Below are some ideas for research/writing projects that you can consider as possibilities for extra credit or honors work in mathematics. Check with your instructor.

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 BTC-510, telephone/voicemail 734-462-4400 ext. 5290, e-mail rschwart@schoolcraft.edu).

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).

The Writing Fellows website at http://www.schoolcraft.edu/fellows/ has some useful guidance in how to go about doing this.

1. 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 resources include—

   • Jones, Philip S., “From Ancient China ’til Today!” [on the Chinese Remainder Theorem], pp. 346-349 in Frank J. Swetz, ed., From Five Fingers to Infinity: A Journey through the History of Mathematics (Chicago, IL: Open Court Publishing Co., 1914).
   • 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). Original 1913 version is available in full text at http://name.umdl.umich.edu/ACD4271.0008.001
   • Needham, Joseph, Science and Civilization in China. Vol. 3, “Mathematics and the Sciences of the Heavens and the Earth” (Cambridge: Cambridge University Press, 1959).
   • Shen Kangshen, John N. Crossley and Anthony W.-C. Lun, 1999. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press.
   • Siu Man-Keung, “An Excursion in Ancient Chinese Mathematics”, pp. 159-166 in Victor J. Katz, ed., Using History to Teach Mathematics: An International Perspective (Washington, D.C.: Mathematical Association of America, 2000). Bradner Library Call # QA 12 .U85 2000
   • 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/
   • Straffin, Philip D., Jr., “Liu Hui and the First Golden Age of Chinese Mathematics”, Mathematics Magazine 71:3 (June 1998), pp. 163-181.
   • Swetz, Frank J., three articles “The Evolution of Mathematics in Ancient China”, “The Amazing Chiu Chang Suan Shu”, and “The ‘Piling Up of Squares’ in Ancient China”, pp. 318-345 in Frank J. Swetz, ed., From Five Fingers to Infinity: A Journey through the History of Mathematics (Chicago, IL: Open Court Publishing Co., 1994).
   • Tillema, Erik, “Chinese Algebra: Using Historical Problems to Think About Current Curricula”, Mathematics Teacher 99:4 (November 2005), pp. 238-245.
   • Yăn, Lĭ, and Dù Shírán, Chinese Mathematics: A Concise History (Oxford, UK: Oxford University Press, 1987).
   1. Chinese counting rods

      Small rods made of bamboo, painted black and red, were used for calculations in China in ancient times. How were these counting rods used to represent numbers? Describe how arithmetic operations were carried out, and give some of your own examples. What were the strengths and weaknesses of this counting-rod system? How does it compare to other systems for representing numerals, such as tallies; written Chinese numerals; Roman numerals; Hindu-Arabic numerals; and the abacus?

   2. The abacus

      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?

   3. Right triangles in ancient China

      The ancient Chinese were very adept at applications involving the right triangle, and especially in using the principle that became known in Europe as the Pythagorean Theorem. Their mastery of the subject was such that mathematics historian Frank Swetz facetiously asked, “Was Pythagoras Chinese?” The ninth and final chapter of the famous manuscript Chiu Chang Suan Shu or “Nine Chapters on the Mathematical Art” (see below), was devoted to this subject, as was the Haidao Suanjing or “Sea Island Mathematical Manual”, a surveying text that is often appended to the Nine Chapters. In what sorts of problems did right-triangle applications arise? Did Chinese scholars ever attempt to rigorously prove the Pythagorean Theorem? How did the techniques of Chinese geometry compare with Greek and other European ones? As part of your paper, give examples of Chinese solutions of such problems.

      Some resources—

      • Shen Kangshen, John N. Crossley and Anthony W.-C. Lun, 1999. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford: Oxford University Press.
      • Swetz, Frank J., and T. I. Kao, Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China (Reston, VA: National Council of Teachers of Mathematics, 1977).
      • Swetz, Frank J., “The ‘Piling Up of Squares’ in Ancient China”, in Frank J. Swetz, ed., From Five Fingers to Infinity: A Journey through the History of Mathematics (Chicago, IL: Open Court Publishing Co., 1994).
      • Zitarelli, David E., “Was Pythagoras Chinese?”, pp. 41-48 in Amy Shell-Gellasch, ed., Hands-On History: A Resource for Teaching Mathematics (Washington, D.C.: Mathematical Association of America, 2007).
   4. Qin Jiushao: A Chinese Mathematician

      Qin Jiushao (1202-1261), also known as Ch’in Chiu-Shao, is one of 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.
   5. Seki Kōwa: A Japanese Mathematician

      October 24, 2008 marks the 300th anniversary of the death of Takakazu Shinsuke Seki, also known as Seki Kōwa (1642-1708), who anticipated many of the leading discoveries of Western mathematics. He was born in the same year as Isaac Newton, to whom he is frequently compared. 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.
   6. Japanese sangaku puzzles

      Beginning in the early 1600’s, there arose a custom of posting mathematical puzzles on the walls of Shinto shrines and Buddhist temples in Japan. The ingenious puzzles, most of them geometric in nature, were inscribed on colorfully-illustrated wooden tablets. They were called san gaku, which translates as “mathematical puzzle”. The sangaku tradition thrived during the Tokugawa shogunate (1603-1868), a period in which Japan was isolated from overseas influence. Who wrote these tablets? Where did the problems, and the mathematical methods used in their solution, come from? Give a few examples of the puzzles and solutions. Why were the sangaku posted in sacred places? And why did this tradition wither away?

      Some resources—

      • Bogomolny, Alexander, “Sangaku: Reflections on the Phenomenon”, http://www.cut-theknot.org/pythagoras/Sangaku.shtml
      • Fukugawa, Hidetoshi, and Tony Rothman, Sacred Mathematics: Japanese Temple Geometry (Princeton, NJ: Princeton University Press, 2008)
      • Rothman, Tony, “Japanese Temple Geometry”, Scientific American 278: 5 (May 1998), pp. 85-91.
2. MATHEMATICS AND VOTING

   The theme for National Mathematics Awareness Month (MAM) in April 2008 was “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 as well as many Congressional and other officials. 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:

   1. Voting Systems

      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.

      Some resources—

      • Guterman, Lila, “When Votes Don’t Add Up: Mathematical Theory Reveals Problems in Election Procedures,” Chronicle of Higher Education (3 November 2000)
      • Jameson, Marie K., and Gregory Minton and Michael E. Orrison, “Borda Meets Pascal”, Math Horizons (Mathematical Association of America) 16:1 (Sep. 2008)
      • Mackenzie, Dana, “May the Best Man Lose,” Discover (November 2000), pp. 85-91
      • MAM theme essays at http://www.mathaware.org/mam/08/essays.html
      • Saari, Donald G., “Suppose You Want to Vote Strategically,” Math Horizons (November 2000), pp. 5-10
      • Saari, Donald G., “Symmetry, Voting and Social Choice,” The Mathematical Intelligencer 10:3 (1988), pp. 32-42.
      • Saari, Donald G., Chaotic Elections! A Mathematician Looks at Voting (Providence, RI: American Mathematical Society, 2001).
   2. Election Polling

      There was a surprise result in the 2008 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. How are polls conducted? Why can pre-election polls be wrong? How can statistics be used to study the problem?

      Some resources—

      • Ash, Arlene, and John Lamperti, “Florida 2006: Can Statistics Tell Us Who Won Congressional District-13?” (and accompanying articles by Joseph Lorenzo Hall and Walter R. Mebane, Jr.), Chance 21:2 (Spring 2008), pp. 18-27. Also available at http://www.amstat.org/publications/chance/pdfs/199.featured.pdf
      • Link, Richard F., “Election Night on Television”, in Judith M. Tanur et al., eds., Statistics: A Guide to the Unknown, Third edition (Belmont, CA: Brooks/Cole, 1989)
      • MAM theme essays at http://www.mathaware.org/mam/08/essays.html
      • Ratledge, Edward C., “The Anatomy of a Preelection Poll”, in Roxy Peck et al., eds., Statistics: A Guide to the Unknown, Fourth edition (Belmont, CA: Brooks/Cole, 2006).
   3. Electoral College

      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.

      Some resources—

      • Brams, Stephen J., The Presidential Election Game, Second edition (London: A. K. Peters, Ltd., 2007)
      • Stout, David, “How Winner of the Popular Vote Could Lose After All,” New York Times (3 November 2000) gives a brief explanation of possible scenarios.
      • U.S. Electoral College website (http://www.archives.gov/federal-register/electoral-college/), maintained by the National Archives and Records Administration, summarizes the relevant laws and procedures, lists state-bystate 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.
   4. Congressional apportionment

      In addition to choosing among presidential candidates, voters in the 2008 election will also choose among candidates for Congress. By the U.S. Constitution, each state is entitled to the same number (2) of Senators, but the number of representatives in the House varies from state to state, depending on the relative populations of the states. Congressional apportionment is the science of determining the correct number of Representatives for each state based on the latest U.S. census figures. It involves lots of mathematics and also lots of controversy and rival theories. Summarize the following methods of Congressional apportionment—

      • the Hamilton-Vinton formula
      • the Webster formula
      • the Huntington-Hill formula
      • the Dean formula
      • the Adams formula
      • the Jefferson formula.

      A good source is—

      • Huckabee, David C., The House of Representatives Apportionment Formula (Washington, D.C.: Library of Congress [Congressional Research Service], August 10, 2001), available at http://www.rules.house.gov/archives/RL31074.pdf