In undergraduate seminars the students teach themselves a topic by using the literature as a resource: students read a book &/or articles in the literature and give lectures to each other on the material.

### Guidelines for undergraduate mathematics seminars at MIT

The M.I.T. Department of Mathematics offers about 10 undergraduate seminars on different topics. These seminars have the following characteristics.

- The enrollment is limited to 15 and is handled by lottery. Preference is given to mathematics majors. If further discrimination must be made it is on the basis of year and progress toward completion of MIT’s undergraduate communication requirement.
- A preponderance of the classes are given over to presentations by the students. As a rule the instructor should not lecture, except perhaps in rotation. The seminar leader should provide specific feedback and suggestions for improvement of oral presentations.
- There is a single focus subject. Generally speaking the level should be at an upper class level, with a prerequisite of one specified departmental subject.
- Appropriate homework may be required.
- At least one coherent mathematics paper of approximately ten pages will be required. It must be written in some form of tex. It must be due (and therefore assigned) early enough in the term to allow at least one revision (and preferably more).
- There are no traditional exams. [Student-created quizzes have been used successfully in some seminars].
- An undergraduate seminar course in the Mathematics Department satisfies the CI-M requirement in Mathematics, which means it is subject to the following descriptive guideline:
“CI-M subjects (Communication Intensive in the Major) teach the specific forms of communication common to the field’s professional and academic culture. Students may write in teams, prepare and present oral and visual research reports for different audiences, learn audience analysis and peer review, or go through the experience of proposing, writing, and extensively revising a professional journal article.

“Communication Intensive subjects (CI) in the major should:

- require at least 5000 words of writing including one mandatory revision, an equivalent amount of oral presentations, or an equivalent combination of the two;
- include substantial instruction and feedback on student work;
- integrate writing and speaking assignments that relate to the professional discourse in the major field; and
- count communication-intensive activities as a substantial portion of the final grade (> 25%).”

- A seminar is a 12 unit class. The grading will be based on:
- Classroom participation, especially the oral presentations.
- Paper or papers.
- Homework. Generally homework should play a supporting role only. It might be used for example to get clear on some of the mathematics which will be incorporated into a paper.
- Degree of improvement.

- Seminar classes provide one of the best ways for a mathematics major to get to know a faculty member well enough to ask for a letter of recommendation. The seminar leader should work to get to know the students well enough to feel comfortable writing for them, and make this potential known to them.

### One format: topics seminar

One approach to running an undergraduate seminar is to have students present on a range of (semi-) related topics.

#### Pros:

- Diverse subject material allows student to focus on their interests.
- Presentations are more engaging when students choose a topic of interest.

#### Cons:

- Students have a hard time gauging level of presentations because of varied backgrounds.
- Some subject matter requires more time and machinery to build up to interesting results.

#### Examples:

- Seminar in Discrete Mathematics, Fall 2010, Sami Assaf, M.I.T.
- Seminar in Physical Applied Mathematics, Fall 2009, Pedro Reis, M.I.T.

### Another format: cumulative seminar

Another approach is to pick a good text and to have the students take turns lecturing on successive sections.

#### Pros:

- Building common vocabulary helps students aim presentations.
- Presentations can build to deeper results than can be reached in a Topics Seminar.

#### Cons:

- Timing can be tricky when student presentations build on one another.
- The course has little flexibility in adjusting the pace since students need to know in advance what they will be preparing.

#### Examples:

- Seminar in Mathematical Logic, Spring 2009, Mia Minnes, M.I.T.
- Seminar in Number Theory, Spring 2010, Scott Carnahan, M.I.T.
- Seminar in Computation Algebra, Spring 2010, Steven Kleiman, M.I.T.

### Schedule and grading

As you decide how to schedule the semester, consider including some of these recommended activities and assignments for teaching mathematical communication.

The primary place students run into time management difficulties in a CI-M seminar is with the final project. To “encourage” them to get started early and to stay on track, you could divide the final project into smaller tasks that have deadlines throughout the semester. For example, you may want to require them to hand in any/all of the following:

- Project proposal
- Project outline
- Draft of introduction
- Draft of proof of main theorem
- Full first draft
- Second draft incorporating revisions
- Final version

Steve Kleiman reports on his use of incremental drafts.

When deciding how many drafts to require students to submit, you may want to weigh the value of revision against the time it takes you &/or peer reviewers to read and comment on each draft.

##### Grading

Some things to keep in mind when deciding on a grading scheme:

- Part of the grade should depend on attendance at other students’ presentations (i.e., grade either attendance or peer critique): it is hard for students to practice giving talks if they don’t have an audience!
- Students sometimes feel uneasy about their standing in the course if too much of the assessment is qualitative or delayed until the end of the term. On the other hand, supplying frequent quantitative grades for communication can cause students to focus so heavily on grades that they aren’t comfortable experimenting with ways to improve their communication. A balance can be reached by giving students a midterm grade in addition to qualitative feedback on each presentation.

- Make sure the grading scheme reflects your assessment of the relative importance of components in the course.

### Assign homework?

Some feel that homework is counter to the spirit of seminars, in which students are supposed to be self-motivated learners working together to learn the material. Others find that homework makes the audience more attentive during lectures and helps students to learn the material. Experienced seminar leader Steven Kleiman provides his perspective:

“A communication-intensive math course is supposed to be, first and foremost, a math course — just one with a more significant component in communication. The point is to promote the idea that communicating math is an important part of learning and developing math, not just in disseminating it. We learn by doing, and doing homework is an essential part of a serious math course. Devaluing homework devalues the course.

“Further, written homework provides an opportunity to help students learn the first principles of format, style, and LaTeX. In my seminar, this spring, I ask my students to email me the TeXscripts of their homework, which I compile, paying attention to the error messages and warnings issued by the TeX compiler and paying attention to obvious errors of formatting and style. Then I send my students an email explaining how to deal with these issues.”

One way to include homework while encouraging students to take ownership for their own learning is to have students assign the homework. Scott Carnahan provides some insights from his seminar, in which students assigned the homework to each other for the first half of the term and had no homework for the second half. Sami Assaf used student-developed quizzes instead of homework.

### Examples

The following undergraduate seminars are reported on this website:

- Topology–Andrew Snowden
- Computational Algebra–Steven Kleiman
- Number Theory–Scott Carnahan
- Mathematical Logic–Mia Minnes
- Physical Applied Mathematics–Pedro Reis
- Discrete Mathematics–Sami Assaf

### Some undergraduate seminars in the literature

*PRIMUS Special Issue: The Undergraduate Seminar in Mathematics*, Vol. 11, No. 3 & 4 (Sept & Dec 2001). This special issue contains the following articles:- The Undergraduate Seminar in Mathematics: Opportunity and Challenge
- A Mathematics Colloquium for Sophomores
- From Calculus to Topology: Teaching Lecture-Free Seminar Courses at All Levels of the Undergraduate Mathematics Curriculum
- The Senior Seminar in Mathematics at Union University
- Banach-Tarski in Senior Seminar
- A History of Mathematics Course as a Senior Seminar
- First-Year and Senior Seminars: Dual Seminars = Stronger Mathematics Majors
- The Senior Seminar: Preparation for Life After College
- Practice Makes Almost Perfect: A Seminar Experience
- Ten Years of Change: The Evolution of a Senior Seminar
- Joining Students and Faculty: A Seminar for Both

- “Hilbert at Vassar: An Undergraduate Seminar,” by John A. Feroe,
*The American Mathematical Monthly*, Vol. 85, No. 8 (Oct., 1978), pp. 669-672 - “An Undergraduate Seminar Emphasizing Oral Presentation of Research Mathematics,” by Deborah S. Franzblau,
*PRIMUS*, v2 n1 p16-32 Mar 1992

Abstract: Details the planning and the outcomes for an undergraduate seminar course that introduced students to journal articles and graduate level texts pertaining to theoretical computer science. Included are useful guides for students that provide basic techniques for reading mathematical papers, and that lead students through the process of generating oral presentations. - “Integrating Mathematical Ideas through Reading, Writing, and Speaking: A Senior Seminar in Mathematics and Computer Science,” by T. H. Barr,
*PRIMUS*, Vol 5, No 1, Jan 1995, pp. 43-54. - “Incorporating Current Research, Wikis, and Discussion Lists in a Mathematics Capstone Course,” by R. Narasimhan,
*PRIMUS*, Vol 19, No 1, 2009, pp. 29-38.

Abstract: This article shows how current mathematical research and innovative internet technologies such as wikis and email discussion lists can enliven a senior seminar or capstone course. We give example of assignments as well as examples of how new technologies can enhance a course. This paper grew out of the author’s experience of teaching a senior seminar for many years at Kean University. - See also articles on using the Modified Moore Method with undergraduates.