In the spring of 2005, I taught a course entitled "Discrete Mathematics for Teachers," originally designed to be a part of a master's degree program at the University of Tennessee for improving the mathematical content knowledge of high school math teachers. However, during the past few years, it has also served as a core course in the ACCLAIM doctoral program designed for rural mathematics education students. I had some students from each program, but the majority were in the ACCLAIM group.

The average mathematical level of these students was about the same as math majors during their junior or senior year, with some students at the sophomore level and some at the second year graduate school level. Some of the students were faculty members at small colleges who had taught courses in discrete mathematics, while other students had struggled to finish the sophomore-level linear algebra course the previous semester.

Deciding how to organize the course for the students who were also scattered across five states was a tremendous task, especially since I wanted to design the course for the students and the unique learning environment provided by the online setting. To get some ideas, I read "Review of Distance Education Literature" (Mayes, 2004), which summarizes several suggestions for designing an online course. I also was able to view current courses in the program to get some ideas of what my class would look like.

One of the decisions when organizing a course is what mathematical topics to cover. One of the primary methods of selecting topics is to choose a textbook. Since I would be teaching discrete mathematics for secondary and college teachers, I decided that the standard discrete textbooks would not be appropriate. Instead I chose *Discrete Mathematics for Teachers* (Wheeler & Brawner, 2005). While the book covered all of the topics that I wanted, the authors designed it for pre-service elementary school teachers, so I decided that it needed enhancement with outside sources and more difficult problems.

I learned some new (to me) terminology early in my preparation. Online courses are taught in two different ways, *synchronous* and *asynchronous*. Synchronous technology allows for students and teacher to be online at the same time and interacting with one another, while asynchronous technology allows students and teacher to interact at other times. In a traditional course setting, classroom lectures are synchronous, while homework is asynchronous. My course would have two hours a week of synchronous class time, and the rest of the interaction with the students would be asynchronous.

Since synchronous class time was limited to two hours per week, the students would have to spend time outside of class working on homework. This was inevitably going to produce large amounts of homework to grade and comment on. To ease this process and decrease turnaround time, I required students to turn in typed homework via the University of Tennessee Blackboard system.

Another planning issue was how to compute the grades and whether there would be any tests in the class. Since the purpose of the course to teach the students how to think mathematically and to introduce them to the realm of discrete mathematics, I decided to have the homework count for the majority of their grade. Also, due to the distance education nature of the course, I decided that frequent tests would not be appropriate. However, I left open the option of either a final exam or a final project.

I decided to make the synchronous class time as interactive as possible by asking questions that the students would answer either vocally or in the text chat, encouraging them to ask questions at any time, and to have them work on problems on their own or in groups during the class. Upon the suggestion of others who had taught in this format, I scheduled a 10-minute break after an hour of class, and I provided more interaction as the night went on to keep the students awake.

The synchronous part of the course revolved around software called Centra, which is designed for online teaching and training. It includes features such as being able to

- talk with my students,
- write on a "white board" that they could see,
- share applications from my desktop,
- break up into discussion groups, and
- have a running text chat during the class.

The software also has video capability, but I decided not to use it, since many of my students would be on slow dial-up connections. Other similar software packages include Macromedia^{®} Breeze or Microsoft^{®} Live Meeting.

The Centra software ran on servers at the Institute for Mathematical Learning (IML) at West Virginia University, one of the partners in the ACCLAIM program. When our class was scheduled, we "attended" the session through a web interface. For more information, see the section Synchronous Class Time on the next page.

For homework, students sent me files in either Microsoft^{®} Word or PDF formats. I converted the Word files into PDF and made comments on all of the submissions using Adobe^{®} Acrobat. These files were transferred from and to the students via the University of Tennessee Blackboard web site. For more information about the homework and Blackboard, as well as examples of student work, see the Homework section on the next page.

I used Microsoft^{®} PowerPoint to create outlines for my lectures and then uploaded these to Centra. During class, I filled in the missing parts by writing on the slides or on other blank slides.

The hardware requirements for students were a Windows computer with a connection to the Internet, a microphone, and either speakers or a headset. I often used a Sympodium interactive pen display, which uses SMART board technology, and which allowed me to write on my prepared notes much like writing on an overhead transparency. This technology made it possible to write proofs, create graphs, and highlight certain points in a similar fashion to a using a chalkboard or white board.