Synchronous Class Time
To prepare for a class meeting, I prepared a Power Point presentation and emailed it to the students so they could write on the slides during the class session. Since the purpose of the course was to develop students' thinking skills and not just their knowledge base, my slides contained definitions, theorems, and problems, but not many examples, proofs, or solutions. To view a sample presentation, click here.
During class, I used the Power Point slides as the foundation for the discussion. To fill in the details, I asked students for inputs and then wrote or typed on the screen using Centra. I cannot include the actual broadcasts due to privacy concerns, but I have included some screen shots. Click on the small (unreadable) images to see the fullsize (readable) images.
Centra enabled students to "raise their hand" by clicking on a hand icon, which signified to me that they would like to speak. I then chose whether to allow them to speak by "giving them a microphone." This allowed me to control the conversation while also encouraging some good interaction between everyone in the class.
Another useful tool in Centra is the text chat window, in which students and instructor can type messages to one another. I often had different students type an answer to a question that I asked or allowed several people to comment at the same time. The text chat was particularly useful when a student's microphone would stop working, which happened about once every other week due to some of the students' Internet providers.
I could also assign students to separate "classrooms" to work together in smaller groups. I then moved among the discussions and interacted with the groups on a more personal level. In many ways, Centra allowed me to run the online course in much the same way that I run traditional courses.
Homework
At the beginning of the term, I required students to turn in their homework in a Word or PDF format. I allowed the Word format because I knew everyone would have access to Microsoft Word or some similar program. To encourage students to try LaTeX, I also allowed PDF files. Unfortunately, only one of my students took the time to learn LaTeX. Also, some of the students worked around having to type their homework by scanning their written work into either Word or PDF.
I found through the course of the semester that typing homework, even though it took twice as long, was time well spent, because the quality of typed homework  a final draft instead of the usual rough draft  was far superior. I have included at the right links to two typed assignments and two handwritten assignments for comparison. Not all of my students agreed  three students commented in course evaluations that typing the assignments was the aspect of the class that most detracted from their learning. But one of these students also stated, "I admit, once I invested the time, they were stronger."
Once the students turned in their homework via Blackboard, I chose to grade by making comments on the files themselves in order to quickly send them back to the students. To simplify the process, I decided to use only one format for comments. Since some of the files were in PDF, and it is easier to convert from Word to PDF than to convert from PDF to Word, I converted all the files into PDF. PDF also has an easy way to make comments on the files  but this works only if you have the full version of Acrobat on your computer, not just Acrobat Reader.
The Group Project
As the course progressed, I decided that the students were doing an excellent job on the homework and were learning the material well. Therefore, to move them to a higher level of learning, I decided to have the students complete a group project instead of a final exam.
For the project I gave them only these three instructions:

Tackle a real life problem using the material covered in our class.

Present the projects professionally, as if you were hired to do the project by an outside company.

Be creative.
Initially the students met such vague instructions with hesitancy, but they eventually took the projects and ran with them.
One of the projects was to design bus routes for a rural school district that reduced the travel time for all students from up to four hours per day to less than two hours per day. In fact, one of the students from this group has published a description of the project in the Rural Mathematics Educator (Britt, 2006) and the entire project has been published as an ACCLAIM Working Paper (Belcher, et al., 2005). Another group designed delivery routes for a rural milk company, as well as locating the ideal location for a production plant. The third group created a design for a supermarket that maximized profit by placing key items in the best locations. Click here to view the report with the names omitted.
The first issue that arose involved students forming groups. I decided not to assign groups, since I wanted the students to work on projects that would be interesting to them. Instead, I had the students come up with ideas for projects and to share these ideas and talk about them on a threaded discussion board. During the following two weeks, students shared many different ideas, and the groups formed naturally around three projects with about an equal number of people in each group.
Once the students belonged to groups, I faced the issue of helping these students communicate with each other. Since the students were scattered over several states, getting together in one location would be difficult. Instead, at different times during the semester, I took the last few minutes of class time and had the groups move into breakout rooms so they could discuss their projects using live audio and writing on the screen. I then took that opportunity to move between the groups answering questions that arose and making sure they were on the right track. Then, later in the semester, the students set up times outside of class when they could all log onto the system in order to complete their projects and practice their presentations.
In the end, these projects turned out to be a central portion of the course, with the outcomes going beyond my expectations. In fact, one of the students said, "Replacing the final with a final project was excellent. I learned more by doing the project." Another stated, "I really was able to pull a lot of the ideas together in our group project."