{"id":838,"date":"2015-07-01T00:01:56","date_gmt":"2015-07-01T04:01:56","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=838"},"modified":"2015-07-01T10:46:12","modified_gmt":"2015-07-01T14:46:12","slug":"number-theory-and-cryptography-a-distance-learning-course-for-high-school-students","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2015\/07\/01\/number-theory-and-cryptography-a-distance-learning-course-for-high-school-students\/","title":{"rendered":"Number Theory and Cryptography: A Distance Learning Course for High School Students"},"content":{"rendered":"

By Matt Baker<\/a>,\u00a0Professor of Mathematics,\u00a0Georgia Institute of Technology<\/em><\/p>\n

Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech. \u00a0Fourteen high school seniors from metro Atlanta took the course in Fall 2014, and overall I would say it was a big success. \u00a0We will be offering the course again in Fall 2015 and are expecting roughly double the number of students. \u00a0After describing the structure of the course, I will relate some of my experiences and describe some of the things I learned along the way. \u00a0I hope this article stimulates others to think outside the box about using technology in education without necessarily following the standard \u201cMOOC\u201d model.<\/p>\n

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. \u00a0However, I know that most public high school students don\u2019t 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.<\/p>\n

Students qualified for the course by successfully completing Georgia Tech\u2019s Distance Calculus program<\/a> (which covers integral calculus, linear algebra, and multivariable calculus) by the end of their junior year in high school. \u00a0My 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. \u00a0A detailed syllabus can be found here:\u00a0http:\/\/www.math2803.gatech.edu\/wp-content\/uploads\/2014Math2803Syllabus.2.pdf<\/a><\/p>\n

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). \u00a0Final projects included a calendar calculation quizzer for Android phones, a Power Point presentation on Furstenberg\u2019s \u201ctopological\u201d proof of the infinitude of primes, and a video on elliptic curve cryptography.<\/p>\n

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. \u00a0They also had weekly homework and reading assignments (from either the course textbook or supplemental handouts). \u00a0There were three hour-long midterm exams, proctored by teachers at the students\u2019 high schools, plus a final project. \u00a0Final projects were presented at an end-of-the-semester, day-long \u201cmini-conference\u201d, in which the students and I got to meet one another in person. \u00a0Students received Georgia Tech credit for the course, and their tuition was fully paid through a state-funded financial aid program (ACCEL<\/a>). \u00a0I held weekly video office hours, and there were extensive discussions on the course Piazza<\/a> 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<\/a> site hosted by Georgia Tech.<\/p>\n

Most of the videos for the course were filmed during a \u201cpilot\u201d offering of a similar course on campus in Fall 2013. \u00a0All lectures for that course were videotaped, and my postdoc Greg Mayer helped me edit all the video footage. \u00a0We broke up the material into roughly 5 minute chunks, uploaded the resulting videos to Videopress<\/a>, and added descriptive captions below each video. \u00a0The videos were then grouped into 14 weekly lessons, each (except for the last one) with an accompanying reading assignment and homework assignment. \u00a0Sometimes it was necessary to add subtitles, for example when a student asked a question which wasn\u2019t audible, or when I referred to something from a previous lecture which got edited out. \u00a0Click here<\/a> or here<\/a> for some sample videos from the course.<\/p>\n

The Piazza discussions were productive, as students asked a lot of questions and often answered each other\u2019s questions before I had a chance to respond myself, which is great! \u00a0Video 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. \u00a0I 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. \u00a0I will probably eliminate that aspect of the course next semester. \u00a0On 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. \u00a0I plan to incorporate more such activities the next time around. \u00a0Students 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). \u00a0This worked reasonably well — much better than the usual office hours — though I don\u2019t know how this approach would scale.<\/p>\n

Overall I think the above format worked quite well, and I\u2019m planning to use the same basic structure this coming Fall when I offer the course for a second time. \u00a0We conducted extensive surveys before, during, and after the course and students seemed to enjoy the course and reported having learned a great deal. \u00a0Georgia Tech\u2019s 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. \u00a0Many students preferred the asynchronous format, and none expressed a strong preference for the synchronous format. \u00a0Most students reported spending an average of 6-9 hours per week on the course. \u00a0A typical comment was \u201cI didn\u2019t expect to spend so much time on the assignments, but they were interesting, so it was worth it.\u201d<\/p>\n

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. \u00a0I 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.<\/p>\n

Two other issues that I continue to wrestle with are the sustainability and scalability of this course model. \u00a0To 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. \u00a0(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.) \u00a0In 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. \u00a0So the ultimate level of scalability is still unknown.<\/p>\n

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\u2019s 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. \u00a0I also had help from Georgia Tech\u2019s School of Mathematics and College of Sciences, as well as from our CEISMC (Center for Education Integrating Science, Mathematics, and Computing) program. \u00a0I 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.<\/p>\n

Note: This article is also posted at\u00a0https:\/\/mattbakerblog.wordpress.com\/2015\/07\/01\/number-theory-and-cryptography-a-distance-learning-course-for-high-school-students\/<\/a>.<\/p>\n

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By Matt Baker,\u00a0Professor of Mathematics,\u00a0Georgia Institute of Technology Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech. \u00a0Fourteen high school seniors from metro Atlanta took the course in Fall … Continue reading →<\/span><\/a><\/p>\n

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