18.783 | Spring 2021 | Undergraduate

Elliptic Curves

Instructor Insights

Instructor Insights

Below, Dr. Andrew Sutherland describes various aspects of how he converted 18.783 Elliptic Curves to an online course during the COVID-19 pandemic.

Note: Dr. Sutherland has also provided insights on teaching 18.783 in person in 2019.

OCW: What was your overall approach to redesigning the course for online teaching during the COVID-19 pandemic?

Andrew Sutherland: I used a combination of slides and hand-written proofs on paper using a document camera. Slides are good for summarizing facts and data, but hand-writing is better for detailed discussions, especially proofs.

"I thought it would be useful to take advantage of the online medium to present additional content that I really couldn’t present as effectively in class."
— Dr. Andrew Sutherland

OCW: How does allowing students to download your lecture slide files change the teaching-and-learning dynamic for you and your students? What do the lecture slides provide that the lecture notes don’t?

Andrew Sutherland: The slides summarize content in a way that is easier to digest in real time than the lecture notes. Pretty much everything on the slides is also in the lecture notes—and if I were lecturing on a blackboard in a physical classroom, pretty much everything I wrote on the board would also be in the lecture notes.

OCW: You decided to include a bonus video (on root-finding and polynomial factorization) among the course materials for this iteration of 18.783. What was your motivation for creating this video, and how does it fit in with the rest of the material in the course?

Andrew Sutherland: I thought it would be useful to take advantage of the online medium to present additional content that I really couldn’t present as effectively in class. Watching someone using a computer algebra system works better online, especially when you can follow along with your own copy of the notebook, try things for yourself, and pause the video whenever you need to.

Read More/Read Less

OCW: In the first half of the course, you make extensive use of notebooks on a cloud computing platform. Can you describe the technology you used, and what some of its advantages were for you as an instructor?

Andrew Sutherland: We used the CoCalc platform and the Sage computer algebra system, both of which are open-source resources, freely available under a GNU public license; the source code for them can be downloaded from the public GitHub repositories at https://github.com/sagemathinc/cocalc and https://github.com/sagemath/sage.

If an educator wanted to run their own CoCalc server they could. I find it more convenient to use the online hosting provided at https://cocalc.com/, which saves me the hassle of managing my own server and ensures that the notebooks that are part of the course materials will stay available for anyone to use free of charge indefinitely. If I were running my own server, at some point I would tire of maintaining it!

The argument for using CoCalc is that you are making your content permanently available to the public in an easy-to-find place. Furthermore, the notebooks on CoCalc can be run in the context they were created, using the same version of Jupyter, Sage, etc., so that their computations should be completely reproducible. This is not necessarily true of software posted on other platforms, since you might be compiling and running it in an environment that is different (OS version, library versions, etc…).

OCW: What advice do you have for other educators faced with the challenge of adapting their graduate-level math classes to an entirely online setting?

Andrew Sutherland: Accept the fact that you won’t be able to cover as much content live as you could in a physical classroom covered with blackboards. Try to take advantage of opportunities for asynchronous delivery of material whenever you can, and leverage the fact that there are multiple media you can use online (video, slides, notes, worksheets, etc.).

OCW: You expect future iterations of this course will be taught in person again. Which of the adaptations you made to the course in response to the COVID-19 pandemic do you expect to retain when teaching in person, and why?

Andrew Sutherland: That remains to be decided. I’ll be teaching 18.783 again in the spring of 2022. I can certainly imagine making more bonus videos—I had planned to make more this spring, but I just didn’t have enough time.

Curriculum Information


A course in algebra covering groups, rings, and fields (including Galois theory) at the level of 18.701 Algebra I or 18.702 Algebra II.

Requirements Satisfied

18.783 can be applied toward a Bachelor of Science in Mathematics, but is not required.


Until this iteration of 18.783, the course was offered every other spring semester; due to increasing enrollment it is now offered every spring.


Grade Breakdown

The grade was derived by averaging the student’s ten highest grades from among the twelve problem sets, with additional credit points given for participation in Zoom polls held in class.

Student Information


28 students

Breakdown by Year

About 3/4 advanced undergraduates, 1/4 graduate students.

Breakdown by Major

About 2/3 pure mathematics, 1/3 applied mathematics.

How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

In Class

  • Met 2 times per week for 1.5 hours per session; 26 sessions total; mandatory attendance.
  • Several of the lectures included interactive sessions using Sage. The Sage worksheets are listed in the lecture notes section.

Out of Class

Course Info

As Taught In
Spring 2021
Learning Resource Types
Problem Sets
Lecture Notes
Instructor Insights