Mathematics is such a vast subject that figuring out what to include in a one-semester course is a challenging task. Our goal will be to explore a series of connected topics that give a coherent glimpse into contemporary mathematics, while also bringing out core principles essential to the mathematical way of thinking. The outline of the course is given below. (Chapters to be covered are roughly 18,21,22,19.1,23,5-8,19.5 for now.)
1. Gentle Introduction: We'll begin with a power point presentation to put you in the mood for the class. I will survey your math background and review some basic concepts (e.g. functions).
2. Precision and Scale: A key core principle of math is its precision. We'll start Chapter 18 with this in mind - to get into the habit of making precise statements and giving precise answers. Paradoxically, for math to be actually used in practice, one has to make approximations, so we'll also learn when this is appropriate. The concept of scale leads to some interesting explanations of why things are as they are in the world.
3. Growth rates: Section 18.6 deals with various power functions, which we will explore, along with such kinds of growth as logarithmic, quadratic, and so on. One of the key uses of math is to quantify relations, and all these different functions can be used as needed to model different applications. We continue with selected topics from Chapters 21-22, which give us an introduction into exponential growth, and also help us practice precision and basic math skills in a very useful context. (This will be the most computational part of the course.) Exponential growth also introduces us to the key concept of infinity, which, as we will see, is one of the most important ideas in math, and ties together most of the topics in this course.
4. Population growth and golden ratio: Problem 4 of Chapter 19 on rabbits multiplying will lead us to the so-called Fibonacci numbers. We will make a brief detour into Section 19.1 to explore these and the golden ratio. We will tie this to the exponential growth we have investigated before.
5. Logistic growth, dynamical systems, and chaos: Some of the most exciting discoveries in mathematics over the last decades have occurred in dynamical systems and chaos. We will explore these through selected topics in Chapter 23, as well as additional material. Computer simulations will assist in our investigations, giving us a taste of how mathematicians use experiments to gain intuition.
6. Statistics: We will look at another type of experimentation, that of sampling and data analysis. For this, we will cover selected topics from Chapters 5-8 (the last on probability). The Central Limit Theorem in Section 8.6 once again shows how all these techniques depend on an understanding of infinity. A knowledge of statistics is quite essential in today's quantification-driven world.
7. Fractals and nature: While some natural phenomena (e.g. the ones dealing with scale we saw in Chapter 18) are deterministic in nature, a number involve a high degree of randomness. We will see how these lead to fractal manifestations (in particular, through some class exercises involving probability). Fractals are covered briefly in Section 19.5, which we will augment with additional material. This topic will help us understand how mathematical processes shape the universe.
8. Infinity and other topics: In the final section of the course, we will examine further topics in mathematics, that will range from the philosophical (e.g. the nature of infinity) to the practical (e.g. further student-requested applications of contemporary mathematics).
The primary goal of this course is to gain knowledge and appreciation of contemporary mathematics. More specific goals, together with topics that help attain them, are listed below.
1. Gain an understanding of how mathematical processes shape our world. (Topics 1,2,4,5,7)
2. Enhance precision, computational skills, and other core mathematical qualities. (Topics 2,3,6)
3. Acquire content- and technique-based knowledge (e.g. statistics) that can be practically applied outside this course. (Topics 3,6)
4. Appreciate the use of the computer as an experimental tool (Topics 5,6,7)
5. Learn to think mathematically. (Topics 1 to 8)
6. Learn about broader issues and open questions in mathematics. (Topics 1,8)
Keeping up with the material is going to be essential. Also, being an active participant in class will enable you to do well grade-wise. Please be informed that some of the topics to be covered are not adequately presented in the book - another reason to make sure you attend (and stay alert in) class. Study groups are encouraged, as is discussion of homework. However, turning in work copied from another student is unacceptable, and will be considered cheating (see "Academic Conduct" below).
Please don't be late for class. If you expect to be more than five minutes late, please clear this with me beforehand. Similarly, don't leave until the class is over (unless cleared with me beforehand). No hand-held devices or texting. No chatting amongst yourselves.
Wed, Feb 8 is the last date to drop a class without a W on your transcript. Mon, Apr 16 is the last date to drop this class with a grade of W. Please do not hesitate to talk to me if you need some guidance on how to proceed regarding these dates.
By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full Student Academic Conduct Policy, consult the UMBC Student Handbook, the Faculty Handbook, or the UMBC Policies section of the UMBC Directory.
If you require accommodations for this class based on disability, please make an appointment to meet with me to discuss your SSS-approved accommodations. Please see http://my.umbc.edu/groups/sss/documents/838 for more information.