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. We will do this by exploring topics not usually covered in typical high school math courses including numerical patterns, codes, infinity, graph theory, and if time allows either symmetry or the mathematics of elections. We will see how these topics are related to our everyday lives.

Though you 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. (For example, a mathematical formula can be thought of as a compact, precise statement of a pattern.) 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 reasoned arguments that make the patterns that we discover begin to rise to the level of theorems.

Many students take Math 110 to fulfill their quantitative reasoning requirement. While it is natural to focus on the quantitative aspect of the course, more important in the long run is the reasoning component. The subtitle of our text is "An Invitation to Effective Thinking" and such thinking is an important goal of the course and your entire college education. As we learn new mathematics we will examine the process by which we do this mathematics. The section on infinity will be especially significant. We will be forced to reason very carefully about abstract concepts. The goal is to learn new mathematical concepts by discovering and verbalizing your thought processes rather than by simply memorizing facts or formulas. 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!"

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 using careful reasoning. 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. We will be able to develop many concepts from scratch using only elementary arithmetic and careful thinking. (Even so, some math majors would be unfamiliar with certain topics.) If you come to class with an interest in learning about mathematics and a willingness to ask questions, experiment, work hard, and think clearly, 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 often involve group work and discussions in which you are expected to participate actively. Early in the term, the small group exercises will come from your text, so please bring The Heart of Mathematics to class each day for the first week or two. It is unlikely that we will finish all of the group work during class, so you will need to continue to work on the exercises outside of class.

At the end of each section of our text there is a series of "Mindscapes" which ask you to explore and extend the concepts that have just been discussed in the text. You will need to be an active reader with a pencil and paper close at hand. Doing these Mindscape exercises is the single most important aspect of the course and is largest component of your final grade.

Assessment

Collected Exercises: About once a week there will be an assignment consisting of selected Mindscape exercises to hand in. Problems will be assigned after each class and will be posted at our course website: http://math.hws.edu/~mitchell/Math110S10/index.php. You should bookmark this website and check it frequently for updates.

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. Many exercises will require a written explanation of the process you used in order to answer the question. 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 time posted on line, generally on Fridays at 3 PM at my office. Of course, you turn them in early, say at class on Thursday. Late homework will not be accepted. Homework will count for 27% of your final grade, the single biggest factor.

Homework Bonus: You will have the opportunity to earn extra credit toward your homework grade through work on certain additional specified problems. Additionally The Mathematics and Computer Science Department has seminar talks regularly. Each seminar talk you attend is worth 4 bonus points. (A maximum of three seminars may be attended for bonus points.)

Journal Work: Each week journal exercises will be assigned. Journal exercises should be done in a composition book (preferred) or a spiral notebook (no looseleaf binders). Reading and journal exercises will be posted online at the course website after each class. I encourage you to work with a partner or two on these problems. Your journal should not contain class notes or collected homework sets---please keep those separate. Follow these guidelines for your journal:

Complete the exercises in the order in which they were assigned.
Label each exercise by section and exercise number.
Use a highlighter to mark each label, so that it is easily identified.
Briefly describe the problem and/or draw a diagram if helpful or required.
If you are unable to completely solve a problem, show your attempts and explain why they did not work.
Record your ideas, questions, and thoughts. These notes will be useful if you see me for help.
Come for help if and when needed.
If you worked with anyone, list each person's name.

I will collect your journals at each exam. Journal work will be most effective if done consistently for each class. 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 the class material and prepare you for groupwork, homework sets, and exams. Your journal grade will be based largely on evidence of your overall effort at exploring topics and creatively searching for solutions. Your journal work will be 10% of your course grade. As an additional incentive to work on your journals, I intend to include one or two of the journal problems on each exam.

Project: You will be required to complete a group project on codes (groups of three). Details of the project will be discussed after the first exam and it will be due Friday, March 26 by 3pm. The project is 10% of your course grade.

Exams: There will be two in-class exams, one on Thursday, February 25th and the other on Thursday, April 8th. Each exam will be worth 15% of your course grade. The final exam (see beginning of syllabus for schedule) is worth 18%. It is impossible to construct fair makeup exams in mathematics. Thus for your protection, my policy is that there are no makeup exams. Write these dates in your calendar. You must be present for all exams.

Participation: Because of the nature of this course, its assignments, and our group work in class, your attendance and active participation are crucial. Participation will count as 5% of your course grade. Missing more than two classes will negatively 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. February 25:    15% 
Exam 2. April 8:        15%
Homework:               27%
Project. Due March 26:  10%
Journal:                10%
Participation:           5%
Final Exam:             18%

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. At the course website http://math.hws.edu/~mitchell/Math110S10/index.php I will post information about the course and the assignments.

Academic Integrity

I strongly encourage you to discuss the reading and journal exercises with each other as well as coming to see me during office hours. In verbalizing your questions or explaining mathematical ideas to others will increase your understanding of the material. Listening carefully to others explain their ideas will help you sharpen your own conceptions. However, you should not feel free to copy someone else's work or make your work available to someone else. Copying constitutes plagiarism, a violation of academic integrity which could result in failure in the course. Obviously no collaboration or use of outside resources is allowed on exams. Violation of the Colleges' Principle of Academic Integrity may result in a report sent to your dean and/or an appearance before the Committee on Standards.

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.

Please direct questions about this process or Disability Services at HWS to David Silver, Coordinator of Disability Services, at silver@hws.edu or x3351.

Assignment 1: Autobiographical Essay and Mindscapes

This assignment counts as part of your homework grade.

Write a full one-page, typed autobiography. Discuss your reasons for taking this course and what you hope to get out of it. Discuss your major and minor (or what you think they will be) and what you hope to do after college. In one or two sentences, describe what you think mathematics is. Do you like math? What do you like and/or dislike about it? What are your favorite and least favorite memories of previous mathematics courses or experiences? Briefly discuss your hobbies or anything else about yourself that I should know.

Your essay should be typed using usual font size and margin widths. Your grade on this assignment will be based on whether you address all the topics requested, as well as the quality of your writing, including grammar and typography.

Note: If you are a senior in the 10:20 section, please tell me whether you have another final exam scheduled for Saturday, May 8, 2010 at 1:30-4:30.
There are also two Mindscape exercises due as well. See http://math.hws.edu/~mitchell/Math110S10/index.php