In the world beyond school, new technologies are fundamentally changing the mathematical and scientific understanding that practitioners need in fields that use mathematics, science, and technology. This study looks at one such field, architectural design, and at the nature of problem-solving in this real life situation. In particular, I ask: What does it mean to “understand” in the context of architectural design? How does this form of understanding develop? And what kinds of experiences facilitate the development of this form of understanding?

This paper is part of a small but growing literature on how professional practices and professional training can inform the creation of learning environments for younger students (Cossentino, 2002; Cossentino & Shaffer, 1999; Erickson & Lehrer, 1998; Hmelo, Holton, & Kolodner, 2000; Jacobson & Lehrer, 2000; Kafai, 1996; Kolodner, Crismond, Gray, Holbrook, & Puntambekar, 1998; Penner, Schauble, & Lehrer, 1998; Resnick & Ocko, 1991; Schon, 1985; Shaffer, 1997c, 2002a, 2002b, 2003, 2004b; Stevens, 2000). The paper describes one particular project in this genre, Escher’s World, as an occasion to explore one of the theories that underlies such work, the theory of pedagogical praxis (Shaffer, 2004a). Pedagogical praxis suggests that new technologies make it possible to take pedagogies developed in the context of professional training–pedagogies that typically emphasize participation in meaningful projects in epistemologically rich contexts and adapt them for younger students. That is, pedagogical praxis suggests that the practices through which professionals are trained can provide constructive models for helping students learn from participation in personally relevant projects using computational microworlds. In the discussion that follows, my goal is to use Escher’s World as an occasion to examine some of the mechanisms that underlie learning through pedagogical praxis. My intent is not to prove that Escher’s World was a ‘successful intervention.’ It would hardly come as a surprise that students would learn some mathematics after participating in 56 hours of design activity in a mathematical microworld, and previous work (Shaffer, 1997b, 1997c, 2002b) has already shown that students can and do learn mathematics through microworld-based design activities similar to those in Escher’s World. Rather, my aim here is to uncover some of the processes through which middle school students developed mathematical understanding in a computational microworld while engaging in activities based on the practices through which designers are trained. I focus particularly on the interactions among three precursors to the development of mathematical understanding in Escher’s World: (a) enactment of specific participant frameworks from the design studio, (b) the autoexpressive properties of the computational tool being used, and (c) the articulation and transformation of students’ own interests through their design work.

In this article, I propose a theory of pedagogical praxis. Pedagogical praxis begins with the premise that under the right conditions, computers and other information technologies can make it easier for students to become active participants in meaningful projects and practices in the life of their community and suggests that professional practices such as architecture, mediation, and journalism can provide constructive models for helping students learn from such experiences. In this vision, new technology reinvigorates Dewey’s (1915) idea of linking school with society. Technology builds a bridge that allows young people to participate in the learning practices of professionals; in the process, they develop epistemological frameworks that organize the skills, habits, and understandings they need to thrive in a complex, postindustrial society. Although further work needs to be done to explore the processes through which such learning can take place, studies suggest that this perspective may be a productive avenue for continuing research. This article presents an overview of the theories and methods that inform such work.

Building on the theory of islands of expertise developed by Crowley and Jacobs (2002), in this paper I develop the concept of epistemic frames as a mechanism through which infusion environments can help students use experiences in one context to help them deal with new situations. I describe epistemic frames as the ways of knowing, of deciding what is worth knowing, and of adding to the collective body of knowledge and understanding of a community of practice (Lave & Wenger, 1991). I use data from two design experiments to extend the concept of islands of expertise, showing how the ability of students to incorporate epistemic frames into their identities suggests a mechanism through which infusion experiences and other rich learning contexts may support activity in novel situations.

Pedagogical praxis sets out a rich and ambitious agenda of research and reform. The goals of this paper are (a) to describe the basic theoretical claims of pedagogical praxis, (b) to explain the experimental methods that explore these claims, and (c) to present examples of technology-based learning environments that exemplify this process in action. As such, this paper serves as a prolegomenon: an overview of a larger argument. The examples and evidence are intended not to validate the theory of pedagogical praxis, but rather to clarify the kinds of claims it implies and the methods through which these claims are being investigated: to describe the shape and scope of this line of inquiry and to suggest how and why it may be a fruitful area for continuing research.

Images of pedagogy are often formed with images of practice in mind. In recent years, design and the practices of designers have been used as models for experimental learning environments and progressive curricula (see, e.g., Cossentino & Shaffer, 1999; Erickson & Lehrer, 1998; Greeno & the Middle School Mathematics Through Applications Project, 1997; Hmelo, Holton, & Kolodner, 2000; Jacobson & Lehrer, 2000; Kolodner, Crismond, Gray, Holbrook, & Puntambekar, 1998; Loeb, 1993; Perkins & Blythe, 1994; Shaffer, 1996). As in many endeavors, though, images of design work and actual enactment of design practices are not necessarily the same: what we see on casual observation is not always the same as what we understand from closer study. Accordingly, interest in design as a pedagogical model has produced a number of studies of design practices, particularly of architectural design, which is often seen as a canonical exemplar of the design process.

The analysis in this paper looks at how design practices were enacted in one design studio, focusing on the relationships among surface structures of the studio, the activities that those surface structures supported, and the view of knowledge that those activities fostered. The study is thus a structural ethnography rather than a hypothesis-driven or micro-genetic account of learning: it attempts to describe phenomena that are local, rather than directly generalizable, but that operate at a relatively broad scale–and thus appear through observation at a more intermediate level of analysis. This is ethnography closer in spirit and practice to Geertz’ (1973a) well-known analysis of conventions for time and identity in Bali than it is to Cobb’s (1986) case study of the emergence of abstract mathematical thinking in the concrete activities of one learner.
In a field such as design, where much is already known about specific cognitive and pedagogical processes, such a structural analysis is useful in extending our understanding of the systemic nature of activity. The analysis of the Oxford Studio that follows suggests that expression and expressive activity are a significant underpinning of the design studio system. To the extent that elements of design practice are interconnected, the study also suggests key features and relationships of the studio that should ideally be preserved in any adaptation of design to the creation of learning environments in other fields.

Since the writings of Francis Parker and John Dewey (Dewey, 1915; Parker, 1994/1969), educators have been excited by the possibilities of learning through design activities. The introduction of computational media to education has made this idea only more appealing, as educators see how computers make it possible to explore more areas of human understanding in an open-ended, design-based environment (Kafai & Harel, 1991; Noss & Hoyles, 1996; Papert, 1980, 1993; Resnick & Ocko, 1991; Wilensky, 1995). One important issue in the open-ended approach of learning-by-design is the need to provide students with skills to regulate their learning activities effectively (see Dewey, 1938).

The studies discussed here have been presented elsewhere (see Cossentino & Shaffer, 1997; Shaffer, 1996, 1998, in press). These earlier discussions have looked at the “success” of linking mathematics and design, and at the role that new media play in creating such a link. Here the focus is on the pedagogy of the design studio, and in particular on the connections between autonomy and collaboration in design learning. The question is whether the systems of the deign studio and the power of new media can help mathematics students take control of their mathematical development and successfully mobilize the social resources they need to understand deep mathematical ideas.