Below, Professor Yufei Zhao describes various aspects of how he taught 18.217 Graph Theory and Additive Combinatorics.
Yufei Zhao: This is the second iteration of a course I first taught two years ago. My goal was to put together a cohesive introduction to an area of modern combinatorics that I actively work in. I wanted to showcase some of the most attractive mathematics that I have learned starting as a beginning graduate student. In particular, I wanted to explain a beautiful and extensive connection that bridges graph theory and additive combinatorics, two seemingly different areas of mathematics.
I am very happy to see that several of the students who took the first iteration of this course have gone on to produce excellent research in this area. One common theme in the feedback that I received from students in the class is that they appreciated how the subject touches upon a broad range of mathematics, including graph theory, combinatorics, number theory, analysis, geometry, group theory, topology, linear algebra, etc.
Yufei Zhao: Some of the materials that I taught in this course were discovered only fairly recently, so they can be hard to find outside of research papers. I wanted to create a set of notes, eventually turning into a new textbook, that would cover the material in an inviting way and be accessible to non-specialists.
I built on a popular practice of asking students to “scribe” notes for individual lectures. Making good notes takes a lot of work and requires feedback from the lecturer.
To streamline this process, I created a single document on Overleaf, an online collaborative LaTeX editor. I asked students to sign up to cover lectures. The document was carefully set up on Overleaf so that students could add their lectures as new sections into the larger document. Other students in the class could then edit and correct these notes as the semester goes on.
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The first iteration of this course resulted in a useful set of course notes, though with variable quality of writing. I found it difficult and time-consuming to provide useful feedback on writing and to request further edits.
This time I added some helpful follow-up steps. I asked each student to schedule a half-hour appointment with me after finishing writing a polished version of the notes. During the meeting, I would go through the writing carefully line by line and share my comments on the writing with the student. I always began the session saying something like “I’m going to be very critical in my comments. Don’t take it personally, as I do the same to everyone in these sessions. Treat it as a writing tutorial.”
I tried to explain to students why I suggested certain changes, rather than just telling them what needed to be fixed. For example, many students write in their proofs something along the lines of “Thus A=B ⇒ A=C.” They grew up using the “⇒” symbol and they don’t understand why I insist on not using it. In these one-on-one sessions, I would explain to the student, “Well, does the sentence mean ‘Thus A=B, and hence A=C,’ or does it mean that you have deduced the logical statement ‘A=B implies A=C’? Do you see the potential ambiguity?”
While it was a labor-intensive process, I found the in-person meetings to be more enjoyable and meaningful than simply giving written feedback, and I think the students felt the same way.
Students were motivated to do a good job, as their notes would be useful to other current and future students. This writing assignment is not simply some rote classroom assignment where the submission is read by an instructor/grader and then forever buried in an academic graveyard for nobody else to see.
Yufei Zhao: Yes. To maintain consistency in the document, I wrote a style guide with “how to” instructions on everything from theorem formatting to citations commands. I also wrote the first lecture myself as an example of the style.
It was also important to set deadlines. The student covering a given lecture was asked to upload a rough version of the notes on Overleaf by the end of the day after the lecture, and a polished version within four days of the lecture, at which point the student should email me for an appointment to discuss the writing. After our meeting, the student had to finish the revisions within three days and then send the revision to me for additional comments. It’s important that the students do the writing and revisions while their memories are still fresh.
Yufei Zhao: As with the course notes writing assignments, students appreciated the meaningful nature of the assignment, in that they were contributing to a resource that would be useful to others (including themselves). I motivated students to engage with the assignment by telling the class that for most of us, whenever we first encounter a new term or concept, our gut instinct is to type it into Google, and often one of the top results is the Wikipedia entry. This makes Wikipedia an important entry point into knowledge, so it’s important that we do a good job to make it useful.
Writing good Wikipedia entries is challenging. One common problem with student contributions is that the articles end up too technical to be useful to non-specialists. I asked the students to imagine a reader who is a slightly younger version of themselves, before they properly learned the subject. One especially good example of a Wikipedia page created by a student in the class is the one on Sidorenko’s conjecture.
Yufei Zhao: Teaching this course was an incredibly enjoyable experience. It was a lot of work but very rewarding. I would certainly be happy to do it over again!
There are no formal prerequisites for 18.217, but the course presupposes mathematical maturity at the level of a first-year math graduate student.
18.217 can be applied toward a doctorate degree in Pure or Applied Mathematics, but is not required.
18.217 is taught every fall semester, but with a different topic each year.
The final grade was determined by the minimum of the student’s performance in two categories:
- 6 problem sets
- Writing assignments, consisting of (1) course notes and (2) Wikipedia contributions.
Breakdown by Year
Mostly graduate students and advanced undergraduates
Breakdown by Major
Primarily Mathematics and Electrical Engineering / Computer Science majors
Typical Student Background
The subject tends to appeal to students who enjoy mathematical problem solving.
During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:
Met 2 times per week for 1.5 hours per session; 26 sessions total; mandatory attendance.
Out of Class
Outside of class, students spent their time solving problems from the five assigned problem sets and completing the two writing assignments.