Exploring Mathematical Patterns
Instructor: Jennifer Galovich
Office Hours: 1 - 2:00 every day. Please feel free to make an appointment at another time if this doesn’t work for you. (I will post my schedule at the beginning of every week so you can see when I am available.)
Phone: Office: 363-3192
Text: D.W. Farmer and T. B. Stanford, Knots and Surfaces
You will also need two notebooks -- one for notes you want to take in class, and the other (THE Notebook) for work to turn in.
The goal of this course is to provide you with an opportunity to practice doing mathematics. “Doing” mathematics includes not only the use of the traditional tools--proof and formal manipulation--but also the variety of ways in which mathematicians think about their subject -- experimentation, educated guessing, etc. This course will emphasize mathematical thinking by challenging you to take these steps for yourselves.
The context for our explorations is (low dimensional) topology. “Topology” is a kind of geometry; low dimensional topology focuses on objects that “live” in two or three dimensions. The good news is that you (yes, you!) can solve many interesting problems without having to know a lot of specialized vocabulary or techniques ahead of time. On the other hand, you will need to be persistent yet open-minded--willing to try new approaches when old ones don’t work, but not giving up too easily or too soon..
Since the goal is for you to do as much mathematics as possible, there will be very little lecture and LOTS of discussion. At the beginning of the semester you will be assigned to a group. On most days, you will be spending class time working with your group on tasks from the text, and/or other problems I will suggest. (Of course, I also expect you to work on these and other problems outside of class.) I will supplement with lecture and commentary as needed.
Expectations and Requirements
Evaluation will be based on your notebooks, class participation, quizzes and a final project.
Notebooks will be collected six times during the semester (dates below). You are encouraged to work together on problems, but you must write up your own solutions in your notebook, giving credit as is appropriate to the others with whom you have discussed the problem. Each problem in your notebook will be evaluated mostly on the basis of your progress toward a solution, but also on your ability to make an effective argument and to use language precisely and appropriately. I will give you lots of feedback (and a preliminary grade), and I expect that you will continue to work on the problem and/or rewrite as needed.
Class participation will be evaluated on the basis of your effectiveness as a collaborator and communicator. You will be expected to be an attentive and critical listener, to contribute regularly to the work at hand through questions and suggestions, and to respect the ideas of others.
NOTE: Because of the structure of this course, attendance and participation are critical!
Quizzes will be given on the days that notebooks are due. The quizzes will be short – about 10 - 15 minutes -- and will emphasize definitions and examples.
Final Projects: In mid-November you will be assigned to a team of three people and given a final project topic which will occupy you for the last two weeks of the semester. During the last two days of class, each team will give present its work in a poster session. A written report on the project (to which each member of the team will contribute) will be due at noon on Wednesday, December 18, 2002. (There will be no final examination.)
Here’s how the various components are weighted:
Class participation: 10%
Notebooks due: September 11, 25; October 11, 25; November 8, 22
Quizzes: September 11, 25; October 11, 25; November 8, 22
Poster Sessions: December 9, 11
Project write-up due: December 18
I really want each of you to do your best in this course. Please don’t hesitate to consult with me at any time.
More About Notebooks
1. Acknowledgments: At the end of every problem -- whether it’s solved, partially solved, or still a mess, be sure to give credit to those with whom you have worked (struggled?) on that problem, and/or from whom you have gathered pearls of wisdom. You do not lose any credit for listing these acknowledgments.
2. Number all the pages in your notebook consecutively. (This is useful when you want to refer to earlier drafts.)
3. Before you turn in your notebook, go through your work and label each write-up with
a) a box -- if you think you’ve got a pretty good solution worked out
b) a wiggly box -- if you’ve gotten started but you haven’t got all the details yet.
c) lots of question marks -- if you’re stuck and want some suggestions.
(Or use your own system--just be clear about the status of your work.)
4. For each notebook submission I expect that you will have completed all the “tasks” for the sections of the text we have covered. In addition, I expect you to have completed (at least) three problems which you have chosen from the additional problems list (to be provided). You will be tempted to put the work off until the last minute. Don’t.
Note: Writing up a solution to a problem is very much like writing a paper. You can expect to go through several drafts, particularly for more challenging problems. In a final draft I expect your work to be legible and organized, written in complete English sentences with correct and appropriate use of mathematical notation. I will give you feedback about these issues of exposition, as well as comments and suggestions about the mathematical correctness of your work. I will be happy to give informal feedback in between notebook dates whenever you feel it would be useful.
5. I will give preliminary and final scores based on a scale from 1 - 10. A 10 means the write-up and solution are both just fine — at most minor errors. A grade of 4 or below suggests that you are on the wrong track and/or the exposition is sloppy.