James J. Kaput

The Early Development of Algebraic Reasoning: The Current State of the Field

The main aim of this chapter is to argue that an early start to algebra education is possible and of great relevance for mathematics education because it provides a special opportunity to foster a particular kind of generality in our students’ thinking. To argue this, we map the various views on algebra education found historically, and trace how the perceptions that mathematics educators hold about children’s thinking and learning have changed. Overall, a great realisation that children can do more in mathematics than was previously believed leads to the adoption of more ambitious objectives for the initial years of school, and to the development of new classroom approaches to algebra education in the early grades. That does not mean teaching the same old school algebra in the same usual way to younger children, but rather to introduce them to new algebraic ways of thinking and immersing them in the culture of algebra. The chapter ends with a research agenda to further developments in this particular sub-field of mathematics education.

Mathematics and Virtual Culture: an Evolutionary Perspective on Technology and Mathematics Education

This paper suggests that from a cognitive-evolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to ‘represent’ events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a ‘virtual’ culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills.

Rethinking calculus: Learning and thinking

1. REMODELING CALCULUS THE INSTITUTION. Surely the renewal of Calculus is a good idea, one good enough to attract the attention and energy of many good people. But this is Calculus the Institution that peculiarly American academic event and all its supporting structures and expectations. Professor Knisely, however, barely hints at matters of institutional implementation, so I conclude that he is addressing Calculus, the System of Knowledge and Technique. As such, his paper is, perhaps, a warm-up exercise to a deep and long overdue reconsideration of the appropriate intellectual content of Calculus, one that has been postponed while we attempt to remodel Calculus the Institution. This remodeling has proven to be an arduous task for two reasons: (1) the renovation is taking place whilst the owners and stakeholders continue to inhabit the institution (a constraint applying to most educational reform); and relatedly (2) we have left all the larger structural features of the institution intact, including those features that connect it to the outside world, e.g., to the rapidly changing K-12 education. The basic architecture and its place in the larger world are untouched. I suggest that we embark on the more fundamentaJ rebuilding towards which Knisely points. In so doing we need to come to terms with the relations, existing and possible, between Calculus the Institution (C-INST) and Calculus the System of Knowledge and Technique (C-KNOWL). And we need to look more deeply and critically at the assumptions, largely tacit, that hold the status quo in place and provide some concrete, implementable alternatives.

Exploring Difficulties in Transforming between Natural Language and Image Based Representations and Abstract Symbol Systems of Mathematics

Students who do very well within mathematics proper often have great difficulty in connecting the formal mathematics with the wider world of experience, as is required in doing “word problems” in algebra. These difficulties seem not to be rooted in their quantitative understanding of events or relationships in the natural world, but rather in the translation between those content-based representations and the formal systems of mathematics.

On the Development of Human Representational Competence from an Evolutionary Point of View

algebra began to emerge. Over the space of a few centuries, mathematics was loosening its tethers to material reality. Paradoxically, at the same time, of course, mathematics was being used to create an entirely new set of extraordinarily powerful models of the material world. This divergence of purpose gradually led to the fissure separating mathematics from science, and was an instance of the knowledge specialization that has marked western science since the Renaissance. But within this newly freed mathematics, the idea of a logically consistent system independent of any kind of reality took hold, and, indeed, a notion of mathematics as a formal system defined only by logically consistent actions on symbols was put forth by Hilbert and others around the turn of the century—the formalist view. While the logical foundations of the formalist view of mathematics as a whole were undermined by Goedel’s work, the idea of formalism and of a formal system not only survived, but has become an essential feature of the mathematical landscape. The idea that one could define well-formed formulas and explicit rules for their transformation set the stage for the idea of a computer program, made explicit in somewhat different ways by Turing and von Neumann (Von Neumann, 1966, Turing, 1992). While the idea of universal (as opposed to numerical) computing machines and logic machines goes back to Leibniz and even earlier, the underlying intellectual infrastructure was not available to render it viable until well into the twentieth century. Of course pragmatic factors, both military and commercial, as always seems to be the case, drove the actual physical realization and early applications of computers. But now the computations could be designed by a human, but executed independently of a human! (It should perhaps be pointed out that Von Neumann conceived of computers that could design themselves, and, more recently in the 1970’s, John Holland (1995) developed the idea of genetic algorithm, wherein the program modifies itself across iterations by way of random mutations of its operation strings, yielding a new level of processing autonomy.) The human could now interact with the model, even change it “on the fly,” but its underlying computations could be executed autonomously of the biological mind rather than in direct partnership with the biological mind as was the case with the previously discussed action notation systems. Moreover, the success of mathematics as a means of modeling aspects of experience—not merely the physical world—had validated not only the utility of many different mathematical systems (e.g., non-euclidean geometries), but the idea of an abstract, formal model itself, one with no necessary connections to anything else. Once computers were available within which to instantiate those systems, the freedom to construct and explore such systems led to an explosion in the use of computer models, especially simulation models, and deep changes in the nature of the scientific enterprise (Casti, 1996). Space limits discussion of the kinds of models now possible, but we must acknowledge that, particularly through the exploration of dynamical systems, an entirely new view of the world is emerging (Heim, 1993; Cohen & Stewart, 1994; Hall, 1994; Holland, 1995; Kauffman, 1995; Casti, 1996; Resnick, 1994). Two other, related, innovations feed the process of creating a vitrual culture. One is the connectivity revolution, currently in the form of the World Wide Web and in local networks, but soon to take the form of more flexible “just-in-time connectivity.” This allows the widespread sharing of data, analyses, and, most especially, models and simulations—including the collaborative manipulation of such models, and a rapid distribution of new insight and modifications. The second innovation involves the feeding back upon itself of the computation processes to form new visual means for the presentation of models and simulations and new ways to interact with them. In particular, it is now possible to design and build human-computer interaction systems that take advantage of the highly sophisticated physical and perceptual competence of human beings. Hence it is possible to create manipulable worlds with increasingly arbitrary “reality”—but without the constraint of physicality (Kaput, 1996), particularly with freedom from the time and size scales of the physical world. The nature of modeling has both changed and been democratized in the sense that one need not be a programmer or mathematician to use models and simulations profitably. In the face of these changes, we are being forced to reexamine the ideas of mathematical abstraction, idealization, and even the psychological idea of abstraction (see Nemirovsky 1998; Noss and Hoyles (1996; and Wilensky (1991). Briefly, as these authors variously suggest, we may need to make room in our notion of mathematical understanding for a kind of “concrete abstraction” that builds mathematical meaning “additively” as an active web of meaningful associations rather than “subtractively” by deletion of elements and features. 4.4. Comparisons to Prior Stage-Transitions The hominds and their episodic mind were of their world. They did not model it in any explicit way and changes were extremely slow because they depended on physical evolution. The mimetic mind, millions of years later, began the process of building autonomy, a separation from their world that was both the basis of symbolic reference and the beginnings of self-initiated practice with the means of modeling actions and experiences, and communication. The possibility now existed for feedback cycles within which the individual could intervene. With spoken language and the mythic culture, ever more comprehensive narrative stories about the world became possible, and with them appeared new forms of experience and meaning, new ability to effect change in others and in the physical world, and new forms of knowledge. Change became even more rapid as feedback cycles tightened and more knowledge could be shared more widely. The move to writing broke the limits of the biological mind, provided external resources for mental activity, both memory and processing. Even the process of thinking, at least the stylized oral aspects of it, could be externalized and made available to be shared and improved by others and even across generations, enabling even more rapid cumulativity and reshaping of knowledge than that begun by the Greeks. At the same time, the process of demythologizing and secularization of human experience into the theoretic culture continued and continues today. The “heavenly bodies” became celestial objects that move according to human-specifiable rules, the earth became just another celestial object, the human body became a subject of study and the heart an organ, humans were recognized to be yet another species, the mind became subject of study, the societies we live in became subjects of study, the idea of “life” has become yet another formalism, and even the process of knowledge building, even model building, became a subject of study. The induction into the symbolic forms and the products of the use of those symbolic forms became an increasingly important part of individuals’ development, requiring new institutions and methods—the idea of education. Importantly, education, while mediated by written material, maintained its goal of producing sophisticated speakers for more than 3000 years—should we be surprised that changes in mathematics education require generations, and that we seem to be educating people for the past? Within the past 300 years, change has accelerated. In particular, the focus of education shifted from narrative and the classics to the new products of the theoretic culture. As our means of understanding—rapid, shared modeling and simulation, for example—become incorporated into the processes of education, we can expect change to accelerate even more. The book will be supplemented by the simulation as the primary intellectual objectand the learning feedback loop will be both enriched and tightened. The reader is also invited to examine Shaffer and Kaput (in press) for more detailed discussions of the implications of these changes for mathematics education. In the plenary discussion I will offer some concrete illustrations. 4 This characterization was offered by David Shaffer (personal communication)

Behavioral Objections: A Response to Wollman

That behaviorist horse, although badly flogged, continues to stalk among us. The well-executed Wollman (1983) study provides a clear example of how psychological and educational behaviorism remains at work influencing both research objectives and methods-even among researchers who are patently not behaviorists. Now, however, one must look beneath the patina to find its effects. Wollman used a mix of interview and written measures in a series of six