This page is a brief introduction to the theory, methodology, and results of the Escher's World project.

To learn about our most recent projects, go to the Epistemic Games website.

Escher's World
The development of mathematical thinking through design activity

For more detailed discussions of all of these issues, please consult the online papers from the project.

Since the turn of the century, educators have understood that the kind of expressive activities that characterize arts learning can play a fundamental role in education--and the idea of bringing "expression" to mathematics is appealing to students, parents, and others who feel disenfranchised by more "traditional" mathematics. And in recent years mathematics educators have called for the introduction of extended projects, group work, and discussions among students--elements of learning environments widely used in design or architecture.

Recent experiments with computational media and mathematics learning suggest that arts-based design activities, supported by appropriate computational tools, do provide a rich context for learning mathematics (Papert 1991, Willett 1992, Loeb 1993, Shaffer 1997). But this body of work has not yet addressed the basic issues of how design activities help students understand mathematics, and what role computational environments play in turning artistic explorations into mathematical insight.

"Escher's World" is a series of recent studies into design and mathematics learning at the MIT Media Laboratory. These studies brought middle- and high-school students to MIT to learn mathematics through design activities in a digital design studio. The goal of the research has been to understand how computer-aided design activities can make mathematics more accessible to students.

Framework

Escher's World brings together a number of theoretical perspectives on cognition and learning, including the long tradition of exploring how students can learn through creative activity. Parker and Dewey, for example, argued that self-expression is an intrinsic part of learning through exploration (Parker 1894/1969, Dewey 1958), and in more recent years, Papert has extended this line of thinking, suggesting that new media provide a particularly rich context for building understanding in the context of expressive activity (Papert 1980).

Escher's World grounds this theoretical perspective in the pedagogical tradition of the architectural design studio. The project explores how the structures of design learning can support the use of computational media to help students develop mathematical understanding. The design studio can trace its roots back more than a century to the Ecole des Beaux-Arts in France (Chafee 1977), and the focus of a designer's training still follows this tradition of open-ended projects and a variety of structured, collaborative conversations that culminate in a public presentation of work. From a cognitive perspective, Escher's World combines work on the social construction of knowledge with a constructivist perspective on individual development (see Vygotsky 1978, Gruber and Voneche 1995, Bruner 1996). Escher's World explores how expressive activity lets students draw on social resources (experts and peers) to help them build their own personal understanding of new phenomena through design activity.

Data

Escher's World has conducted a series of workshops in mathematics and design, the most recent being a four-week Summer Program for middle-school students leading up to the design and installation of an exhibit for the MIT Museum on mathematics and art. In all of these workshops, students have learned important concepts in transformational geometry through projects in graphic design in a technology-rich environment modeled on a traditional architectural design studio.

Data for the Escher's World projects has been collected in pre-workshop, post-workshop, and follow-up interviews, and in clinical interviews throughout the participants' design work Recordings of participants' design conversations were used to produce an illustrated "design record" for each participant, giving a detailed account of his or her work over an extended period of time.

The strength of the Escher's World results clearly depends on the richness of this qualitative data, but by dividing the participants' "design records" into individual "design episodes" and coding these episodes, it has been possible to use logistic regression to support the qualitative results with statistical analysis, and to explore patterns in mathematical learning across participants.

Results and Conclusions

A number of important practical and theoretical results have emerged from the Escher's World project thus far. More information about these results can be found in various published papers about the project.

The most recent work in Escher's World has shown that three features of a digital studio are particularly significant in supporting mathematics learning:

There is clearly more work to be done in identifying the important features of the studio as a learning environment, and in understanding how these features relate to one another, but these results suggest that expressive activities are a promising venue for thinking about deep mathematical ideas, and that the design studio is a useful model for structuring such activities.

References

Bruner, J. S. (1996). The Culture of Education. Cambridge, MA, Harvard Univ Press.
Chafee, R. (1977). The Teaching of Architecture at the Ecole des Beaux-Arts. The Architecture of the Ecole Des Beaux-Arts. A. Drexler. New York, Museum of Modern Art.
Dewey, J. (1958). Art as experience. New York, Capricorn Books.
Gruber, H. E. and J. Voneche, Eds. (1995). The essential Piaget. New York, Basic Books.
Jackiw, N. (1995). The Geometer's Sketchpad. Berkeley, Key Curriculum Press.
Loeb, A. (1993). Concepts and Images: Visual Mathematics. Boston, Birkhauser.
Papert, S. (1980). Mindstorms: children, computers, and powerful ideas. New York, Basic Books.
Papert, S. (1991). Situating constructionism. Constructionism. I. Harel and
S. Papert. Norwood, NJ, Ablex Publishing.
Parker, F. W. (1894/1969). Talks on Pedagogics. New York, Arno Press.
Shaffer, D. W. (1997). "Learning mathematics through design: the anatomy of Escher's world." Journal of Mathematical Behavior 16(2).
Vygotsky, L. S. (1978). Mind in Society. Cambridge, MA, Harvard University Press.
Willett, L. V. (1992). The efficacy of using the visual arts to teach math and reading concepts. 73rd Annual Meeting of the American Educational Research Association, San Francisco.