Week #1: What is mathematics (for)?: Analysis and Generalization

• What is an analytical solution?
• Finding mathematical solutions and/or showing that such solutions exist.
• The mathematics of art galleries?: Klee's theorem.
• Generalizing mathematical results: The Seven Bridges of Königsburg, Leonard Euler, and translating and expanding mathematical findings into different mathematical domains
• How do mathematical statements compare with those of science and history? A comparative look at "proof", "truth", and "evidence".

Week #2: The mechanics of mathematical proof.

• Deductive methods in number theory and Euclidean geometry.
• Mathematical induction, and why mathematicians love it.
• Applying deductive and inductive methods to a range of problems from the theory of graphs, computational geometry, and the theory of numbers.

Week #3: The mathematics of counting.

• Sets and Permutations on sets.
• Combinations and their applications.
• Binomial and multinomial coefficients.
• From Counting to Statistics

Week #4: An introduction to the mathematical theory of probability.

• How can there be a mathematics of chance??
• From counting to calculating probabilties.
• The fundamental concepts of discrete probability.
• Integrating combinatorics with probability.
• A final exam (of course)

Notebooks: Each student is required to keep a separate notebook for this course, and as the final will be an "open book" exam, it is a good idea to make these notebooks as complete as possible.

Readings & Writings: During the course we may read an article or two and the article(s) will be given to you in class. In addition, student teams will be responsible for reviewing one of the mathematics resources on the World Wide Web.

Homework: You will be assigned two problem sets during the course. Your participation in class, your performance on the homework sets, and your grade on the final exam will be (in that order) the major considerations in my evaluation of your work. To help with homework, and to discuss further material covered in class or at the Web site, I will keep office hours in the Lewisohn computer lab, Tuesday-Thursday from 12 until about 1 (depending on whether or not anyone shows up).