Math 110: Discovering in Mathematics

About Math 110

The goal of this course is to increase your understanding and appreciation of the beauty of mathematics. While you likely have had 11 or 12 years of mathematical training, you may not have a clear idea of what mathematics is or what mathematicians do. Mathematics is about discovering patterns. You do this sort of discovery, at least in a low level way, all of the time. Mathematicians, however, state these 'patterns' in a formal way as 'theorems' using a very precise language. Theorems differ from mere opinion or observation in that they require 'proof,' that is, they should follow logically from some given set of basic assumptions. While we won't be doing lots of formal mathematics, we shall be very interested in developing reasons or arguments that make the patterns that we discover rise to the level of theorems.

As the title of our text suggests, we will use geometry as the primary area of our inquiry. For the most part, this will involve questions quite different from those that you covered in your high school geometry course, though we will need some basic facts that you encountered there. For example, we will explore topics such as the fourth dimension when we read Flatland. A more mundane topic that we will encounter is wallpaper patterns. Despite the thousands of wallpaper designs that you can find at the Home Depot, there are actually fewer than 20 truly different wallpaper patterns. We'll see why. We will also think about a few non-geometric topics. For example, in this presidential election year, we will examine issues about voting. There's a lot of interesting 'mathematics' surrounding fair voting, more than simply tallying votes for each candidate. We might also spend some time thinking about infinity.

Our focus, however, will not be just on learning new material or memorizing formulas, as you might have experienced in earlier courses. Rather we will also explore the process by which we do mathematics. The section on four-dimensional geometry will be especially significant here. We will be forced to reason about such objects by analogy and by extrapolating from patterns we observe in one, two, and three dimensions. So, yes, you will learn new mathematical concepts, but the goal is for you to do so by discovering and verbalizing your thought processes rather than by simply taking lecture notes. As one of my colleagues puts it: "Think of yourself as an explorer setting off in your ship to discover new lands . . . and being required to write reports back to your homeland about your discoveries and how you found them!" In fact, this is exactly what the narrator of Flatland, the other text we will use, does in that book.

I hope you will enjoy this course. Since we have no set agenda, we can stop to enjoy a particular topic as long as we wish. Even if you think of yourself as a non-mathematical person at the beginning of this class, I hope you will become excited about discovering patterns and proofs on your own. By the end of this course, I hope you will have encountered some topics that interest you and that you will be more attuned to 'mathematical patterns' and the idea of discovering mathematics.

Expectations and Prerequisites

First the good news: There are no formal prerequisites for this course. One of the reasons that I have chosen to focus primarily on geometry is that we will be able to develop many concepts from scratch using only elementary arithmetic and careful thinking. If you come to class with an interest in learning about mathematics and a willingness to ask questions, experiment, and work and think hard, you should do well.

HOWEVER, I will expect you to generate your own approaches and solutions to questions and you will be required to do as much writing as calculating. Although there will be some lectures, class meetings will consist mostly of group work and discussions in which you are expected to participate actively. Much of the class will be spent working in small groups on the questions in the text. Unlike most mathematics texts you've previously encountered, this text does not have questions at the end of each section. Rather, the questions are integrated into the text. If you do not stop to answer the questions, the subsequent material will not make sense. In other words, you need to be an active reader with a pencil and scratch paper close at hand. Doing the exercises in the text is the single most important aspect of the course.

Assessment

Much of our class time will be spent working in small groups on exercises in the text or other handouts and then discussing some of our results or 'discoveries.' Please bring Symmetry, Shape, and Space to class each day. It is unlikely that you will finish each assignment during class, so you will need to continue to work on the exercises outside of class.

I will ask you to hand in some of these exercises each week. While you may work with a group in class, ultimately you will be responsible for your individual work. The exercises will be divided into two types: collected and uncollected. It is to your advantage to work carefully and thoroughly through all of these exercises before the next class period. Daily work on the exercises will help you understand what is being covered in class and prepare you to do well on the exams. In particular, I am likely to choose some problems from your uncollected exercises to be included on your exams. I highly recommend using a loose leaf or spiral notebook or composition book in which to complete all your uncollected problems.

Collected Exercises: About once a week there will be an assignment (about 20 to 30 points) consisting of selected exercises to hand in. Solutions should be written neatly or typed, and stapled if you have more than one page. You must show all work and indicate your reasoning to receive credit. Although I encourage you to discuss ideas for most exercises with other class members, your write up must be your own individual work. Some exercises will require a written explanation of the process you used in order to answer the question. You will be required to list anyone with whom you discussed the homework. Since illness or other unavoidable circumstances occasionally occur, I will drop your lowest homework grade. Assignments are due at the beginning of class and late homework will not be accepted. Homework will count for 40% of your final grade.

Homework Bonus: You will have the opportunity to earn extra credit toward your homework grade. The Mathematics and Computer Science Department has seminar talks regularly. Each seminar talk you attend can earn you five bonus points. Another opportunity to earn bonus points is to attend the events sponsored by Geneva Concerts. I will let you know when these are. For the concerts I may ask you to type a half-page reaction to the performance. (There is a maximum total of fifteen possible bonus points.)

Project: You will be required to complete one project, which will likely involve working on one of the sections in the text that we do not cover. However, there will be other options available. Details of the project will be discussed after the first exam and it will be due on Monday, November 25 (unless I notify you that the project is due later). The project will be worth 10% of your course grade.

Exams: There will be two in-class exams, one on Friday, October 3rd and the other on Friday, November 7th. Each exam will be worth 15% of your course grade. The final exam is scheduled for Wednesday, December 17th. It is impossible to construct fair makeup exams in mathematics. Thus for your protection, my policy is that there are no makeup exams. Write the dates above in your calendar. You must be present for all exams.

Participation: Because of the nature of this course, its assignments, and its assessment, your attendance and active participation are crucial. Participation will count as 5% of your course grade. Missing more than three classes will severely affect your participation grade. If you must miss a class for some reason beyond your control, talk to me about it in advance.

Your final grade will be determined as follows.

Exam 1. October 3:       15% 
Exam 2. November 7:      15%
Homework:                40%
Project. November 25:    10%
Participation:            5%
Final Exam. December 17: 15%

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Office Hours, Website

My office is Lansing 305. My extension is 3619. My e-mail address is mitchell@hws.edu. Scheduled office hours are listed at the top of this sheet. I am often in my office much of the day; drop in to get hints or help with course assignments or just to chat. The course website http://math.hws.edu/~mitchell/Math110F08/index.html will also contain information about the course as well as links to most handouts and assignments.

Rough Guide to Weekly Readings

This outline will be adjusted as necessary during the term, depending on how deeply we pursue certain topics.

A Note about the Center for Teaching and Learning (CTL)

Hobart and William Smith Colleges encourage students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their study skills, writing skills, or have other academic inquiries should contact the CTL. You may visit the CTL web site to learn more about the services and programs that are available. http://www.hws.edu/academics/ctl/index.aspx.

If you are a student with a disability for which you may need accommodations, you are required to register with the Coordinator of Disability Services at the CTL and provide documentation of the disability. Services and accommodations will not be provided until this process is complete. The web site for information pertaining to registration with the CTL and documenting disabilities is: http://www.hws.edu/studentlife/stuaffairs_disabilities.aspx.

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