Concerns for the instructor

Technology Additional crypto topics Reference material for the course Pedagogy Attendance Scaling up? A few orphan problems

Here are remarks about some issues which might concern an instructor trying to use this course material.


Is technology a bane and or a blessing? I encountered the following technology issues.
  1. Differing versions of Maple
    The student system (Eden) and my home machine had more recent versions of Maple than the Math Department systems. There were syntactical differences and differences in the user interface. Many students in this course need to be taught that programs will not break if you don't type in the correct syntax, and get easily discouraged if things don't "work" immediately. So eliminating as many initial roadblocks as possible is a good idea. Always check programs and homework on the specific machines and systems students will be using. Don't hope or expect that things will work in the same way without checking for yourself.
  2. Do the darn machines work?
    I naively expected after taking a training session that the machines in my "smart" classroom would behave like the ones I learned with. At the beginning of the second semester, the videotape machine wouldn 't "obey" me. I assumed the problem was mine, not the machines. Only weeks later I learned that something was broken. I always examine a classroom before I teach in it, to see how large I need to write, to see how loud I need to speak, to check it all out. Now I know that checking it all out means playing with the machines enough to make sure they behave the way I think they should.
  3. A laptop that works and displays
    It would be wonderful to have a laptop working and displaying the appropriate software (Maple, Mathematica, Derive: whatever) in the classroom. I didn't have that but wanted it. Also access to "the web" during the class would be useful. Class webpages could be displayed, as could other web references.
  4. Reporting on the course
    I spent much more time writing and constructing these web pages than one might imagine. I probably dedicated the equivalent of two solid weeks of work to this effort. I hope this is too much work.


Additional crypto topics

I did not discuss polyalphabetic systems. They were mentioned by Jim Reeds in his presentation when he discussed Trithemius, but students really didn't think about them. These systems have a marvelous history, and much effort was devoted to attacking them.

I did not discuss shift registers. These are fundamental to much of modern crypto. The text has good information and problems, and the theory of shift registers is complete, unlike, say, the "theory" of block ciphers. The mathematical attractiveness of unbridled theory probably made me very cautious, perhaps too much so, because nice problems could have been given. And shift registers are fast and easy, and do exist in the world. They would fit in nicely if there had been time to discuss more about bitstream generators.

Mr. Radomirovic observed correctly that we spent a great deal of time on the asymmetric public key systems -- they are seductive because they allow a great deal (the prison scenario, digital signatures, anonymous e-cash) that could scarcely be envisioned classically. But (to continue his observation) in fact symmetric systems (usually block ciphers) are much more commonly used. We should have paid more attention to them. I think this is correct, and in a future incarnation of the course would try to create some interesting assignments, in and out of class, as has already been discussed.

I wrote the following in the prospectus for the course.

They will learn the basics of combinatorics such as various ways to count. Additional topics of discrete mathematics, including basic graph theory, may be studied. In addition, parts of probability (such as Bayes' Theorem) will be discussed. Depending on time and choice of topics, a bit of group theory may also be included.
I covered almost none of this material. There really are quite exciting things which can be done on a quite elementary level (see, e.g., the lovely "Kid Krypto" work of Fellows and Koblitz) nor did I have a chance to discuss information theory or coding theory, which are serious uses in, for example, biology, as well as crypto.


Reference material for the course

Not much information about the topics discussed in this course can be found in books. Frequently more recent developments have superseded what there is in texts. It is very difficult to keep up-to-date.



I believe that adapting the pedagogy traditionally used in the humanities and social sciences (class discussions, essays, small group work) helped a great deal. Many of the students had problems with traditional math instruction and math content, and I believe that the "friendly" pedagogy was useful and might have decreased unpleasant associations that had formed. But it takes a while to become comfortable with these techniques: I'm not there yet!

My attempt to run a "chat room" consisted of having a web page of questions and comments. It isn't clear to me how helpful this was. Certainly if the class were larger, this would have helped me decrease the amount of e-mail I answered. In a typical week, I think I received about 1 e-mail message per student in this course. This is an average: a few students never sent me e-mail outside of mandated homework, and some were far more "verbose" (e-verbose?). I know that students e-mailed each other quite a bit, and I certainly encouraged them to work together on several assignments. I also said they could work together on other assignments provided they indicated that the work was a joint enterprise. Again, some students were "lone wolves" and only worked with others when forced to. Other students were rather derelict in joint projects, and several times students spoke to me about how to compensate for uncooperative teammates. My lack of experience showed and I was not as helpful as more informed instructors might have been.



Much of the material in the course can't be read about in books. I grew irritated at people who didn't attend class, because I was quite sure there was no other source (readily) accessible to them which could give them the information that I wanted to give them.

This was all seemed correct until I contrasted the attendance in the course with that of students in calculus courses. In most of our calculus courses, attendance after the middle of the semester would tend to be 30% to 50% of nominal enrollment. In this course, attendance usually seemed to be about 80% to 90%.

I had told people at the beginning of the semester that attendance was important. I could have used 1 point quizzes to reward their attendance. It isn't clear what would have changed but at least I would be doing "something" and would have, in addition, a nice way to gauge how each class meeting went.

Again, correct student behavior should be rewarded. And stimulating behavior that is desired can be done by "rewards". I don't think students participated enough in the presentations. I think I should have made clear that I was somehow "scoring" participation. I think I also should have said that a question on the final exam would reflect some aspect of student presentations, and then followed though. That almost surely would have encouraged more attention be paid to presentations and the ensuing debates.


Scaling up?

It is not clear to me if this course could be "scaled up". The standard class size for Math 103 at Rutgers is 40 students. The customary instructor for the course does not have substantial long term ties to the university. The pedagogy is non-standard, and most math instructors would need to think about it carefully and commit themselves to some significant effort. The content is not been taught in standard courses and many instructor candidates would need to learn the material. Both the policy and technical aspects are changing rapidly so some effort is needed to represent current practice.


A few orphan problems

There were also some homework problems which did not survive from the first semester. In fact, some didn't even survive (to be given in class even once) after they were written! They are recorded here since inventing them required some work and maybe someone will find them useful. [PDF|PS|TeX]