In this section, Prof. Demaine describes the open-problem sessions, and also the benefits and challenges to offering such sessions to the students.
Even in the first lecture video, several open problems were presented alongside an overview of the course.
Open-problem sessions are offered for two hours in the evening once a week, though the exact day of the week might shift depending on students’ schedules. Early in the semester, people are curious about what this open-problem solving is about, and they come to see what they can contribute. Some of them get scared off, I think, which is too bad. There are always some really amazing students who come to these sessions. They can be a little bit intimidating to other students, but the attendees usually converge to a set of regulars. Not everyone comes every week because of other constraints, but there is a core group that shows up every time. Even then, some people from architecture might come in and say, “Oh, what are they working on?” or “What is this open-problem solving thing?” because it is a completely different world from their field or major. It is nice to be able to share that experience and to show others what mathematical research is like.
Because the course content covers active areas of research, the idea is to get students engaged and interested in pushing the frontiers of research and trying to solve problems to which no one yet knows the answers. To me, this is the most exciting part! It is a lot of fun, and it works really well in the context of a course because every week, students know the latest and greatest techniques, tools, and results in a particular subfield. We immediately attack some of the problems in that subfield and see if we can do a little better using the techniques that are fresh in the students’ minds. This back-and-forth between learning new material and trying to create new material has worked pretty well. Certainly not all students come to the optional problem-solving sessions; we usually get around fifteen students, which is a nice size for the group.
A session works like this: We convene in a classroom, and I pose an open problem or two that I believe are unsolved and close enough to the course material that the students may be able to make progress. From there, it is just group brainstorming. Sometimes we divide into groups, and other times we all listen to one person present his ideas in a “there are no bad ideas” atmosphere. Everyone is encouraged to say, “Could we do this?” and it doesn’t matter if the answer is “no.” Any beginning of an idea still might inspire someone else to say, “Maybe not that, but we could do this instead.” This conversation allows students to build on each other’s ideas. Such sessions have actually led to solving a bunch of open problems, and we end up writing several papers every time the course is taught.
Writing and publishing papers does not necessarily happen within the timeline of the course; in the cases when we have formed a good group, it is nice to continue to work together. It does not usually last forever—maybe one or two semesters or perhaps a summer. Though, some of the groups have gone on for a while. This is especially true for the groups that have resulted from 6.849 when viewed in context of the groups formed during the open-problem sessions of other courses I teach. Sometimes I pick up UROPs and even graduate students that I sort of “discover” in the context of these open-problem sessions. These students are solving lots of problems and really displaying their skills in this area. It becomes quite natural to work together.
In all cases, we start with results that are worthy of research papers. These proofs still have to get written, and even if a student writes up the results as a final project, there is always more work to turn it into a publishable paper. This follow-up happens after the semester is over. My usual model in terms of authorship is that whoever thinks he or she has contributed in any form to solving a problem can be an author. Everyone is able to decide autonomously though not everyone is involved in the actual final writing process, primarily because there are not that many words to write. Nevertheless, I think it is nice to recognize all the people who came together to solve the problem. Even if some people did not contribute as authors, they are usually mentioned in the acknowledgements with something such as, “This paper was written in the context of this problem session in this class at MIT, and thanks to everyone for making it a fun atmosphere.”
Some students enter 6.849 having taken one of my courses before, and so they have experience working in a collaborative research setting. But for most students, I think this is their first exposure to research. It is an honor to be able to introduce this kind of research. To me, it illustrates a particular style of research that is super-collaborative, which I learned very early on in my career. My first exposure to research was similar to something like this; it was a one-week conference at McGill University with twelve people, and we just worked together on open problems for the entire week. That experience shaped my career early on and led me to lots of collaboration with many different individuals. I am trying to spread that spirit and show how collaboration is a really effective way to work. You can solve problems in two hours that would have been impossible to do by yourself.
The hardest concept to get across during collaborative sessions is that everyone should speak up regularly. Oftentimes, the more senior the students, the less they want to speak up. In the open problem sessions, it is really critical that people are always stating their ideas instead of thinking about them and writing them down on their own papers. That introspective approach has its place, but it is much better to err on the side of over-sharing, as long as the ideas are succinct and short, and hope that they inspire something. A lot of students will start out just listening, and the few brazen students will lead. The hope is that by the end of the semester, everyone is equal in that spirit of sharing ideas. It can be challenging, which is probably why some students choose to stop coming along the way. But that is okay too because research is not for everyone. The goal is that they get exposed to doing research in this way and can better evaluate whether they truly enjoy it.