Women and Minorities in Mathematics:

Incorporating their Mathematical Achievements into School Classrooms

Dr. Sarah J. Greenwald and Dr. Andrew Nestler

Research has shown that incorporating the mathematical achievements of women and minorities into school classrooms is beneficial to all students. Some conclusions:

While there are a number of references on activities incorporating multiculturalism for the classroom [2, 4-6, 9-10], except for [7], sources that discuss the mathematical achievements of women and minorities do not include activities for inclusion in the classroom.

Carolyn Gordon is very well known for her research, and she presents it at conferences all over the world. She has received numerous grants and awards, such as the American Mathematical Society Centennial Fellowship. She is currently the Cheney Professor of Mathematics at Dartmouth College.

Much of what scientists know about our world comes from indirect observations. For example, X-rays, CAT scans, and other medical imaging techniques are indirect ways of seeing inside the body. Mathematicians are also concerned with indirect ways of observing the world around them. In 1966, the Polish-American mathematician Mark Kac asked if one can always hear the shape of a drum. The problem challenged researchers for nearly three decades. Finally, in the spring of 1991, Gordon and her fellow researchers came up with the answer: No. They found two drums that have different shapes, yet sound exactly the same when they are played [12].

Carolyn Gordon and husband David Webb with their drums in 1991. At one point while they were working to find them, they filled up their living room with huge paper models! Reprinted with permission of Carolyn Gordon and Washington University in St. Louis.

Kate Okikiolu is also well known for working on hearing the shape of a drum. She is the first person of African descent to have won one of the most prestigious awards for young mathematics researchers, a Sloan Research Fellowship. She is a professor of mathematics at the University of California at San Diego. Okikiolu [17] explains her research: "The sound of a drum changes as its shape changes. In fact, listening to the drum very carefully tells me how to change its shape to make it sound like a round drum and when it sounds like a round drum, it is indeed round!" Okikiolu "has developed a mathematical procedure called 'moving to the music' that links sounds to shapes… You're blindfolded, listening to drum music, and someone asks you to draw the drum." [16] While Carolyn Gordon's mathematics tells us that we cannot always distinguish between drums of different shapes, Kate Okikiolu's mathematics succeeds in distinguishing the shapes of certain types of drums.

Kate Okikiolu Dancing to the Music in Higher Dimensions poster. Reprinted with permission of Pamela Davis Kivelson. Copyright 1999 P. Davis Kivelson. May not be reproduced without permission. Posters can be purchased by calling the Discovery Place in Charlotte, NC at (800) 935-0553 x419 or x418. More Information on the Women in Science Poster Project can be obtained at http://www.physics.ucla.edu/scienceandart/pdavisposters/

Now we offer activities designed to introduce ideas related to the research of Carolyn Gordon and Kate Okikiolu to students. Sample classroom worksheets can be found at [11]. Slight modifications make these activities appropriate for use at a variety of grade levels.

Explain to students that one can hear the size of a drum, and ask the students to bring drums into the classroom. Ask the students to close their eyes and listen to the different drums being played, and ask them to draw the shape of the drums they hear. See if they can guess that larger drums have lower pitches. Wrap a string around the outside of the drums to compare perimeters. Fit small square tiles onto the drum heads to compare areas. Use a ruler and mathematical formulas to find the perimeter and area of each drum.

A two-piece band is a union of two drums. Given a known example [13] of two distinct two-piece bands that produce the same sounds, students can verify that each band has the same total perimeter and area. They can attempt to discover other such examples as well, first with restrictions, such as using only rectangles. Using solid materials, they can build physical drums to form their own two-piece bands.

Since one can hear the area and perimeter of a drum, if students were looking for two soundalike drums, then these drums would have to have the same area and the same perimeter. The following represent some of the many possible student explorations of perimeter and area of circles and polygons.

Students can explore why two circles of different sizes can never have the same area or the same perimeter. Using algebra to solve two equations with two unknowns, they can show that a circle and a square can never have both the same area and perimeter. The solution yields a radius and side both of length zero, so this could lead to a discussion that a zero radius or side length means that the figure is a point.

To show that a circle and a rectangle can never have both the same area and perimeter, students can solve equations relating the area and perimeter for the radius of the circle, r. Let x and y be the lengths of the sides of the rectangle. One obtains the equation after equating the radius conditions and then simplifying the resulting equation. Students can use a computer algebra program such as Maple to algebraically solve this equation in order to see that there are no non-trivial solutions. In addition, for x and y larger than 0, they can create a three-dimensional plot of the surfaces on each side of the above equation. They can then rotate the plot and examine different views of the surfaces in order to see that they never intersect [11].

Since Carolyn Gordon was looking for soundalike drums, she had to find two drums with the same area and perimeter that also sounded the same when they were played. Introduce students to the pictures of the drums labeled in [12] as the original pair of soundalike, or isospectral, drums discovered by Gordon and Webb. Explain that Carolyn Gordon proved that her drums sounded the same without actually testing them out in real life and explain why a mathematical proof does not need to be constructive. Since the students will be worried about this, explain that some physicists created the drums and tested them in real life. They found that the drums sounded nearly the same with an error attributed to the experimental procedures [14].

Have students cut along the boundaries of the drums. Then they can see whether the figures were drawn to scale by measuring the perimeters by using either a piece of string or a ruler and seeing whether they match. Have students divide the figures into rectangles and triangles so that they can determine the areas. Finally, they can determine the areas to see if they match.

While many of the ideas that we describe here can be nicely visualized in a dynamic geometry software program such as Geometer's Sketchpad, problems arise with accuracy. For example, for the kettledrum activity, students can use Sketchpad to draw regular polygon drumheads (see [11] for scripts and examples). Then, students can use the Measure command to have the program calculate the area and perimeter of each figure. By clicking at the appropriate place and dragging, they can rescale the figures so that they each have area one. They will have trouble succeeding due to the discrete nature of such software and may be tempted to say that this is impossible to achieve. In fact, one may wish to start these activities with Sketchpad for motivation, by asking if it is possible to construct regular polygons of equal areas, in order to highlight the dangers of relying only on visualization software. Aside from this accuracy issue, Sketchpad beautifully illustrates the relationships between the number of sides, the perimeter and the area of regular polygons and circles.

We hope that these activities and references are informative and enjoyable, and we welcome any feedback regarding the use of these or similar activities in the classroom.

4. Edwards, Carol. Changing the Faces of Mathematics: Perspectives on Asian Americans and Pacific Islanders, National Council of Teachers of Mathematics, 1999.

5. Karp, Karen et al. Feisty Females: Inspiring Girls to Think Mathematically, Heinemann, 1998.

6. Ortiz-Franco, Luis. Changing the Faces of Mathematics: Perspectives on Latinos, National Council of Teachers of Mathematics, 1999.

7. Parker, Marla. She Does Math! Real-Life Problems from Women in the Job, Mathematical Association of America, 1995.

8. Rosser, Sue. Female-Friendly Science: Applying Women's Studies Methods and Theories to Attract Students, Pergamon Press, 1990.

9. Strutchens, Marilyn. Changing the Faces of Mathematics: Perspectives on African Americans, National Council of Teachers of Mathematics, 2000.

10. Trentacosta, Janet. Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity, National Council of Teachers of Mathematics, 1997.

11. Greenwald, Sarah, "Incorporating the Mathematical Achievements of Women and Minorities into Schools: Classroom Activity Sheets"

http://www.mathsci.appstate.edu/~sjg/ncctm/activities

http://www.ams.org/new-in-math/hap-drum/hap-drum.html

http://www.ams.org/new-in-math/hap-drum/fig15.html

14. Peterson, Ivars. "Ivars Peterson’s MathLand: Drums that Sound Alike" http://www.maa.org/mathland/mathland_4_14.html

15. Weintraub, Steven. "What’s New in Mathematics -- June 1997 Cover" http://www.ams.org/new-in-math/cover/199706.html

17. McDuff, Dusa et al. "Poster Project, Dancing to the Music in Higher Dimensions Poster" http://www.math.sunysb.edu/posterproject/materials/dancing-to-music/dancing-to-music.html

18. McDuff, Dusa et al. "Okikiolu" [biography of Kate Okikiolu] http://www.math.sunysb.edu/posterproject/biographies/okikiolu.html

19. Okikiolu, Kate. [Homepage] http://math.ucsd.edu/~okikiolu/

Dr. Andrew Nestler, Santa Monica College, [email protected]