MATHEMATICAL PATTERNS
IN
AFRICAN AMERICAN HAIRSTYLES

by GLORIA GILMER
MATH-TECH, MILWAUKEE

BACKGROUND.
The discipline of mathematics includes the study of patterns. Patterns can be found everywhere in nature. See Figure 1 with two bees in a beehive. Often these patterns are copied and adapted by humans to enhance their world. See the pineapple in Figures 2a and the adapted hairstyle in Figure 2b. Ethnomathematics is the study of such mathematical ideas involved in the cultural practices of a people. Its richness is in exploring both the mathematical and educational potential of these same practices. The idea is to provide quicker and better access to the scientific knowledge of humanity as a whole by using related knowledge inherent in the culture of pupils and teachers.

 fig1 TWO BEES IN A BEEHIVE FIGURE 2a PINEAPPLE TESSELATING HEXAGONS

FIGURE 2b GIRL

Going into a community, examining its languages and values, as well as its experience with mathematical ideas is a first and necessary step in understanding ethnomathematics. In some cases, these ideas are embedded in products developed in the community. Examples of this phenomena are geometrical designs and patterns commonly used in hair braiding and weaving in African-American communities. For me, the excitement is in the endless range of scalp designs formed by parting the hair lengthwise, crosswise, or into curves.

INTRODUCTION.
The main objective of my work with black hair is to uncover the ethnomathematics of some hair braiders and at the same time answer the complex research question: "What can the hair braiding enterprise contribute to mathematics education and conversely what can mathematics education contribute to the hair braiding enterprise?" It is clear to me that this single practical activity, can by its nature, generate more mathematics than the application of a theory to a particular case.

My collaborators include Stephanie Desgrottes, a fourteen year old student of Haitian descent, at Half Hollow Hills East School in Dix Hills, New York and Mary Porter, a teacher in the Milwaukee Public Schools. We have each observed and interviewed hair stylists at work in their salons along with their customers. Today's workshop for middle school teachers will focus on the mathematical concept of tesselations which is widely used and understood by hair braiders and weavers but not thought of by them as being related to mathematics.

TESSELATIONS.
A tesselation is a filling up of a two-dimensional space by congruent copies of a figure that do not overlap. The figure is called the fundamental shape for the tesselation. In Figure 1, the fundamental shape is a regular hexagon. Recall that a regular polygon is a convex polygon whose sides all have the same length and whose angles all have the same measure. A regular hexagon is a regular polygon with six sides. Only two other regular polygons tesselate. They are the square and the equilaterial triangle. See Figures 3a and 3b for parts of tesselations using squares and triangles. In each figure, the fundamental shape is shaded. To our surprise two types of braids found to be very common in the salons we visited were triangular braids and box braids which describe these tesselations on the scalp!

 FIGURE 3a TESSELATING SQUARES FIGURE 3b TESSELATING TRIANGLES; FIGURE 4 TESSELATING A NON-STANDARD FIGURE.

Box Braids.
In the tesselations we saw, the boxes were shaped like rectangles and the pattern resembled a brick wall starting with two boxes at the nape of the neck and increasing by one box at each successive level away from the neck. The hair inside the box was drawn to the point of intersection of the diagonals of the box. Braids were then placed at this point. You may notice in Figure 3a that braids so placed will hide the scalp at the previous level in the tesselation. In this style, the scalp is completely hidden. In addition, we were told that braids so placed are unlikely to move much when the head is tossed.

Triangular Braids.
In the tesselations we saw, the triangles were shaped like equilateral triangles and the pattern resembled the one shown in Figure 3b. The hair inside the triangle was drawn to the point of intersection of the bisectors of the angles of the triangle. Again, this style allowed hair to move less liberally than hair drawn to a vertex and then braided.

Tesselations can be formed by combining translation, rotation, and reflection images of the fundamental shape. Variations of these regular polygons can also tesselate. This can be done by modifying one side of a regular fundamental shape and then modifying the opposite side in the same way. See Figure 4.

CLASSROOM ACTIVITIES.
1. Draw tesselations using different fundamental shapes of squares and rectangles.

2. Draw a tessalation using an octagon and square connected along a side as the fundamental shape.

3. Draw tesselations with modified squares or triangles.

4. Have a hairstyle show featuring different tesselations.

REFERENCES.

Eglash, Ron, "African Fractals."New Brunswick, New Jersey: Rutgers University Press, 1999. ·

Gerdes, Paulus, "On Culture, Geometrical Thinking and Mathematics Education." Ethnomathematics:Challenging Eurocentrism in Mathematics Education edited by Arthur B. Powell and Marilyn Frankenstein, Albany, N.Y.:State University of New York Press, 1997. ·

Gilmer, Gloria F. Sociocultural Influences on Learning, American Perspectives on the Fifth International Congress on Mathematical Education (ICME 5) Edited by Warren Page. Washington, D.C.:The Mathematical Association of America, January 1985. ·

______"The Afterward. "Ethnomathematics:Challenging Eurocentrism in Mathematics Education edited by Arthur B. Powell and Marilyn Frankenstein, Albany, N.Y.:State University of New York Press, 1997. ·

______"Making Mathematics Work for African Americans from a Practitioner's Perspective." Making Mathematics Work for Minorities: Compendium of Papers Prepared for the Regional Workshops, Washington, DC.:Mathematical Sciences Education Board, 1989. pp. 100-104. ·

______ and Mary Porter. "Hairstyles Talk a Hit at NCTM. " International Study Group on Ethnomathematics Newsletter, 13(May 1998)2, pp. 5-6. ·

_______ "An Ethnomath Approach to Curriculum Development," International Study Group on Ethnomathematics Newsletter, 5(May 1990)2, pp. 4-5. ·

_______ and Williams, Scott W. "An Interview with Clarence Stephens." UME Trends. March 1990. ·

Sagay, Esi, African Hairstyles, Portsmouth, New Hampshire; Heinemann Educational Books Inc. USA, 1983.

 Dr. GLORIA GILMER is a mathematician, mathematics educator, consultant and the founding president of the International Study Group on Ethnomathematics. She also Chairs the Commission on Education of the National Council of Negro Women who for the last decade have established standards and awarded, with Shell Oil support, more than seventy teachers who have distinguished themselves by developing outstanding African American students. This paper was prepared for presentation at the 77th annual meeting of the National Council of Teachers of Mathematics in San Francisco, California, USA. click for a profile on Dr. Gloria Gilmer

Dr. Gloria Gilmer Math-Tech Milwaukee
9155 North 70th Street
Milwaukee, WI 53223-2115
Phone: 414-355-5191
Fax: 414- 355- 9175
E-mail: ggilme@aol.com

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