M302 SyllabusINTRODUCTION TO MATHEMATICS
Prerequisite and degree relevance: Three units of high school mathematics at the level of Algebra I or higher. The Mathematics Level I test is not required. It may be used to satisfy Area C requirements for the Bachelor of Arts degree under Plan I.
M302 is intended primarily for general liberal arts students. It may not be included in the major requirement for the Bachelor of Arts or the Bachelor of Science degree with a major in mathematics. In some colleges M302 cannot be counted toward the Area C requirement nor toward the total hours required for a degree. Only one of the following may be counted: M302, 303D, or 303F. A student may not earn credit for Mathematics 302 after having received credit for any calculus course.
Responsible Parties: Kathy Davis and Mike Starbird, August 9 2007
Course Description: Introduction to Mathematics is a terminal course satisfying the University's general-educationrequirement in mathematics. Topics may may be chosen from: Fibonacci numbers, number theory (divisibility, prime numbers, the Fundamental Theorem of Arithmetic, gcd, Euclidean Algorithm, modular arithmetic, special divisibility tests), infinity, geometry (Pythagoean Theorem, Platonic Solids, the fouth dimension, rubber sheet geometry, the Moebius band), chaos and fractals, probability (definition, laws, permutations and combinations), network theory (Euler circuits, traveling salesman problem, bin packing), statistics, game theory, voting paradoxes. Some material is of the instructor's choosing.
Texts: For all Practical Purpose or The Heart of Mathematics, Second Edition
There is a broad spectrum of students who take M302. Some are quite good at math and may even have had some calculus in high school. These, however are greatly outnumbered by the students who have weak math skills and poor backgrounds. It is not at all uncommon for the students to exhibit a fear of and dislike for math and most have very low self-confidence about their ability to succeed in a math class. In answer to this, the goal of the course should be to demonstrate that math is not about memorizing formulas, but is rather a process of thinking which is relevant to them on a daily basis. The two recommended books, both are geared toward this type of course. For All Practical Purpose emphasizes applications of math in today's world such as scheduling problems and consumer finance models, for example. The heart of Mathematics, while dealing with more theoretical topics such as number theory and topology, emphasizes that the problem solving stategies used to solve mathematica problems are universal and can be applied to solving day-today problems. Both texts have proven to be successful at engaging this population of students and giving them new appreciation of math as well as boosting their self-confidence.
The topics to be covered will depend on the choice of text. Both texts cover probability and statistics and at least 3 weeks of the course should be devoted to this topic. The coverage in For all Practical Purpose is more thorough, especially in area of statistics. If this is the chosen text, then the syllabus should include chapter 5 and 7. Chapter 6 can be covered lightly, if at all, and chapter 8 should be considered optional. If The Heart of Mathematics is the chosen text, then all of chapter 7 should be covered.
Sample Syllabus for "For All Practical Purpose":
- Chapter 1 Street Networks All sections (3 days)
- Chapter 2Visiting Vertices(omit Minimum cost spanning trees)
- Chapter 3 Planning and Scheduling(omit Bin Packing) (4 days)
- Chapter 5 Producing Data All sections(4 days)
- Chapter 6Exploring Data Cover lightly (2 days)
- Chapter 7 Probability All sections (5 days)
- Chapter 10 Transmitting Information (supplement the modular
arithmetic and cover cryptography
only) (4 days)
- Chapter 15 Game Theory All sections (5 days)
- Chapter 20 Consumer Finance Models All section (time
permitting) (6 days)
Notes: For all Practical Purposes
Chapter 1 is an introduction to graph theory and is a good chapter for establishing the course as one which is not "formula-based." Chapter 2 and 3 then follow up with some applications of graph theory.
As mentioned above, Chapters 5 and 7 should be covered thoroughly and Chapter 6 lightly. Chapters 9 and 10 introduce the concept of modular arithmetic with applications toerror detecting codes cryptography. Students tend to find the arithmetic challenging, but in general they enjoy the ideas in these chapters.
Chapter 13 on Fair Division is fun to do, however it is difficult to get the ideas across. Students tend to get lost in the logic and may end up simply memorizing procedures.
Chapter 15, Game Theory, also gives the students a work-out in the area of following a logical argument and again they tend to memorize algorithms for finding good strategies. This chapter does give a chance to revisit expected value and they also appreciate the real-world applications of the "Prisoner's Dilemma" problems.
apter 20 deals with compound interest and annuities.The relevance of this material to their lives makes it one of the most widely-appreciatedchapters on the part of the students.
Sample syllabus for "The Heart of Mathematics":
Chapter 1: Fun and GamesAll sections (3 days)
- Chapter 2: Number Contemplation 2.1-2.3; 2.6-2.7(light) (6 days)
- Chapter 3: Infinity Sections 3.1-3.3 (4 days)
- Chapter 4: Geometric Gems 4.1, 4.3, 4.5, 4.7 (6 days)
- Chapter 5:Contortions of Space 5.1-5.3 (4 days)
- Chapter 6: Chaos and Fractals 6.1, 6.3 (2 days)
- Chapter 7:Taming Uncertainty 7.1-7.3, 7.5-7.7 (8 days)
- Chapter 8: Deciding Wisely 8.1, 8.4 (3 days)
Notes: The Heart of Mathematics:
Chapter 1 is excellent for setting the tone of the class and illustrating some problem-solving strategies. The puzzles also tie in with the material from later chapters.
Chapter 2 covers some topics from number theory and gives an appreciation of number theory as an ancient area of mathematics. Section 2.5 can be summarized but should probably not be covered in deail.
Chapter 3, on infinity, is guaranteed to provoke lively discussions as well as controversy.
Chapter 4 contains some nice sections on geometry. The section on the Pythagorean theorem give the students several examples of geometric proofs. In the section on the Platonic solids, the students are encouraged to build the solids and explore the concept of duality. The section on the fourth dimension gives them the opportunity to experience an abstract idea through the process of generalization. The Moebius Band is a nice, concete application.
Chapter 5 deals with some ideas from topology. The section on rubber sheet geometry has some fun and surprising results, but the students will probably need a model to convince them that the results are indeed true. The section on the Euler characteristic ties in with chapter 4's section on Platonic Solids.
Chapter 6 deals with chaos and fractals. The key point here is for students to understand iterative processes,, and how they relate to fractals.
Chapter 8 is really about understanding expected value, and applying that to voting paradoxes.