Number Theory and Cryptography: A Distance Learning Course for High School Students

The following post was originally published on the AMS Blog “On Teaching and Learning Mathematics”.  I have reproduced it here with the permission of the AMS.

Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech.  Fourteen high school seniors from metro Atlanta took the course in Fall 2014, and overall I would say it was a big success.  We will be offering the course again in Fall 2015 and are expecting roughly double the number of students.  After describing the structure of the course, I will relate some of my experiences and describe some of the things I learned along the way.  I hope this article stimulates others to think outside the box about using technology in education without necessarily following the standard “MOOC” model.

I was motivated to create this course because I benefited tremendously from a Saturday course on Linear Algebra and Differential Equations when I was a senior in high school — this was one of the key formative experiences which eventually turned me into a successful mathematician.  However, I know that most public high school students don’t have that kind of opportunity available through their school systems, and I wanted to see if I could use modern technology to deliver an experience comparable to the one I had without requiring students to show up in person once a week.

Students qualified for the course by successfully completing Georgia Tech’s Distance Calculus program (which covers integral calculus, linear algebra, and multivariable calculus) by the end of their junior year in high school.  My course covered basic number theory (e.g., modular arithmetic, primality testing, discrete logarithms, and quadratic reciprocity) and applications to cryptography (e.g., Diffie-Hellman key exchange, RSA, and El Gamal), plus some fun applications to topics like calendar calculations, music theory, and card tricks.  A detailed syllabus can be found here:

Students also learned to write proofs and to program in SAGE, and many of them learned LaTeX as well (all homework solutions had to be typed).  Final projects included a calendar calculation quizzer for Android phones, a Power Point presentation on Furstenberg’s “topological” proof of the infinitude of primes, and a video on elliptic curve cryptography.

The course followed an asynchronous distance learning model, meaning that students watched videos on their own time rather than participating remotely in a live classroom.  They also had weekly homework and reading assignments (from either the course textbook or supplemental handouts).  There were three hour-long midterm exams, proctored by teachers at the students’ high schools, plus a final project.  Final projects were presented at an end-of-the-semester, day-long “mini-conference”, in which the students and I got to meet one another in person.  Students received Georgia Tech credit for the course, and their tuition was fully paid through a state-funded financial aid program (ACCEL).  I held weekly video office hours, and there were extensive discussions on the course Piazza page. (Piazza is an integrated online discussion platform designed specifically for academic courses.) The videos, homework assignments, syllabus, and other course materials were organized through a password-protected WordPress site hosted by Georgia Tech.

Most of the videos for the course were filmed during a “pilot” offering of a similar course on campus in Fall 2013.  All lectures for that course were videotaped, and my postdoc Greg Mayer helped me edit all the video footage.  We broke up the material into roughly 5 minute chunks, uploaded the resulting videos to Videopress, and added descriptive captions below each video.  The videos were then grouped into 14 weekly lessons, each (except for the last one) with an accompanying reading assignment and homework assignment.  Sometimes it was necessary to add subtitles, for example when a student asked a question which wasn’t audible, or when I referred to something from a previous lecture which got edited out.  Click here or here for some sample videos from the course.

The Piazza discussions were productive, as students asked a lot of questions and often answered each other’s questions before I had a chance to respond myself, which is great!  Video office hours were less successful: usually only a few students would join in and they typically did not want to turn their cameras on, so these became chat sessions which could have been conducted just as efficiently via Piazza.  I tried having a few discussion activities during the semester, with students working together in small groups via Piazza, but somehow this did not end up being one of the more effective aspects of the course.  I will probably eliminate that aspect of the course next semester.  On the other hand, there were some Piazza-facilitated interactive homework assignments which worked very well, such as an activity where students posted RSA public keys to Piazza and then encrypted and decrypted messages to one another.  I plan to incorporate more such activities the next time around.  Students got extra credit for learning how to calculate (in their heads) the day of the week given a date, and I tested them by video chat (using Adobe Connect).  This worked reasonably well — much better than the usual office hours — though I don’t know how this approach would scale.

Overall I think the above format worked quite well, and I’m planning to use the same basic structure this coming Fall when I offer the course for a second time.  We conducted extensive surveys before, during, and after the course and students seemed to enjoy the course and reported having learned a great deal.  Georgia Tech’s Distance Calculus program is conducted in a synchronous format, meaning that the lectures are broadcast live, and since all the students in my course had previously taken part in that program, I thought it would be interesting to ask them which format (synchronous or asynchronous) they preferred.  Many students preferred the asynchronous format, and none expressed a strong preference for the synchronous format.  Most students reported spending an average of 6-9 hours per week on the course.  A typical comment was “I didn’t expect to spend so much time on the assignments, but they were interesting, so it was worth it.”

I did encounter some problems. At the start of the course there was only one female student enrolled, and she dropped just before the first midterm exam. That left me with 14 men and no women.  I could speculate on the reasons for this, but the bottom line is that I want to see a better gender balance in the future, and I will be discussing how to achieve this with our admissions office.

Two other issues that I continue to wrestle with are the sustainability and scalability of this course model.  To help make the course more sustainable, in Fall 2015 I will teach the course both as a high school distance learning course and as a flipped course for undergraduates in the Georgia Tech Honors Program.  (By flipped, I mean that students will watch videos and read content before class, and then I will meet with them face-to-face once a week in a workshop-style format; the high school and Honors Program sections will interact through Piazza, but not face-to-face except for the end of the semester mini-conference, to which both sections will be invited.)  In theory I would like to eventually offer the course to students outside the state of Georgia, but that would probably involve more grading than I can do by myself, and it also raises currently unresolved issues regarding tuition, proctoring of exams, etc.  So the ultimate level of scalability is still unknown.

I received a lot of assistance with this course, including help from Greg Mayer, who built the WordPress site in addition to editing the videos; from Nick Culpepper, an undergraduate student who graded all the homework assignments; from Georgia Tech’s Professional Education department, which handled proctoring and mailing of exams as well as the videotaping of the pilot course; and from the Admissions Office, which handled accreditation, registration, and tuition payment.  I also had help from Georgia Tech’s School of Mathematics and College of Sciences, as well as from our CEISMC (Center for Education Integrating Science, Mathematics, and Computing) program.  I would not recommend embarking on a project like this without a helpful and professional team like I had — they really made the whole experience quite enjoyable and (relatively) painless.

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