Mathematical Research for (Undergraduates and) High School Students (11/10/2001)

prepared by:

Joseph Malkevitch

Mathematics and Computing Department

York College (CUNY)

Jamaica, New York 11451-0001

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

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Students under 18 who have conducted advanced research in mathematics might be interested in the Davidson Fellows Scholarship program.

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Many high school students are interested in trying their hand at doing mathematics research (Appendix IX). This effort is hampered by the fact that even 12th graders, not to mention 9th graders, do not have a huge collection of mathematical tools at their disposal. Furthermore, many high school mathematics department libraries and public libraries do not have books that are of value to students who are interested in doing research. High schools can help support their mathematics research programs by systematically attempting to acquire the resources listed in the appendices. Despite impediments there are surprisingly many options by which high school students can get started. One important source of ideas for high school students to do research on is to extend the problems that occur in Mathematics Team competitions to more general settings. Students can attempt this with problems they have been challenged by in recent competitions. Books which contain such problems appear in some of the lists shown in the appendices. Below are some suggestions for getting students going on research projects. Working on mathematics research problems is exciting and rewarding.

There is no better introduction to a variety of very rich mathematical problems and ideas explained at a level that can be understood by high schools students than the works of Martin Gardner. (see Appendix I)

Stewart's monthly column (dropped by Scientific American recently) has problems of current interest that are accessible to high school students. (Stewart has also published some excellent books. See Appendix VIII)

This magazine is written specifically for high school students. It is published by Springer-Verlag. (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010-78580. Phone: 1-800-SPRINGER) Quantum should be part of every high school mathematics department library. (Unfortunately, Quantum has ceased publication. Some libraries have the issues that were published.)

This series of nearly 40 volumes is written specifically for high school students. The books in the series cover a wide range of topics from graph theory and number theory to game theory. (see Appendix II)

This series of expertly written monographs has wonderful materials not easily available elsewhere. The level is comparable to that of the New Mathematics Library. (See Appendix III.)

The NCTM (National Council of Teachers of Mathematics) publishes a wonderful journal primarily for high school mathematics teachers. This journal has many articles that can be read by students and that can suggest interesting research problems.

This series of books, (see Appendix IV) are published by the American Mathematical Society. Many of the volumes consist of materials developed in other countries that have been translated into English.

These two magazines, published by the Mathematical Association of America, are available through the Mathematics Department office of many high schools. They contain many articles that can be read by high school students.

This series of modules (see Appendix V) is aimed at high school teachers and students and treats many topics at the high school level that have been traditionally only discussed at higher levels. The series is published by COMAP (Consortium for Mathematics and Its Applications). COMAP's quarterly newsletter CONSORTIUM has articles that may be of value, and COMAP also has available both in CD-ROM and print form hundreds of applications-oriented modules (aimed for college students), many of which are accessible to high school students.

This relatively inexpensive journal has many nifty articles dealing with quick-starting topics in mathematics (number theory, geometric dissection problems, etc.) and a problem section. (Published by Baywood Publishing Company, 26 Austin Avenue, Amityville, New York 11701. Phone: 1-800-638-7819.)

NCTM (National Council of Teachers of Mathematics) publishes yearly volumes devoted to a specific theme. Some of these volumes contain pointers to ideas that lead quickly to research investigations.

Many resources to assist with high school mathematics research are available on the web. These include being able to access the card catalogues for major libraries, access the home pages of the mathematics professional societies, access the home pages of particular individuals that have research problems in various areas of mathematics, and look at materials related to the history of mathematics, etc. Appendix VI lists some web pages that may be of use.

Some miscellaneous books and resources can be found in Appendix VII and Appendix VIII In particular in Appendix VII you can take a look at a collection of projects that I generated for students to work on.

1. The Scientific American Book of Puzzles and Diversions.

2. The Second Scientific American Book of Mathematical Puzzles and Diversions.

3. New Mathematical Diversions.

4. The Numerology of Dr. Matrix.

5. The Unexpected Hanging.

6. The Sixth Book of Mathematical Games from Scientific American.

7. Mathematical Carnival.

8. Mathematical Magic Show.

9. Mathematical Circus.

10. Wheels, Life and other Mathematical Amusements.

11. Knotted Doughnuts.

12. Time Travel and other Mathematical Bewilderments.

13. Penrose Tiles to Trapdoor Ciphers.

14. Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine

15. Entertaining Mathematical Puzzles.

16. Mathematics Magic and Mystery.

17. The Last Recreations.

18. The Colossal Book of Mathematics.

Other books by Martin Gardner:

Aha! Gotcha

Aha! Insight

The Ambidextrous Universe

Fads and Fallacies in the Name of Science

The Incredible Dr. Matrix

Logic Machines and Diagrams

The Magic Numbers of Dr. Matrix

The Relativity Explosion

Riddles of the Sphinx

Note 1: Many of these books consist of edited versions of articles Gardner wrote for Scientific American Magazine for many years.

Note 2: Gardner's books have been published by a variety of publishers including W. H. Freeman. Recently, the Mathematical Association of America has been republishing new versions of some of the older books with corrections and some updating.

Note 3: I know this list is not complete.

Note 4: Carl Lee (University of Kentucky) has a web page giving where in Martin Gardner's works one can find various topics.

Several books which though not written by Martin Gardner either honor him or are very much in the spirit of what he wrote. Here are some examples:

Klarner, D., (ed.), The Mathematical Gardner, Wadsworth, Belmont, 1981. This book was recently reprinted by Dover Press

Gale, D., Tracking the Automatic Ant, Springer-Verlag, New York, 1998.

Mathematical Association of America (MAA)

1529 Eighteenth Street, N.W.

Washington, DC, 20036-1385

Phone: 1-800-331-1622

This series was originally begun by Random House and subsequently taken over by MAA. The books are specifically designed to be readable by secondary school students. Of course, they are of interest to other audiences as well.

1. Niven, I., Numbers: Rational and Irrational, 1961.

2. Sawyer, W., What is Calculus About?, 1961.

3. Beckenbach, E. and R. Bellman, An Introduction to Inequalities, 1961.

4. Kazarinoff, N., Geometric Inequalities, 1961.

5. Salkind, C., The Contest Problem Book I, 1961.

6. Davis, P., The Lore of Large Numbers, 1961.

7. Zippin, L., The Uses of Infinity, 1962.

8. Yaglom, I., Geometric Transformations I, 1962.

9. Olds, C., Continued Fractions, 1963.

10. Ore, O., Graphs and Their Uses, 1963.

11. Rapaport, E., (trans.), Hungarian Problem Book I, 1963.

12. Rapaport, E., (trans.), Hungarian Problem Book II, 1963.

13. Aaboe, A., Episodes from the Early History of Mathematics, 1964.

14. Grossman, E. and W. Magnus, Groups and Their Graphs, 1964.

15. Niven, I., The Mathematics of Choice, 1965.

16. Friedrichs, K., From Pythagoras to Einstein, 1965.

17. Salkind, C., The Contest Problem Book II, 1966.

18. Chinn, W. and N. Steenrod, First Concepts of Topology,

19. Coxeter, H. and S. Greitzer, Geometry Revisited, 1967.

20. Ore, O., Invitation to Number Theory.

21. Yaglom, I., Transformation Geometry II, 1968.

22. Sinkov, A., Elementary Cryptanalysis, 1968.

23. Honsberger, R., Ingenuity in Mathematics, 1970.

24. Yaglom, I., Transformation Geometry III, 1973.

25. Salkind, C. and J. Earl, The Contest Problem Book III, 1973.

26. Polya, G., Mathematical Methods in Science, 1977.

27. Greitzer, S., International Mathematical Olympiads, 1959-1977, 1978.

28. Packel, E., The Mathematics of Games and Gambling, 1981.

29. Artino, R. and A. Gaglione, N. Shell, The Contest Problem Book IV, 1982.

30. Schiffer, M. and L. Bowden, The Role of Mathematics in Science, 1984.

31. Klamkin, M., International Mathematical Olympiads 1972-1986.

32. Gardner, M., Riddles of the Sphinx, 1988.

33. Klamkin, M., U.S.A. Mathematical Olympiads 1972-1986, 1988.

34. Ore, O., Graphs and Their Uses, Revised and updated by R. Wilson.

35. Engel, A., Exploring Mathematics with Your Computer.

36. Straffin, P., Game Theory and Strategy, 1993.

37. Honsberger, R., Episodes in Nineteenth and Twentieth Century Euclidean Geometry, 1995.

38. Berzsenyi, G. and S. Maurer, The Contest Problem Book V, 1997.

Mathematical Association of America (MAA)

1529 Eighteenth Street, N.W.

Washington, DC, 20036-1385

Phone: 1-800-331-1622

1. Honsberger, R., Mathematical Gems, 1973.

2. Honsberger, R., Mathematical Gems II, 1976.

3. Honsberger, R., Mathematical Morsels, 1978.

4. Honsberger, R., (ed.), Mathematical Plums, 1979.

5. Eves, H., Great Moments in Mathematics (Before 1650), 1980.

6. Niven, I., Maxima and Minima Without Calculus, 1981.

7. Eves, H., Great Moments in Mathematics (After 1650), 1981.

8. Barnette, D., Map Coloring , Polyhedra, and the Four -Color Problem, 1983.

9. Honsberger, R., Mathematical Gems III, 1985.

10. Honsberger, R., More Mathematical Morsels, 1991.

11. Klee, V. and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, 1991.

12. Halmos, P., Problems for Mathematicians Young and Old, 1991.

15. Boas, R., Lion Hunting and other Mathematics Pursuits, 1995.

16. Halmos, P., Linear Algebra Problem Book, 1995.

17.Honsberger, R., From Erdös to Kiev, 1995.

18. Konhauser, J. D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, 1996.

19. Honsberger, R., In Polya's Footsteps: Miscellaneous Problems and Essays, 1997.

20. Bashmakova, I., Diophantus and Diophantine Equations

21. Halmos, P., and S. Givant, Logic as Algebra

22. Dunham, W., Euler: The Master of Us All, 1999.

31. Frank E.Burk, A Garden of Integrals

32. Steven G. Krantz, A Guide to Complex Variables

33.Keith Kendig, Sink or Float? Thought Problems in Math and Physics

36. Claudi Alsina and Roger Nelsen, When, Less Is More

37. Gerald B. Folland, A Guide to Advanced Real Analysis

38. Steven G. Krantz, A Guide to Real Variables

40. Steven G. Krantz, A Guide to Topology

41. Underwood Dudley, A Guide to Elementary Number Theory

42. Claudi Alsina and Roger Nelsen, Charming Proofs

43. edited by Joseph Gallian (editor), Mathematics and Sports

44.Steven H. Weintraub, A Guide to Advanced Linear Algebra

45. Claudi Alsina and Roger Nelsen, Icons of Mathematics

46. Keith Kendig, A Guide to Plane Algebraic Curves

48. Fernando Q. Gouvea, A Guide to Groups, Rings, and Fields

American Mathematical Society

P.O. Box 6248

Providence, RI 02940

Phone: 1-800-321-4267

1. Tikhomirov, Stories About Maxima and Minima, 1990.

2. Shashkin, Y., Fixed Points, 1991.

3. Sadovskii, L. and A.Sadovskii, Mathematics and Sports, 1993.

4. Prasolov, V., Intuitive Topology, 1995.

5. Farmer, D., Groups and Symmetry: A Guide to Discovering Mathematics, 1995.

6. Farmer, D. and T. Stanford, Knots and Surfaces: A Guide to Discovering Mathematics, 1995.

7. Fomin, D. and S. Genkin, I. Itenberg, Mathematical Circles (Russian Experience), 1996.

8. Kodaira, K., (ed.), Mathematics 1: Japanese grade 10, 1996.

9. Kodaira, K., (ed.), Mathematics 2: Japanese grade 11, 1996.

10. Kodaira, K., (ed.), Algebra and Geometry: Japanese grade 11, 1996.

11. Kodaira, K., (ed.), Basic analysis: Japanese grade 11, 1996.

12. Varadarajan, V., Algebra in Ancient and Modern Times, 1997.

13. Stahl, S., A Gentle Introduction to Game Theory, 1998.

The Amercian Mathematica Society also publishes the ongoing series of monographs by Barry Cipra devoted to new developments in the Mathematical Sciences:

What's Happening in the Mathematical Sciences, Volume 1, 1993.

What's Happening in the Mathematical Sciences, Volume 2, 1994.

What's Happening in the Mathematical Sciences, Volume 3, 1995-1996.

What's Happening in the Mathematical Sciences, Volume 4, 1998-1999.

Also noteworthy from AMS:

Knuth, D., Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, American Mathematical Society, Providence, 1997.

The American Mathematical Society recently started the Student Mathematical Library, aimed at undergraduates. The books in this series differ greatly in level but many of the books are self-contained and are accessible to high school students. Here are the titles in this series. The first book in the series was Charles Radin's miles of tiles, but rather than list all of the titles here is the link to the AMS page where the books are listed in reverse order of publication.

1. Radin, C., Miles of Tiles, 1999.

Titles in the American Mathematical Society's Student Mathematical Library

The American Mathematical Society also publishes the DIMACS Series in Discrete Mathematics and Theoretical Computer Science. (DIMACS is housed at Rutgers University and sponsors research and education programs related to Discrete Mathematics and Theoretical Computer Science.) Although books in this series are designed for researchers, some of the articles in the books are survey papers that can be read by novices. These books also pose many research problems some of them easily understood by individuals with little technical background.

COMAP

57 Bedford Street

Lexington, MA 02173

Phone: 1-800-772-6627

Himap

1. Froelich, G. and J. Malkevitch, The Mathematical Theory of Elections.

2. Cozzens, M., and R. Porter, Recurrence Relations-"Counting Backwards."

3. Zagere, F., The Mathematics of Conflict.

4. Crowe, D., Symmetry, Rigid Motion and Patterns.

5. Growney, JoAnne, Using Percent.

6. Cozzens, M., and R. Porter, Problem Solving Using Graphs.

7. Brown, S., Student Generations.

8. Bennett, et al., The Apportionment Problem: The Search for the Perfect Democracy.

9. Bennett, et al., Fair Division.

10. Francis, R., A Mathematical Look at the Calendar.

11. Djang, F., Applications of Geometrical Probability.

12. Martin, W., Spheres and Satellites.

13. Francis, R., The Mathematician's Coloring Book.

14. Kumar, G., Decision Making Math Models.

15. Rogers, J., A Uniform Approach to Rate and Ratio Problems.

16. Meyer, R., Explore Sorts.

17. Metallo, F., The Abacus: Its History and Its Applications.

18. Malkevitch, J., and G. Froelich, D. Froelich, Codes Galore.

19. Lucas, W., Fair Voting: Weighted Votes for Unequal Constituencies.

20. Sriskandarjah, J., Optimal Pays: An Introduction to Linear Programming.

21. Chavey, D., Drawing Pictures With One Line.

22. Malkevitch, J. and G. Froelich, Loads of Codes.

Geomap

1. Malkevitch, J., Graph Models.

2. Crisler, N. and W. Meyer, Shortest Paths.

3. Servatius, B., Rigidity & Braced Grids.

4. Arvold, B. and P. Cromwell, Knottedness.

5. Crisler, N., Symmetry & Patterns.

6. Mallinson, P., Tessellations.

7. Johnson, A., Geometric Probability.

8. Dickerson, M. and S. Drysdale, Voronoi Diagrams.

Although the web sites below do not in all cases have research problems for high school students, they have a rich variety of information about mathematics, as well as many links to resources that will help high school students interested in doing mathematical research.

History of Mathematics

A large collection of biographies of mathematicians and lots of other useful information can be found here.

Mathematical Ancestors

An attempt is being made to list all individuals who ever received a doctoral degree in Mathematics together with each individuals doctoral thesis advisor, dissertation title, and list of doctoral students. A lot interesting history can be gleaned from this project.

Mathematics Forum

The Mathematics Forum at Swarthmore College provides a broad array of resources for people interested in mathematics. The site provides extensive searchable archives and well as many links to other sites dealing with mathematics.

**Professional Societies:**

American Mathematical Society (AMS):

AMS

*The American Mathematical Society is the major professional society for research mathematicians.*

The Ameican Statistical Association (ASA):

ASA

*The American Statistical Association is one of the major professional socieites for statisticians.*

Association for Computing Machinery (ACM):

ACM

*The Association for Computing Machinery is one of the major professional societies for computer
scientists.*

Consortium for Mathematics and Its Applications (COMAP):

COMAP

*The Consortium for Mathematics and Its Applications develops innovative materials
for high schools students and undergraduates. These materials emphasize recent applications
of mathematics.*

Institute for Electrical and Electronic Engineers (IEEE):

IEEE

*The Institute for Electrical and Electronic Engineers provides lots of information about mathematical support for computer science and engineering, in particular, how mathematics is being used to help create new technologies.*

Institute for Operations Research and Management Science (INFORMS):

INFORMS

*The Institute for Operations Research and Management Science is the major professional
organization in the areas of operations research and management science. These two
areas of knowledge aim at improving the efficiency with which business and governments deliver services internally and to the public.*

Mathematical Association of America (MAA):

MAA

*The Mathematical Association of America is the major professional society for college
mathematics teachers.*

National Council of Teachers of Mathematics (NCTM):

NCTM

*The National Council of Teachers of Mathematics is the major professional society
for high school mathematics teachers.*

Society for Industrial and Applied Mathematics (SIAM):

SIAM

*The Society for Industrial and Applied Mathematics' name tells the story.*

David Eppstein

David Eppstein's Home Page

David Eppstein's Geometry Pages

Eppstein's Geometry Junkyard

David Eppstein maintains a staggering collection of pointers to exciting ideas and results in geometry and the application of geometry. His home page also has materials related to recreational mathematics, combinatorial games and number theory, etc.

George Hart

George Hart's Home Page

George Hart has assembled a vast collection of pages dealing with polyhedra, rich in images, dealing with all aspects of polyhedra. This is an invaluable resourse for people interested in polyhedra.

Joseph Malkevitch

Research Projects for High School Students (and other interested parties!): SET I

D'Angelo, J., and D. West, Mathematical Thinking: Problem Solving and Proofs, Prentice-Hall, Englewood Cliffs, 1997.

This book is designed for undergraduates and deals with a wide variety of basic tools (e.g. logic, functions, graph theory, probability, etc.) and proof techniques (e.g. mathematical induction, parity, etc.). However, it serves as a good introduction to these ideas for high school level students.

Cofman, J., What to Solve? Problems and Suggestions for Young Mathematicians, Oxford, New York, 1990.

A collection of interesting combinatorial problems and methods to solve and extend them.

Frederickson, G.N., Dissections: Plane & Fancy, Cambridge U. Press, New York, 1997.

This lovely book contains a wealth of accessible research problems and a lot of exciting mathematics.

Gale, D., Tracking the Automatic Ant and Other Mathematical Explorations, Springer-Verlag, New York, 1998.

This book contains a series of wonderful brief explorations in the spirit of Martin Gardner's writings.

Schoenfeld, A., Mathematical Problem Solving, Academic Press, Orlando, 1985.

This book surveys methods of problem solving techniques. (This book is aimed at teachers but it is readable by students as well.)

Stewart, I., The Problem of Mathematics, Oxford U. Press, New York, 1987.

A nice survey of important ideas in mathematics.

Stewart, I., Game, Set and Math, Penguin, New York, 1989.

A lovely collection of short articles on many varied topics.

Stewart, I., Another Fine Math You've Got Me Into..., W. H. Freeman, New York, 1992.

A compendium of some of the articles that Stewart wrote for Scientific American Magazine. Most of them are very well done.

American Scientist

Published by Sigma Xi, the honor society for those interested in Science (and Mathematics), this journal has superb survey articles dealing with science and mathematics. It also features excellent articles by Brian Hayes dealing with the interface between mathematics and computer science.

College Mathematics

Published by the Mathematical Association of America (MAA) and originally aimed for students and faculty at Two Year (Community) Colleges, this journal now specializes in topics primarily treated in the first two years of college.

Geombinatorics (Direct subscription inquiries to: Professor Alexander Soifer, Geombinatorics, UCCS, P.O. Box 7150, Colorado Springs, CO 80933, USA)

This journal has interesting problems and articles in the areas of geometry and combinatorics. Soifer also publishes several collections of problem oriented books.

Math Horizons

Published by the MAA (Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, DC, 20036-1385; Phone: 1-800-331-1622) for undergraduates this quarterly has information about careers in mathematics, articles about mathematicians and mathematics, and a section with problems for solution.

Mathematical Intelligencer

Published by Springer-Verlag (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010-78580. Phone: 1-800-SPRINGER) this journal although it has many articles at an advanced level, also has materials dealing with the history of mathematics, mathematical biographies, book reviews, and a lovely column entitled Mathematical Entertainments.

Mathematical Spectrum

This journal published in Australia has many interesting articles that could inspire high school student research.

Mathematics Magazine

This journal is published by the Mathematical Association of America (MAA) and treates all of undergraduate mathematics. However, there are often historical and survey articles that can be read by a wide variety of students.

Pi Mu Epsilon Journal

This journal is published by Pi Mu Epsilon which is an honor society for undergraduates interested in mathematics. It contains research articles written by undergraduates, problems, and surveys.

Quantum

Published by Springer-Verlag this journal is aimed specifically at high school students. Many articles are translations from the Russian journal Qvant. In Qvant articles for high school students written by many of Russia's most emminent mathematicians appeared. The articles are a wonderful mix of classical and recent topics in mathematics. Unfortunately, publication of this journal has been discontinued, but old issues are still available in some libraries.

Scientific American

This journal has survey articles about mathematics and science of the highest calibre. A regular feature is a column by Ian Stewart about recreational mathematics. Recently, Stewart's column has been dropped.

SIAM News

This journal has wonderful expositiory articles about applications of mathematics and applicable mathematics. I regular feature is a wonderful column of Philip Davis and the columns of Barry Cipra are excellent.

Spectrum

Published by IEEE this wonderful magazine features articles about engineering and how science and mathematics support new developments in engineering.

UMAP Journal

Published by COMAP this journal has articles related to mathematical modeling, book reviews, and articles of general interest about mathematics.

The phrase "mathematics research" has many interpretations and it may be useful to say a few words about what might constitute mathematics research in a high school context. The remarks that follow deal with "typical"students, not the mathematically precocious.

The range of mathematics that students have been "officially" exposed to in high school varies a lot from high school to high school, and certainly 9th graders have seen much less than 12 graders. Furthermore, students vary greatly in their knowledge of mathematics and "ability" in a given grade, and the amounts of work that they are willing or are able to commit to doing "research." However, independent of "ability" and "knowledge" mathematics research begins with student

Some of this work was prepared with partial support from the National Science Foundation (Grant Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning(LICIL) (administered by SUNY at Stony Brook, Alan Tucker, Project Director).

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