*In this section, Prof. Haynes Miller and Susan Ruff describe the enrollment and structure of the course.*

## Student Enrollment

*Project Laboratory in Mathematics* is offered every semester, and it is open to all MIT students. Because the course is the only mathematics course that fulfills MIT’s laboratory requirement , and it also fulfills MIT’s CI-M requirement , the demand for the course is generally high; usually 40 to 50 students register to take the course each semester. However, because the course is labor-intensive to run, we limit enrollment to 27 students per semester. Mathematics majors are guaranteed that they can take the course as their lab subject, and priority is given to students who most urgently need the course for their graduation requirements. Unfortunately, there is rarely enough capacity to accommodate as many underclassmen and non-majors as we would like. Typically, about 75% of the students are seniors, and 25% are a mix of juniors and sophomores.

The students in the course come from our full range of math majors; only about 10% of undergraduate mathematics majors from MIT go on to become research mathematicians.

## Prerequisites

Students are required to have taken at least two advanced mathematics courses (courses numbered 18.100 or above).

## Course Components

Students are expected to spend 12 hours per week on the course. Their activities are outlined below. Each activity is explained in more detail in other sections of this course site.

### Class Sessions

The semester begins with several class sessions during which students learn about the course and engage in workshops. In the Spring 2013 semester, there were four class sessions at the beginning of the semester: an introductory lecture about the philosophy and structure of the course, a teamwork workshop, a presentation workshop, and a writing workshop.

### Projects

Students work in teams of three on three open-ended research projects. The project cycles are deliberately overlapped to allow time for revision in response to staff feedback. During the overlap, students are expected to begin their next projects while the course staff reads and gives feedback on the draft papers of the prior projects. Each project cycle lasts five to six weeks and includes the following:

**Topic selection**: Each team selects a project topic from a list of over 40 options. Each project topic typically can be chosen by at most one team per semester.**Mathematical work**: Students work together in their teams to explore their project topics. Students may work out examples, run computer simulations, read relevant literature, refine or redefine the focus of the project, make conjectures, and attempt to construct proofs.**Mentor meetings**: For every project, each student team is assigned to one of the three mathematics instructors for the course. Each team meets with its mentor once a week for the duration of the project. The course also has a communication instructor who works with all of the teams.**First draft**: Each team submits a first draft of the paper 1.5 weeks before the final draft due date. The paper is then read and commented upon by the team's mentor and sometimes also the communication instructor.**Debriefing meetings**: The team meets with a group including the lead instructor, the team's mentor, the team's mentor for the next project, and often the communication instructor. The team gives a brief presentation of their research, and the whole group discusses both the mathematics and the writing.**Final draft**: Students finish a project cycle by revising their paper following instructor feedback and submitting a final draft.

### Presentations

Each team delivers a presentation to the class on one of their projects during the semester. The presentation is always preceded by a practice presentation, which is attended by course instructors and which includes extensive feedback. The team then refines its presentation before presenting to the entire class.

## Is this “Undergraduate Research”?

There is no expectation that the teams will discover previously unknown mathematics, though sometimes they do. Rather, the goal is for teams to discover mathematics previously unknown to them. Thus, *Project Laboratory in Mathematics* is a mathematics course, not what mathematics educators would typically call an "undergraduate research experience." This perspective is essential in creating a course that is accessible to a wide range of students. It also enables the course to be repeatable, since projects can be reused from year to year.

## Why Three Projects?

We ask students to work on three projects over the course of the semester. This quick turn-around is consistent with the rhythm of students’ experiences in their other courses, which often involve weekly homework assignments and monthly exams.

Having three iterations also gives students fresh opportunities to apply their learning: to choose more appropriate projects, conduct better research, and write better papers. We can see the evolution of these teams as the term goes on.

In Spring 2013, for example, at the first meeting to discuss their first projects, several teams said, “We have a sketch of how this paper is going to look.” They thought they knew what their conclusion was going to be before they had done any research! By the second project, they had learned that research is a process, there are surprises along the way, and there are often unanticipated discoveries. Throughout the course, many students progress from trying to guess what the professor had in mind in proposing a given project, to thinking about the mathematics in an exploratory fashion, coming up with conjectures, and by the end, proving some of those conjectures and writing them up in correct style. It is really exciting to see teams grow over the course of the semester.

As another example of team maturation, one very strong team chose their first project knowing that it was highly prescriptive. They made the check marks, solved some of the proposals made in the project description, and left it at that. We were unhappy about that performance because it did not come close to their potential, and we let them know clearly that this was not the point of the course. We gave them a shockingly poor grade for their effort, and they listened to that. Their second project was much more challenging and quite frustrating for them. They reached a point where they were facing questions that were completely unsolved or deeply embedded in current research papers. They talked to other faculty members about the problem. We think that it was a great experience for them to have come up against the frontier of research within three weeks of starting the project.