Writing/Research Projects for Math
Below are some ideas for research/writing projects for which your math instructor might give you extra credit.
We continue to publish the best projects in The Right Angle, monthly newsletter of the Mathematics Department. You should feel free to contact the editor, Randy Schwartz, who can offer advice and sources of information such as those mentioned below (office LA-563, telephone/voicemail 462-4400 ext. 5290, e-mail email@example.com).
In writing your paper, you shouldn’t be satisfied with a general description of the topic. You should actually dig into it, explain what is involved, and get into some actual mathematics using examples that help clarify it. Otherwise, what you and your readers learn will be superficial.
Also, remember that writing a research paper, even a brief one, doesn’t mean splicing together phrases and sentences that you find in other people’s books, articles, and websites. Of course, you’ll need to gather information and to list the sources where you found it, but you’ll also need to:
- analyze the information (break the facts and ideas down into their parts in order to see how they relate to one another)
- synthesize the information (put facts and ideas together in a systematic way in order to get a comprehensive understanding of the topic)
- organize and write the paper (figure out the best way, using your own words and understanding, to unfold what you’ve learned so that the readers get a clear understanding, too).
MATHEMATICS IN EAST ASIA
In conjunction with the Focus East Asia project this year at Schoolcraft College, we invite you to write about one of the following topics, which help show the historical contributions made to mathematics by people in Japan, China, and other countries. Some good general resources include—
- Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, New Edition (Princeton, NJ: Princeton University Press, 2000).
- Katz, Victor J., A History of Mathematics: An Introduction. Second Edition (Reading, MA: Addison-Wesley Publishing Co., 1998).
- Martzloff, Jean-Claude, A History of Chinese Mathematics (New York, NY: Springer-Verlag, 1996).
- Mikami, Yoshio, The Development of Mathematics in China and Japan (New York, NY: Chelsea Publishing Co., 1974).
- Smith, David Eugene, and Yoshio Mikami, A History of Japanese Mathematics (Chicago, IL: Open Court Publishing Co., 1914). Available in full text at http://books.google.com/
- Yăn, Lĭ, and Dù Shírán, Chinese Mathematics: A Concise History (Oxford, UK: Oxford University Press, 1987).
The abacus has been used in East Asia for about 2,000 years. How does it work? How did the device originate and evolve? How is the traditional Japanese abacus different from the Chinese one? What have been the uses and limitations of the abacus?
Qin Jiushao: A Chinese Mathematician
Qin Jiushao (1202-1261), also known as Ch’in Chiu-Shao, is one of the history’s greatest mathematicians. As part of your paper, describe his use of at least one of the following methods:
- a technique, later called the Ruffini-Horner method in Europe, for solving polynomial equations of high degree
- a matrix technique, later called Gaussian Elimination in Europe, for solving simultaneous linear equations
- a technique, later called the Chinese Remainder Theorem in Europe, for solving simultaneous integer congruences.
Seki: A Japanese Mathematician
Takakazu Seki-Kowa (1642-1708) anticipated many of the leading discoveries of Western mathematics. As part of your paper, describe his use of at least one of the following methods:
- a technique, later called the Ruffini-Horner method in Europe, for solving polynomial equations of high degree
- a technique, usually called the Newton-Raphson method in Europe, for finding roots of equations
- a calculation, later called the “determinant” in Europe, for solving simultaneous linear equations.
WOMEN AND MATHEMATICS
In conjunction with Women’s History Month, which is celebrated every March, we invite you to investigate the life and contributions of an important woman mathematician. Some examples include:
- Gabrielle Emilie du Châtelet (1706-1749) was famous for her work on the dynamics of fire, and for her translation of Newton’s work in physics from English into French. There have been a number of biographies, including two recent ones: David Bodanis, Passionate Minds: The Great Love Affair of the Enlightenment, Featuring the Scientist Emilie Du Châtelet, the Poet Voltaire, Sword Fights, Book Burnings, Assorted Kings, Seditious Verse, and the Birth of the Modern World (New York: Crown Publishers, 2006) [Bradner Library PQ 2103 .D7 B63 2006]; and Judith P. Zinsser, La Dame d’Esprit: A Biography of the Marquise Du Châtelet (New York: Viking, 2006).
- Mayme I. Logsdon (1881-1967) was a specialist in algebraic geometry who taught at the University of Chicago between 1921 and 1946. A starting point for reading about her is the chapter by Judy Green and Jeanne LaDuke, “Contributors to American Mathematics”, in Gabriele Kass-Simon, Patricia Farnes and Deborah Nash, eds., Women of Science: Righting the Record (Bloomington: Indiana University Press, 1990).
- Olga Taussky-Todd (1906-1995) was raised in Jewish Eastern Europe but immigrated to the west, doing most of her work in England and the United States.her main area of excperties was linear algebra, but she made contributions to several other branches of mathematics as well. Three remembrances of her are contained in the Mathematical Intelligencer 19:1 (Winter 1997), which can be accessed full-text via EBSCOhost at Schoolcraft’s Bradner Library; and there is an essay in Donald J. Albers and Gerald L. Alexanderson, eds., “Olga Taussky-Todd: An Autobiographical Essay”, Mathematical People: Profiles and Interviews (Boston: Birkhäuser, 1985), pp. 309-336.
- Anneli Cahn Lax (1922-1999) was a professor at New York University and distinguished editor at the Mathematical Association of America. Start with the remembrance written by Mark Saul in Notices of the American Mathematical Society 47:7 (August 2000), pp. 766-769, available at http://www.ams.org/notices/200007/mem-lax.pdf
- Ingrid Daubechies (born approx. 1954) is a Belgian-born Princeton University mathematician who has been a pioneer in the new field of wavelet analysis. Wavelets are being applied wherever masses of data must be compressed (such as in high-definition television), or where visual or audio patterns must be recognized (such as in fingerprint identification). The best place to start is an annotated interview of Daubechies published by Deana Haunsperger and Stephen Kennedy, “Coal Miner’s Daughter”, in Math Horizons (April 2000, pp. 5-9, 28-30).
Your report should answer these questions:
- What were the mathematical contributions made by this woman?
- What motivated her to pursue mathematical work?
- How did she surmount all of the obstacles that traditionally barred women from pursuing advanced mathematical study?
MATHEMATICS AND VOTING
The theme for National Mathematics Awareness Month in April 2008 is “Math and Voting”. After all, this is a presidential election year, and voting is uppermost on many people’s minds. Candidates vie for attention, polls are taken, debates held, blogs written, primaries conducted, and, ultimately, a general election will lead to the naming of the next president of the United States. Most people wonder at some point: Does my vote matter? Is the election process fair? Are the votes being counted correctly? What is the probability that my vote could be decisive? These questions are very complex, but mathematics and statistics provide the means to answer them. In the words of mathematician Don Saari, “An election outcome can more accurately reflect the choice of a voting system, rather than the voters’ wishes.” Saari and others have applied many mathematical methods to understand and solve complex issues involving voting.
Write a paper that addresses one of the following topics:
Many elections use a straightforward voting method: one person, one vote; the most votes wins. In races with more than two candidates, explain how this voting method might encourage some voters to cast their ballots “strategically” rather than for their favorite candidate. Explain why the top vote-getter might not be the most popular candidate running. Discuss several other voting methods that have been proposed as alternatives. Good sources include the MAM theme essays at http://www.mathaware.org/mam/08/essays.html; and articles by Donald G. Saari, “Suppose You Want to Vote Strategically,” Math Horizons (November 2000), pp. 5-10; Dana Mackenzie, “May the Best Man Lose,” Discover (November 2000), pp. 85-91; Lila Guterman, “When Votes Don’t Add Up: Mathematical Theory Reveals Problems in Election Procedures,” Chronicle of Higher Education (3 November 2000); and Donald G. Saari, “Symmetry, Voting and Social Choice,” The Mathematical Intelligencer 10:3 (1988), pp. 32-42.
There was a surprise result in the recent New Hampshire Democratic presidential primary. All of the major polling organizations— including Reuters/C-Span/Zogby, Rasmussen, CNN/WMUR/UNH, American Research Group, Marist and CBS News— predicted that Barack Obama would win the vote by about 10 percentage points, yet Hillary Clinton ended up taking the victory. Why can pre-election polls be wrong? How can statistics be used to study the problem? Good sources include the MAM theme essays at http://www.mathaware.org/mam/08/essays.html.
The 2000 election marked the fourth time in U.S. history that a president was elected without receiving a majority of the popular vote. Why was the Electoral College procedure incorporated into the U.S. Constitution? What have been some of its implications and results? How is the number of presidential electors for each state determined, and how are the votes of these electors decided? Explain the mathematics of how George W. Bush was able to secure a majority of these votes without securing a majority of the popular vote. The National Archives and Records Administration maintains the U.S. Electoral College website (http://www.archives.gov/federal-register/electoral-college/), which summarizes the relevant laws and procedures, lists state-by-state outcomes of the popular vote and the Electoral College vote for every presidential election up through 2000, and includes an Electoral College Calculator that you can play with experimentally. A brief explanation of possible scenarios is given in David Stout, “How Winner of the Popular Vote Could Lose After All,” New York Times (3 November 2000).
MATHEMATICS AND THE ENVIRONMENT
Earth Day on April 22 reminds the world of its responsibility to protect the Earth, and to conserve and wisely manage its natural resources. We invite you to investigate one of the following ways in which mathematics is used to study these environmental resources.
Climate change and global warming
How have earth climates changed over geologic time? Are greenhouse gas emissions and other human industrial activities responsible for an overall warming of the earth’s atmosphere in modern times? Answering this controversial question is difficult, in part because climate varies naturally over time, even without human interference, and in part because the underlying mechanisms of climate change are very complex. Mathematical modeling is an important part of answering the challenge of global warming. University of Michigan professors of geological sciences Henry N. Pollack and Chris J. Poulsen are nationally known experts in this field. They and their colleagues have gathered geothermal temperatures at sites around the world and have developed theoretical climate models to reconstruct the history of Earth’s surface temperatures and to help predict future trends. For information, see:
- Univ. of Michigan Paleoclimate Research Group publications page, http://www.geo.lsa.umich.edu/paleoclimate/publications.htm
- Henry N. Pollack and Shaopeng Huang, “Underground Temperatures Reveal Changing Climate”, Geotimes 43:8 (August 1998).
- Univ. of Michigan press release, “New Evidence of Global Warming”, San Diego Earth Times, Jan. 1998, available at http://www.sdearthtimes.com/et0198/et0198s2.html.
- Henry N. Pollack and David S. Chapman, “Underground Records of Changing Climate”, Scientific American June 1993, pp. 16-22.
The importance of biodiversity
Why does life in a given area become precarious when the number and variety of species fall below a certain point? Mathematics helps us answer this question. Here are some resources to investigate:
Lord Robert May, formerly of Oxford University, studies communities of interacting plants and animals, especially their food chains, then applies mathematical concepts to understand how species diversity affects the stability and complexity of these ecosystems. Articles about his work are available at:
Dr. G. David Tilman at the University of Minnesota uses mathematics extensively in analyzing the results of his field experiments in biodiversity. The New York Times has carried many articles about his work, such as William K. Stevens, “Ecologist Measures Nature’s Mosaic, One Plot at a Time”, New York Times 6 October 1998 (Science section).
Models of environmental pollution
Environmental analysts use a variety of mathematical methods to assess the dangers of, and losses caused by, air, water, and land pollution. These methods include cost-benefit analysis, the environmental Kuznets curve, and quantitative risk assessment (QRA).
- An article that summarizes the use of cost-benefit curves and the environmental Kuznets curve to model the relation between societal wealth and pollution, and the appropriate level of sensitivity to environmental threats, is S. W. Pacala et al., “False Alarm over Environmental False Alarms”, Science 301 (Aug. 29, 2003), pp. 1187-1188.
- A textbook that explains algebraic and computer techniques in investigating the threat of groundwater contamination, air pollution, and hazardous material emergencies is Charles Hadlock, Mathematical Modeling in the Environment (Washington, D.C.: Mathematical Association of America, 1998).
- The EPA’s use of quantitative risk assessment to gauge the risk of cancer from pollutants emitted in the air by the coke ovens of Ostrava (Czech Republic) is discussed in Jeff Wheelwright, “The Air of Ostrava”, Discover 17:5 (May 1996), pp. 56-67.
Sustainable harvests with linear algebra
Models based on matrix algebra are used to predict the growth rate of a population and the long-term trends for its age distribution. This can allow forest and livestock rangers, and the managers of fish, game and wildlife preserves, to design sustainable harvesting policies. A discussion is provided in Howard Anton and Chris Rorres, Elementary Linear Algebra: Applications Version, 9th edition (New York: John Wiley & Sons, Inc. 2005), pp. 648-657, 743-761. [This is the textbook used for Math 230 at Schoolcraft. Similar discussions are provided in other Linear Algebra textbooks]
Balancing predator and prey with differential equations
In 1925-26, population ecologists Alfred J. Lottka and Vito Volterra independently formulated an important model to describe the population fluctuations between a predator species and a prey species, like foxes and rabbits, or bass and sunfish, or ladybugs and aphids. The Lottka-Volterra model is a system of two nonlinear differential equations, and it predicts that in many cases, the populations of predator and prey will mutually stabilize by cycling between two extremes. See if you can address these questions in your paper:
- What practical problems prompted Lottka and Volterra to formulate their model?
- Explain the assumptions that were made and the variables and equations that were used to define the model.
- Describe the different mathematical solutions of the equations. What ecological behavior is predicted by these solutions?
- See if you can implement this model numerically or graphically with a graphing calculator or a computer.
- What are some of the limitations of this model? How has it been extended to apply to other situations?
Among the many good discussions of this work are:
- John Vandermeer, Elementary Mathematical Ecology (New York: Wiley & Sons, 1981), pp. 158-233.
- C. Henry Edwards, Jr., and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Third edition (Upper Saddle River, NJ: Prentice-Hall, 2004), pp. 393-406. [This is the textbook used for Math 252 at Schoolcraft. Similar discussions are provided in other Differential Equations textbooks]
- Judith R. Goodstein, The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician, 1860-1940 (Providence, RI: American Mathematical Society, 2007)
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