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:
``Although evolved to attract men of color, women, and others underrepresented in the pool of scientists, these techniques have been shown to be as successful -- or more so -- for white males." 
``Not only are more women and people of color likely to become scientists, but also more men may choose careers in science, since these improved methods have been shown to also be attractive to white males." 
``Males who perceived mathematics teachers as treating males the same as females tended to take more mathematics." 
While there are a number of references on activities incorporating multiculturalism for the classroom [2, 4-6, 9-10], except for , sources that discuss the mathematical achievements of women and minorities do not include activities for inclusion in the classroom.
This is the inaugural column of a semi-annual "Women and Minorities in Mathematics: Incorporating their Mathematical Achievements into School Classrooms" series. This article explores the inclusion of the research of Carolyn Gordon and Kate Okikiolu on hearing the shape of a drum into school classrooms. This is a great topic since it is fun and easy to make drums of different shapes and listen to them in class. It is also an entertaining way to demonstrate to studentsthat there is not always just one conclusion that can be reached from a complete set of measurements. Sample classroom activity sheets can be found at .
Carolyn Gordon - Groundbreaking Geometer
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 .
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 - Outstanding Young Mathematician
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  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."  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/
Classroom Activities and Worksheets
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 . Slight modifications make these activities appropriate for use at a variety of grade levels.
Familiar Real-Life Drums in the Classroom
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.
You Cannot Hear the Shape of a Two-Piece Band
A two-piece band is a union of two drums. Given a known example  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.
Circular vs. Polygonal Drums
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.
A kettledrum can be tuned to play notes of different pitches by adjusting the tension on the skin that makes up the head of the drum. Imagine that we have a set of five special kettledrums whose heads have the shape of regular polygons and that we also have an ordinary drum whose head is circular in shape. If all the heads of these drums have the same area and the same tension, then students can discover that the pitches will decrease as the number of polygonal sides increases. Assume that we are given the following drums and frequencies (in vibrations per second): equilateral triangle (146), square (136), regular hexagon (132), regular octagon (131), circle (130). Students can explore whether it is possible to have similar regular polygonal drums with pitches of 125, 134 or 140. In addition, if all of the drums have an area of one square foot, students can use area formulas to calculate the side lengths in order to algebraically confirm the intuitive statement that the side length must decrease as the number of sides increases. They can then explore what happens as the number of sides gets large. If instead, all of the drums all have a perimeter of one foot, students can use perimeter and area formulas to discover that the area increases as the number of sides increases, and that this approaches the area of the corresponding circle. In other words, a circle is the shape that maximizes the area of an enclosed regular region when using a given amount of fencing.
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 .
Drums in the Photograph of Carolyn Gordon and David Webb
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  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 .
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.
Using Geometer's Sketchpad
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  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.
General References on Multicultural and Gender Equity in the Mathematics Classroom
1. Armstrong, Jane M. "National Assessment of Women In Mathematics." Susan F. Chipman, Lorelei R. Brush, Donna M. Wilson (eds.). Women And Mathematics: Balancing The Equation, Lawrence Erlbaum, 1985.
2. Campbell, Mary Ann and Randall Campbell-Wright. "Towards a Feminist Algebra."Sue Rosser (ed.). Teaching the Majority: Breaking the Gender Barrier in Science, Mathematics and Engineering, Teachers College Press: Athene Series, Columbia University, New York and London, 1995.
3. Davis, Cinda-Sue and Sue Rosser. "Program And Curricular Interventions." Cinda-Sue Davis et al. The Equity Equation: Fostering the Advancement of Women in the Sciences, Mathematics, and Engineering, Jossey-Bass Publishers, San Francisco, 1996.
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.
Reference on Classroom Worksheets
11. Greenwald, Sarah, "Incorporating the Mathematical Achievements of Women and Minorities into Schools: Classroom Activity Sheets"
References on Carolyn Gordon and Materials for Further Study
12.Cipra, Barry. "You Cant Always Hear the Shape of a Drum"
13.Cipra, Barry. "You Cant Always Hear the Shape of a Drum - Figure 4"
14. Peterson, Ivars. "Ivars Petersons MathLand: Drums that Sound Alike" http://www.maa.org/mathland/mathland_4_14.html
15. Weintraub, Steven. "Whats New in Mathematics -- June 1997 Cover" http://www.ams.org/new-in-math/cover/199706.html
References on Kate Okikiolu and Materials for Further Study
16. Kivelson, Pamela Davis. Dancing to the music in higher dimensions poster, 1999Reprinted 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/.
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/
20. Williams, Scott. "Mathematicians of the African Diaspora"[personal biography] http://www.math.buffalo.edu/mad/PEEPS/okikiolu_katherine.html
Dr. Sarah J. Greenwald, Appalachian State University,email@example.com and
Dr. Andrew Nestler, Santa Monica College, firstname.lastname@example.org