MATH 405 Introduction to Mathematical Research (3)

Catalog Description:

Critical review of papers in current undergraduate mathematical journals. Introduction to and development of the techniques involved in researching and preparing a paper on a mathematical topic. May be repeated for additional credit as the topic/study changes. (Offered as needed.)


Prerequisites:

MATH 211 Calculus III and MATH 261 Linear Algebra

While no particular topic may be an absolute prerequisite, it is expected that the student will have strong algebraic skills, a thorough conversance with the calculus and some significant experience with axiomatic mathematics.

 

Required Course Materials:

This course has no required textbook as it is meant to defy the confinement which might be implied by any designation of a text. However it should be noted that the student is expected to have and use his texts from previously completed mathematics courses as references when appropriate.


Course Coordinator:


Lamarr C. Widmer, Ph.D., Professor of Mathematics

Course Audience:


Juniors and seniors majoring in mathematics, especially those considering graduate level study.

Course Objectives:

  1. To significantly sharpen problem solving skills by tackling a variety of new problems for which even the instructor may not know a solution.
  2. To experience the interplay between attempts to find a proof and attempts to find a counterexample which eventually leads to determination of the validity of a conjecture
  3. To develop skill in concise, clear and logical presentation of problem solutions and proofs both orally and in writing.
  4. To carefully read a journal article of moderate length and difficulty, justifying steps of the argument to our own satisfaction, constructing examples of our own to illustrate definitions and theorems, checking references and possibly building from it to further investigations of our own.
  5. To personally experience the process of submitting a problem solution to a journal for publication.
  6. To experience the collective "brainstorming" process of problem solving with a group of peers.
  7. To prepare and deliver a short formal mathematical talk to an audience other than the members of this class.
  8. To establish and encourage e-mail communication with former Messiah College students now attending graduate school or working in the mathematical profession.
  9. To select and use technology, typically computers, when appropriate as an aid in forming and testing conjectures and in checking computations.
  10. To gain some familiarity with and appreciation for the mathematical professional establishment: professional organizations, journals and the submission/refereeing process, use of CML, e-mail communication, Mathematical Reviews, etc.

Topics:

Papers and problems will be chosen according to interests and background of the students and instructor. The instructor intends for them to mostly be chosen from very recent issues of journals. In addition, students may choose problems or projects connected with material from some other course they are or have been enrolled in. Ideally there should be no repetition of specific problems from one offering of this course to the next. In the past, course material has typically been related to the following topics:

  1. Sequences and series.
  2. Mathematical induction.
  3. Polynomials.
  4. Theory of equations.
  5. Number theory.
  6. Inequalities
  7. Abstract algebra.
  8. Existence and uniqueness of solutions.
  9. Numerical analysis.

 

Revised: May 2006

Return to Course Index