Undergraduate Mathematics Education
Cryptography: The History and Mathematics of Codes and Code-Breaking
Clicker Questions for Statistics
Mathematica Activities for Teaching Calculus
Excel Worksheets for Teaching Probability
Pre-Calculus Review Chapter on Polynomials
Mathematics Talks for Undergraduates
Cryptography: The History and Mathematics of Codes and Code-Breaking
During the summer of 2009, I taught a class titled “Cryptography: The History and Mathematics of Codes and Code-Breaking” in Vanderbilt University’s Master of Liberal Arts and Science program. This was a new course for me, but the students were great and the course went very well. When I looked for syllabi and other materials from similar courses online, I couldn’t find any. Other cryptography courses I found online tended to focus on the mathematics of cryptography or on modern computer security issues. None had the blend of math and history I used in my course. So I thought I would share some materials from my course here.
- Here’s the course syllabus [PDF] and week-by-week schedule. And here’s the course blog, where I posted course information, weekly problem sets, and other resources (including PowerPoint slides and Excel files).
- Our textbook was The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography by Simon Singh (Anchor Books, 1999). I was quite proud to have a textbook one could purchase for $11 on Amazon! On the course blog, I posted weekly pre-class reading questions. Students responded to these questions on the blog as part of their class participation grade. You can read their responses on the blog to get a sense of the kinds of discussions we had in class.
- Another way students could contribute to their class participation grade was to tag Web resources in the social bookmarking service Delicious. Here are the 157 resources they (and I) tagged during the course. Most of the students tagged at least one resource, and a few students really got into it and tagged many resources. (I have continued tagging resources using the “cryptography” tag since the course ended.)
- Since there have been so many codes and ciphers used throughout history–far more than we had time to discuss in class–I assigned an “expository project” to my students in which they were to select a code or cipher not covered in class and explain it to their classmates. Most of the projects took the form of five-page papers, although two students selected the video option I gave them and produced short online videos about their topics. You can see all the expository projects on the course blog. I asked students to read and comment on two of their peers’ projects since it seemed a shame that I would be the only one to read them, and you can see their responses on the blog, too.
- The students also completed a “big questions” paper at the end of the course in which they selected one of the more controversial questions we considered during the course about the use of cryptography, chose a side, and defended their side using evidence from the course. I didn’t require my students to share these papers on the course blog, but four of my students chose to do so. You can see their “big questions” papers here. Several students referenced their peers’ expository projects in their final papers, indicating to me that having students share their expository projects on the blog was a good idea.
- One course component I added too late to be fully utilized was a cryptography timeline built collaboratively by the students in the course. A brief Google search didn’t turn up any existing online timelines of the history of cryptography, so I followed Brian Croxall’s tutorial and set up a collaboratively-generated timeline using Google Spreadsheets and Javascripts from MIT’s SIMILE Project. Here are the instructions I gave my students for contributing items to the timeline, and here’s the finished product. As I said, I created the timeline rather late in the course. Next time I teach the course, I’ll build this assignment in earlier, perhaps as another way to earn class participation points. Hopefully then we’ll end up with a much more complete timeline, one that should be of great use to students as they research their “big questions” projects.
- Therese Huston interviewed me about this course for her article “Teach as You Learn” in the NEA Advocate‘s Thriving in Academe series. Her focus in the article is on teaching courses for the first time. Huston’s article was based on her new book, Teaching What You Don’t Know, which includes an interview with me about some of my other courses.
- Also, the course received some favorable coverage from Vanderbilt Public Affairs and the Vanderbilt undergraduate newspaper, The Vanderbilt Hustler.
Clicker Questions for Statistics
This set of multiple-choice questions was used during my spring 2008 offering of Math 216: Introduction to Probability and Statistics for the Engineering Sciences. I asked these questions of my students during class using a classroom response system (“clickers”). The questions are organized according to the sections in Navidi’s Statistics for Engineers and Scientists. Each question includes a little commentary on that question’s answer. Please let me know if you have questions about these clicker questions or if you find them useful in your courses.
Mathematica Activities for Teaching Calculus
Listed below are several Mathematica-based computer labs I designed and used in my accelerated calculus courses while a graduate student instructor at Vanderbilt University (1999-2002). To view the files marked (.nb) below, you need either Mathematica or MathReader, the latter available for free download.
- Volumes of Solids of RevolutionI designed this lab to help my calculus students visualize the process of rotating the area between two functions around an axis to generate a solid of revolution. The Mathematica notebook features animations of such rotations.Lab Instructions
Mathematica Notebook (.nb) - Parametric and Polar CurvesThis lab was designed to help my calculus students develop some intuition about the graphs of curves given in parametric or polar form. Included in the Mathematica notebooks are routines for animating the graphing of such curves.Lab Instructions
Mathematica Notebook: Parametric (.nb)
Mathematica Notebook: Polar (.nb) - The Tangent ProblemThis Mathematica lab is a self-paced tutorial on the tangent line problem, featuring an animation of the secant line limiting process.Mathematica Notebook (.nb)
- The Velocity Problem, The Derivative, and the Derivative as a Function
This lab is a three-part follow-up to the Tangent Problem lab. The lab uses Mathematica to expedite computations in order to build conceptual understanding through trial and error. It also draws on Mathematica’s plotting ability to illustrate the graphical connection between a function and its derivative.Mathematica Notebook (.nb)
Excel Worksheets for Teaching Probability
Listed below are two illustrative Excel worksheets I designed and used in a course on the theory and practice of teaching probability I co-taught for the Harvard Extension School (2004).
- Buffon’s NeedleThis Excel worksheet runs a Monte Carlo procedure which gives an estimate for the value of pi via the classic Buffon’s needle problem. The worksheet takes advantage of Excel’s random number generator.Instructions (.pdf)
Excel Worksheet (.xls) - The Monty Hall ProblemThis Excel worksheet runs a Monte Carlo procedure which sheds light on the classic Monty Hall problem. The worksheet takes advantage of Excel’s random number generator.Instructions (.pdf)
Excel Worksheet (.xls)
Pre-Calculus Review Chapter on Polynomials
The text From Here to Infinity: A Foundation for Calculus was used circa 2002 in the Vanderbilt University Department of Mathematics for algebra and trigonometry review during first-semester calculus courses. In addition to editing the TeX files used to typeset the book, I wrote Chapter 4: Polynomials and Factoring.
Mathematics Talks for Undergraduates
I have given a number of mathematics talks for undergraduates in the past. Below are descriptions of several of them. Venues for these talks include the Vanderbilt Undergraduate Seminar in Mathematics, the Furman University Department of Mathematics Colloquium, and the Harvard Math Table.
- How to Win at Monopoly
March 2006
Vanderbilt Undergraduate Seminar in MathematicsModeled the game of Monopoly with Markov chains as way to determine which properties are likely to be landed on most often.
- Wavelets: Uniform and Otherwise
November 2004
Furman University Department of Mathematics ColloquiumPresented basic wavelet and data compression ideas, including an introduction to my research on nonuniform wavelet bases. I also presented this talk at the Harvard Math Table in October 2004.
- The Incredible Shrinking Data
March 2004
Harvard Math TablePresented basic Fourier analysis and data compression ideas. The talk featured data compression examples of both audio and image data. I also presented this talk at the Vanderbilt Undergraduate Seminar in Mathematics in October 2001 and April 2003. See these overheads from the October 2001 version.
- Two’s Company, Three’s a Conundrum
March 2002
Vanderbilt Undergraduate Seminar in MathematicsPresented material on Cardano’s solution to the general cubic polynomial equation and the Four Color Theorem as part of a lecture on famous math problems in which “one more” or “one less” makes a significant difference in difficulty.
- And the Winner Is…?
February 2002
Vanderbilt Undergraduate Seminar in MathematicsPresented material on voting methods and basic social choice theory. At the talk the week before, the undergraduate attendees were directed to a web site where they could vote on their Oscar picks. During the voting theory talk a week later, this voting data was used to illustrate how different voting methods can produce different results from the same set of voter preferences. The interactivity and inclusion of pop culture helped keep the students interested. See these overheads for more information.
- Huh?
November 2001
Vanderbilt Undergraduate Seminar in MathematicsPresented nonintuitive mathematics results in basic probability and traffic flow to a general undergraduate audience.
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