MAT 268
Introduction to Research in Mathematics
Spring ****
NOTE: This syllabus was created prior to implementing the Teacher-Scholar Program.
Instructor: Dr. Saad El-Zanati E-mail: saad@ilstu.edu
Office: STV 304 Phone: 438-5765 (Home: (309) 376-3776)
Office Hours: **:**--**:**; and by appointment.
University Withdrawal Policy: Withdrawal through January ** yields no withdrawal grade. Withdrawal through March ** with instructor’s signature yields a WX grade (does not count in GPA).
Disability Concerns: Any student needing to arrange a reasonable accommodation for a documented disability should contact Disability Concerns at 350 Fell Hall, 438-5853(voice), 438-8620 (TDD).
Course Overview:
The course is meant to afford undergraduate students the opportunity to explore several topics drawn from diverse areas of mathematics with emphasis on experimentation, conjecture, careful justification, and clear, precise reporting. The topics may be drawn from number theory, algebra, analysis, combinatorics, dynamical systems and statistics. Frequently the topic involves the application of computational techniques to gain further insight into the problem. The computer software package Maple, as well as other existing software, will be used as exploratory tools in some of the projects. The instructor will introduce each topic, giving the mathematical and perhaps historical background, as well as some simple exercises. This will require one or two classes. The instructor will then pose several substantial questions for exploration with suggestions to aid students. The students will then generate examples, either by hand or using the computer, to explore the questions, and perhaps raise new questions. By accumulating information from these examples, students will observe regularities, real or accidental. They will then make conjectures based on these observations, testing their conjectures with further examples. Students will be asked to present ideas, conjectures, or examples to the class. Finally, each student will write a clear, precise report describing the student's own investigations, and giving careful justification for the resulting conjectures.
Student Objectives:
There are three general goals for the course:
1. To immerse students in the exploration of mathematical phenomena in order to develop habits of mathematical thinking and writing.
2. To expose students to potential research problems in mathematics.
3. To learn how to use technology, such as the software package Maple, to assist in the exploration of mathematics.
Texts used:
There will be no required text for the course. The instructor will distribute handouts on the various topics.
There will be three or more topics for exploration in the course. These topics will vary from semester to semester (according to the instructor’s preferences). A good topic will require only first year mathematics, use ideas likely to recur in advanced courses, require some aspect of computing, and raise substantial, yet accessible, questions for investigation. The topics will be chosen from several areas to display the breadth of mathematics.
Some possible topics drawn from Discrete Mathematics:
History of Math
The Greeks and the idea of “proof”
Logic and the different types of proof
Basic Number Theory
Prime factorization
Greatest common divisors
Modular arithmetic
Basic Concepts in Graph Theory
Cycles, paths and trees
Hamiltonian factorizations
Introduction to Maple
Basic commands and syntax
Graph Labelings and Graph Decompositions
Definitions of labelings
Applications of labelings
Search for new labelings
The Oberwolfach Problem
Known results (and methods)
Variations on the problem
Search for new results
Partitions of Vector Spaces
Partitions into linearly independent sets
Partitions into subspaces
Applications in combinatorial design theory
Search for new results
Other possible topics include: Iterations of Linear Maps in the Plane, Iteration of Quadratic Functions and Chaos, Public Key Encryption and the RSA Algorithm, Balanced Incomplete Block Designs, and Tree Decomposition of Regular Graphs.
Specific Goals:
The course will provide experience in formulating and testing conjectures, and then justifying the conjectures based on empirical observations and mathematical analysis. Such experiences motivate the necessity of proof in mathematics.
Students will see mathematics as a work-in-progress, and they will gain a sense of how mathematicians produce mathematics.
Students will see more of the breadth of mathematics at an early stage in their undergraduate career.
Students will (hopefully) experience the mystery of an unproved conjecture and the joy of discovering some mathematics on their own.
The course will introduce students to Maple, which is a computer software package for doing numerical, symbolic, and graphical mathematics. Maple also contains its own programming language. Students can learn to do some fairly sophisticated computations in Maple with only a modicum of training. Furthermore, students will find immediate applications for Maple in their other classes, especially calculus and linear algebra.
Each student will write a final report on their research results, stating clearly and precisely the results of their investigations. Such careful writing and reasoning are precursors to the writing of proofs in advanced courses.
During the last week of class, students will give a final presentation of their work in a seminar format open to the entire department. Students will also be expected to present their work at the ISU Undergraduate Research Symposium (March **) and at appropriate symposia for undergraduate research. If results are of wide interest, publication in a journal for undergraduate research or in a regular research journal may also be appropriate.
Required student tasks:
For each topic a student will write a report of about three typed pages. Reports will normally consist of five sections: an introduction, the experimental strategy, experimental results, analysis of data, and mathematical analysis of conjectures. The introduction will describe the topic under investigation, providing background information and terminology. The experimental strategy section will motivate the questions the student asked, the order in which they were asked, and the choice of examples, including any strategy for generating examples. The experimental results section will contain a careful organization of data, using tables, graphs, or pictures where appropriate. The analysis of data section will discuss patterns found in the data, including real or accidental regularities, the formulation of conjectures based on the data, the testing of conjectures by choice of examples, and the empirical justification of these conjectures. The mathematical analysis section will provide, when possible, theoretical justification of the conjectures.
Students will be able to accumulate up to 100 total points in this course. Grades will be based on two tests (a Midterm and a Final Exam) worth 40 points total, on short student reports related to the topics (worth 15 points), on group and individual work assignments (worth 25 points), on the presentation (worth 10 points), and on originality, class participation and contributions (worth 10 points). The standard 10 point-scale will be used to determine the letter grade (90-100 points = A, 80-89 points = B, 70-79 points = C, 60-69 = points D, 0-59 points = F). Generous amounts of extra credit will be given for significant contributions to the covered topics. Reports will be judged on clarity and precision of the writing and cogency of the justifications. Students will be expected to distinguish between examples and general statements, as well as between conjectures and proven results.