• Make clear and well-motivated mathematical presentations.
• Write mathematical papers of professional quality.

Nicholas J. Higham, Handbook of Writing for the Mathematical Sciences, Society for Industrial & Applied Mathematics (SIAM), 2nd edition/August 1998, ISBN: 0898714206.

Donald Ervin Knuth, Mathematical Writing: A Manual for Authors, Mathematical Association of America (MAA), July 1989, ISBN: 088385063X.

Steven G. Krantz, A Primer of Mathematical Writing: Being a Disquisition on Having Your Ideas Recorded, Typeset, Published, Read & Appreciated, American Mathematical Society (AMS), 1991, ISBN: 0821806351.

1. The paper is well-organized.
   1. Include a title page and a bibliography and use standard scientific formatting throughout.
   2. Limit the main body of the paper to the required length, using good judgment about what to include in the paper.
   3. Begin the main body of the paper with an introduction designed to capture reader's interest and to make clear the objectives and approach selected by the author.
   4. Following the introduction, have an identifiable and well-organized body of the paper that focuses on main points and logical transitions between them. Sub-headings may be used to help the reader better follow the paper’s structure.
   5. Make the relationships among ideas clear by developing clear transitions.
   6. End the paper with a conclusion that accentuates the structural plan and, as appropriate, identifies related questions or directions for future development.
   7. Appropriately use citations throughout the paper.
   8. Number all theorems, figures, and any important formulae appropriately.
2. The author exhibits an acceptable level of understanding of the mathematical material.
   1. If a part of the paper emphasizes the formal statement of definitions and theorems and standards of proof, then it is anticipated that the paper should communicate key definitions and results accurately and should include appropriate examples to illustrate them. The paper should also demonstrate understanding of the way definitions and prior results are applied in the development of a proof.
   2. If the paper is to communicate an overview of the entire topic through a careful selection of definitions, theorem statements and examples with central concepts and results being stated formally and illustrated, then the paper should explain key definitions and results accurately and include appropriate examples to illustrate them.
3. The paper is readable at a level appropriate for the intended audience.
   1. Assume the reader has solid mathematical reasoning skills and has been exposed to the ideas of calculus and the fundamentals of sets and proving, but do not assume any additional background.
   2. Be aware of the readership. Use good judgment in distinguishing between concepts and results known to readers versus those that require review or introduction and development.
   3. Be free of the kind of grammar, word usage, and mathematical notation errors that interfere with the clarity of communication.
   4. Use standard notation, layout, and style appropriate for the intended audience.

1. The presenter exhibits a clear structural plan for the presentation.
   1. Begin with an introduction designed to capture audience interest and to make clear the objectives and approach selected by the presenter.
   2. Following the introduction, have an identifiable body of the presentation that focuses on main points and logical transitions between them.
   3. Make the relationships among ideas clear by developing clear transitions.
   4. End the presentation with a conclusion that accentuates the structural plan and, as appropriate, identifies related questions or directions for future development.
   5. Adhere to the given time limit.
2. The presenter exhibits awareness of the audience.
   1. Prepare your presentation in such a way as to assure the understanding of your audience.
   2. Assume the listener has solid mathematical reasoning skills and has been exposed to the ideas of calculus and the fundamentals of sets and proving, but do not assume any additional background.
   3. Use good judgment in distinguishing between concepts and results known to audience members versus those that require some degree of review or introduction and development.
   4. Using note cards, overhead transparencies, PowerPoint slides and other forms of support as appropriate, and speak to members of the audience instead of reading the paper.
   5. Maintain eye contact during the presentation making an effort to include everyone in the audience.
   6. Dress appropriately.
   7. During the question and answer period:
      1. Seek feedback when responding to a question by maintaining eye-contact and inviting follow-up questions or comments;
      2. Treat all questions and questioners with respect.
3. The presenter exhibits an acceptable level of understanding of the mathematical material.
   1. If a part of the presentation emphasizes the formal statement of definitions and theorems and standards of proof, the presentation should communicate key definitions and results accurately and use appropriate examples to illustrate them. The speaker should demonstrate understanding of the way definitions and prior results are applied in the development of a proof.
   2. If the presentation is to communicate an overview of the entire topic through a careful selection of definitions, theorem statements and examples with central concepts and results being stated formally and illustrated, then the speaker should explain key definitions and results accurately and select appropriate examples to illustrate them.
   3. If the presentation is to communicate an overview of the whole topic, but the mathematical treatment is more informal, then the speaker should introduce central concepts and results through examples and informal statements designed to stimulate intuitive understanding.
   4. Respond appropriately to questions during the question and answer period.
4. The presenter delivers the presentation with sufficient clarity and professionalism that main points can be understood by most audience members.
   1. It is crucial that the audience understand the main points of the presentation. This is far, far more important than coverage of a large amount of material from the paper. Use good judgment in narrowing the scope of the paper for the presentation to achieve this purpose. Your guideline should be this question: Can the audience go home and complete a reasonable homework assignment--at the level you would be able to do after a class lecture?
   2. Use good judgment in deciding how to best present the ideas inherent in your presentation, and use forms of media support, including projected computer output, as appropriate. Weigh the advantages and disadvantages of using an overhead slide, a physical model, a PowerPoint presentation, etc.
   3. Use of the chalkboard, while not prohibited, should be kept to an absolute minimum.
   4. Prepare overhead transparencies or PowerPoint slides that are:
      1. easily read from any place in the seminar room;
      2. simple, uncluttered and designed to help communicate, review and relate main points.
   5. Be free of the kind of serious grammar, pronunciation, word usage and mathematical notation errors that interfere with the clarity of communication.