### Project Laboratory in Mathematics: A Taste of Research

What is *Project Laboratory in Mathematics*? This video provides a peek into how students explore mathematics and share their findings in this non-traditional, highly participatory course.

### Course Introduction

University courses are generally designed to teach students what is known in a subject in a structured, clear, and efficient tour through the intellectual domain. This approach is often reasonable for courses aimed at introducing students to a specific content area. But a student can easily leave a course with little understanding of how the field developed and with the mistaken conclusion that everything worth knowing has already been discovered.

*Project Laboratory in Mathematics* is designed to provide undergraduate students with some experience of the mystery, frustration, and thrill of discovering new mathematics. Students work in teams on three open-ended projects. They generate and examine data, find regularities, attempt to explain them mathematically, and write about and present their results. The learning goals are entirely experiential, and consequently, there is no set content for the students to master. For this reason, this OCW course site does not include a syllabus; there simply isn’t one.

The course is taken by a broad range of math majors at MIT, from students who go on to become professional mathematicians to students who pursue careers in completely unrelated fields. For many undergraduates, *Project Laboratory in Mathematics* is the one course where they see mathematics as a living, evolving field to which they can contribute.

### Site Introduction

This course site is intended for a specific audience: *educators* interested in creating math research-type experiences for their students. The type of experience described here is accessible in some form to students in a range of environments, from high school to undergraduate and early graduate mathematics programs, as well as teacher education programs.

This course site is built upon descriptions and commentary from two key instructors of the Spring 2013 offering of the course: Prof. Haynes Miller, the lead instructor, and Susan Ruff, the communication instructor. In each section, they describe not only how the course operated but also why various decisions were made, successful and unsuccessful policies, and the students’ experiences. These sections, which can be accessed via the navigation panel along the left side of this page, include Logistics, Calendar, Staffing the Course, Mathematical Work, Teamwork, Writing, Presentations, Grading, and Related Resources. Wherever possible, course materials are included for reuse or adaptation.

### Course History

All MIT undergraduates are required to take a laboratory course . For many years, there was no lab course in mathematics and so mathematics majors had to take lab courses in other departments. While there are virtues to this breadth of experience, some mathematics faculty felt that many students were forced to take courses that were not very meaningful to them and that an important educational opportunity was being undervalued.

In the early 2000s, Prof. Haynes Miller and Prof. Michael Artin created *Project Laboratory in Mathematics* as a way for mathematics majors to have a “lab” experience that is relevant to their mathematical studies. It sometimes surprises people to hear that there is a laboratory subject in mathematics. They wonder, “How can students take a lab course when there is no lab? Where are the test tubes?” In fact, the course adheres closely to the scientific method in much the same way that a conventional lab course does. Students look at some mathematical situation and frame a hypothesis, called a *conjecture* in mathematics. They may test the hypothesis by doing numerical experiments, verify the hypothesis beyond doubt by means of a *proof*, or refute the hypothesis with a *counterexample*. They then write up their findings; the writing itself is an integral part of the research process. Thus, like other lab courses, *Project Laboratory in Mathematics* steps carefully through the scientific method. The difference is just the mathematical context.

First taught in 2004, the course has been offered over 17 times. It has evolved over time and continues to develop through changes in course structure, material, and teaching approaches.

### Introducing the Course to Students

During the first class session of the Spring 2013 semester, Prof. Haynes Miller delivered an introductory lecture about the structure and philosophy of the course. He described the cycle of work, the importance of teamwork, the roles of the staff members, and the experiential goals of the course. The expectations of the course are quite different from others that students have taken: not only are there no “right answers,” but there aren’t even any “right questions.”

This video documents most of the first class session from Spring 2013. Not included are the last 15 minutes of class, when students met with their teammates to select their first project topics.