one that will exist in the future.

In mathematics education, the question is how much of the first-generation technology world will remain in place during the coming generations? My glib answer is "less than most of us expect." The computational medium alters the growth of mathematical content, changes which content is important and for whom, changes the means by which it can be known, taught or learned, changes the socio-cultural milieu in which teaching and learning occur and in which the institutions of education live, changes the relations between schooling and living - indeed, finding aspects of education not deeply affected by the computational medium is not an easy matter, unless, of course, one looks in schools, or in schools of education.

The phrase "Virtual X" (supply your favorite values for X), reveals that our very notions of object and relation are changing. Put differently, computational technology is shifting our ontology, shifting the nature of "reality." I believe that this wider frame is necessary to make judgments regarding the roles of long division in arithmetic, simplifying radicals in algebra, two-column proofs in geometry, or integration by parts in Calculus. Failure to reconsider our content and practice in a sufficiently inclusive framework endangers even more the significance of school mathematics, perceived and actual, to life and work.

Efficacy and Research Questions: Is Fire Good?

The role of view-frame is also critical in questions of efficacy, especially comparative with/without technology questions. More broadly, the value of research regarding the appropriate roles and impacts of first or second generation technologies is acutely sensitive to frame of reference. Imagine that we are in the first days of the Model A Ford (which followed the Model T). How should we read studies of the traffic patterns, dangers, and advantages of the Model T Ford? (There were serious conflicts with horses, dangers of broken wrists due to backfires while turning the starter-crank - unless you were careful not to wrap your thumb around the crank and didn’t set the spark setting too high, etc. See Kaput (1992) for an elaboration of this analogy.) And what do such studies tell us about appropriate design of autos, roads, driver-education, traffic laws, and so on, for the people, young in the 1920’s, who would be living and driving on interstate highways in the 1960’s? And did the question "Are cars better than horses?" ever make sense as a research question? Or how about "Are computers (or calculators) the panacea for education?" as a policy question?

One way to help deal with the difficult challenges of doing and using research in early versions of rapidly changing technology is to be as explicit as possible about the frames of reference and assumptions of such research, especially regarding what the technology at hand cannot or should not do. One goal of this paper is to help identify tacit assumptions associated with much of our work to date through the device of discussing explicit alternatives to those assumptions - assumptions regarding notation, curriculum structure, learning, content, and the technologies themselves. But before beginning, I will make explicit my own foundational assumption.

We Need Extraordinary Achievement from Ordinary People

"Extraordinary" is a relative term. Our grandparents, as youngsters early in this century, would find extraordinary the daily achievements of ordinary people today - in such fields as transportation, communication and medicine. We routinely travel at more than two orders of magnitude greater speed than we could without technological support. We can use our ordinary voices to speak across continents and oceans. But of course, it is a commonplace to note that our grandparents would not find today’s classrooms out of their ordinary. I believe that our next generations of students absolutely require changes in educational achievement similar in scope and magnitude to the changes that this century has brought in such fields as transportation, communication, manufacturing, and is beginning to offer in medicine. If we in education rethink our enterprise deeply enough and are successful in our reforms, the achievements of our grandchildren will appear extraordinary by our standards, but ordinary by theirs.

We face a dual challenge in mathematics, reflecting the acceleration of trends that have been underway for centuries: The need to teach much more mathematics to ever more people than ever before in human history. The percentage of the 18 year old cohort taking AP Calculus today is 3.5. This is also the percentage of students who graduated from high school a century ago! What percentage of the population did we expect to learn Calculus, in any form, a century ago? Perhaps a tenth of one percent? (Roughly 10 percent take some kind of Calculus today.) How many did we expect to learn algebra? Maybe 2 percent? In the face of calls for "algebra for all," have we changed either curricula or pedagogies to an extent that might offer promise to increase the percentage of students learning (some form of 21st-century) algebra from 2 towards 100 - students infinitely more diverse than the initial 2 percent?

Let’s turn from automobiles to the printing press. For our purposes here, I draw attention to two of the many consequences of the printing press: (1) The widening of literary forms (e.g., as reflected in the invention of the novel), and (2) The democratization of literacy. Will there be analogous consequences of the computer? On #1 we have an easy affirmative answer. As discussed below, we are seeing a dramatic widening of mathematical forms in computational media. On #2, will there be a democratization of mathematical reasoning? The answer will very much depend on how educational technology is used in our society, whether the transformative effect of information technologies will impact schools and schooling, and how we rethink what it means to learn and use mathematics. For example, must all mathematics, to be legitimate, be written in character strings? Should we accept as given the layered curricular structure that acts to filter out so many students from life-opportunity? Or are other, more inclusive, strands-oriented approaches possible? By way of caution, we should note that it was centuries before the new "vulgar" literature was allowed inside universities along side the classics as a proper topic of study.

Goal and Plan for the Paper

The goal of the paper is to provide illustrations of technology uses and technology implications that (a) challenge prevailing and frequently default assumptions and images, yet (b) appear likely to have a major impact during the period when Standards 2000 is the operant NCTM policy document. The plan of the paper is to address several interlinked challenges that involve, respectively, (1) representations and modeling, (2) curriculum structures and prerequisites, (3) content shifts due to the computational medium, and (4) the move to ubiquitous, heterogeneous, connected technologies. In each case my account leaves out much more than it includes, and my choice of examples is both biased and informed by recent research and development. I suggest that all the innovations described below are massively implementable in the next 5 to 10 years.

(1) Beyond Linking Representations and "The Big Three" to Creating Phenomena

Importing Data - The Direct Modeling of Phenomena

Much enthusiasm and effort has been invested in the promise of multiple linked representation systems since they became available in the mid 1980’s, and they indeed provide a step beyond the use of isolated notation systems, both as pedagogical and as conceptual tools. The basic idea was that the technological linkage could be used to create a conceptual linkage between the notations. Prior to their introduction, technology had been used to perform computations within a given symbol system, or used to link symbol systems in a uni-directional serial manner (e.g., input the character string for a function and then have the computer graph it). However, teachers and researchers have come to see the shortcomings of learning contexts where the notation systems only represent each other, and do not refer to anything beyond each other. Although it was not intended as such, the marvelously detailed analyses by Schoenfeld and colleagues of a student’s behavior in a multiple linked (linear) function environment (Schoenfeld, et al. 1994), one that included a version of the classic Green Globs game (Dugdale, 1982), provide a definitive study of the fragility and emptiness of learning when notations do not represent anything of significance for the student other than each other. Despite her maturity and sophistication, and her close tutorial support, she shows every indication of learning failure because her experience of the linked notations is not grounded in her experience - they do not represent anything of significance for her.

In the past few years, the pedagogical importance of data has grown and has been technologically supported by the availability of microcomputer-based laboratory (MBL) equipment (invented in the late 1970’s by Tinker & Thornton) that allows one to import physical data into a computer, and, more recently and less expensively, into a calculator (CBL). Now, the representations can be about something, especially something that the student has first hand experience with. In most cases, the student learns about both the phenomena represented and the mathematics used to represent quantitative aspects of it, reflecting the connectedness of mathematics with experience and its power as a sense-making tool. The level of directness in these cases is much greater than when data is provided in text and a "cover story" likewise is provided in text.

Creating Physical and Cybernetic Phenomena: Closing the Epistemological Loop

Two additional innovations expand opportunity to create referential meanings for mathematical notations and configurations of them. One of these, pioneered by Nemirovsky and colleagues at TERC turns the idea of MBL around (and hence they sometimes call it LBM - "line becomes motion") and enables one to use mathematically defined functions to control physical devices such as motorized toy cars running on tracks, pumps moving fluid in and out of containers, laser-pointers, and so on. Here, phenomena are being created and modeled at the same time. The directness of data connection provided by MBL is augmented by the mathematically-based control that students have of the phenomena being modeled. Whereas the student previously could move the toy-car by hand, or roll it down an inclined plane, and see a velocity or position graph of the resulting motion, or try to match a given graph by producing physical motion, they can now create a mathematical function and observe its physical realization. They can also attempt to match mathematically produced motions through physical actions, for example, by moving one car by hand while a mathematically driven car moves on a parallel track. In these ways, students deepen their experience of the mathematical ideas involved to include kinesthetic, visual, auditory, and so on (Nemirovsky, 1994; Nemirovsky, Kaput & Roschelle, 1998; Nemirovsky, et al., in press). Current intensive work in research classrooms will yield widely available commercial products within 2-3 years, and the quality of the data probes will improve as their price continues to fall (Banesche, 1998).

In addition to creating and studying physical phenomena, we can create cybernetic phenomena, or simulations (Casti, 1997). Simulations have the degrees-of-freedom advantage of not being constrained by physicality, but the obvious disadvantages of decreased perceptual concreteness and diminished richness of interaction. This freedom can be exploited in many ways to pursue ambitious pedagogical and curricular objectives not attainable without the technological support and the freedom to "suspend," temporarily, the physical constraints that complicate our mathematics - in some sense, a strong version of the commonly used strategy of idealization.

For example, in order to take advantage of students’ ability to do whole number arithmetic (and to provide additional interpretations of multiplication as area), we can create simulation motions whose velocities are step functions, as is the case for the velocity graph on the left in Figure 1. This velocity graph (labeled "Red") controls the elevator on the left side of the building (which is color-matched to the graph, but depicted here in gray-scale). Such graphs allow us to pursue the basic connections between velocity and position descriptions of motion while preserving computational tractability. Students in upper elementary and middle grades work effectively with such graphs and quickly become adept at predicting the motion of the elevator (not merely its ending-point) based on a velocity graph and starting floor. Such motion, with discontinuous velocity, is, of course, impossible, but this is not problematic for the students until we make it so, which we do through a variety of means. For example, we can use MBL experiences in which students attempt to create such graphs physically, we can make acceleration an issue by having the students create motions for vehicles under various constraints (for example, a wheelchair van that is to carry your grandmother, who has an injured neck), we can introduce discontinuous position vs time graphs similar in appearance to the velocity vs time graphs and study the analogy between them, or we can approximate such step functions with linear functions, as is the case in the velocity graph on the right side of Figure 1 where the graph titled Blue drives the "blue" elevator to the right of the "red" one. (The two coordinate systems are juxtaposed here to save space and would not typically appear on the same screen, although multiple graphs can appear in the same coordinate system for comparison/contrast purposes.)

What is the behavior of the red elevator? What is the behavior of the blue elevator? How far apart are they at the end of the trip, or after 2 seconds? Note that these graphs can be directly adjusted by simple dragging, so it is an easy matter to improve the blue graph’s approximation by dragging it upward, say. Further, we can enforce snap-to-grid, so that the calculations can remain relatively simple - in effect, we are controlling the number system used to model/create the situation. Suppose we ask the question: Does there exist a constant velocity for Blue so that it gets to the same floor as Red in exactly the same? The answer (which is the average velocity of Red, of course) depends on whether snap-to-grid is on, or, put differently, it depends on whether there are numbers where we need them (the issue of continuity again). We can even adjust scaling and labeling so that students are required to deal with fractions or decimals. More on the curricular issues below.

Figure 1

As is the case with most learning environments instantiated in dynamic interactive media, cycles of conjecture and feedback, or intention and feedback are very tight, although delays can be imposed, as can explicit predictions.

New Forms of Action-Notations

Although it cannot be apparent from the static picture of graphs in Figure 1, the fact that these graphs are visually editable in very fluid ways means that they, in some measures, accomplish what character-string notations did previously. While the ability to write a function or even a function-family compactly using a character string is enormously powerful, the civilization-defining power of algebra that evolved beginning in the 15th century follows from its syntactical manipulability (Bochner, 1966). The fact that the forms of expressions can be changed and/or combined in systematic ways, that these forms themselves can carry important information and structure, and that they can yield representations of potential entities or relationships before these are concretely realized - this is one of the great Pierian springs of mathematics’ magic. Some of this power, for constrained situations, is available through graphically editable systems such as briefly illustrated in Figure 1. One can create such functions by simply dragging in pieces, dragging and stretching them, etc. to produce functions that would be very difficult to express using conditionals applied over intervals. Even the simple velocity function Red would be a challenge for most students to express algebraically.

Further, we should note that linked representations are still available, but in extended ways. In particular, we can link any of the "Big Three" (extended to a fourth - natural language) representations of, say, velocity, a "rate description" of motion with any of the standard representations of position, a "total description." Some of the essence of what we are discussing is expressed in Figure 2, which puts the phenomena in a central role as the subject of any of the other descriptions. We will return to this diagram below.

Figure 2: The New "Big Three"

The above discussion, tightly focused on the Mathematics of Change and Variation, has barely introduced a complex set of semiotic perspectives associated with the representational affordances of computational media, and leaves untouched how such affordances can be brought to bear in the learning of other major mathematical strands, e.g., in the mathematics of space and the mathematics of uncertainty. A major message, however, is that we need no longer be confined to the inherited notation systems and finding ways to teach them more effectively. After all, these notation systems, with very few exceptions, were designed under the constraints of static, inert media by and for a tiny knowledge-producing elite who used them intensively for the bulk of their adult lives. Now we must find ways to make major mathematical ideas learnable by the large majority of people, infinitely more diverse than the geniuses who originally built them. Fortunately we are no longer constrained by static, inert media - unless we choose to be.

(2) New Curricular Options and Prerequisite Structures: The Case of Calculus

Currently, a bit more than ten percent of the population survives the long set of algebraic prerequisites to take some kind of formal Calculus, this despite the fact that the bulk of the core curriculum can be regarded as preparation for Calculus, and despite the fact that understanding growth and change, and more generally, the Mathematics of Change & Variation (MCV), is an essential ingredient of education for life in a rapidly changing democratic economy - the complexities of choices in modern life, personal, economic, political, require understanding its core ideas, especially for that 90% whose MCV needs are largely unserved by inherited curricula.

The discussion above introduced the notion that the major ideas of Calculus might be approachable by students in the upper elementary and middle school levels. We began in the context of the Mathematics of Motion (where the Calculus originated historically in the 14th and 15th centuries (Claggett, 1968)) with a brief illustration of how one might begin to understand how a varying quantity (velocity) accumulates over time yielding total displacement and hence position. In formal terms, this is the idea of integration. Its inverse, determining derivatives - rates of change - can be built from similarly concrete starting points and similarly drawing on student cognitive, linguistic and kinesthetic resources. Sample curricular materials and technologies developed in the SimCalc Project illustrating these assertions in more detail are available at http://www.simcalc.umassd.edu and are in preparation for commercial distribution from a variety sources and on a variety of hardware platforms (see the last section below).

The importance of the key ideas of Calculus, the reversible connections between rates and totals of varying quantities, the different categories and mixes of variation, and the supporting conceptions of continuity and approximation, cannot be overestimated in our culture. Calculus, and the algebraic systems of thought and technique that co-evolved with it, are the foundation for most of the science leading into the 20th century and hence lie behind the technologies that we live by (Kline, 1972). Hence it should be no surprise that our curriculum beyond arithmetic seeks to lead students towards Calculus, one of the defining masterpieces of Western Civilization (MacLane, 1984). But, given the notational factors alluded to earlier, it is also no surprise that few students actually reach the Promised Land despite our best, and sometimes almost heroic, efforts.

However, stepping back for a broader view, we can see that while large amounts of curricular capital are invested in teaching numerical, geometric and algebraic ideas and computational techniques in order that the formal symbolic techniques of Calculus might be learned, the ways of thinking at the heart of Calculus, including and especially those associated with the Fundamental Theorem, do not require those formal algebraic techniques to be usefully learned. Indeed, by approaching the rates-totals connections first with constant and piecewise constant rates (and hence linear and piecewise linear totals), and then gradually building the kinds of variation, we have seen the underlying relations of the Fundamental Theorem become obvious to middle school students. Perhaps as important, however, many of the other mathematical ideas that we expect students to learn along the way are contextualized, motivated and organized by studying the MCV, ideas such as rate, ratio, proportion, slope, area of geometric figures, arithmetic of signed numbers, fractions, and so on, including the key ideas of function algebra, particularly the algebraic representation of the basic classes of functions. Students also learn basic ideas of the Mathematics of Motion as well as new kinds of heuristics that involve comparing and distinguishing virtual and actual data - critically important in an increasingly virtual culture. But, importantly, mainstream students are also in a position to learn the newer MCV, especially that involving nonlinear dynamical systems as discussed below. Finally, that minority of students who need to learn certain formal techniques of Calculus now have some sense of what those character string manipulations are about.

Summary and Implications

The above thumbnail sketch points to a broader effort now underway and needs fuller testing to be justified as a major alternative to currently organized curricula - although the CorePlus high school curriculum (1997) embodies many of its elements, and the Connected Mathematics middle school curriculum (Lappan, et al., 1997) embodies some of them as well. Nonetheless, clinical and classroom prototyping in a variety of (predominantly economically poorer) settings show that mainstream grade 6-10 students and academically disadvantaged college freshmen (with weak or nonexistent algebra backgrounds) can learn the key ideas in numerical and graphical forms at levels higher than we typically see for more elite populations of older students. Further, work with both pre- and in-service teachers shows that mainstream teachers who have not had, or have had limited formal Calculus training can learn the requisite material, often with considerable enthusiasm (Doerr & Bowers, in preparation).

An implication of the above work is that more than small adjustments in curricular structure can be enabled by representationally deeper uses of technology than we have seen to date, most of which leave the global prerequisite structures and underlying assumptions in place. Prerequisite structures may be and should be substantially revised for the coming century, eliminating the algebra bottleneck that prevents the great majority of students from reaching such important ideas as those underlying Calculus. Further, such an approach centered on building big ideas over an extended period of time and across many contexts, both within and outside mathematics, can simultaneously enhance the learning of much additional mathematics as well. We now turn to the growing need to learn much more mathematics of the sort that thrives in the computational medium and cannot live without it.

(3) New Content That Grows Naturally in the Computational Medium: The Case of Nonlinear Dynamical Systems

 

If you knew the algorithm and fed it back say ten thousand times, each time there’d be a dot somewhere on the screen. You’d never know where to expect the next dot. But gradually you’d start to see this shape, because every dot will be inside the shape of this leaf. It wouldn’t be a leaf, it would be a mathematical object. But yes. The unpredictable and the predetermined unfold together to make everything the way it is. It’s how nature creates itself, on every scale, the snowflake and the snowstorm. ... The future is disorder. A door like this has cracked open five or six times since we got up on our hind legs. It's the best possible time to be alive, when almost everything you thought you knew is wrong.--Valentine in (Stoppard, 1993, p. 48).

Kaput & Romberg (in press) used this quotation to introduce a section of a paper that elaborates the assertions below with concrete examples and more detail, drawing also on the inspirational work of Robert Devaney, which is also represented in this collection and which will offers additional concrete illustrations. While Romberg and I used Stoppard’s poetic and youthful character as a dramatic scene-setter rather than as an authority representing our scientific analyses, we feel that he is pointing in the right direction.

The mathematics and science of the next century will be more different from today’s than today’s are different from that of the 18th century. My confidence in this assertion increases steadily as I see the explosive growth of new kinds of mathematics and science in the computational medium. This medium, acting as a kind of combined growth hormone and nutrient, supports the development of new forms of mathematics in two ways: (1) through rapid iteration of algorithms (numerical and otherwise), and (2) through dynamic, interactive visual displays (Stewart, 1990). Just as most mathematics of the 18th and 19th centuries is now subsumed by the mathematics of today as a few special cases of more general and abstract ideas, the mathematics of the next century will subsume today’s as a subset. Profound changes in content are underway. We are moving:

From the mathematics of closed-form solutions of highly constrained problem-situations

To the mathematics of numerical and graphical solutions to ever more realistic situations

From modeling situations with few, weakly or linearly interacting parts as was the case in classical mathematics’ foundations in the physical sciences

To modeling situations with many, strongly & non-linearly interacting parts as is inevitably the case in the social & life sciences (any living thing or ecology is comprised of many strongly interacting subsystems)

From the extraordinarily powerful tools of classical physics & engineering

To the extraordinarily powerful iterative & graphical tools of non-linear science.

These changes in the ways we understand our world are as foundational as those that accompanied the birth of mathematics-based science in the Enlightenment (Glass & Mackey, 1983; Pagels, 1989). However, the change conceptually is less a matter of discarding than augmenting ways of thinking. And the new ways of thinking require computational tools, just as modern astronomy requires telescopes and biology requires microscopes (and as, indeed, both these fields now require computational tools).

Our students will need to deal with dynamical systems embodying new kinds of properties, descriptive languages and ways of thinking: they will need to understand differences between deterministic and non-deterministic chaos, between stable vs unstable systems, attractor vs repellor states; they will need to understand fractal dimension, sensitivity to initial conditions and predictability (Kellert, 1993), orbits, periodicity; and period-doubling; they will need to learn the new visual languages of multidimensional phase-space descriptions of complex phenomena, ideas of closeness and neighborhood once the province of specialists (although they were originally defined by Poincare to understand the phenomena of dynamical systems a century ago) and more generally, become adept with highly plastic means of viewing complex relationships. Students will need to understand the powerful new ideas of emergence (Holland, 1995) and self-organizing systems (Kauffman, 1995), the ideas of data structure and algorithm, the ways that locally defined behaviors yield global structures (emergence again), the different forms of complexity, information coding & flow within physical, social and biological systems, neural networks and cellular automata (Kelso, 1995). And their "process skills" will need to include the ability to distinguish a simulated system from a physical one, and simulated from "real" data - a matter that becomes especially difficult in social science, e.g., economics. They will need greatly increased capacity for uncovering assumptions built into models as those models become ever more realistic and support immersive interactions - in the sense of Jackiw and Finzer (this volume).

Changing Reality

Just as Galileo’s telescope changed the ontological status of "moon" from the unique heavenly orb that circles the earth, and whose 28 day changes provided a time-marker for most of human history, to one of a class of astronomical objects, such fundamental processes that define "life" can now be abstracted to a class of behaviors that are shared by artificial life systems (Adami, 1998). In a similar way, our students will need to learn not only that data can have structure, but that one structure for data can be put alongside another and compared. They will not need to learn algorithms to do algorithms, but rather to learn what algorithms are as executable objects, objects that occur in various families based on their properties and purposes, objects that can be compared and contrasted, for example, with respect to efficiency, or convergence.

Millennia ago, humans learned to create external record systems to augment the limited memory capacity of the mind and to accumulate knowledge across generations in what Merlin Donald (1991) refers to as the "theoretic culture." A few hundred years ago, we learned how to create notation systems that, in partnership with an appropriately skilled human, could provide external support for computation, both numerical and algebraic. And, after a century of developing an understanding of the idea of formally definable systems and symbols, we found a way to externalize procedures on abstract symbols entirely, through the invention of the computer program and its instantiation in electronic hardware. The emrging result is what Shaffer & Kaput (in press) call the "virtual culture" (in keeping with Negreponte, 1996) that embodies the kinds of changes now underway that will dominate our students’ century. I believe that we need to maintain this larger perspective as we attend to the details of technology-forced change.

Immediate Need for Research

The kinds of mathematics and science alluded to above are not possible in static inert media. Nor are they currently represented in mainstream curricula, although moves towards inclusion are being made in a few exceptional curricula (e.g., CorePlus) and by curricular pioneers such as Devaney and Choate. We know precious little about how to render this material learnable. We do know, however, something about how this knowledge is being developed - through a complex mix of theorizing, empirical study, and modeling, usually employing simulations. The remarkable power of iterative models and simulations, together with their analytic and visualization tools, have made few inroads in education or in education research, either as methods or as topics.

(4) New Mixes of Connected, Heterogeneous Technologies in the Classroom

Historical Trends Towards Distributed, Ubiquitous Computing and Connectivity

The move in the 1970’s and 1980’s from mainframe computers to widely distributed desktop computers reflected a move towards broader distribution and more integrated use of what was previously a scarce, expensive and highly regulated commodity, computing power. The desktops initially (think "Apple II") were unconnected to each other and did not print easily, except text. During the past ten years since the NCTM Curriculum and Evaluation Standards were written, desktops were joined by laptops and hand-held devices, especially scientific and graphing calculators in schools. And, with extraordinary abruptness reminding us of the unpredictability of technology, the Internet became the World Wide Web, leading to changes all across our society and economy. The changes involve the relations between schools and the resources of the wider society, changes in the forms of education possible (e.g., distance learning, virtual schools, etc.) and changes in the ways that educators might be supported in reform. In the last year or two, as the cost of computing power continues to fall and connectivity grows, new classes of small devices have appeared that amount to miniature computers that accept stylus input, can be connected to computers and hence the WWW, and in some cases, have phone and email functionality. The overall trend is away from the dominance of general purpose computers and towards greater distribution of computing power in more varied kinds of specialized devices, but with a premium on connectivity that provides access to resources and at-hand communication.

School and classroom versions of connectivity have taken the form of client-server systems to allow easier control and maintenance and growing access to the WWW. But the hand-held devices have remained unconnected to each other, and, except in limited ways, to computers. This connectivity limitation is about to be overcome.

Classroom Connectivity, the Next Major Step

Until the recent advent of Flash ROM technology into graphing calculators that allows input of software, the hand-held devices used in grades 6-14 have been stand-alone, closed boxes. In their opened up form, they are now analogous to the Apple II computer, but with a smaller screen. Indeed, as of this writing, much Apple II software is being rewritten for use on these devices. However, the larger change, just around the corner, involves linking hand-helds to each other and to computers inside the classroom. Previously, computers and graphing calculators had acted as competing technologies - classrooms were equipped with one or the other, but not both. But now they need to be viewed as cooperating technologies.

Now, imagine a classroom in which the teacher has a powerful laptop computer with a classroom display, and each student has a hand-held device that, under various circumstances mainly controllable by the teacher, can communicate with the teacher’s computer and with any other student’s device. The students may work as individuals or in small groups. The potential impact of the combination of new classroom activity structures, increased interaction intensity, and support for assessment and homework may be greater than the impact of connectivity outside the classroom. After all, this is change precisely where teach and learning take place.

I now list various categories of activity structures that exploit this configuration of computing resources. You are invited to use your imagination to fill in your favorite kinds of topic details.

 

  1. Publicly contribute to a shared object on the teacher’s computer
  1. Participate as an agent in a mathematically structured system or game (Stroup, Wilensky, in preparation) as described below
  2. Actively control a ‘remote’ resource (LBM, viewscreen, etc.)
  3. Publicly respond to downloaded challenges (uniform across the class or targeted to particular students) such as:

7. Guess my:

A variety of interactions between the computer and hand-held devices is possible, such as:

 

  1. Fancy simulation on the classroom display computer vs schematic simulation on the students’ hand-helds
  2. Data-generating computer simulation vs data-analysis on hand-helds
  3. Offloading the graphing or viewing of complex data from a hand-held for viewing or enhancement on a computer
  4. Grabbing data or other material (including curriculum) off a near or distant server with a computer and downloading it to the hand-held (includes filtered internet access)
  5. Assessment - downloading tests/quizzes and then uploading student data and capturing student activity
  6. Polling the class on thought-revealing questions as a base for discussion (i.e., ClassTalk-like work, where the class’s aggregated choices become the starting point for class discussion)
  7. Homework: downloading, collecting, grading
  8. Printing
  9. Classroom management

While some of the above are direct facilitations of common activities, others are new. While we do not yet know how this connectivity will play out in real classrooms (obviously, research is urgently needed), we can suspect that qualitative changes will result.

A Vignette Illustrating Connectivity, Heterogeneous Devices, and New Curriculum

Figure 3 offers a snippet of one particular activity for middle school students involving an extension of a typical MBL activity, where students import physical measurements of their own motion (using a sonar-based sensor) into a computer where they can be graphed and manipulated, and so on. In SimCalc MathWorlds software, one can use either a computer or a calculator version of this technology not only to graph the motion of a student (as in the position vs time graphs in the lower left part of the Figure), but to attach the imported motion data to a character in a motion-simulation - the Froggie character in this case. He can repeat the physical motion, say walking back and forth at varying velocities. But in addition, now imagine that each student in the class has a hand-held device on which she can create a graph for the motion of her own character, say a clown as in the Figure, where that character is to move in some relation to the Froggie, say in a Marching Parade (or perhaps a dance). The student can then upload that data to the teacher’s workstation where it can be aggregated with each other student’s motion data and displayed for the whole class to see. Each student is now represented by a character in the Parade! We have done this activity with middle and high school students using computers rather than calculators. The level of engagement and intensity is much higher than in a typical classroom situation since the students are participating in a shared object - the Parade - and they are using sophisticated mathematical representations to do so.

The cost of such a classroom setup, about $7000, is much less than a computer laboratory (in excess of $40,000): $75-100 per hand-held device (which in some cases the students purchase for themselves, reducing the classroom cost to perhaps $4500), $125 for the motion measurer, $1000 for the classroom network, $2000 for the teacher’s laptop, and $1500 for the display. Critically, such a setup, while an order of magnitude less in cost than a computer lab, can be part of everyday instruction in the teacher’s regular classroom. Moreover, the teacher’s workstation can be connected to whatever other networks are available, including the Internet, so material - including homework, quizzes, other data - can be passed between the students’ devices and the Net via the

Figure 3: A CBL Parade

teacher’s workstation-laptop (which, presumably, she can take home as needed). And student work can reported to the wider world - other students, parents, others.

The Issues of Interaction-Intensity and Integration with Practice

What is the level of interaction-intensity now in typical classrooms (with or without interactive, but unconnected, technologies)? My experience is that, by and large, it is much lower than it could be. But what would it be if a teacher were looking over students’ shoulders at the contents of their screens, as might be the case at least some of the time? Or what if the teacher could publicly display any student’s screen? Once everyone is connected in a classroom, the classroom becomes a different kind of entity, much more of an organism or system (Stroup & Wilensky, in preparation).

However, many unanswered questions arise, many of which will be answered in different ways as circumstances vary for a given teacher and across teachers and schools:

It is entirely possible that the real potential of technology in classrooms will only become apparent when the devices in classrooms are conveniently linked. It is worth recalling that the potential of technology in the business world to increase productivity was realized only after connectivity was achieved - when sales, marketing, distribution, accounting, management and perhaps even manufacturing, were linked in ways that supported full integration of activity (Sichel, 1997). Similar observations are possible regarding the worlds of communication and transportation. Think, for example, of the extraordinary mixes of highly integrated technologies associated with the airline transportation industry, most of which are barely visible: reservations, ticketing, payment, route scheduling, maintenance, traffic routing, balancing loads for take-off and landing, the actual operation of the aircraft, and so on - and all these are integrated into the other communication and economic systems operating in the country (and abroad). Contrast this level of integration with that in education. The difference is striking.

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