Illinois State University Mathematics Department
MAT 305: Combinatorics Topics for K-8 Teachers
Dr. Roger Day (firstname.lastname@example.org)
Interest in computer science and the use of computer applications, together with connections to many real-world situations, have helped make topics of discrete mathematics more commonplace in school and college curricula. A topic of widespread application and interest is combinatorics, the study of counting techniques.
Enumeration, or counting, may strike one as an obvious process that a student learns when first studying arithmetic. But then, it seems, very little attention is paid to further developments in counting as the student turns to "more difficult" areas in mathematics, such as algebra, geometry, trigonometry, and calculus. . . . Enumeration [however] does not end with arithmetic. It also has applications in such areas as coding theory, probability, and statistics (in mathematics) and in the analysis of algorithms (in computer science). [Ralph P. Grimaldi, in Discrete and Combinatorial Mathematics, 1994, p. 3]
Combinatorial Analysis is an area of mathematics concerned with solving problems for which the number of possibilities is finite (though possibly quite large). These problems may be broken into three main categories: determining existence, counting, and optimization. Sometimes it is not clear whether a problem has a solution or not. This is a question of existence. In other cases solutions are known to exist, but we want to know how many there are. This is a counting problem. Or a solution may be desired that is "best" in some sense. This is an optimization problem. [John A. Dossey, Albert D. Otto, Lawrence E. Spence, & Charles Vanden Eynden, in Discrete Mathematics, 1987, p. 1]
Documents that support the reform of school mathematics education suggest the need for increased attention to topics in discrete mathematics as well as in probability and statistics. The topic of combinatorics--counting--is mentioned in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics (1989) and in Principles and Standards for School Mathematics (2000). The Mathematical Association of America's recommendations for teacher preparation identify combinatorics as a topic area worthy of study by middle school and high school teachers.
In this course we will study and apply
combinatorial techniques in a variety of settings. In doing so, we
will make connections to algebra, probability, geometry, number
theory, and many other topics in mathematics. During the course, we
may also study the process of proof
by induction, the use of recursion, and
the generation of polynomials.
The course text is Mathematics of Choice by Ivan Niven, published in 1965 by Random House. The bookstores have copies of the text. You may also purchase the book online from the Mathematical Association of America (ISBN 0-88385-615-8, List: $18.50 MAA Member: $14.95 Catalog Number: NML-15/W).
Office Hours & Contact
Room 329B Stevenson Hall
Mondays 5p - 6p; Wednesdays and Thursdays 10a - 11
I am available at other times as well. Please see me to schedule an appointment.
course URL: http://www.math.ilstu.edu/day/courses/current/305
Course Requirements and
Quizzes and Homework (10%)
These will be occasional quizzes during the course. Some homework assignments may be collected and graded.
Tests (3 @ 20%)
Three exams will be given during the course.
The test dates will be announced during class sessions. Part or all of one or more exams may be a take-home exam. Specific restrictions for such take-home portions will be stated when the take-home portion is distributed.
Final Examination (30%)
A comprehensive course exam will be completed Monday, 5 May, beginning at 5:30 pm.
Course grades are based on these requirements. I will use a distribution no more severe than A: 92-100%, B: 84-91%, C: 76-83%, D: 68-76%, F: <68%.
Please refer to the grade guidelines for additional information on grade determination and expectations. There also is information on my expectations for your submission of work in the course.
Course materials are available electronically through a World-wide Web home page for the course: www.math.ilstu.edu/day/courses/current/305.
STV 302 is open regularly for student use. There
are multiple copies of many books, booklets, pamphlets, and journals.
You may use the materials there or check them out on a limited
Attendance, Make-up Work, and Extra Credit
Your active involvement--individually, in small groups, and with the entire class--is an important way for you to help meet the course objectives. For you to be involved, you must be present. If you are now aware of attendance conflicts or should you become aware of such conflicts, please let me know of them as soon as possible.
The deadlines, due dates, and test dates described in the written course requirements and announced during class are just that. Plan ahead to complete the required tasks on time; be prepared for quizzes, tests, and the course exam. Make-up work is negotiated on an individual basis only under emergency situations.
The course requirements are designed to meet the objectives of the course. Unless announced otherwise, there is no extra credit as a substitute for successful completion of the required components of the course.
I invite and appreciate your comments and suggestions for the course as it unfolds. Please share with me in person or in writing your reactions and perceptions. Your contributions will serve to enhance the course for you, your classmates, and future students.