Mathematical Research for (Undergraduates and) High School Students (11/10/2001)
Mathematics and Computing Department
York College (CUNY)
Jamaica, New York 11451-0001
Email: firstname.lastname@example.org (for additions, suggestions, and corrections)
Students under 18 who have conducted advanced research in mathematics might be interested in the Davidson Fellows Scholarship program.
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.
1. Martin Gardner's books
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)
2. Ian Stewart's column in Scientific American Magazine
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)
3. Quantum Magazine
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.)
4. The New Mathematics Library
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)
5. Dolciani Mathematical Expositions
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.)
6. Mathematics Teacher
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.
7. Mathematics World
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.
8. College Mathematics and Mathematics Magazine
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.
9. HIMAP and GEOMAP Modules (High School Mathematics and its Applications Project
and Geometry and its Applications Project)
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.
10. Journal of Recreational Mathematics
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.)
11. NCTM Yearbooks
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.
12. World Wide Web
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.
13. Miscellaneous Resources
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.
Appendix I (Martin Gardner's Books)
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
15. Entertaining Mathematical Puzzles.
16. Mathematics Magic and Mystery.
17. The Last Recreations.
18. The Colossal Book of Mathematics.
Other books by Martin Gardner:
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.
Appendix II (New Mathematical Library)
Mathematical Association of America (MAA)
1529 Eighteenth Street, N.W.
Washington, DC, 20036-1385
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,
38. Berzsenyi, G. and S. Maurer, The Contest Problem Book V, 1997.
Appendix III (Dolciani Mathematical Expositions)
Mathematical Association of America (MAA)
1529 Eighteenth Street, N.W.
Washington, DC, 20036-1385
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
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.
Appendix IV (Mathematical World (and other goodies))
American Mathematical Society
P.O. Box 6248
Providence, RI 02940
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,
7. Fomin, D. and S. Genkin, I. Itenberg, Mathematical Circles (Russian Experience),
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. Here are the titles in this series:
1. Radin, C., Miles of Tiles, 1999.
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.
Appendix V (Himap Modules and Geomap)
57 Bedford Street
Lexington, MA 02173
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.
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.
Appendix VI World Wide Web Locations
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.
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.
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.
American Mathematical Society (AMS):
The American Mathematical Society is the major professional society for research mathematicians.
The Ameican Statistical Association (ASA):
The American Statistical Association is one of the major professional socieites for statisticians.
Association for Computing Machinery (ACM):
The Association for Computing Machinery is one of the major professional societies for computer
Consortium for Mathematics and Its Applications (COMAP):
The Consortium for Mathematics and Its Applications develops innovative materials
for high schools students and undergraduates. These materials emphasize recent applications
Institute for Electrical and Electronic Engineers (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):
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):
The Mathematical Association of America is the major professional society for college
National Council of Teachers of Mathematics (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):
The Society for Industrial and Applied Mathematics' name tells the story.
Appendix VII Special sites:
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'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.
Research Projects for High School Students (and other interested parties!): SET I
Appendix VIII Miscellaneous Resources
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
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
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
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.
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.
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.
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.
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.
This journal published in Australia has many interesting articles that could inspire
high school student research.
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.
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.
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.
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.
Published by IEEE this wonderful magazine features articles about engineering and how science and mathematics support new developments in engineering.
Published by COMAP this journal has articles related to mathematical modeling, book reviews, and articles of general interest about mathematics.
Appendix IX What does mathematics research in high school mean?
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 self-confidence and intellectual curiosity
. The student with confidence and curiosity can begin by dabbling into one of the
resources listed above (and many others not listed) and learning something that is
"intriguing." This "intriguing something" may be a mathematical question suggested
by what has been read but more often is a variant of such questions. The student is then likely
to try a variety of experiments with pencil and paper or with the assistance of a
computer to explore the "intriguing something." This exploration will either show
that the original idea (to the student) seems to lead to a valid "conjecture" or shows
that the original idea has to be modified. (Students may subsequently learn that
the idea they thought they had invented is not new. This can be demoralizing in the
short run but often quickly leads to euphoria that some related question is really new.)
In either case described, the student has learned something and is on the road to
doing "research." High school student research more often than not consists of generating
new questions than in proving new mathematical theorems. This is not surprising since
not only is elementary mathematics heavily picked over by former high school students,
undergraduates and professionals, but high school students have only a limited amount of knowledge and proof techniques with which to attempt proofs! Student researchers
should take pride in being able to have had the courage to look at something new
to them, explore it a bit, and come up with a new conjecture, theorem, problem, or
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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