M.I.T. professors Michael Artin and Haynes Miller developed the Project Lab in Mathematics, which was offered for the first time in 2004 and which has been refined by subsequent instructors.

Students work in teams of three on problems such as “If the reciprocal of a prime has a repeating decimal expansion, what can you say about the period with which the decimal repeats?” Each team works on three such problems during the term, collaboratively writing up their results from each project and presenting one of the projects to the class as a whole.

### Semester schedule

The first class or two are devoted to assigning teams and choosing first projects. Some subsequent class meetings provide guidance for teamwork, writing, presenting, conducting research, etc., but the bulk of classes during the semester are devoted to student presentations.

Teams spend about 5 weeks on each project, with a week of overlap at the transition between projects. During this week of overlap, course staff read and provide feedback on a draft of the preceding project’s paper while students start thinking about the next project. Students then revise the paper before turning all of their attention to the next project.

### Logistics

There are typically 9 teams and 3 team mentors: each team meets with their mentor once a week and is expected to meet on their own as well. Teams submit a draft of each paper to receive feedback and revise before submitting the final paper for that project. Similarly, each team gives a practice presentation and receives feedback before presenting a project to the class. The feedback is provided by the team mentor and by a lecturer from M.I.T.’s Writing, Rhetoric, and Professional Communication.

To simplify logistics, the class is advertised as meeting three times a week. The entire class usually meets on only one of these days; the others are reserved for practice presentations (in the classroom), for team meetings, and for schedule flexibility in case of (e.g.) snow days.

To ensure good presentations, a particularly strong team should be chosen to give the first presentation and set the tone for the term.

### Support for teamwork

Each term, at least one of the 9 teams is likely to experience some team friction. To help teams through such rough spots, various strategies are used to support teamwork.

In addition to the teamwork workshop at the beginning of the term, the following strategies have been used with some success:

- At the start of the term, we assign teams based on the following criteria:
- Teammates should be interested in working on the same projects: the primary predictor of team success is shared commitment to the team’s goal.
- Teammates should have compatible degree of expertise (as measured, e.g., by the number and difficulty of the mathematics classes they’ve taken)
- Teammates should have roughly compatible “procrastination indices.” Each person’s “procrastination index” is the answer to the following question (idea by David Jerison):

If you have a 10-page paper due at midnight on Thursday night, when are you likely to start?- 1: the preceding weekend.
- 2: Monday or Tuesday.
- 3: Wednesday.
- 4: Thursday before 7pm.
- 5: Thursday after 7pm.

- A third of the way through the term, after the completion of the first project, teams have completed a short team communication exercise at the end of class. Each student writes “A Wish and A Star”: one thing that the student wishes the team did differently, and one star for something that the team is doing well. Students then read their teammates answers and have time to talk about them if they like.
- When a team mentor becomes aware of significant issues with the dynamics of a teams, the mentor consults with the lead instructor and the communication instructor to develop a strategy, and then works with the team to resolve the issues.

### Publications

- M.I.T.’s Project Lab in Mathematics had been published on MIT OpenCourseWare:

Haynes Miller, Nat Stapleton, Saul Glasman, and Susan Ruff.*18.821 Project Laboratory in Mathematics, Spring 2013*. (Massachusetts Institute of Technology: MIT OpenCourseWare) - A description of M.I.T.’s Project Lab in Mathematics is available on page 7 of the 2009 newsletter of M.I.T.’s Department of Mathematics.
- “A Laboratory Course in Mathematics” by Kathy Lin and Haynes Miller from
*Directions for Mathematical Research Experience for Undergraduates*

edited by Mark A Peterson and Yanir A Rubinstein, World Scientific 2016, pp 33-52. - S. Greenwald and H. Miller, “Computer-assisted explorations in mathematics: Pedagogical adaptations across the Atlantic,” in
*University Collaboration for Innovation: Lessons from the**Cambridge-MIT Institute*, edited by D. Good, S. Greenwald, R. Cox, and M. Goldman, Sense Publishers, 2007, pp. 121–131.