Discovering in Mathematics, Section 1Department of Mathematics and Computer Science Hobart and William Smith Colleges Spring 2008. Instructor: David J. Eck. Monday, Wednesday, Friday, 9:05 to 10:00 AM. Room Napier 201.
About The Course
Although everyone has some experience with mathematics of one form or another, most people have only vague or even mistaken ideas about what mathematicians do. At its core, mathematics is about discovering, proving, and applying theorems. A theorem is a statement that follows logically from some given set of basic assumptions, and to prove a theorem is to show that it does, in fact, follow from the assumptions. Mathematics is powerful because the assumptions that it makes are often true statements about the world -- and in that case the theorems that are derived from them are also true.
For many people, doing mathematics just means applying mathematical facts and rules that they learned in school and have memorized. The question is, how were those facts discovered in the first place? Furthermore, when you want to apply mathematics to some problem, how do you discover which of its many facts and rules should be applied? Discovering mathematics or using it in a novel way is not something that can be done by following memorized rules. It requires creativity.
In this class, we look at some examples of how great ideas in mathematics have been discovered historically -- often with great difficulty over a large span of time. We will also do some mathematical discovery of our own, on a much smaller scale, in the form of problem solving. In fact, every time you are faced with a mathematical problem that you can't solve by rote application of some memorized rules, you are being asked to perform a creative act of mathematical discovery. Let's get creative!
There are four required books for the course. Each is available as a paperback, at a cost of about $15 each. The first three books are "popular math books." That is, they were written for non-mathematicians. I reviewed a number of such books and picked out the ones that I thought would be most interesting and most productive of ideas for the course. The fourth book is a textbook on mathematical problem solving that was written for use in courses like Math 110. We will use only a part of each book. The books are:
- The Math Instinct, by Keith Devlin (ISBN 156025839X). This book makes the argument that every person (and many animals) are born with a lot of built-in mathematical abilities, and it discusses why so many people nevertheless have so much trouble with school mathematics. You need to get this book immediately, since there is a reading assignment from it in the first week of class. We will skip over a big chunck out of the middle of this book.
- ZERO: The Biography of a Dangerous Idea, by Charles Seife (ISBN 0140296476). People used numbers for a long time before they thought of having a number to represent zero. This book traces the history of zero and shows why the idea of zero (nothingness; the void) was "dangerous" and hard to accept. It also discusses infinity, which is seen as a kind of twin of zero. We will skip the last three chapters, which talk about zero in physics. The first reading assignment from this book will be in the week of February 25.
- The Poincaré Conjecture: In Search of the Shape of the Universe, by Donal O'Shea (ISBN 0802716547). This is the most mathematical of the three popular math books. In fact, parts of it are rather difficult and will require some additional explanation. The Poincaré Conjecture is a famous mathematical theorem that was first conjectured to be true in the nineteenth century by the mathematician Henri Poincaré. A proof of this theorem was not found until 2002. This book discusses the conjecture, its mathematical background going back some 2500 years, and its proof. We will skip several historical chapters that deal mainly with the biographies of some famous mathematicians. The first reading assignment from this book will be in the week of April 7.
- Problem Solving Through Recreational Mathematics, by Bonnie Averbach and Orin Chein (ISBN 0486409171). "Recreational mathematics" refers to math problems that people work on for fun. (Imagine that!) Of course, those people don't necessarily know that they are doing math, since the problems are often presented as puzzles, brain teasers, or games. Each chapter of this book presents some sample problems, discusses mathematical techniques that can be used to solve them, and provides the reader with a set of similar problems to work on. We will use material from this book throughout the term, starting with Chapter 1 in the second week of the course. In general, I will ask you to read the expository material in a chapter, and I might lecture on it and work through some of the problems in class. Then I will assign some problems from the chapter for group work in class or for homework. Note that I will also bring readings and problems from other sources into the course.
There will be a variety of graded assignments in this course, including both individual and group work. These assignments will count for 36% of your final grade for the course.
Each week, part of at least one class will be devoted to group work, usually some type of problem-solving, and often this work will continue outside of class. The work will be collected and graded in some form. Sometimes, I will ask for a single report from the group as a whole; in that case, everyone in the group will get the same grade. Sometimes, I will ask each person in the group to write up one or more problems, which will be graded individually.
There will be some short writing assignments such as reports on web research or reactions to readings. These will generally be no more than a page in length. (If I get the idea that people are not doing the readings before we discuss them in class, I might start collecting written reports on the reading before the discussions.) There will also be some individual problem-solving assignments.
In addition to the regular weekly assignments, you will complete one larger project, which will be due towards the end of the semester. This will not be a huge project, and it will count for only 10% of your final grade for the course. A paper of some sort, probably in the range of 5 to 7 pages, is one possible type of project, and probably the most common. However, other types of project will be possible. For example, you might create some artwork using techniques or ideas that we cover in the course. Or you might choose several of the harder problems in Problem Solving Through Recreational Mathematics and write up solutions for them. More information for the final project will be available by the middle of the term, along with some specific topic suggestions.
There will be two in-class tests, which will be given on Friday, February 23 and on Friday, April 4. There will also be a third test, which will be given as a final exam in the scheduled final exam period, Saturday, May 10, at 1:30 PM. The final exam will not be longer than the other tests, and all three tests will have the same weight in your final grade.
Each test will cover one-third of the course. The tests will not be cumulative, except to the extent that is forced by the material. (That is, questions on the second and third tests will be about material that is new since the previous test, but you might need to remember some of the older stuff when the new material builds on older material.)
I will check attendance in every class. I expect you to attend every class, but I understand that there might be extraordinary circumstances that make this impossible in some cases. When you have a good reason to miss class, please discuss it with me in advance if possible. If it's something that you don't know about in advance, let me know as soon as you can.
The Mathematics and Computer Science Department Colloquium is a series of afternoon and evening lectures that are presented on an irregular schedule. As part of your mathematical experience for this course, you are required to attend at least one of this term's colloquium lectures. This will count for 3% of your final grade for the course (which could be as much as one-third of a letter grade). There will be an attendance sheet for the class at each lecture. To get credit, you should sign the attendance sheet and you should write and turn in a short essay about the lecture. The essay can be as short as one-half page.
(If it is absolutely impossible for you to attend any of the colloquium talks this term, you should talk to me about designing some other way to get the 3% credit.)
Your numerical grade for the course will be computed as follows:Assignments: 36% Final Project: 10% Colloquium Attendance: 3% First Test: 17% Second Test: 17% Final Exam: 17%
I reserve the right to adjust your grade downwards if you miss more than two classes without good excuses.
Office Hours, E-mail, and Web
My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. My regular office hours are Monday, Wednesday, and Friday, 10:30 to 12:00, but I will often be there at other times. Office hours are times when I promise to try my best to be in my office. I do not generally make appointments during my office hours, since they are times when I am available to students on a first-come, first-served basis. When necessary, we can make appointments for meetings outside my scheduled office hours at a mutually agreeable time.
My e-mail address is email@example.com. E-mail is good way to communicate with me, since I usually answer messages within a day of the time I receive them.
There is a short Web page for this course at http://math.hws.edu/eck/math110_s08/. I will post weekly readings and assignments on that page.
Here is a schedule of tests and of weekly readings. You should always complete the reading before class on Friday of the week when it is assigned (or Wednesday in the week of March 10, when there is no class on Friday). Note that there will also be readings and problems from other sources, including but not limited to Problem Solving through Recreational Mathematics
Week Of Reading or Test Jan. 21 The Math Instinct, Chapters 1, 2, and 3 (pages 1--38). Jan. 28 The Math Instinct, Chapter 10 (pages 165--198). Feb. 4 The Math Instinct, Chapter 11 (pages 199--238). Feb. 11 The Math Instinct, Chapters 12 and 13 (pages 239--264). Feb. 18 Test on Friday, February 22. Feb. 25 ZERO, Chapters 0, 1, and 2 (pages 1--62) Mar. 3 ZERO, Chapters 3 and 4 (pages 63--104) Mar. 10 ZERO, Chapter 5 (pages 105--130) Spring Break, March 14 through 23 Mar. 24 ZERO, Chapter 6 (pages 131--156) Mar. 31 Test on Friday, April 4. Apr. 7 The Poincaré Conjecture, Chapters 1, 2, and 3 (pages 1--31) Apr. 14 The Poincaré Conjecture, Chapters 4 and 5 (pages 32--56) Apr. 21 The Poincaré Conjecture, Chapters 8 and 10 (pages 88--105 and 122--135) Apr. 28 The Poincaré Conjecture, Chapters 12, 13, 14, and 15 (pages 151--200) May 5 Classes end on Monday, May 5. Final Exam, Saturday May 10, 1:30 PM