ABSTRACT. 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.

My task is to reflect on the papers in the two Special Issues on "Representations and the

Since the publication of this chapter, Jim Kaput was killed in a tragic road accident. We have all lost an energetic, visionary and dedicated colleague. Mathematics Education research has lost one of its greatest exponents, a researcher who not only understood the world of education, but knew how to change it. His theoretical work on the evolution of notational systems – much of it presented by him in this article – as well as his practical contribution, most recently through SimCalc, is living testimony to the importance and impact of his work. And we have lost a friend: a friend with a wonderful sense of humour and wit, a vibrant sense of fun, and an inspired intelligence. We will miss him. CH/RN Not for the first time we are at a turning point in intellectual history. The appearances of new computational forms and literacies are pervading the social and economic lives of individuals and nations alike. Yet nowhere is this upheaval correspondingly represented in educational systems, in classrooms, or in school curricula. As far as mathematics is concerned, the massive changes to

If design research involving technology-based tools is going to impact educational settings, the design process must be extended beyond the tool itself to encompass a broader range of factors such as the classroom social structures (e.g., beliefs about learning and knowledge, learning activities and participant structures, configurations of both physical space and cyberspace). Although prior research has underscored the importance of classroom social structures in technology integration, it has failed to specify the critical design variables that must be taken into account. Only by understanding the critical variables involved is it possible to develop a deep understanding of how and why things work. The Social Infrastructure Framework systematically frames the critical design elements in terms of 4 dimensions: (a) cultural beliefs, (b) practices, (c) socio-techno-spatial relations, and (d) interaction with the “outside world. ” This article details the design issues associated with each dimension based on examples drawn from a range of educational technologies. This article also describes how the framework can serve to advance the methodology of design research by serving as a tool for both design and analysis. Since the 1990s, starting with papers by Brown (1992) and Collins (1992), there has been a growing movement to develop a new methodology for carrying out studies of educational interventions under the labels design research, design experiments, anddesign studies. Various researchers have contributed to the effort

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

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