Sarah Inkpen, Professor
Seneca College of Applied Arts and Technology,
http://www.oise.utoronto.ca/~sinkpen/SARAH.html


CYBERSPACE CALCULUS CARNIVAL

"I would rather be a cyborg than a goddess." --Donna Haraway




Abstract

Rapid developments in technology have resulted in major changes in society, both in what is done and how it is done. Educators need to be challenged, to sense the excitement and opportunity of the digital age and to be exposed to visions of what may be possible. Educational technology, more specifically virtual reality, holds great promise in the quest of enhanced learning. As visual images, texts and sounds circulate in cyberspace, we may expect a thorough exteriorization of knowledge and a dramatic transformation in curricula and in instructional processes. By integrating html, vrml and java script into the environment many students can actively inhabit an inclusive computer-generated environment. The presentation will include a prototype of a virtual environment designed to have students interact with basic calculus concepts such as rotation of solids, and centre of gravity. This assimilation, transforming previously abstract mathematical concepts into dynamic and manipulable objects , will give students an opportunity to create a firm foundation for further discovery and experiential learning.


In the realm of education, technology offers educators the opportunity to move away from instructional strategies that focus on presentation of abstract information to the passive learner to an active process in which meaning is developed on the basis of experience. In the constructivist view, the learner is building an internal representation of knowledge and a personal interpretation of experience. No meaningful construction nor authentic activity is possible if all relevant information is pre specified. What is meaningful is the development of learning environments which encourage construction of understanding multiple perspectives. This is in contrast to the typical school environment where the goal is to transfer knowledge to the learner in the most efficient, effective manner possible. The mind does not "process" information, it constructs it based upon past experience and ongoing interactions in the world. In school we are generally taught what to think rather than how to think. Instruction does not cause learning!


Creativity or the building of thought patterns is the ability to use visual or previous experience to solve problems never encountered before. Integration of technology into the calculus curriculum has totally changed what and how we teach. Previously, math had become a series of templates and algorithms with very little relevance to the world outside of school or in the advancement of higher order thinking skills. With our rapidly changing society, it is essential that educators empower students to be life long learners; aiding them in knowledge building, and encouraging collaborative work. Inclusion of interactive math labs, graphing calculators, and multimedia animation's have all been towards this goal of visualization and knowledge building.

Virtual reality, an environment created by the computer in which the user feels present, is an emerging computer-based technology that offers promise as a learning tool. It is a medium for communication and creative expression as well as a tool for simulation and model building. Because virtual reality emphasizes multisensory, multidimensional information-presentations, it offers a wider range of representational and presentation tools, bridging many disciplines and providing a more powerful synergistic learning tool. A virtual learning environment designed to have students interact with basic calculus concepts will help facilitate understanding, transferability and knowledge building. This assimilation, transforming previously abstract mathematical concepts into dynamic and manipulable objects , will give students an opportunity to create a firm foundation for further discovery and experiential learning in calculus. The difficulty will be that the students will have to override the learned experience of using algorithms or memorized patterns and trust in their knowledge building skills.

The 'Cyberspace Calculus Carnival' is truly the bare foundation for future work. The first pavilion, The HoloMirrors, is a virtual reality environment filled with two dimensional graphs, axis, and solids of rotation. This prototype will give students the opportunity to 'experience' not only the newly formed shapes but the concept of volume found by filling them with disks, shells or washers.




CYBERSPACE CARNIVAL
The choice of a math carnival, as the theme, is deliberate. If we hope to show the power and applicability of calculus we need to put it in a different, fun context. Here are some possible additions:

* A Mathematics Reference Room, including biographies of famous mathematicians, internet resources, software resources, relationships to other areas such as art, philosophy, and music.

* A House of Horrors where major math errors could be displayed--for example canceling over addition or integrating over a discontinuous function.

* The midway could have thrilling rides defined by parametric equations -- some with gravity and some without.

* Permutations and combinatorics could be illustrated by numerous games of chance and shooting galleries could be designed to require that the 'bullet' travel a tangent line in order to hit the target.

* A playground including slides following trigonometric curves, teeter totters using centre of gravity concepts and swings to illustrate related rate problems and pendulum problems.

Although limited by the restrictions of the software, students could still see the numerous applications to mathematics in the real world. Students could even define their own space with their own given rules. For example, William Bricken's abstract algebra where blocks are assigned certain attributes. Many of these concepts have been difficult for most students to comprehend. Virtual reality may give a concrete example which will aid in further development of these theories.

CONCLUSION

Technology cannot effortlessly transform education. "It is clear the technology in and of itself is not a magic remedy for current failures in educational effectiveness." [Herman, 1994] The Futurist, David Thornburg, addresses this issue in his book entitled Edutrends 2010. "How do we plan for the future when technology is advancing by leaps and bounds? Our answer to this question is that we first identify the solid curricular and pedagogical ideas that we believe in..." [Thornburg, 1992] It is from this vantage point, that I view Virtual Reality's contribution to mathematics education. It allows the students to go beyond 'book knowledge' and actually develop a feel for advanced mathematical concepts and relationships.

"As virtual reality evolves in its ability to comprehensively engage a user in an immersive reality embedded with valuable lessons, it will help to build a new means of learning by which humans and technology are optimized according to the nature of the lesson and the type of learning to be realized. As an ultimate learning technology, virtual reality will also serve to help redefine the role of educators." [Traub, 1992, p.10]

Virtual reality environments may make it possible to discover more about the very process of learning. However educators must participate in its development so as to guide the growth of the technology and perhaps influence the course of educational change. How do we as educators harness this powerful new technology to support learning?

BIBLIOGRAPHY

Animated Gif's

Duderstadt, J. (1994).
The University of the Twenty-first Century. In Key Note Address, American Society for Information Science, Portland, Oregon.


Haraway, D. (1985).
A Manifesto for Cyborgs: Science, Technology and Socialist Feminism in the 1980's. Socialist Review, 80, 101.

Herman, J. (1994).
Evaluating the Effects of Technology in School Reform. In B. Means (Eds.), Technology and Education Reform (pp. 133-167). San Francisco: Jossey-Bass.

Thornburg, D. (1992).
Edutrends 2010 Restructuring, Technology and the Future of Education. San Carlos: Starsong Publications.

Traub, D. (1992).
The Promise of Virtual Reality for Learning. In ICAT'92 The Second International Conference on Artificial Reality and Tele-Existence, Tokyo.