## Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education (1999)

Venue: | Educational Studies in Mathematics |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Shaffer99mathematicsand,

author = {David Williamson Shaffer and James J. Kaput and David Williamson Shaffer and James J. Kaput},

title = {Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education},

booktitle = {Educational Studies in Mathematics},

year = {1999},

pages = {97--119}

}

### OpenURL

### Abstract

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.

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Citation Context ... individual and social experience, and new cultural forms. Nor can we reliably predict the dynamics of evolution that will occur in the virtual culture, and what new representational forms may arise (=-=Dyson, 1997-=-)—after all, who really predicted the WorldWide Web? It seems clear, however, that in a virtual culture students will have new ways of sharing new forms of mathematical experiences, mathematical repre... |

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Citation Context ...nt artistic expression within formal systems (as, for example, in image-manipulation tools such as Adobe PhotoShop) means that students can explore mathematical and aesthetic concerns simultaneously (=-=Shaffer, 1997-=-). Part of the power of a computer is that a sufficiently fast formal process can take on artistic dimensions—just as a sufficiently fast static image can take on dynamic qualities, as in a motion pic... |

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Citation Context ...scribe, much less investigate. Computational media also create new representational forms, such as dynamic geometry environments or manipulable Cartesian graphs separate from algebraic notations (see =-=Kaput & Roschelle, 1998-=-). Such new representations enable students to work on problems using different cognitive modes, allowing students, for example, to take more concrete approaches to abstract problems (see Turkle & Pap... |

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Citation Context ... calculator. Or use iterated estimation with a spreadsheet. Or use a geometric representation in a dynamic geometry environment. Or use a motion simulation. Or use symbolic manipulation software (see =-=Fey, 1989-=-). This representational pluralism (see Turkle & Papert, 1990; Papert, 1993) suggests that one of the key changes in mathematics education will be a move away from the traditional focus on algorithmic... |

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Citation Context ...go. The recordkeeping needs of commerce and astronomy drove the creation of external symbol systems (p. 333ff), of which mathematical notations were probably the first (Donald, 1991, p. 287; see also =-=Schmandt-Besserat, 1978-=-, 1992, 1994) 5 The existence of external representations, according to Donald, made it possible for humans to begin reflecting on the interrelationships among recorded ideas in an analytic fashion. D... |

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Citation Context ...t were difficult or impossible to model using non-computational methods. In particular, computational tools provide visual, manipulable models of dynamical systems (Heim, 1993; Cohen & Stewart, 1994; =-=Hall, 1994-=-; Holland, 1995; Kauffman, 1995; Casti, 1996). These models approach complex, often non-linear phenomena with iterative, dynamic representations—representations that can only produce meaningful result... |

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Citation Context ...d sensory information would surely complete a “virtual” culture. A point of view strongly arguing for the importance of more direct experience and less heavily mediated experience is offered by Reed (=-=Reed 1996-=-). 13 Given the relative referential “senselessness” of the intermediate steps in an algebraic solution, the process of algebraic manipulation usually requires the temporary suspension of referential ... |

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Citation Context ...ng numerals. The base-ten placeholder system of numerals ands12 DAVID WILLIAMSON SHAFFER AND JAMES J. KAPUT the algorithms build upon it have been a critical factor in the development of our culture (=-=Swetz, 1987-=-). A prodigious advance in the development of mathematics was the creation of another, more general and therefore more powerful set of algorithms for representing and manipulating quantitative relatio... |

3 |
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Citation Context ...n—that come about as a result of the development of externalizable algorithmic thinking.s16 DAVID WILLIAMSON SHAFFER AND JAMES J. KAPUT 6.1. New Representational Forms Papert and Kaput (Papert, 1980; =-=Kaput, 1992-=-) among others have written about the way in which computational media make it possible to externalize algorithms and thus make processes of thinking available as explicit objects for reflection. Whet... |

3 | Technology, curriculum and representation: Rethinking the foundations and the future - Kaput - 1996 |