Math 368 - Introduction to Abstract Mathematical Structures

TR 9:35 -10:50am, CE 136

Instructor: Dr. Kathryn Nyman
Office: Milner 221
Office hours: Mon. 10:00 - 11:30 am,  Tues. 3:00 - 4:00 pm, or by appointment.
E-mail:  nyman@math.tamu.edu                         Phone:  845-7636
URL address:  http://www.math.tamu.edu/~nyman

Course Description:  We will cover topics in logic, set theory, number theory, relations, modular arithmetic, groups and cryptography (as time permits).  The course is designed for K-8 education majors who wish to be math specialists for grades 5-8.  The purpose of the course is to deepen the students' understanding and knowledge of mathematics (especially in regards to the "whys" of mathematics).  Active participation in class is expected.    Prerequisites:  Math 131, 166, 365, 366 and 367.

Textbook:  Edward R. Scheinerman, Mathematics - A discrete introduction, Brooks/Cole, 2000.

Grading:  There will be three exams given during the semester.  The time and subject matter will be announced at least a week in advance.  In lieu of a final examination, you will have a final paper (see below).  In addition there will be semi-regular quizzes, homework and collected in-class work and writing assignments. 
3 Exams @ 100 points                   = 300 points                   
Final paper                                       = 100 points                   
Quizzes/homework/class work:     = 150 points

Paper:  Your paper is due on Friday, May 7 by 5:00pm (the first day of finals).  The paper will be on a topic of mathematics, preferably one that interests  you.  A list of possible topics is provided.  Feel free to suggest a topic not on the list, but you must get my permission before proceding.  You will be required to find materials both in books and on the web.  The best papers will contain some original work (such as examples that you created, applications, etc).  The main purpose of the paper is for you to learn some new mathematics on your own so that you will know that you can learn mathematics without a professor.  This is very important for teachers as curriculums and requirements change over time and you will likely need to learn something new (as well as learn new ways to look at old math) as you teach.

Homework:  In order to do well on quizzes and tests it is imperative that you practice the material we cover in class.  Homework will be collected on Thursdays in class.  Late homeworks will not be accepted except in extraordinary situations (contact me before homework is due in such cases).  I encourage you to form study groups and discuss homework problems with each other, however every student must write up their assignment independently.

Help:  Ask questions in class!  Come to my office during office hours!  Work with each other!  The Learning Skills Center (845-4427) and Support Services for Students with Disabilities (845-1247) may also be of help.

Make-up Policy:  Make-ups for missed exams will only be allowed for a university approved excuse in writing.  Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam.  Otherwise they forfeit their right to a make-up.

Scholastic Dishonesty:  Copying work done by others, either in class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy.  For more information on university policies regarding scholastic dishonesty, see University Student Rules.

Copyright Policy:  All printed materials disseminated in class or on the web are protected by US Copyright laws.  Multiple copies or sale of any of these materials is strictly prohibited.

Math Paper  
Instead of a final this semester you will be researching a mathematical topic on the web and in books, and any other way you wish, and then writing a paper on the topic.  The best papers will contain something *you* created (such as examples to demonstrate understanding).  The level of the discussion and examples should be for college students (you are not writing materials for teaching middle school).  I am happy to discuss the project with you, however you are not to discuss the topic with anyone else working on the same project.
Possible topics:
1. The golden ratio                                          18. Magic squares
2. Fibonacci numbers                                       19. Latin squares
3. Triangular numbers                                      20. Fractal patterns
4. Pascal's Triangle and its applications             21. The history of Zero
5. The mathematics of calendars                        22. Non-transitive dice
6. The pigeon hole principal
7. Rational election procedures (Can you set up a voting system that cannot be manipulated?)
8. Tilings (e.g., the chess problem of the knights)
9. Fair cake cutting or other apportionment problems
10. The abacus - different kinds and how does it work
11. The mathematics of piano tuning
12. Ciphers or cryptography (secret codes) other than RSA code (we will do this in class)
13. The 5 color problem (all planar graphs can be colored using at most 5 colors so that no 2 countries sharing a border have the same color (4 colors are actually sufficient, but that is beyond this course))
14. Investigate and explain a paradox -
      a. The prisoner's dilemma (how to get the smallest sentence for both)
      b. The surprise exam paradox (can there be an announced surprise?)
      c. Newcomb's paradox (a variant on the Monte Hall problem)
      d. Zeno's paradoxes
15. P-adic integers
16. The Koenigsburg bridge problem (or Eulerian circuits)
17.  Sizes of infinity and arithmetic of infinity