# Guidelines for Solutions to Problem SetsMath 301 Real Analysis, Fall 2019

One of the central goals of the course is that you improve your ability to communicate mathematics clearly, both in writing and verbally. Learning to write precise and complete mathematical arguments and proofs is a challenging endeavor, and this process will not only aid your mathematical development but can also greatly improve your clarity of thought in other disciplines as well.

I firmly believe that one of the best ways to build your understanding of mathematics is to explore the ideas with other students. Therefore, you will work on the Problem Sets in groups of two (or possibly three), and each group will turn in a single set of solutions. I will randomly assign new groups for every problem set.

### Guidelines for Working in Groups

• You should think about each problem on your own before you get together with the other members of your group, and then you can compare approaches and work through the harder problems together.
• Each member of the group is responsible for the content of the entire problem set and the quality of the writeup.
• All members of the group should sign the Honor Code statement.
• You will evaluate your partner's willingness to be an effective member of the team by filling out a confidential form on onCourse by 4:00 pm on the day the Problem Set is due.
• I will share your average evaluation from your peers with you at the end of the semester, and I will multiply the total points you have earned by your average evaluation to determine your Problem Set grade.

### Format for the Write-ups

I have high expectations for the organization and presentation of your Problem Sets. I am not being unnecessarily annoying about this. Learning to carefully organize your thoughts and clearly communicate them is one of the most important skills you will learn in college.

• You and your partner should put your name and date on the first page of each assignment.
• Group the Focus Problems together, in order, at the beginning of your problem set, and begin the set of Basic Problems on a new sheet of paper.
• Clearly label each problem with the chapter and exercise number.
• Leave enough empty space on your solutions for me to make comments.
• Your writing must be clear and legible. Use complete sentences to justify your arguments where appropriate.
• Do not turn in the first draft of your solutions. Expect to rewrite and polish your Problem Sets. I strongly suggest you write up your solutions using pencil so that you can more easily correct mistakes, but if you do use a pen, there must be no scratch-outs. You will earn no credit for any problem with scratch-outs.
• Do not turn in paper torn from a spiral notebook with ragged edges.
• All of this may be summarized by what I have come to identify as the transitive property of happiness in grading: The neater and better organized your assignment is, the happier I am while grading it, and therefore, the happier you will be when you get it back.
• The only outside resources that you may use are the text, your class notes, other students currently enrolled in our Real Analysis class, and me during office hours. Remember that we are operating under the Wheaton Honor Code. You must cite your sources, and if you are ever uncertain if something is allowed, then please ask me!

### Evaluation of Problem Sets

You should always have a target audience in mind whenever you write. Here is a good rule of thumb to follow for your Problem Sets:

Write your solutions so that you could hand them to a student who previously took Real Analysis and she would be persuaded that your solution is correct and that your conclusions are believable.

There are three types of errors* that frequently occur:

• A computational error occurs when a mathematical computation is carried out incorrectly, either by hand or by computer. For example, solving $$x^3 = 8$$ to obtain $$x=3$$ is a computational error.
• A conceptual error occurs when one of the concepts from the course is applied incorrectly or the solution/proof is not complete.
• An error in communication occurs when the solution is not well-organized or fully justified for the target audience. In particular,
• The solution should not omit any parts that contribute to the solution and would not be obvious to the target audience. If you are in doubt about what counts as "obvious" relative to the target audience, please ask me!
• The solution should be concise and not include any information that is not relevant to the solution.

* Thanks to Robert Talbert at Grand Valley State University for this classification of types of errors.

Each Focus Problem will be graded on a scale of 0-7 with the following criteria:

• 7: Exceptional. All areas perfect.
• 5: Essentially complete with no significant errors of any of the above kinds and the number of minor errors is minimal. That is, a small number of minor errors can be tolerated as long as they do not cast doubt on your understanding of the concepts.
• 3: Significant conceptual or communication errors.
• 1: Needs substantial improvement in all areas.

Remember that the purpose of all of the assignments and activities in the course is to help you learn Real Analysis and develop as a mathematician!