Transitions Projects

Project Topics List
Final Papers (4-5pages).
Your paper should consist of proving a theorem or two, and showing a number of proven examples step-by-step. Students are expected to create their own examples when possible, not just copy examples from sources. The style should be approachable so that a fellow peer student (a peer who is qualified to take this class) can learn your topic by reading your paper. The page length is considered from material in your project that we have not already covered during class time. You can certainly include things we have covered in your paper but it does not count towards your page count, and you may want to condense or minimize those items we’ve already covered.
UPDATE: Here is a which explains margins and font size requirements, as well as some other tips and suggestions.

ANOTHER UPDATE: Here are of f math research papers, to give you a sense of balance between equations, diagrams, and text.

Final Presentations (10-12 mins) during class time about your project.

- partial orders and power sets
- cardinal number and Schroder-Bernstein theorem (see Ch 11)
- properties of the Cantor set
- compactness property of various sets
- special numbers in number theory (there are many different types)
- metric spaces and open/closed sets and examples
This list is not exhaustive. If you find something of a similar difficulty feel free to suggest it! Also, two students cannot have the same topic. The Prof. will help you to make sure your topic is not the same as another student. Check this document frequently and the Prof will leave some notes regarding what each student is researching.
DISCLAIMER: Paper topics should not overlap with a paper you did for another class, and it should not cover material already covered in our required class material. You can certainly state something we covered in class for the purposes of setting up another theorem, result, or example.
Daphcar - Schroder-Bernstein Theorem
Davon - compact sets
Jonathan - Rational Roots Theorem
Mark - partial orders and power sets
Nicholas - Cantor Sets
Stephan - special numbers (Mersenne, Amicable, etc.)
Joshua - open , closed sets and metric spaces
Dec 6 Presentations
1. Jonathan
2. Mark
3. Daphcar

Dec 8 Presentations
4. Nicholas
5. Davon
6. Joshua
7. Stephan

Presentation order was assigned randomly. If students wish to switch spots with each other, it must be a mutually consenting switch! You can talk amongst yourselves and decide to switch spots. Just let the Professor know if you agreed with another student to switch. The Prof will update the list above when any switches agreed.

Support your peers! All students must attend all student presentations!
There will be two participation components to the presentations for the audience members.
i) After each presentation, the audience will ask questions. Each student in the class should plan to ask at least TWO questions total, for all the presentations. .
ii) After each presentation class, the Professor will collect . Each student will be assigned TWO presentations to provide feedback. Students will receive anonymous feedback after all the presentations are finished.
Avoiding Plagiarism
Do not submit in any draft math that you don’t understand or examples copied verbatim from any source. All parts of your project, at all stages, for each and every draft, must be submitted in your own words and based on your personal understanding of the subject using notation and phrasing you crafted. Maintain the mathematical rigor and technical details, but explain in your own words — there is a balance. Each individual draft will go through plagiarism checks by the Prof. Some questions to ask yourself when reading from your sources:
- How could I explain this better so students like me can understand it?
- Are there gaps in the source’s explanation?
- How would I teach this topic to a peer in my own words?
- How can I maintain mathematical rigor while still using an understandable approach?
- Is the source’s explanation repetitive anywhere or could be simplified?
These questions will help you to craft your own narrative and avoid plagiarism. Think of the style like teaching to another student... how would you explain this topic? and how would you explain this correctly? Also do not copy examples verbatim especially if it’s feasible to create your own examples. Students are expected to create their own examples whenever possible. At all stages of the project references must be provided. No source you are using should be kept hidden from the Prof.
Common Mistakes
- Not defining words or symbols before using them (define words first, then use them).
- Including something in your paper you don’t understand (don’t do this).
- Thinking you need to include everything from a source (you don’t need to).
- Using a lot of energy to redo something we already covered in class (shorten these things).
- Not fixing feedback provided in a previous draft (please fix feedback items).
- Using letters just because a source did that are conflicting with other letters you use (develop your own consistent notation throughout your paper).
Reacting to Feedback
At each stage of the feedback process, students are expected to act on and resolve the feedback provided — not doing so may indicate the project is stalled or not progressing towards its goals. If you don’t understand feedback from the Prof. that should be discussed during office hour appointments where the Prof. can help you by discussing your topic one-on-one.
Timeline and Due Dates
- Friday Nov 4th topic due: Email the Prof. saying what topic you are thinking of doing for your paper. Don’t just say the title or topic name. Explain in a short paragraph with at least 5-10 sentences what example you found that looks interesting that you will compute, and what theorem you found that you will prove. Provide two or three references (formal or informal) that you plan on using for your project. Explain why you picked the topic and whether you have any special knowledge about it (if not that’s ok, just curious!). The Prof. will reply making some suggestions to your topic, and she will make sure that topics are not overlapping among students in the class. Check back at the top of this webpage and a list of student topics will appear once they are determined.
- Friday Nov 11th outline due: Relatively complete outline of theorems/examples/plan for whole paper due. Include section titles, statements of theorems, show which examples you will do and how they will work, and explain which proofs you will do and roughly how they will work. Present some math you have figured out for yourself even if it’s not completed. At least 2 or 3 formal references must be identified at this point. P.S. PLEASE “PRINT TO PDF” FROM YOUR WORD DOCUMENT TO AVOID ANY VERSIONING ISSUES, THANKS!! Schedule an office hour appointment this week to discuss math questions you have regarding your project (how a proof works, how to create your own examples, etc.).
- Friday Nov 18th first draft due: First Draft due with proofs mostly completed, examples mostly done, math mostly done, perhaps English sentences unfinished, and all feedback from the previous deadline fixed and incorporated into the project. Students should have at least 3-4 pages at this point and 3-4 formal references. Also email Prof. explaining your plan for the presentation part. Will you use powerpoint slides or the whiteboard? What is your plan / outline for what to cover during the presentation? Your presentation should be a subset of your paper, and the presentation should perhaps at a minimum include a proof of your main result, and perhaps one main example. P.S. PLEASE “PRINT TO PDF” FROM YOUR WORD DOCUMENT TO AVOID ANY VERSIONING ISSUES, THANKS!! Let the Prof. know how it’s going and whether you have questions. Schedule an office hour appointment this week to discuss your math questions.
- Monday Nov 28th second draft due: Second Draft due with math done and English sentences done, and all feedback from the previous deadline fixed and incorporated into the project. Should have at least 4-5 pages at this point and 4-5 formal references. Also email Prof. explaining your latest plan for the presentation part, what new ideas have you had since the last update? P.S. PLEASE “PRINT TO PDF” FROM YOUR WORD DOCUMENT TO AVOID ANY VERSIONING ISSUES, THANKS!! Let the Prof. know how it’s going and whether you have questions. Schedule an office hour appointment this week to discuss your math questions.
- Optional Deadline Monday Dec 5th third draft: Get final additional feedback, if you want.
Presentations during class time Dec 6th and Dec 8th.

Friday Dec 9th, ALL papers due, final deadline
(remember print to pdf)

Writing Tips
The tone of your paper and your presentation should be aimed at teaching a fellow peer student, at the level of this class. Think of the person reading your paper (your audience) as a qualified math student. In other words, you don’t need to explain what the contrapositive or converse means, or what an integer is, or what a real number is (that’s all prerequisite info). You should explain how the logic of your proof works in a “teachable” way that identifies the method of proof and clearly explains all steps, in mathematically correct language. Your proof steps should be in your own words and phrased in such a way that you think can help another student learn your topic. Think about writing a textbook and teaching a qualified student your topic ー that’s the tone your project should take. :)
- Read the paper out loud to yourself to test out the phrasing and language.
- Do not include anything you yourself don’t actually understand.
- You should be able to field reasonable questions from the audience. Try to anticipate questions and build that into your paper/presentation so the narrative is as helpful as possible.
- Just like homeworks, all proofs should be self-contained. Don’t refer to a theorem unless you stated the theorem. Don’t use theorem numbers/labels from the book (or from any source), create your own labels so that your paper is self-contained.
- If you find yourself copying a proof from any source ask yourself: do I really understand this? how will I improve this proof? what steps are not clearly explained? which parts are too short that I can expand on? which parts are too long that I can simplify?
Presentation Tips
Preparation and practice is key
If you don’t practice, then you might run way over time or way under time. Do a few trial runs to test your timing. You might be surprised how much different it is to speak math out loud versus thinking it through mentally or writing it down on paper. Speaking to a class verbally is an entirely different skill that takes practice. When you write something down on paper or think about it mentally, you are usually trying to convince yourself. When you are explaining something verbally, you are usually trying to convince another person. It’s an important distinction to think about when preparing.
You can try recording yourself and watching the recording for a practice run. While this is not the most pleasant activity (trust me, as a Prof I’ve made a lot of videos! lol), it really does help to critique your own explanation when you see yourself played back to watch.
P.S. Don’t worry, the class session will NOT be recorded when you do your actual presentation!
Anticipating questions from the audience
You cannot anticipate every question, but you can prepare yourself for “obvious” questions. Examples include: What does that symbol mean? How did you get to that step? Would your math still work with a different number or a different type of quantity? If you think about similar questions while preparing your presentation this will help you to manage audience questions better in the moment. Sometimes fielding questions can be stressful, but you can significantly reduce stress by being well-prepared. Preparedness is the #1 stress reducer for presentations !
It’s OK if you don’t know everything
It’s easy to feel “judged” or defensive by a person asking a question for which you don’t know the answer. That’s the most common initial emotional response ー we are only human! :) Part of what you will learn after doing many college-level presentations is learning how to stay calm and collected while addressing questions from the audience. Don’t be afraid to respond saying something was outside of the scope of your project (that’s a diplomatic way of saying “I don’t know”!). Audience questions might also give you an opportunity to talk about something you thought was cool but didn’t make it into your official paper. The more you view audience questions as helping students learn the more fun the interaction will be! Don’t think that every single tiny mistake is necessarily huge and embarrassing ー quite the contrary, mistakes are inevitable! Part of teaching is learning to embrace the process of learning and mistakes are part of the learning too! Sometimes a mistake can be helpful to explain something !

Informal References
Informal references include blog posts from an unidentified author, wikipedia pages, or tutorial websites. If you use resources from one of these informal references you can include them in a separate Informal References section at the end of the document, and they cannot be listed in the Formal References list. If you are using a bunch of informal references, look for where these websites may provide formal references and look into using those formal references as well. For example, wikipedia pages have references at the bottom of the page, you could look into reading those published books so that your information is coming from the most reputable sources.
Formal References
For a formally written mathematical paper, appropriate formal references include: published articles, published books, or lecture notes written by a Professor at an accredited college. Your paper should have at least 3-5 formal references. The formatting of references can follow the format supplied in the paper examples provided in the Class Google Drive (see below). If you prefer a different formatting it’s OK just keep it consistent throughout your references.
NOTE: If any reference has a URL associated to it, you can include the URL link at the end of the reference line. All provided links must actually work and cannot require any login or special account. If you need special access to get to your link, then print out the document (to pdf, or physically) and submit the document entirely for this reference to the Professor.
See Brightspace → Links menu → Class Google Drive → Project Resources Folder
References formatting from “Paper Example - Taco”
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References formatting from “Paper Example - Volumes”.
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Grading Standards
A participation and Peer Feedback component will provide +5 extra credit points. The Prof. will email you details before the presentation dates about this.
I. Oral presentation (20 points out of 100)
a) Content
• Appropriate choice of material given the time constraint
• Mathematical concepts (examples, theorem statements, etc.) explained correctly
• Sufficient definitions, illustrations and/or motivation is given for teaching peer students
• The mathematical material is at a level appropriate for peer students to understand
b) Clarity and delivery
Board writing or presentation format is clear and large enough, including diagrams/pictures.
Speaking is clear and well-paced, and presenter faces the audience when possible.
• Concepts are clearly and concisely explained and can be followed by a qualified peer student.
• The presentation appears well-prepared (rehearsed but not rote).
• The presentation is within the time limit.
Note: it is okay – in fact I encourage you – to have some written notes with you while you present. But ideally, you should not have to look at them very much!
II. Written report (80 points out of 100)

a) Mathematical content
Is the mathematics presented both internally consistent and actually correct? Is it at a level of difficulty or sophistication appropriate for this class? Are topics and ideas introduced with sufficient explanation? If there are pictures, figures or examples, are they accurate, appropriately used and do they support the text?
b) Breadth and Depth
Is detailed knowledge of the topic presented? Does the paper move past summarizing or just stating facts into analysis, proof, and evaluation of those facts? Does the paper show an ability to apply critical thinking and independent thought to new situations? Does the paper develop a clear and unique voice from the student author?
c) Clarity of mathematical exposition
Are topics presented in a logical order? Does the paper achieve an appropriate balance of conciseness and explanation? Are complicated parts/proofs (if any) broken into logical and clear steps?
d) Style
Is the paper clearly written with appropriate narrative in paragraph form? Is the grammar, spelling, and sentence construction correct? Are equations built into the sentence structure in a way that is grammatically correct when read aloud? Do the introduction and conclusion each serve their purpose? Is the paper readable, does it flow?

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