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| Bureaucracy | Teaching help | The students | Pedagogy | Technology | Text | The paperless course |
The bureaucracy of the course
At Rutgers-New Brunswick, the appropriate academic niche for a
mathematics course directed at non-majors having minimal mathematical
background is Math 103, called "Topics in Mathematics for the Liberal
Arts". This is a 3 credit course which has two 80 minute meetings each
week for 14 weeks. Since the honors program always seems to
have a shortage of appropriate courses for their students, and since
the course I was proposing to teach definitely might call for
resourcefulness beyond what is standard for students, i decided that,
at least in its pilot stages, my proposed course on cryptography
(crypto) would be nice as an honors version of Math 103.
Teaching help
NSF support allowed me to have a graduate student assigned as a
teaching assistant each semester, which is highly unusual in Math
103. I gave the course twice during the 1999-2000 academic year. A
first-year mathematics graduate student, Nina Fefferman, with
substantial experience using crypto in the commercial world and
supported by NSF funds through the VIGRE
program, worked with the course in the fall semester. This was
her fall VIGRE
rotation project. Another first-year graduate student from
Switzerland, Sasa Radomirovic, whose senior thesis at ETH was about
secret sharing, worked with the spring version of the course.
Both of these individuals contributed a great deal to the course. Ms. Fefferman may have suffered more, because she was present the first time, and helped students who may have been exposed to some half-baked pedagogical ploys. Ms. Fefferman and Mr. Radomirovic usually attended one meeting each week of the course during the semesters they each were associated with it. They each contributed several class presentations, and helped me construct homework assignments, deal with small group work in the classroom, and generally plan what to do.
In the fall, Ann Stehney helped with several classes, discussing the Zodiac killer's ciphers and other topics. In the spring, Jim Reeds lectured in one class. Matt Blaze spoke one evening on some of the policy aspects of the course. I had hoped to get someone from the Rutgers computing "establishment" to discuss some of the crypto realities in maintaining privacy but was unable to get any of these busy people to come.
After the fall semester was over, NSF funding allowed me to get a detailed review of the course from an undergraduate who had been a student in the course. M. Ruedy, majoring in art history, English, and history (!), carefully read my course diary and looked over all of the material that had been handed out during the fall semester. She wrote an excellent critique, with detailed student reactions, and pointed out some problem areas which needed attention.
The students
Almost all of the students were from college honors programs. The
overwhelming majority were from the Rutgers College honors program.
| Semester | Total number | Stated graduation year | |||
|---|---|---|---|---|---|
| First-year (03) | Sophomore (02) | Junior (01) | Senior (00) | ||
| Fall 1999 | 20 | 6 | 5 | 4 | 3 |
| Spring 2000 | 26 | 0 | 12 | 11 | 3 |
The students' declared (or intended, for first-year students) majors were amazingly diverse to an instructor whose usual auditors are almost all either engineers or science/math majors. Many students were attempting to satisfy the requirements of two majors, or in several cases, three. Detailed lists for both fall and spring are available, but some idea can be gotten from this list:
| # of students | Majors |
|---|---|
| 9 | English |
| 8 | Computer Science, Political Science |
| 7 | Psychology |
| 5 | Economics, Biological Sciences (various) |
| 4 | Art History, Business, Communications, History |
| 3 | Anthropology, Mathematics |
| 2 | American Studies, Latin American Studies, Sociology, Spanish |
| 1 | Administration of Justice, Classics, Film, Geology, Labor Studies, Linguistics, Religion, Theater Arts, Undeclared, Urban Studies, Visual Arts |
One could therefore at some point be encouraging an art history major that interesting and relevant ideas on intellectual property would be studied late in the course, while attempting to reconcile various political science majors to the limited amount of time we would spend on government crypto policies: interesting games. This course, unlike almost all courses taught in math departments, is not a prerequisite for any other. I felt somewhat less pressure to "cover" all the material. Also, I didn't (and don't!) exactly know what "all the material" means in this case.
Students filled out an information sheet [PDF|PS|TeX] on the first day of class each semester. The question "Briefly, why are you taking this course?" was most frequently answered as I had anticipated: "... because it fulfills a requirement ..." I will add that a considerable portion of the students indicated some interest in the subject matter and its social and political interactions. But, generally, the answer best representing majority sentiment might have been the one which began: "Lesser of all evils. I am excited to see if I can take a math class without HATING IT the whole time. ..." I'd like to also give the most deflating response, which ended: " ... and this class fit into my schedule." Ah, well, scheduling classes at Rutgers-New Brunswick, with students traveling among four campuses separated by several miles and lots of traffic, is often most infuriating and sometimes nearly impossible. I will admit that there was a "smiley" after the end of the last-quoted sentence.
I should note finally that the remarkable diversity of Rutgers students was a great asset in the course. For example, students from Europe, Asia, and Latin America were in both classes. They served to give an additional perspective when we considered, for example, the communication policies of governments, just as the varied majors studied allowed students to consider the problem of medical record privacy from many points of view.
Pedagogy
I needed to realize that a certain portion of these students are
frightened of mathematics. Even after I got to know each class, and
realized that they were interesting and intelligent human beings, each
with their own ration of seriousness and silliness, I saw that when I
turned to the board and wrote 10100 or (worse)
AB that some of those in the room would blink nervously or
laugh or look blank with few other cues. Ingrained in these
intelligent students was a fear or distaste for math as an object of
intellectual interest. I don't know whether this was some inborn trait
or if it was a result of previous educational experiences, but great
care and gentle persistence must be used. They must be inveigled to
study math. Much of my professional instructional career has been
devoted to teaching engineering students. There almost the opposite is
true: many of the students are fanatics about internalizing as much
math as possible. They are correctly sure that math is a principal key
to professional success. Some students in Math 103 are almost as
mournfully sure that math is their personal fate, with unpleasant
connotations.
I resolved to try to be different. Of course, I cannot change my entire professional approach to my subject, so the differences were not a complete change, but I hoped they were significant. I declared at the beginning of each class that the students' job was to attend class and do some work. It was my job to be as persuasive as possible about the importance and interest of mathematics, but I would seek to show this to them by embedding the subject into what I considered natural concerns of anyone in our complex civilization. I raised themes of privacy of medical and financial records, of security of personal communications, of government intrusion and limitation of safekeeping of secrets, and of intellectual property in the digital era. I deliberately did not choose to name mathematical topics we would study. I also stated that students would need to write papers and make presentations: natural behavior in many social science and humanities courses. There would be standard homework, but I would try to keep it interesting and not repetitive. Any exams would be "open book, open notes."
I wrote the following in the prospectus for the course.
| The mathematical topics will rarely be taught systematically, but will be supplied on a "just in time" basis ("infused into the curriculum") as some of the topics listed below need them. Traditional lecturing will be supplemented and, where possible, replaced by various active learning strategies. We hope to have writing assignments which will explore both social and mathematical issues. |
Before this course I had never tried deliberately monitoring and fostering discussions in class. Here my two teaching assistants were quite valuable. In effect we "rollplayed" discussions, trying to see what would happen. I don't think I quite have the knack (yet?) of doing this well, but working at it helps. I tried to write comprehensive questions on the board, to steer the discussion and not to let certain students or points of view dominate.
More routine (after dealing with workshop calculus for several years, especially our EXCEL program) was splitting the class up into small groups to work on problems in class, and then on occasion, having students report on this work. I had done similar things in various courses (the advanced calculus and the first three semesters of calculus) and and generally felt comfortable with such activities. Most classes featured some in-class activity for students. Very few classes were straight lectures.
Technology
Students could use calculators at any time in the course. The types of
computations necessary in modern cryptography are not really
facilitated by a pocket calculator, except in rare cases. Stronger
help is needed. For example, it would not be unusual for a modern
crypto person to ask what the remainder of 234567 is after
division by 89 (it is 67). I used Maple for that computation. Maple is
a large program currently installed on most Rutgers computer
systems. It is similar to Mathematica and Derive. It has "arbitrary"
arithmetic capabilities and can do algebraic manipulation and draw
graphs. I resolved that these students would use Maple, and they would
treat it like a large "silicon slave". Then they'd be able to work
with protocols like DIffie-Hellman and RSA.
I've rarely used any sort of technology in the classroom. Here I would at least show some videotapes. I would reference Maple computations and rue my lack of ability to show them live in class. I'd try to bring my teaching style into the (late) twentieth century, even as society as a whole was advancing into the twenty-first! Getting a "smart" classroom with a TV, videotape recorder, etc. was useful. Having a properly equipped laptop would have been even better.
Outside of the classroom, I wanted students to be able to use e-mail and to do research in libraries and on the web. Rutgers-New Brunswick is spread over several campuses in and around New Brunswick. Students sometimes cannot conveniently visit faculty members. In this course, with much of the material quite new and not in accessible textbooks, being able to interact with the teaching assistants and me by e-mail was important. A number of homework assignments required computer use and web work.
Text
I felt I made the best choice possible at this time:Cryptology
by A. Beutelspacher. However, I didn't use the book very much and
should probably have used it more. The text has more math than I
wanted to do, and less discussion of social/political/legal issues. It
also did not cover secret sharing, hashing, and discussion of
algorithms. But it had good treatments of public key encryption and
such consequences as digital signatures, and also covered topics like
shift registers and digital cash which are good candidates for
inclusion in such a course. Other references were used where
appropriate in the course.
The paperless course?
I debated about how much paper I'd hand out in the course. I decided
to be cowardly and generally used paper handouts, supplemented by web
material that I created. Some material (e.g., the RSA assignment) was never given out on paper
because the blocks of numbers involved would just need to be returned
to the computer for further work. Clearly we're evolving towards a
course with minimal physical handouts, though.
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