Figure 29. 1. Webct modules at the University of Pretoria by year.

The Handbook of Blended Learning

TABLE 29.1. NUMBER OF WEBCT MODULES CREATED BY FACULTY DEPARTMENTS.

A key factor in the successful adoption of Web-supported learning at the Uni­versity of Pretoria is the support offered to lecturers by the Department of TLEI. Instructional designers are available to design and develop modules in WebGT or in combinations of IGTs. Together with the department's education consultants and faculty, instructional designers determine the most appropriate combination of instructional methodologies to be used within the flexible learning environ­ment. The team is experienced in developing different types of multimedia, including tutorials, simulations, case studies, and collections of visually rich resources. Several campuswide projects are currently in development, including vitally important e-portfolio and electronic assessment systems.

TLEI also provides training to academic staff to facilitate student success in the online learning environment. The training focuses on how to effectively use WebCT (technology training) as well as the pedagogy of teaching online. Lec­turers may choose to attend only basic training to facilitate their courses, or they can choose to follow the advanced designer training in order to take full control of the development and maintenance of their online courses.

At the university, the Web is widely used in combination with contact sessions; in effect, blended learning is the norm. The teaching model naturally varies to accommodate student needs and program requirements. Various new programs at the postgraduate level have been designed with the possibilities of the Web in mind. Examples of higlily successful programs, with good learning facilitation and student participation, are the master's degree in early childhood intervention (Alant, Dada, Fresen, & Marx, 2002) and the master's of philosophy in wildlife management. Our experience has been that new programs provide the possibility

Blended Learning in Undergraduate Mathematics at the University of Pretoria 403

to design appropriate teaching and learning strategies for the Web from the start, whereas the adaptation of existing face-to-face activities to the Web environment is usually more challenging.

One of the concerns regarding ongoing innovation is that the higher educa­tion sector in South Africa has been restructured. The new funding formula has had a detrimental effect on the University of Pretoria, in particular, which may affect the availability of resources to support and fund educational innovation in the future.

In the sections that follow, we use an undergraduate mathematics course as an example of how the Web can be integrated into the teaching model. We reflect on this model through the all-encompassing lens that blended learning provides.

Online Teaching and Learning in Undergraduate Mathematics

In mathematics, the century-long practice of chalk-and-talk teaching has been resilient and resistant to change until recently. Mathematics is a conceptual sub­ject, and common opinion is that face-to-face contact with demonstrated exposi­tion is necessary for conveying these concepts. The technology revolution experiences during the past two decades of the previous century offered the first real challenge to this tradition. Resistance tentatively began to crumble as the advantages of a more visual and experimental approach, with increased ease of calculations, became widely apparent. Traditionally presented mathematics was reformed to incorporate the use of technology for a more hands-on and visual approach (Hughes-Hallett et al., 1994). This revised approach was initially received with varying degrees of enthusiasm but gradually infiltrated almost every aspect of mathematics. In so doing, the first step to a more blended approach was taken, in this case, blending educational technology (such as CDs, computer-assisted instruction software, hypermedia, and multimedia) into the traditional approach. Having barely adapted to change brought about by this onslaught, the second wave of change, the Internet, emerged to offer even more hands-on, exploratory learning opportunities for mathematics education.

Again change came slowly, and mathematics courses have certainly not been at the pioneering front of Internet course development. This slow pace of acceptance and curriculum change can perhaps be blamed on the strong role of tradition in teaching mathematics. At the same time, it might be more accu­rately pinned on the problems surrounding symbol presentation in HTML, the language commonly used for Internet presentation. Yet mathematics could not withstand the onslaught of opportunity made possible by the Internet either. In fact, in recent years, mathematics courses on the Internet are no longer a rarity,

The Handbook of Blended Learning

whether these are fully or partially Web based. Well-designed mathematics material on the Web offers a multitude of visual facilities and exploratory opportunities, often animated for enhanced effect. Bookbinder (2000) argues, " Enhancing the mathematics curriculum with Web-based technology takes time and effort, but the effort is well worth it" (p. 8).

The mathematics teaching fraternity has taken notice of the Internet, but in many cases fear of the unknown and skepticism still manifests itself in a reluctance to venture into virtual worlds in what was a highly traditionally taught subject matter area. Concerns on how much of material should go on the Web, how to get students to work regularly, and whether it is possible to learn mathematics out­side the chalk-and-talk paradigm are common. Web-based courses have by no means replaced traditionally presented courses and possibly never will, but the Internet has had a definite impact on mathematics teaching. There is no single answer to the question as to how one should harness the Internet in mathemat­ics, but, as noted below, it may simply be a matter of blending. We describe one particular case at the University of Pretoria where blended learning in mathe­matics has been implemented for the past four years.

Historical Background

When, at the beginning of 2000, teachers at the Mathematics Department took the first steps toward designing a calculus course on the Web, a wise decision was made to run the course for a trial semester in parallel with the conventional lecturer-driven course, giving online access to all the mainline calculus first-year students (all one thousand of them). Calculus is the bread-and-butter mathe­matics course taken by all first-year mathematics students. There were definite hiccups to sort out, and this period of grace was invaluable (Engelbrecht & Harding, 2001a). It gave us the necessary experience and confidence to run a blended calculus course in the follow-up semester, mainly Web based but with paper and face-to-face components.

The blended calculus course has a specific target market. Students typically enroll for their first calculus course at the start of the first semester. If successful, they continue with a follow-up course. If not, they can repeat the same course in the second semester with the benefit of not having to skip a semester. This course is then called an anti-semester course. The majority of students enter the anti-semester course as repeaters and are reluctant to attend class, since, from their perspective, they have heard it all before. Given that situation, it seemed obvious that all anti-semester courses should go on the Web. It was an obvious and relatively easy decision to use WebCT to supplement this course since it was the platform implemented at the University of Pretoria and had excellent support.

Blended Learning in Undergraduate Mathematics at the University of Pretoria 405

Since 2000, we have continued offering anti-semester courses in calculus topics using the blended calculus teaching-learning model described later in this section. This model has been adapted somewhat and is in the process of undergoing a major revamp to enrich the course. Most courses in the department (more than twenty to date) have a WebCT Web site with various degrees of blending. On the lower end of the scale are Web sites that serve only as notice boards, with other courses making increasing use of one or more of the many WebCT tools. The model described is on the upper end of the scale.

Course Structure

This section provides details about how we structured our approach to the math­ematics course.

Didactic Approach. There are few detailed examples of mathematics courses conducted over the Web with the continuous guidance and support from the lecturers. As noted by Crowe and Zand (2000) in an extensive survey of existing Internet activities and resources in undergraduate mathematics teaching, " The balance of costs at present is heavily geared towards production of material, rather than maintaining it and supporting its presentation to students" (p. 147). Engelbrecht and Harding (2004, 2005a) report on the full spectrum of under­graduate mathematics courses taught online.

Since we firmly believe that most of our students are not academically mature enough to follow a course that does not structure their learning in some way, we decided to develop a course that falls within the slender category of courses conducted over the Web with continuous guidance and support from the lectur­ers. For example, our approach uses the textbook for the actual subject content through which the students are guided on a day-by-day basis.

Running the Course. The dynamical day-by-day running of the course is done through the Calendar facility or diary, the entry point for the course. Here, we list the study unit of the day, linked to the Study Guide (supplying detailed learning objectives and references to the applicable section in the textbook) and the corresponding short Lecture Notes (supplying a shorter version of what one would normally do in a live class but still acting as facilitator between the Study Guide and the textbook). A number of practice problems are also listed daily. An­nouncements and administrative issues complete the daily communication.

Assessment. We also follow a blended approach for course assessment, employ­ing both online and paper components. The weekly quiz is an online activity completed individually by students on the Web with rapid feedback. Although

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there is no security check, quiz marks do contribute to the semester aggregate, and students soon get to use it as a formative tool and as a fair judge of their progres-The assignments and projects component is a paper activity in which students hand in four hard copy assignments and one project during the semester, completed in groups. Assignments mainly consist of problems selected from the textbook. In contrast, projects consist of application problems requiring the use of mathematical software such as Matlab or Maple.

For the two semester tests as well as the final examination, online and paper assessment are blended. Each of these assessment activities consists of an online section (for which the QUIZ facility of WebCT is used) as well as a paper section, carrying equal weight. The online section is done in a computer lab under supervision, posing no security risk.

Blending assessment modes uses the best of two worlds. Paper questions assea skills such as formulation, exposition, sketching, and the logical development of mathematical thought. Online questions are particularly suited to testing conceptual understanding and visualization. Setting online questions is challenging and also refreshing because it requires the lecturer to think differently. From the lecturer'-side, there is the added benefit of reduced grading and the additional diagnostic features of the online assessment. We are convinced of the possibility of setting ques­tions of a high standard on the Web (Engelbrecht & Harding, 2003, 2004).

Other Components and Activities in the Course. Our experience is that when following a blended approach within an online environment, a premium should be placed on cooperative learning. For this reason, we let the class (normally about 150 students) divide themselves into groups of two to four members each. Group activities then include assignments and projects (Engelbrecht & Harding, 2002). On submitting the finished tasks, students have to make a small declaration that all group members contributed (more or less) equally.

Communication also follows the blended approach. Most communication happens online (via WebCT's Discussion Forum and the E-Mail faculty) with face-to-face office consultation possible but not the rule. A face-to-face hour-long live weekly meeting provides an opportunity for questions and an extra example or two. Attendance is voluntary.

On the technical side, for material containing mathematical symbols, we use Scientific Workplace (a front end for LATEX, the software tool generally used for typing mathematics) and publish this in pdf format (Hutchinson, 1999).

Feedback and Performance

The most significant feedback result, obtained from student questionnaires, was that in the trial run leading up to the first semester of full implementation

Blended Learning in Undergraduate Mathematics at the University of Pretoria 407

(when the Web was optional), only 6 percent of students indicated that they pre­ferred the Web course to a more traditional classroom-based model. However, by the end of the first presentation of the Web-based course, this figure increased to an impressive 58 percent. Perhaps students are stuck in the traditional class­room paradigm mentality, but when they are forced to experience a different way of learning, they find that not only can they cope with it, the majority of students actually prefer it (Engelbrecht & Harding, 2001b).

On the importance of the different activities, students rated the assignments highest, with the projects a close second. Somewhat surprisingly, the live weekly meetings were only moderately successful and were not particularly well attended. This may be because most of the students were repeating the course and did not feel a need to attend a scheduled event a second time.

The pass rates of the Web-presented courses from 2000 to 2003 were consistently higher than those of the preceding four years of live classroom instruction. The av­erage pass rate over the period 2000-2003, when the course was presented over the Web, was nearly 78 percent. Over the period 1996—1999, when the course was still presented face-to-face, the average pass rate was just 64 percent. A contributing factor for this phenomenon could be that students performed particularly well in the online assignments because they can consult and work in groups.

The percentage of student distinctions in the blended version of the courses, however, was somewhat lower than in previous years. The average rate for the four years prior to 2000 was higher than the average percentage for the four years of teaching over the Web: 4.8 percent compared to 3.1 percent. An explanation could be that it is not easy to master the finer points of a conceptual subject such as math­ematics when direct interaction and modeling from an instructor are limited.

A concern often expressed by those who are unfamiliar with Web courses is that the lecturer will be flooded by e-mail enquiries. Our experience does not support this at all. In the example course (group of about 150 students), over a fifteen-week period we received 300 messages in the Discussion Forum and 150 via e-mail, on average about 5 messages per day. It required no more of our time than a typical face-to-face group of students. In addition, we had the benefit of answering all these postings first thing in the morning.

Scope of Blended Learning in Mathematics

Having described one particular model of blended learning in mathematics, we now widen our horizons to look at the scope of blended learning in mathematics as prac­ticed worldwide. We develop a structure to visualize the extent of blending in math­ematics courses. (For a more detailed survey and classification of existing Web sites in undergraduate mathematics, see Engelbrecht & Harding, 2005a, 2005b.)

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We focus on mathematics courses that have Web-based presentation as at least one of its components. The Web component could be an add-on to a mainly lecture-based course, or it could be the main component of the course with little or no face-to-face contact. The latter includes courses that run within a virtual learning environment such as WebCT or within a created and customized structure that facilitates content conveyance, communication, assessment, and other online components.

Taking reconnaissance of existing examples of blended learning is not an easy task. Part of the difficulty is the large number of available online courses world­wide. A second, and more problematic, issue is that many universities have strict security measures on their academic Web sites, reserving access for registered students. As a result, external access is often impossible.

When comparing two blended courses, one should realize that both might be lacking in certain, not necessarily the same, aspects and exceed again in other aspects. In an attempt to visualize at one glance what the scope and extent of blend­ing is and to indicate the associated strengths and weaknesses, a radar chart is pro­posed (see Figure 29.2). Six radials are identified in this chart, each with a question to quantify a measure.

Dynamics and Access: What Is the Frequency of Access Necessary for Success in the Course?

1—Once per term 2—Once per month 3—Once per week

FIGURE 29.2. RADAR CHART FOR THE UNIVERSITY OF PRETORIA COURSES.

Dynamics

Independence

Assessment 5^

Communication

Content

Richness

Blended Learning in Undergraduate Mathematics at the University of Pretoria 409

4—Two to three times a week 5—Daily

Assessment: How Much of the Assessment Is Done Online?

1—Little

2—Almost half of it

3—More than half of it

4—Most of it

5—All of it

Communication: How Much of the Communication Happens Online?

1—Litde

2—Almost half of it

3—More than half of it

4—Most of it

5—All of it

Content: How Much of the Course Content Is Available Online?

1 each for book, course information, course administration, lecture notes, and study objectives, with a maximum score of 5

Richness: How Many Enriching Components Does the Online Part of the Course Have?

1 each for a computer algebra system, graphics, Java applets, slide presenta­tions, video clips, and sound clips; in effect, more than text communication, with a maximum score of 5 components

Independence: How Independent Is Success in the Course from Face-to-Face Contact?

1—Fully contact lecture and tutorial driven; Web site an add-on

2—Contact lectures but Web-based tutorials or assessment

3—Limited regular contact

4—Sporadic contact

5—No face-to-face contact

The area of the radial diagram gives an indication of the extent of Internet use. The larger the area, the bigger the Internet component, and the smaller the

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area, the bigger the face-to-face component. A convex shape, partially filling the chart area, points to a well-blended course.

The first three radials—dynamics, assessment, and communication—could be grouped under a heading interaction. In the radial diagram, they are to the top. A top-heavy radial diagram indicates more interaction over the Web. The sec­ond three radials—content, richness, and independence—could be grouped under the heading material. Radial diagrams of courses with content provided on the Web will therefore be heavier toward the bottom.

As an illustration of this model, we look at several examples. It may be mentioned that in doing the grading for the examples below, the rating was done on what was available from the Web site and approved by the course designers. The three examples illustrate various degrees of blending.

Calculus Course

The calculus course (University of Pretoria) described in the previous section serves as our first case study. The course representation on a radar chart ap­pears in Figure 29.2.

Although this course does not have any formal lectures, it does not run fully online. Students need to access the Web site at least two or three times per week since more than half of the assessment is done online. However, communication happens mostly online but also during contact sessions. Success in the course is dependent on sporadic face-to-face contact. As can be seen from the radar chart, this course lacks richness and could be supplemented by additional multimedia material.

NetMath Project

The second example is the NetMath project of the University of Illinois Urbana-Champaign, which offers various online courses conducted via Mathematica note­books and assisted by a NetMath support team. NetMath grew out of the Calculus& Mathematica project of the University of Illinois Urbana-Champaign. This course made use of the computer algebra package Mathematica and originally was supported by conventional lectures. Due to student feedback, the lec­tures were eventually dropped, and the course became an on-campus distance learning course. As a result of extensive interest from other institutions, the NetMath consortium now consists of the University of Illinois Urbana-Champaign, Ohio State University, the University of Pittsburgh, the University of Iowa, and Harvard University. The courses are also used at a number of other universities in the United States.

Blended Learning in Undergraduate Mathematics at the University of Pretoria 411

FIGURE 29.3. RADAR CHART FOR THE NETMATH COURSES.

Dynamics

Independence

Assessment 5.

Richness

NetMath is a distance education program with communication between student and the NetMath team using e-mail, telephone, and online discussions. Students need to submit a weekly report to NetMath on their progress and prob­lems. Content delivery is with Mathematica notebooks, and cooperative learning is encouraged. Students have a degree of electronic support: courseware and assignments are placed on the course Web site, and a human mentor is available for help by telephone or e-mail. Assignment problems are submitted online and marked by the NetMath team.

As indicated, this program has attracted great interest and has grown to include differential equations, linear algebra, and probability theory. The radar chart representation appears in Figure 29.3. The chart shows that this course is conducted almost exclusively online and addresses both the content and interac­tion radials but is slightly lacking in richness.

Notice Board

The third case explored here is a " notice board" site that mainly contains administrative information such as syllabi, announcements, handouts, reference to homework problems, and past papers. This is a popular way of starting out on the Web, and many courses have grown from this humble beginning. The example we offer here is " Introduction to Calculus Fall 2003" at Stony Brook University in New York, a lecture-based course with a supplemental Web site. Figure 29.4 gives the radar diagram for this course. In this case, the radar chart clearly shows limited blending. As indicated earlier, such courses are fairly common in mathematics

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