I would like to submit the following proposal for consideration of a Mellon Teaching with Technology Fellowship for the Summer of 1998. The focus of this proposal is to develop teaching materials for use in investigating the relationship between symmetry and group theory. I have previously developed two interactive computer tutorials under a Keck Grant for Multimedia in the Sciences. These tutorials use moving images to describe the symmetries of various plane geometric objects. The student is asked to connect these concrete visualizations to the more abstract notions of group theory. These tutorials were used in a mathematics course in 1996 and 1997, and the results were extremely encouraging.

I propose to continue to develop this work. In particular, I will add tutorials concerning three-dimensional symmetry groups, frieze groups and crystallographic groups. Additionally, I plan to include more interactive materials. All of these materials will be formatted in a version suitable for delivery across the World Wide Web.

PREVIOUS WORK

The first of the tutorials introduces the notion of symmetries in the plane, describing the basic geometric ideas behind symmetry groups. The second tutorial gives a thorough treatment of the symmetry group of an equilateral triangle, D3, including the construction of a complete multiplication table and investigation of the subgroup structure. Both tutorials were designed with student interaction as a major component.

The effectiveness of these tutorials can best be communicated by the student responses. Upon constructing the multiplication table for D3 and comparing it to the table for S3 (a group with which they were familiar), the students were able to gain an understanding of the notions of isomorphism and subgroup. They then proceeded to generalize the dihedral group D3 to Dn, for arbitrary n, and discovered for themselves both the structure of the subgroups and the variance of the axes of reflection with the parity of n. The students were clearly developing an intuition into these groups, enhanced by the visual representation of the tutorials. Indeed, after I presented these results at the meetings of the Mississippi Academy of Sciences, one mathematician said he was "fascinated" by the student response. It is with this type of response in mind that I wish to continue to work at enhancing the visual aspects of group theory.

FUTURE WORK

As indicated above, I propose to continue developing materials designed to effectively allow the student to visualize the symmetries of various objects, and thus to better understand the underlying group structures. The plane symmetry tutorials mentioned above will be enhanced with additional interactive materials. For example, the student will be asked to discover the symmetry groups of various planar objects (e.g. pinwheels, snowflakes).Secondly, I would like to develop a fairly comprehensive treatment of three-dimensional symmetry groups. This would include moving images of regular solids such as the cube and tetrahedron, along with various other solids (e.g. soccer balls, prisms). Once again, interactivity and a focus on the relationship between the geometry and the group structure would be emphasized.

The third component of the project would be a geometric investigation of the frieze groups and crystallographic groups. These groups involve symmetries of infinite patterns, and can be studied using wallpaper patterns or Escher-like drawings.

TECHNICAL REQUIREMENTS

The original tutorials were developed using Macromedia's Authorware software package. Moving images were produced using Caligari's Truespace software. These tutorials can be viewed on the World Wide Web (using the Shockwave for Authorware plug-in) at http://www.millsaps.edu/www/keck/shock/wick/wick.html.

I have found the Truespace software to be a suitable package for creating the moving images. I have a working knowledge of this package, and have also identified several students capable of providing assistance in this area.

I am presently considering packaging the materials in HTML format instead of using Authorware. The tutorials would be written in HTML code with links to the video files, which will most likely be converted to RealMedia format. However, I continue to search for more current and effective means of delivery.

I have obtained permission to locate the materials on a local server here at Millsaps, and they will be available to anyone on the World Wide Web.

USE AND ASSESSMENT

The new materials will be first used in Abstract Algebra in the fall of 1998. After the students have thoroughly studied permutations, the notion of an abstract group is presented. It is at this point, about 4 weeks into the course, where numerous examples allow the student to connect the abstract concepts to more concrete objects. It is here that the tutorials are first used. However, understanding of the frieze and crystallographic groups requires more development of the mathematics, and hence these tools will not be used until later in the course.

I have used the original materials in class the previous two years and have devised a student evaluation form specifically for the tutorials. The results of these evaluations have been very positive. This form (or a revised version) will again be given to the students after they have completed the tutorials.

Additionally, professional and peer evaluation will be sought. An evaluation form will be included on the web site, and users will be asked for their analyses. Also, I have received and will continue to seek feedback from my colleagues in the mathematics and computer studies departments.