To
appear: College Mathematics Journal
A Survey of Online Mathematics Course Basics
G. Donald Allen
To create an online course, one must consider the scope of design, software tools, student deliverables, and security as major factors, each with multiple sub-factors. As for any educational publication, style is extremely important, and for the Web, especially so [14]. Each factor should be considered thoughtfully, even though not every single one can be resolved immediately.
The scope of course design must address issues of content and interactivity, among others. Content is the most important. One shortcut used by many authors is to use a print textbook as the sole source of mathematical content. Remarkably, irrespective of the modern mathematics textbook design, extensive editing, and multiply colored production, few are written for self-study. They are written to be the print-companions of the traditional lecture course. Indeed, it is for this traditional lecture-recitation environment that almost all textbook authors write. For the online course, relying on a single textbook to convey mathematics content may well produce disastrous results. One alternative, that of writing the complete contents oneself, may require too much time and of course there is no guarantee of good results. For additional discussion on tools and methods, see [10, 11].
However, a blend of a print text with personally generated supplements can work well. Consider the following three supplements: a news page, additional examples, and practice exams. It is relatively easy to produce and maintain a news-type page with rich mathematical content that is aimed at answering individual inquiries in a more general way, for all students to read. Another supplement can contain complete solutions to homework problems and additional examples that illustrate some particular point, supplying when necessary more than the usual detail found in the text. The third supplement, a surprisingly important one, is practice exams. Students have unusual tenacity about working these problems out. It may be that they feel that while homework has theoretical importance, practice exams have significant and practical importance. A lengthy practice exam may provide the student with an important learning instrument. Having threaded discussions and chat rooms can be helpful as well, though these require continual monitoring, which translates into additional, possibly substantial, instructor time. Below we tabulate these and a few other options for online course management.
Online Course Features
|
Feature |
Description |
Pros |
Cons |
|
News page |
A page devoted to extended discussion of mathematical questions. Topics to include: supplements to the theory, questions and clarification about homework, references to outside reading. |
Useful supplementary information for students; extends the lecture mode. |
|
|
Chat room |
A live session via computer bulletin board. Students can make comments, ask questions, and chat with other students |
Lends immediacy of the online class and gives students a genuine chance to communicate |
Time-consuming |
|
Practice tests, solutions to problems, supplementary readings |
Clear and well written additions to the course. Students tend to read these items carefully. |
Students can see properly constructed solutionsa good learning experience for all mathematics students. Supplements become a focal point. |
Production time, including conversion to math-on-web can be excessive. |
|
Virtual classroom and virtual office hours software for synchronous online learning |
Internet based software that allows a full video/audio/whiteboard connection between professor and student. Professors can give live yet virtual lectures. Sessions can be stored for review by students. Good also for Web meetings, online conferences, and content management. Examples: Centra, Net Tutor |
Second only to face-to-face teaching. |
Expensive and time- consuming |
|
Course managements systems |
An online course management environment, particularly useful for course grades. Fully password protected. Examples: WebCT, Blackboard |
Eases somewhat the management and maintenance of an online course. |
Another software system to learn. |
Web sites: Centra: www.centra.com, Net Tutor: www.link-systems.com/
Interactivity is the Internet buzzword these days. What is it? And how can anyone put it into his or her Web-based course? A broad interpretation would define interactivity as anything that communicates with the student beyond mere text and static images. This includes animations and student response to applets generated through Java, Flash, CAS engines, or Web editors.
Animations, while simple to create for the Web, are not that easy to simulate in class. Most of us can remember trying to show our classes how the secant line may approach the tangent line by “moving” a fixed chalk line to the chalk tangent line. The result is not always convincing. With simple animations, these motions are both possible and effective. If the student cannot attend the lecture to see your simulated animation, he or she can see the real thing on the Internet. Well-crafted animations and brilliantly colored graphics have a certain fascination for many students. Animations are relatively easy to make with tools like Maple, Mathematica, and MATLAB, MathCAD, Flash, Java, and JavaScript. With respect to tools such as Maple, however, some postproduction may be necessary to render the animation more convincing. (The Web site www.academicsolutions.com/workshop gives a brief discussion, demonstration, and tutorial on various ways to create animations for mathematics courses and discusses the post processing of Maple animations. The interested reader may also view Larry Husch’s home page, www.math.utk.edu/~husch/, for some animations using Flash and Java. A general animation tutorial from the Webmonkey site is available at hotwired.lycos.com/webmonkey/multimedia/animation/tutorials/tutorial1.html. It emphasizes general graphic design for commerce and therefore has limited utility for mathematics courses.)
The next level of interactivity requires the student to interact in some meaningful way with the computer. For example, the student may enter the acceleration of gravity and elasticity of a ball to study how it bounces. Such interactivity almost always requires programming. Whether one uses Java, JavaScript, Flash, Mathematica, or MATLAB, special skills or services are needed. Each can be programmed (all in different languages of course) to generate remarkable interactive and truly educationally valuable applets. Java remains the most powerful tool available today. JavaScript, which lacks Java’s powerful graphics API, is also very useful. However, it is Flash (Macromedia, about $100 for educators) that is developing as a strong competitor. The newest version, Flash MX contains a full scripting language. This, together with its vector graphics capabilities and relatively shallow programming learning curve, may give a broad base of instructors a powerful animation and interactivity generator.
For our purposes, we allow interactivity realized only without client costs, beyond of course the Internet service and perhaps downloading a plug-in. Both Mathematica and MATLAB allow such interactivity over the Web. However, the developer needs special Web components and server-side CGI-bin access. To learn about interactivity creation tools, it is strongly recommended that the user take a course. For example, the MAA and ICTCM both offer short summer workshops on various software tools and Web course design.
Lastly, there is one type of interactive applet that everyone can construct, have working, and upload, all in the one day! Using active-X components generated by FrontPage 2000 or 2002, it is easy to create interactive applets involving almost anything possible with a spreadsheet and a graph or graphs based on the spreadsheet. For example, it is easy to create an interactive applet that allows students to see changes in the graph of a quadratic as they enters its coefficients, or its roots. Important drawbacks are that users need Internet Explorer running in a Windows environment and that editing the active-X script is virtually impossible. For example, the code for the graph of a quadratic equation with user-input coefficients is longer than 600 lines—all generated by FrontPage. In general, the more intensive the programming required to produce an interactive applet, the more browser-sensitive it becomes, a daunting drawback which we will not discuss here. We mention, for completeness, two more excellent programs, LiveMath [15] and MathCAD. Both allow some interactivity and ease of use with the Web. Indeed, MathCAD exports documents in HTML format making graphics and mathematics as GIF images.
Finally, it is most important to use the interactivity in some meaningful way. Most students, perhaps already over-stimulated by the world around them, have little use for visual curiosities in their college courses. When designing interactive functionality pay special attention to its purpose, its appearance, and its ease of use. The serious developer should be willing to combine several tools to obtain the animations and interactivity he or she desires. For mathematics courses, using a CAS engine is nearly essential, if only for its graphical imagery.
The third component of the Web-based course is the set of student deliverables. What you ask your distance or local students to deliver will certainly affect your course design. If the course includes lectures, there is wide latitude in what can be done. The most likely assessment methods of choice will be traditional homework, quizzes, and exams, all synchronously ordered and arranged. For a distance or Web-based course, what are or should be the deliverables is not as clear. When the students are all adult learners, such as in a graduate course, the use of homework, class projects, and take-home exams have worked very well. [2, 3, 5, 7]. Windowed exams, online exams available to the student only for a certain window of time, can to be taken over the Web with results delivered to a database via a course management system. Within the WebCT or Blackboard course management systems, the add-on tool Respondus (www.respondus.com) is an effective exam generator.
However, exams must be given for
undergraduates. Local and distance Web-based courses must now be
differentiated. Local courses have on-campus students, making exams,
help, and peer associations easy to arrange. For locally
administered Web-based courses, where the scheduled class period takes place in
a computer lab and there is no lecture, the traditional exam schedule can be
used. However, to keeps the students on task, many more quizzes should be
given each semester. [8]. Some comparative work has been
done on giving both paper and pencil and online examinations within the
traditional class setting. Evidence
seems to be that there results no significant difference in performance [13].
Distance Web-based courses are very much different. With greater class
diversity, the tasks of assessment are more difficult. Exams should be
proctored and must be windowed, so remote proctors are needed. Security
is a key issue here and course management systems are an important tool for
many online courses. Some firms offer
secure exam portals that limit student options, such as email and browser
usage, during the online exam. (See Software Secure, at www.softwaresecure.com/
comp.htm.)
Tools for creating online assessments are available. Within every course management package, such as Blackboard and WebCT, there are extensive tools for online exam creation and online grading. However, all are unsuited to the mathematics environment. Nevertheless, quite a number of freely available tools are available and can be downloaded. For example, D. P. Story of the University of Akron has developed the ExerQuiz package that uses PDF files and JavaScript. (www.math.uakron.edu/~dpstory/webeq.html). Another quiz generator, rather intuitive, Quizmaker 1.4, can be downloaded or used directly from the Web (www.academicsolutions.com/quizmaker/quizmaker142.htm) Neither of these can be described easily, but both allow for mathematics symbols. Quizmaker allows the inclusion of any valid HTML code. So, whether MathType or Scientific Notebook, or even LaTeX2HTML, is used to create the native HTML code for mathematics display, it can be pasted directly into the quiz question, answer, or feedback entry boxes.
Quiz making software
|
Assessment system |
Ease of use |
Remarks |
Cost |
|
Quizmaker 1.4 |
Straightforward, feeback to questions possible |
Multiple choice. Editing after creation can only be done with editor software. |
free |
|
The Manhattan Virtual Classroom |
Powerful but requires some training |
Requires CGI access |
free |
|
Easy Quiz |
Straightforward |
Makes three kinds of computer-based multi-media Quizzes—Match Question and Answer, Type an Answer and Multiple Choice. |
$25 |
A main goal of most online course authors is to use methods that will attract the students to the online materials. The proper medium for readable mathematical content is excellent mathematical typography. While mathematicians can get along with most any “fortran-esque” script, students cannot. It is difficult enough for most to learn mathematics but even harder to compound the struggle with inadequate or obscure mathematical notation. Much has been written about how to do this [1]. In this note we cannot do much more than mention the basic options. They are (1) PDF, (2) Converters, (3) Graphics, (4) Plug-ins, and (5) MathML.
The simplest way to put mathematics online is to use Acrobat files. The Adobe Acrobat portable document format (PDF) is ubiquitous by now. It does the very best job of rendering a mathematics page exactly as it appears in print, meaning that users may focus completely on the creation of the content itself, although one needs to be mindful of font embedding problems. The course creator needs only make a link within the Web. The principal criticism is that PDF documents are more or less static, not yielding simply to dynamic changes.
Among the several converters on the commercial or free market are (Microsoft) MSWord and LaTeX2HTML, though the latter is strictly a converter. Both can be used convert an existing native document to HTML. This has meant in recent years that mathematical constructs are rendered as GIF images with the converter aligning them with the text. Converters have steadily improved by increasing the number of Web features allowed. Full links, bookmarks, tables of contents, and the like are certainly possible. For TeX/LaTeX users, the book by Goossens and Rahtz [12] contains a wealth of information on integrating mathematics into Web pages The MSWord conversion is done with many XLM constructs, making the file size quite large and moreover making post-conversion editing difficult. The Word Perfect conversion yields a somewhat cleaner HTML file. A companion program to Word and Word Perfect is MathType (www.mathtype.com), which has been the full-featured mathematics-typesetting alternative to the native equation editor. Now in version 5, MathType can create output in the GIF image style or in MathML. The instruction manual is one of the very best you will ever read.
Scientific Notebook [16] has been available for several years and has improved steadily. With it, not only is mathematics painlessly simple to create, it can be displayed over the Web and is inherently “live” with the Maple engine running in the background. It is not strictly a plug-in, and clients need the program to view the native TeX files. However, the latest version, 4.0, does allow HTML export. Operating similarly to LATEX2HTML, it renders mathematics as GIFs shown inline with the HTML text portion.
A quite remarkable (and inexpensive) plug-in type product is Tech Explorer. With it, TeX and LaTeX files are rendered accurately within Netscape or IE. Hyperlinks, footnotes, and a number of other normal html features can be inserted directly into the TeX file. Tech Explorer is robust, coming in versions for Windows, Macintosh, Linux, and Unix platforms. It is certainly the first choice (with PDF) for the display of legacy LaTeX files, where little time is available for technical modifications. For mathematicians who need to give presentations and are a bit tired of being shown up by glitzy PowerPoint presentations from nonmathematical colleagues, Tech Explorer offers hope. Within the Windows system, Tech Explorer allows the input of LaTeX and MathML scripts as objects that the user can insert directly into PowerPoint. It operates quite simply: just paste or type the TeX codes from your LaTeX file into an input field and the text is displayed instantly. One can even adjust the colors of the fonts to accommodate various styles. MathCAD files can also be exported to an HTML format for use with Tech Explorer. MathType has also been compatible with Power Point. Other products are also available.
Math Converters
|
Name |
Ease of Use |
Remarks |
Cost |
|
latex2htmlL/tex4ht |
For LaTeX users only |
Available for Windows with some additional program installation, i.e. Perl |
free |
|
Scientific Notebook 4.0 |
Very easy |
LaTeX based but completely icon driven |
$116* |
|
MathType 5.0 |
Easy, high-quality on-screen and print |
Works with MS Word |
$89* |
|
Adobe Acrobat (5.0) |
Easy |
Best rendering of math available. Some TeX to PDF converters are freely available. |
$56* |
*Academic pricing from online vendors
Online courses can reach a vast population of students for whom collegiate mathematics courses are otherwise unavailable. Web-based and Web-assisted courses give students resources not available in the traditional format. It is worth noting that the lecture format is the centuries-old compromise to teach the many by the one. In an age where we have identified verbal, visual, and tactile learners, traditional teaching methods have been abandoned in favor of newer face-to-face teaching methods. There are constructivist and instructivist methods; there are normal and dynamic learning communities. The impact of postmodernism has changed the world of teaching and even the definition of learning. The point we emphasize is that Web-based materials offer another, and possibly very important channel, for information delivery. Moreover, they can offer options and venues beyond the scope of classroom teaching.
To be specific, assume that the Web-based course is offered without a lecture and that some of the students are not on campus. Assume also that all students have been certified to be reasonably independent learners without reading difficulties. Qualifications are important; a sure way to defeat a new instructional enterprise is to admit students with learning difficulties. As we all know, teaching reluctant learners is a formidable challenge and one that many of us fail to meet. The type of course should favor mathematical training as opposed to mathematical concepts. To be sure, teaching Web-based undergraduate courses in advanced calculus, topology, or abstract algebra without a lecture may not work well. On the other hand, when many of the theoretical foundations of such courses as numerical analysis, cryptography, and matrices have been previously learned, these can work well as Web-based courses. Nonetheless, we have only just begun to explore the scope of Web-based learning of mathematics.
To operate a Web-based course requires an active strategy. Communication with students by bulletin board and chat, feedback quizzing, online testing, news, assessment, online help, homework collection and return, and proctoring examinations are many of the considerations. As always, good communication channels to students are key. We consider the critical channel of online help.
Students will have questions; answers must be provided. If the questioner arrives during office hours, normal procedures are in play. When the question comes by e-mail or fax, the instructor’s teaching experience is critical. Here you must interpret the question and decide just what aspect of it needs to be answered. Answers should be relatively complete, with full details supplied. Often the student will simply read a problem wrong, which is easy to handle. Yet all questions—easy, hard or in between—take time.
Special software tools can also transact online lectures and help. NetMeeting (Windows only) is an example. Students and teacher can interact with voice and software tools. While the program is not perfect, acceptable results can be obtained. Other software tools include NetTutor, TutorsEdge, Interwise, Centra, and many others. Here is the way these tools work. Synchronous tutorial hours are arranged. At the appointed time the instructor logs in to a specific URL and students wishing to participate in the day’s session log in as well. Each participant sees the list of those logged in. The heart of the communication is by whiteboard or voice. The instructor has the master controls. He or she can allow or deny privileges such as writing on the white board or speaking. Students can (electronically) raise their hands to ask questions; the instructor answers questions privately or in class format. All can participate in the session. Interestingly, the instructor can give an entire lecture this way, using prepared slides, taking questions, answering questions, and making comments. It’s not too much different than a lecture, except that more preparation is required. Online office hours can be conducted the same way.
The tricky part is how to help with the mathematics, including symbols. Most students don’t write mathematics during office hours. Usually they ask a question about some homework exercise or some derivation. The instructor can respond with words, though often some mathematical explanation is necessary. With the exception of NetTutor these tools do not allow mathematical notation on their white board—the common drawing and writing area. So, using another mathematical preparation tool such as MathType or Scientific Notebook, the instructor typesets the appropriate mathematics, captures it and pastes it to the white board. With the mouse and various drawing tools, the instructor can annotate the mathematics while both students and instructor discuss its meaning.
Mathematical typography is the greatest drawback to this interaction. The instructor must have the skill to create mathematics, virtually on the fly. There are a couple of shortcuts, for example using a commercial whiteboard product (e.g., Aiptek, www.aiptek.com) to write and then capture the handwritten mathematics. Another way is for the instructor to write the mathematics on paper, scan it, and then display the captured graphics file. One good feature of these online communication products is that the online sessions can be saved. Students unable to log in during the session can gain the full benefit of it later. Faculty can refer students to such sessions.
We have presented a brief summary that merely touches upon many facets of online mathematics course development. However, assuming that the reader is already an accomplished mathematics instructor and wishes to continue on to the next steps in online course creation, we offer a few tips from our personal experiences. The overarching consideration is this: The software tools you select will have a direct bearing on the quality of your results. In addition we cite six principles for online course development:
It is not necessary to fully embrace the new technologies in your instruction, but many new tools are available that can supplement and improve even your very best teaching efforts. As a final remark, these methods and ideas will help us through this, the first phase of online education. The next true wave of innovation will come from those that have actually learned mathematics from online sources.
1. Allen, G. Donald, What do we do until MathML? In Math/Science Online Newsletter, Fall 2000. Online, www.math.tamu.edu/ms-online/.
2. Allen, G. Donald, Online calculus, in Using Information Technology in Mathematics Education, D. James Tooke, Norma Henderson, eds., Haworth Press, 2001.
3. Allen, G. Donald, Online calculus, the course and survey results, Computers in the Schools, 17 (2001),17-30.
4. Allen, G. Donald, Online choices for online courses, in Proceedings of the 13th ICTCM Conference, Addison-Wesley, 2002, 11-16.
5. Allen, G. Donald and Michael Pliant, The distance education degree program for the master of mathematics with a teaching option at Texas A&M University, to appear in the Proceedings of the AACE Conference: SITE 2001—Society for Information Technology and Teacher Education International Conference, Orlando, Florida; March 5-10, 2001.
6. Allen, G. Donald and Michael Pilant, The distance education degree program for the master of mathematics with a teaching option at Texas A&M University, to appear in the Proceedings of the AACE Conference: SITE 2001—Society for Information Technology and Teacher Education International Conference, Orlando, Florida; March 5-10, 2001.
7. Allen, G. Donald and Michael Pilant, Michael, Creating a quality online masters program, Computers in the Schools, to appear.
8. Allen, G. D., M. Stecher, and P. Yasskin, The Web-based mathematics course, a survey of the required features for an on-line math course and experiences in teaching one, Syllabus Magazine (Nov/Dec 1998).
9. Allen, G. D., M. Stecher, and P. Yasskin, WebCalC I, a description of the WebCalC project, its history and features, in Proceedings of the Eleventh ICTCM Conference, Addison-Wesley, 1998.
10. Bogley, John A., William A, Dorbolo, Robert O. Robson, Robert and John A. Sechrest, New pedagogies and tools for Web based calculus, Proceedings of the AACE WebNet96 conference, 1996 Online, iq.orst.edu/papers/WebNet96
11. Bogley, William A., Jon Dorbolo, Robert O. Robson, and John A. Sechrest, Pedagogic innovation in Web-based instruction, Proceeding of the Ninth ICTCM Conference, Addison-Wesley (1998), 421-425. Online: iq.orst.edu/papers/ICTCM96.html
12. Goosens, Michel and Sebastian Rahtz, The LaTeX Web Companion, Addison-Wesley, 1999.
13. Hall, R. J., M. S. Pilant, M. S. and R. A. Strader, The impact of Web-based instruction on performance in an applied statistics course. In the Proceedings of the International Conference on Mathematics/Science Education and Technology (M/SET 99), San Antonio, Texas, March 1999.
14. Lynch, P. J. and Sarah Horton, Web Style Guide. Yale University Press, 1999.
15. Tharp, Marcia, Review of LiveMath Maker 3.0, College Mathematics Journal, xx (2001), xx-xx.
16. Wilkin, Jon, Review of Scientific Notebook, College Mathematics Journal, 29 (1998) #1, 62-65.