Pedagogical Principles for Teaching Art in Mathematics Courses

Year: 2005 Authors: Russell Jay Hendel

Core claim

Clock-pattern tutorials and interview-guided teaching strategies improve recognition of frieze symmetries for students with limited mathematical proficiency.

Topics

frieze patterns, symmetry classification, pedagogy, student interviews, cultural pattern analysis

Domains

group theory, symmetry groups, plane patterns, textile patterns, pottery patterns, decorative design

Methods

student interviews, computer tutorial, algorithm design, teaching strategy

Media

clock-motifs, frieze patterns, screen-based tutorial

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Russell Jay Hendel Mathematics Department Towson University 8000 York Road Towson, MD, 21252, USA E-mail: RHendel@Towson.edu

Abstract

We use the one-dimensional frieze patterns to illustrate basic principles of pedagogy in teaching mathematical applications to art. Five pedagogic principles are used: (a) Creating a model of student cognitive process, (b) defining a specific problem domain, (c) developing algorithms, data structures and notations representing this model, (d) creating a teaching strategy based on the algorithms and data structures, (e) requiring problem richness. We show how student interviews facilitate the development of concepts, which are pedagogically useful in enabling students with limited mathematical proficiency to appreciate the mathematics and frieze patterns of various cultures.

1.1. Goals. This paper explores five pedagogic principles useful in facilitating the presentation of applications of mathematics to art. The methods presented were used to develop a two week module in an introductory liberal arts mathematics course taught to non-mathematics majors, some having extremely limited mathematical proficiencies. The art module showed how applying the theory of symmetry classification to cloth and pottery patterns could lead to a definition of cultural diversity [1].

1.2. Review. A set of points, $\mathbf{P}\subset {\mathbf{R}}^2$ is a frieze pattern if $\mathbf{P}$ contains a bounded subset $\mathbf{M}$, the motif, such that $\mathbf{P}$ equals the union of translates of $\mathbf{M}$ by a fixed vector (which we assume is in a horizontal direction). A rigid motion of a pattern is a symmetry of the pattern if the pattern remains invariant under the rigid motion. There are five symmetries possible for frieze patterns: translation, reflection in a horizontal mirror, reflection in a vertical mirror, rotation by $180^{\circ }$ and glide reflection. The collection of symmetries of a pattern is the symmetry group of the pattern. There are exactly seven distinct symmetry groups of frieze patterns.

1.3. Student Cognitive Processes. The five pedagogic principles laid down in [3],[4], and [5] guided the art module of the introductory liberal arts mathematics course, developed over a four-year period. During the first year, interviews were conducted on students who came during office hours for help; we assumed that the other students understood the material. The goal of the interviews was to clearly define problem areas as well as to identify teaching strategies that worked well.

To illustrate the issues, consider a pattern $\mathbf{P}$ consisting of a sequence of adjacent equilateral triangles alternately pointing up and down, and the modification $\mathbf{Q}$ of this pattern that adds dots to two vertices of each triangle, as shown below.

P $△\nabla △\nabla △\nabla △\nabla$

Q $△\nabla △\nabla △\nabla △\nabla$

Q only possesses translation and glide symmetries while P possesses additional vertical reflection and rotational symmetries. We could summarize by saying that the motif of Q contains two themes—the triangle theme and the dot theme, while the motif of P contains just the triangle theme. Multiple-theme motifs increase pattern recognition error in the weaker students. However, student interviews revealed

that coaching to identify the ‘clock positions’ of various pattern items, such as dots and vertices, provided students with a computational tool that they could use to verify the presence or absence of symmetries.

A problem also arose with respect to notation. Although flowcharts for naming the frieze patterns [2] are straightforward to use, non-technical students found use of these cumbersome. They could not, for example, identify several frieze patterns in a minute or two. Student interviews revealed that identification of patterns became quick when the students simply tested each frieze pattern for each of the five possible symmetries.

1.4. The problem domain. The interviews suggested defining the problem domain of clock-motifs and clock-patterns. A clock-motif consists of several rays emanating from a common center, possibly with varying lengths—for example, Y, V, X, I. These produce clock patterns by translation.

1.5. Algorithm. We created a computer tutorial that presented translations of randomly created clock-motifs on a screen and allowed the student to guess its symmetry group an unlimited number of times before proceeding to the next example [3].

1.6. Teaching strategy. We present four useful teaching strategies that arose from the student interviews and resulting model of student cognitive processes. (a) Student identification of patterns was greatly facilitated by identifying clock positions (in minutes) on the patterns. (b) The following computational theorem was provided to check on symmetries: A clock-motif with a ray at minute position $p$ has violated vertical reflection, rotational, or horizontal reflection symmetry respectively, if a ray of equal length is not located at the following clock positions: $60-p$ (mod 60), $30+p$ (mod 60), or $30-p$ (mod 60). (c) To facilitate dealing with multi-thematic patterns the tutorial produced clock-motifs with 4 to 16 rays. (d) To facilitate quick naming of patterns, each tutorial screen flashed the five symmetries with familiar letters or symbols that illustrated the symmetries; for example, vertical reflection (V), horizontal reflection (<), etc. The name of the symmetry group of a pattern was a simple listing of the initials of the symmetries present, in alphabetic order. For example, the patterns P and Q presented earlier have symmetry groups grtv and gv, respectively.

1.7. Problem richness. Clock-patterns are not meant to replace the sophisticated examples from diverse cultures presented in [1] and [2], but rather complement these real-life examples. Student practice with clock-patterns improved their recognition of real-life examples. These observations are anecdotal; no formal statistical measure of improvement was carried out.

References

[1] S. Garfunkle, Project director for COMAP, For all Practical Purposes; Introduction to Contemporary Mathematics, 2nd edition, Arlington, Ma: W. H. Freeman and Company, 1991.

[2] D. K. Washburn and D. W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Seattle, Washington: University of Washington Press, 1988.

[3] R. J. Hendel, Symmetry Tutor for Introductory Liberal Arts Mathematics Courses, Vol. 11, pp. 80-88. 1993.

[4] J. R. Hartley and K. Lovell, The Psychological Principles Underlying the Design of Computer-Based Instructional Systems. In Walker, F. Decker, and R. D. Hess (Eds) Instructional Software: Principles and Perspectives for Design and Use, pp. 38-56 Belmont, CA: Wadsworth. 1977.

[5] S. Ohlsson, Some principles of Intelligent Tutoring, Instructional Science, Vol. 14, pp. 293-326. 1986.