Exploring Mathematical Patterns
Instructor: Jennifer Galovich
Office Hours: 1 - 2:30 every day except Tuesdays, when I must leave at 2:00. Please feel free to make an appointment at another time if this doesn’t work for you.
Phone: Office 363-3192
Course Materials: Honors Math Explorations Problem Collection
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 Combinatorics--the branch of mathematics concerned with describing and exploring the properties of patterns. Sometimes these patterns involve objects which are obviously mathematical, but sometimes not. The good news is that you (yes, you!) can solve many interesting combinatorial 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.
As you (undoubtedly) are aware, this course carries both writing and discussion flags. Therefore, in addition to the “doing” of mathematics, you will also be practicing “writing” and “talking” mathematics. If you’re thinking that Symposium never prepared you for this, don’t worry. We will be working on these skills as part of the course.
Since the goal is for you to do as much mathematics as possible, there will be very little lecture and LOTS of discussion. On most days, you will be spending class time working in groups of at most three on problems chosen by the group. (Of course, I also expect you to work on these and other problems outside of class.) I will introduce each (approximately) two-week section of the course with a new type of problem to add to the collection. At the end of the section each group will give an informal presentation on some interesting (or intriguing, or impossible) problem.
Expectations and Requirements
Evaluation will be based on your notebooks, classwork and your final project.
Notebooks will be collected 5 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 on the basis of your progress and your writing, especially 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.
Your classwork--small group problem solving, discussion and informal presentations--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!
Projects: At the beginning of May you will be assigned to a team of 3-4 people and given a final project topic which will occupy you for the last two weeks of the semester. During the last few days of class, each team will give a formal presentation of its work. The final written report (to which each member of the team will contribute) will be due on Wednesday, May 20 at 5 p.m.
Here’s how the various components are weighted:
Notebooks due: February 13; March 5; March 26; April 16; May 1
Project presentations: May 12, 14, 18
Project write-up due: May 20
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: By 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.
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. You will start work on lots of problems, but some of them may prove to be more interesting and more worthy of pursuit than others. On a separate piece of paper, to be turned in with your notebook, tell me which problems you want me to read and how to find them. That is, tell me which problems are (in your opinion) completed problems (boxed) -- the goal is to end up with as many of these as possible, of course; which problems are partially completed, and on which you’d like some feedback (wiggly box); which ones you want to continue to work on, but need advice to get started (question marks). For each problem that you’d like me to read, give the page number(s) in your Notebook where I can find your work. Do not erase or tear out pages from your notebook -- I will read only what you want me to read, as described above. Also, some of your scratch work may be useful later on.
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.
5. You should aim for a total of 7 - 10 problems in the first two categories for each notebook due date. This means that you should have completely solved about 40 problems by the end of the semester. Therefore I strongly suggest that you resist the temptation to procrastinate.
6. 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.
First Notebook Assignment--Due February 13
You are free to work on any problems you like, but I recommend that you begin with those on pp. 1 - 6.