Summer 2007 |
MTHS 620
Mathematical Problem Solving
Fieldsteel,Adam
06/25/2007 - 08/03/2007
Tuesday & Thursday 01:30 PM - 04:00 PM
Science Tower 139
The notion of "problem solving" as a subject in and of itself is not new. It has attracted attention from various points of view. The distinguished mathematician George Polya wrote several well-known books on the subject. His most famous is probably How to Solve It, which discusses general strategies for solving mathematical problems. The aim of the book is to make the student (or mathematician) conscious of systematic approaches to various types of problems. The analysis of problem solving applies not only to mathematics, but to any field where logical analysis and ingenuity is called for.
Polya wrote several other books on the same subject, which contain more mathematics, and which are a source of many problems that illustrate his ideas. These are the two-volume Mathematical Discovery and the two-volume Mathematics and Plausible Reasoning, especially the first volume of the latter, entitled Induction and Analogy of Mathematics. These books consider issues that extend beyond mathematics, but my intention is to focus on mathematical problem solving, and to use particular types of problems as motivation to introduce ideas from specific mathematical subjects, such as combinatorics, number theory, probability, difference equations, and geometry.
While the mathematical work of all students and mathematicians inevitably involves problem solving, there is a form of this activity that is particularly entertaining. This is the problem solving that is called for in competitive exams and puzzles. There are two very good new books that discuss such problems in a systematic way. Each is more than a compendium of problems or a training plan, but is informed by the same general desire to think about problem solving itself as an intellectual activity that is worthy of study.
Class time will be divided between lecture and discussion, with actual problem solving done individually and in groups. Students will be expected to give correct and carefully written solutions to problems and participate in class discussions.
No mathematical prerequisite is needed beyond a solid foundation of high-school level mathematics.
Adam Fieldsteel (A.B. Brown University; Ph.D. University of California, Berkeley) is professor of mathematics. His research focuses on ergodic theory and topological dynamics, and his recent publications include: (with A. Blokh), "Sets that force recurrence," Proceedings of the American Mathematical Society (2002); (with K. Dajani), "Equipartition of interval partitions and an application to number theory," Proceedings of the American Mathematical Society (2001); (with R. Hasfura), "Dyadic equivalence to completely positive entropy," Transactions of the American Mathematical Society (1998). Click here for more information about Adam Fieldsteel.
ENROLLMENT INFORMATION
Consent of Instructor Required: No
Format: Seminar | Level: GLSP | Credits: 3 | Enrollment Limit: 18 |
Texts to purchase for this course:
G. Polya, HOW TO SOLVE IT: A NEW ASPECT OF MATHEMATICAL METHOD (Princeton University Press), Paperback
G. Polya, MATHEMATICS AND PLAUSIBLE REASONING (Princeton University Press), Paperback
William Briggs, ANTS, BIKES, AND CLOCKS: PROBLEM SOLVING FOR UNDERGRADUATES (SIAM), Paperback
READING MATERIALS AVAILABLE AT BROAD STREET BOOKS, 45 BROAD STREET, MIDDLETOWN, 860-685-7323
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