Jim Gleason is in the Department of Mathematics at University of Alabama.

I was a post-doc at the University of Tennessee when the mathematics department head asked if I would teach the discrete mathematics course designed for secondary mathematics teachers. The majority of the students enrolled in the course were part of the ACCLAIM (Appalachian Collaborative Center for Learning, Assessment, and Instruction in Mathematics) doctoral program in mathematics education, and so my student were scattered across six states, which made a distance learning environment essential.

I discuss here what I learned about online mathematics courses, in terms of both curriculum and technology, and what I would suggest for those about to teach online.

© 2005, Jim Gleason

Published March, 2006

Background of the Course and the Students

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.

Choosing a Textbook

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.

Synchronous and Asynchronous Class Time

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.

Collecting and Grading Homework

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.

Determining Course Grades

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.

Organizing Class Time

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.

Technical Preparation

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.

Synchronous Class Time

To prepare for a class meeting, I prepared a Power Point presentation and e-mailed 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 full-size (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:

1. Tackle a real life problem using the material covered in our class.
2. Present the projects professionally, as if you were hired to do the project by an outside company.
3. 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."

Since I had such a great experience teaching a mathematics course online, I have wondered why more teachers do not want to teach using these methods. I have also wondered what advice I would give to someone about to teach an online course for the first time. Therefore, I have come up with the following list, which is by no means exhaustive.

• Remember that online courses are different from the traditional classroom.

  When teaching an online course, you must view the course as something completely different from a physical classroom. This new teaching environment uses different methods and ideas. You must use computer screens and web cams instead of chalkboards or white boards. You will multi-task in new ways by teaching the class while following the text chat and watching for students to "raise their hand." You have to find new ways to get a feel for how your students are doing, other than looking at their blank faces. Most of all, you must be comfortable using computers and able to work on technical problems as you go. For some ideas on methods to use, talk to other people who are teaching similar classes, get ideas from the education faculty, and read about what others are doing. For additional ideas, I have included some interesting and informative articles on the next page (Kubala, 1998; Taylor & Mohr, 2001; Carnevale & Olsen, 2003; Engelbrecht & Harding, 2005a, 2005b).

• Get some training with the software and hardware.

  Get several hours of training with the software you will use. Then before the class starts, take some time to run through a couple of practice sessions with someone else who will be using the same software. Finally, find a mentor who has been through this before. This will likely involve going outside of your department to the fields of education or business, which have a longer history of using distance education technologies.

• Get to know your technical support.

  You will have problems. If you have already developed a relationship with your technical support team, then it will be easier and faster to get help when the problems occur. Most of the problems that occurred during my course were minor ones involving difficulty with my Sympodium pen or some of my students having trouble with their dial-up connections. However, there was one night that a worm attacked the server on which our class operated. It was sort of like the power flickering on and off in a normal classroom. Thankfully I was able to call our technician at home, and he was able to work on the problem while we continued having class.

• Use text chat during the class.

  This is one area where there is disagreement. Some people have had difficulty paying attention to both the text chat and what they are teaching. However, since many of my students were more comfortable asking questions in the text chat than orally, I found that this extra effort paid off. As an extra bonus, other students were sometimes able to answer those questions without my having to stop class. The text chat also enabled me to get to know my students better, as they were more open and expressive with typing their comments than saying them.

• Have students turn in all homework electronically.

  This will be a pain for your students, but it helps immensely with speeding up the turn around time on grading. When students faxed in their work, I had a more difficult time making comments on their work and sending it back to them. I also found that it helps to limit the number of formats for homework submission. I used only two, Word and PDF. When students used other programs, such as The Geometer's Sketchpad^®, they simply copied and pasted these files into their Word document.

• Provide quick feedback.

  When teaching a distance education course, it is easier to be not as prompt returning homework and answering students' questions as when they stop by your office. Therefore, you need to make a point of promptly returning e-mail messages to answer students' homework questions, so they can finish their assignment, and of grading their homework quickly, so they can use the feedback to make adjustments for future assignments.

• Build in collaborative work.

  When designing an online course, there must be collaborative opportunities built into the course. Some possibilities are

  • Have students work in groups during class time.
  • Maintain a discussion board for students to ask questions.
  • Have separate class times for discussions with the students.
  • Make sure that each student has the e-mail addresses of the other students.

  Since my class was a graduate class with most of the students being part of a cohort program, most of these came naturally for them.

• Make yourself available.

  Students will not just stop by your office, so make sure that you stay on top of your e-mail and that your students have a phone number where you can be reached. Also, with distance education students, most of their homework is done during the evenings or on the weekends when they are not at work. You need to keep this in mind when setting up office hours and due dates.

• Set up an opportunity to meet the students in person.

  Towards the end of the semester I had a chance to meet several of my students at a conference. In the following weeks, I found it much easier to interact with these students and to know how to help them better. It would have been much better if I had been able to meet the students earlier. There are many ways to do this, including having an on-campus class meeting, meeting the students when they begin their program, or traveling and giving the class from the different students' locations.

Most of all, remember to have fun. Realize that you will make mistakes when you are trying something new -- you need to be willing to look foolish in front of your students at times. They will understand and admire you more for trying something new.

Britt, Deborah (2006). Your school bus routes might be rural... if... Rural Mathematics Educator (4)3. Available online at http://www.acclaim-math.com/docs/html_rme/rme11/05.02fea_britt-your_school_bus.html (accessed 02/03/2006).

Carnevale, Dan & Olsen, Florence (2003). How to succeed in distance education. Chronicle of Higher Education (49)40, A31-A33. Available online at http://chronicle.com/weekly/v49/i40/40a03101.htm (accessed 08/03/2005, subscription required).

Engelbrecht, Johann & Harding, Ansie (2005a). Teaching Undergraduate Mathematics on the Internet Part 1: Technologies and Taxonomy. Educational Studies in Mathematics (58)2, 235 - 252. Available online at http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10649-005-6456-3 (accessed 08/03/2005, subscription or fee required).

Engelbrecht, Johann & Harding, Ansie (2005b). Teaching Undergraduate Mathematics on the Internet. Part 2: Attributes and Possibilities. Educational Studies in Mathematics (58)2, 253-276. Available online at http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10649-005-6457-2 (accessed 08/03/2005, subscription or fee required).

Kubala, Tom (1998). Addressing student needs: Teaching on the internet. T.H.E. Journal (25)8, 71-74. Available online at http://thejournal.com/articles/14083/ (accessed 03/06/2006).

Mayes, Robert (2004). Review of Distance Education Literature. ACCLAIM Occasional Papers Series, (6). Available online at http://acclaim-math.org/docs/occasional_papers/OP_06_Mayes.pdf (accessed 08/03/2005).

Taylor, Janet A. & Mohr, Joan (2001). Mathematics for math anxious students studying at a distance. Journal of Developmental Education, (25)1, 30-37. Available online at http://vnweb.hwwilsonweb.com/hww/shared/shared_main.jhtml; jsessionid=YV12JJASBVCMRQA3DIMCFGGADUNGIIV0?_requestid=19390 (accessed 08/03/2005, subscription required).

Wheeler, E. & Brawner, J. (2005). Discrete Mathematics for Teachers. Houghton Mifflin Company.