MATH.5200-201 Mathematical Problem Solving

The main focus of the course will be to solve interesting, challeging mathematics problems. Another objective is to help you develop good habits of mathematics communication (see below).

Here is one of the problems from the 2013 competition:

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them. | |

The following notes serve as the main text for this course.

I'd suggest that you download a copy of this. HOWEVER, your grade will be partially determined on your activity in annotating a copy of this document that is posted on Perusall. So most of your reading should be done on Perusall. You'll be able to access Persuall through Blackboard, or you can create a free account at Perusall.com to access the course once the semester starts.

In addition, there are many resources, including free online ones. Here is a partial list.

- The Putnam Archive
- This is the third in a series of Putnam problem books published by the Mathematical Association of America.
Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil,

*The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary*(MAA Problem Book Series), 2002, ISBN: 088385807X - Michael Penn (Randolph College) has posted many excellent (and entertaining) videos at https://www.youtube.com/c/MichaelPennMath/. Lots of them are Putnam problems or problems from similar competitions.
- The Art of Problem Solving Community is a site devoted to problem solving at all levels, but has some specific resources relating to the Putnam.

To earn a grade in this course you will need to do the following:

**Attend classes.**Unless you have valid medical or legal excuse for missing a class, you will be considered absent. You get one free absence, and then your starting grade will be moved down the list {A, A-, B+, B, B-,...}. Your starting grade will then be the highest possible grade you can earn in the course, assuming you do "A" work for everything.**Participate.**40% of your grade will be based on your participation. This is measured through reading assignments on Perusall.**Work independently on assignments.**There will be problems assigned for work outside of class. Your work on them will represent 30% of the final grade.**Compete**. You must compete in the William Lowell Putnam Mathematical Competition. Competing is 20% of the grade. Competing means working on problems for at least 2.5 hours in each of the the two 3 hour sessions on the day fo the competition. The scores are announced long after grades are due, so everyone who shows up gets an "A" in this part of the course. We will meet on Wednesday, December 8 to "debrief."**Reflect**. The final requirement for the course will be to write up a reflection on the course and the competition. In this 3-5 page paper, you can describe your efforts, preferably highlighting one of the Putnam problems. In addition, you can comment on what you've learned about mathematics and the problem solving process. This will count for 10% of the course and will be due Monday, December 13 at noon.

For each class there is a main topic (e. g., Calculus, Probability, Geometry,...). About a week before each class, you will be given a reading assignment on Perusall. It will consist of some notes on the main topic and a list of problems. I'll expect you to post comments and/or questions on the reading on Perusall. Part of your grade will be based on your engagement in this activity. The problems in the reading assignment will be the ones we will work on in the class meeting for that topic. As part of the reading assignment activity, you can ask questions about the problems. Another way to engage in the this activity is to answer your classmate's questions. I'll answer some, but often I'll hold off answering if I think it's reasonable to expect the answer to come from the class.

If you have a solution to a problem in a reading assignment (partial or complete), you can post a hint or general outline of you solution. In addition, I'll encourage you to be prepared to share the solution in our meeting. We will have document camera in the class, so having the solution on paper would be a quick way to share it. Depending on how much board space we have, you could also write the solution on the board before the class.

Don't worry if you don't have a solution in every class. Many of the problems are difficult, but I've tried to include a few relatively easy problems in each week. I'm not telling you which problem is easy. Also, what is easy to one person isn't necessary easy to someone else. I understand that there are some topics that you might have less familiarity with than others.

After some classes I may assign a problem for the class to turn in using Gradescope.

The first class meeting is not typical. Since there is no required reading assignment although there are notes on Perusall you should take a look at. Feel free to comment on them.

I won't require that you use LaTeX in the course, but I'd like you to give it a try. There will be one assignment that will require using it. Even if you don't get in the habit of using it for full documents, you should be comfortable writing mathematical expressions with it since it's the standard way to do so on sites like Perusall.

One of the objectives of this course is to help you develop your mathematical communications skills. The following are a list of the things you should keep in mind for this course. I'll be looking for you to attempt to follow these guidelines.

- Ask questions. Notice, identify, and clarify sources of confusion. Look for connections and relationships between ideas. Explore topics and ideas deeply.
- Analyze and constructively critique the reasoning of others. Actively listen and summarize key ideas to check comprehension. Test conjectures against examples and potential counterexamples. Work together to find errors and fix flaws. Assess and reconcile various approaches to the problem.
- Explain and justify your reasoning. Communicate and justify your conclusions to others. Indicate the general strategy or argument, and identify the key step or idea(s). Actively listen to the critiques of others. Work together to find errors and fix flaws.
- Attend to precision. Communicate precisely to others. Use clear definitions, and carefully state any assumptions or results used.
- Be clear and concise. Use the appropriate amount of generality or specificity in arguments. Avoid use of any extraneous assumptions, hypotheses, or statements. Indicate any example(s) that you have in mind.
- Use and develop mathematical fluency. Use standard mathematical notations and terms (as discussed in class or demonstrated in course materials). Clearly indicate and explain any use of non-standard shorthand, notation, or tools

Source: Richard Wong, UCLA, Active Learning Exchange on MAA Connect

All students are advised that there is a University policy regarding academic integrity. It is the students' responsibility to familiarize themselves with these policies. Students are responsible for the honest completion and representation of their work. Link to the Academic Integrity Policy

We are a campus that cares about the mental health and well-being of all individuals in our campus community, particularly during this uncertain time. If you or someone you know are experiencing mental health challenges at UMass Lowell, please contact Counseling Services, who are offering remote counseling via telehealth for all enrolled, eligible UMass Lowell students who are currently residing in Massachusetts or New Hampshire. I am also available to talk with you about stresses related to your work in my class.

If you have a documented disability that will require classroom accommodations, please notify me as soon as possible, so that we might make appropriate arrangements. Please speak to me during office hours or send me an email, as I respect, and want to protect, your privacy. Visit the Student Disability Services webpage for further information.

UMass Lowell and your professor value human diversity in all its forms, whether expressed through race and ethnicity, culture, political and social views, religious and spiritual beliefs, language and geographic characteristics, gender, gender identities and sexual orientations, learning and physical abilities, age, and social or economic classes. Enrich yourself by practicing respect in your interactions, and enrich one another by expressing your point of view, knowing that diversity and individual differences are respected, appreciated, and recognized as a source of strength.