A Balancing Act for Distance Education: Mathematics

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A Balancing Act for Distance Education: Mathematics

Irfan Altas & Ken Eustace
School of Information Studies
Charles Sturt University
Wagga Wagga, NSW 2678
Australia
ialtas@csu.edu.au
keustace@csu.edu.au

Abstract
This paper will illustrate the use of on-line hypermedia on the World Wide Web in the teaching of university and high school mathematics. Charles Sturt University (CSU) is a leading distance education provider in Australia where educators have to develop techniques to balancing distance teaching and on-campus teaching methods. In the past, teaching mathematics by both internal and external modes has caused problems in the presentation of dynamic mathematical concepts, to both groups of learners. The new computing technologies of hypermedia and the World Wide Web (WWW), in conjunction with the use of self-instruction learning methods, computer software and new methods in mathematics teaching, can be employed to arrest any imbalance with dynamic teaching methods. We examine the solution of inequalities amongst various calculus concepts to demonstrate the merits of a WWW site and discuss the use of this medium to balance distance and on-campus teaching. The use of the on-line mathematics teaching methods with WWW can not only balance the distance education and on-campus approach, but also provide a link between traditional classroom mathematics and the computerised classroom that makes use of the state of the art software such as Maple or Graphic Calculus. However, the use of technology in learning is often controversial and not favoured by all students. We also discuss these pedagogical issues and explain our approach to these issues in our teaching environment.
Keywords
balancing pedagogic methods, computer graphics, deeper learning, distance education, dynamic teaching, mathematics laboratory, on-line resources, solving inequalities, student perception, World-Wide Web.

1. Introduction
Many educators embrace the idea that students today learn better using the new technologies that can provide a dynamic representation of information and concepts. In teaching mathematics, considerable effort continues to push forward the development of user-friendly systems, important to a deeper understanding of mathematical concepts in undergraduate students. The authors are seeking to extend upon the current work in mathematics teaching by developing hypermedia learning tools [LOO93] to assist in teaching first year mathematics at CSU. As a prototype, the topic called rates of change has been chosen to be developed first and will be available and tested in many forms: CD-ROM, WWW, Disk. The WWW version of the package will be free to access. The development of the project in full will depend upon the results of testing the prototype and funding success.
A combination of mathematics, computer graphics, and communications technology is required [BRU93] to give students a mathematical and technological experience that is a more dynamic way of learning. Students use software tools such as Maple or Mathematica, to test mathematical ideas such as the new methods in calculus and at the same time develop an awareness of the wider learning arena provided by access to the Internet. Hypertext links are used to learn mathematics visually, by experiencing not only the worked examples of the WWW tutorials, but also by motivating the student to experiment with the Maple software in the computer laboratory. The student can use Maple to plot functions and describe mathematical solutions in their own words. Many students feel that traditional mathematics subjects lack application to real situations that are meaningful and relevant to their own ideas. Indeed the on-line teaching of mathematics can inscribe mathematical concepts as part of the student's own experience, improving their overall understanding, logical thought processes, communication skills, confidence and the enjoyment in learning mathematics.
Among the numerous on-line mathematics teaching projects, the Harvard University, Core Calculus Consortium [HUG95] has already made a significant contribution in this area. Project CALC (Calculus as a Laboratory Course) [MOR95] is another project at Duke University that has developed a new curriculum for calculus instruction based on laboratory experiments, discovery learning, real-world applications, cooperative learning, writing, revision of writing, and high expectations of students. The Calculus & Mathematica Distance Education Program (C&M DEP) at the University of Illinois at Urbana-Champaign and The Ohio State University [UHL95], has some features which are closely related to this initiative at CSU. The C&M DEP runs a wide area network connecting rural and inner city high school students to the university and sends top university undergraduates to the schools - forming a collaborative community devoted to mathematics learning. The Calculus & Mathematica Distance Education Team consists of a homogeneous group of high school students, high school teachers, undergraduate students and mathematics faculty members.
In Australia, several National Mathematics Teaching Projects [NAT95] have been under development by various institutions. Among those projects "Alternate Solution Paths for Motivation in Computerised Mathematics" at Griffith University and "Developing an Assessment Program Incorporating Graphics Calculators" at Murdoch University can be stated. There is also a Mathematics Archives in the Internet which contains a modest collection of materials used in laboratories to teach Mathematics. The materials in this archive are available by anonymous ftp and WWW at archives.math.utk.edu.
In Section 2, the technology medium of CSU is presented. Pedagogical aspects of the technology in teaching is discussed in Section 3. In Section 4, we discuss our approach to teaching in the technology environment by demonstrating inequalities concept from calculus. Conclusions are presented in Section 5.

2. The technology medium for on-line education
CSU Remote Dial-In Access Policy is developing equality of network access for all students. The Serial Line Internet Protocol (SLIP) or Point-to-Point Protocol (PPP) both use a modem and a telephone line connection as a serial line extension to a local node, and is the technology medium at the moment for a richer connection to the Internet at 14.4kbs . SLIP access is provided for Windows users by the Trumpet Winsock program and by MacTCP and InterSLIP Control Panels for Macintosh users. Staff and students at remote locations can login to a SLIP server in order to gain access to the host network. Each SLIP user is assigned an Internet Protocol (IP) number or address for the current session by the SLIP server. Each Internet node has to have a unique IP number to run the various tools such as Telnet, ftp and WWW browsers such as Mosaic or Netscape.
A fast modem and SLIP or PPP Access allow connections to our campus-wide network over standard telephone lines. Once connected, students gain access to our WWW file server, on which assignments and lessons are transferred and stored. Assignment submission for binary files can be made using Pegasus Mail software and our POP mail server. Pegasus Mail makes electronic mail easy to send and receive, once configured. Students and lecturers each receive an individual e-mail account on our network servers, which provides regular communication to all Distance Education coordinators and others with an Internet e-mail address.
Maple is the main computational software, where calculations are entered, animations of mathematical models and three-dimensional graphics are created to become part of a student's lesson.

3. Pedagogical advantages of technology in learning mathematics
Kaput (1992) [KAP92] has suggested that the mathematical thinking ability to recognise translations from one representation of a function to another, can be assisted by the use of computers. The use of Maple or Mathematica in combination with the WEB, can develop new strategies and a deeper understanding of the calculus concept. Such software tools can produce a very quick translation between representations when compared to a slower manual method. Hall and Elliott (1993) [HAL93] combined the use of a computer-mediated environment with a metacognitive teaching approach and found that the gains in mathematical achievement are greater in the computer-mediated environments, even when the more traditional learning environments are metacognitively enriched.
The use of technology in learning is often controversial and not favoured by all students and has problems associated with gender, equity and ethics which must be addressed by the project team. A 1993 study by Helen Geissinger (1994) [GEI94] showed that there is not only an imbalance in the level of technology used by distance education students, but also a gender bias towards male students having access to a higher level of computer equipment. The technology strategy at Charles Sturt University is involving staff in the planning process over the next five years with Internet access and the World Wide Web service at the forefront of plans suggested by the CSU Teaching Technology Implementation Committee (TTIC). The second interim report of the TTIC [TTI95] suggests that:
technology could significantly enhance the effectiveness and quality of teaching and learning by providing more independent learning resources and thus more flexible learning options for its diverse student body; and maximising the efficiency and effectiveness of its communication with and between students;
Eustace (1994) [EUS94] states TWO issues in the use of on-line educational resources in university teaching:

Merging distance teaching and on-campus teaching methods requires a delicate balance of computer technology and user interface design, in which the learning medium is not forced upon those who prefer other learning styles, but is made attractive to the learner.[KER92]
Educators need a high level of computer literacy and skills to provide and maintain a domain of interactive CD-ROM and linked Internet services such as the WWW and a multimedia MOO.
The final system will be developed using the combined mathematics and computing skills of the project team. Bates (1994) [BAT94] reveals how the amount of time and effort required by educators using networked multimedia, demands the project team approach. The World Wide Web service will be the backbone of a mathematics learning cycle which will encourage and accept students responses via forms and links to other services at Charles Sturt University such as AussieMOO, [FEL95] and the WWW Bulletin Board [TSA95].
The Computer - based calculus project group [BOG95] at Old Dominion University conducted an evaluation of their Mathcad based mathematics teaching medium with 54 students. They concluded from the responses that:

students found the computer-based calculus interesting and enjoyable;
the windows version of Mathcad was easy to use;
the use of Mathcad improved the learning of calculus;
computer labs improve students' perception of their calculus skills.
Such comments reveal the educational advantage and reward for all the time and effort needed to provide and maintain a computer-assisted learning environment, [ROM93], which will now be further enhanced by the WEB. A similar evaluation of our system will be used by the project team.

4. Dynamic techniques in teaching mathematics: an inequality problem
Visualisation is important for teaching mathematics for all students but those students studying by distance learning methods pose a question of balance. One strategy for the mathematics lecturer is to provide a selection of graphical solutions to appropriate problems, on the network. Distance education students will then have the opportunity to study those solutions without losing advantage to their face-to-face colleagues. Without having a uniform software and resources environment for all students, this exercise will not be of much benefit. The main access method will involve the use of WWW browser software such as Mosaic or Netscape. Let us illustrate the merits of building such a WWW server site with the following mathematical concept.
Let us consider the various ways of the explaining the following inequality to students.?

(x-1) (x+2) (x+1) < 0 (1)

One approach to solve such a problem is to make the following table:
Table: 1 Solution of Equation (1) by using tabular form.
The region where Equation (1) is negative can be read from the last row of the table as ( -, -2) ( -1, 1).
Another approach is to treat Equation (1) as a function, f(x)= (x-1) (x+2) (x+1) and sketch this function. Once you consider this inequality as a function the question is reduced to:

In which region is this function negative?
Students can obtain the same solution domain as the one in the first approach by examining the following Figure 1.
Figure : 1 The functional representation of Equation (1) as f(x)= (x-1) (x+2) (x+1).
The second approach builds up a better understanding based on the use of the technology medium available. Students could make some cognitive connections by using the graph of the function. They can easily sketch such a graph by employing a software package such as Maple and Mathematica. In fact, colour graphing and animation can be employed to distinguish different properties of the function. If we re-sketch Figure 1 as in Figure 2 by employing a different colour for the regions where the function is negative, students can have the same concept further developed. In a computer laboratory environment, this region can be highlighted by flashing out the graph of the function in the negative regions.
Figure 2.  The solution region of Equation (1): f(x)= (x-1) (x+2) (x+1)
These graphic examples in figures 1 and 2 hopefully motivate students towards a deeper understanding better than the first approach using the table method. This approach may provide students with an intuitive sense of what is going on. Building a mathematics teaching platform with these properties for the inequalities concept in a WWW site will create an equal opportunity to our external and internal students. As Charles Sturt University provides the access to its computing facilities freely for both internal and external students, this WEB site will provide the same teaching environment for students without any discrimination due to study mode. Students will voluntarily study this WEB site at their convenience rather than at a compulsory base. While we believe that most students may be motivated and benefit more having such a WEB site, some other students may feel that using such a WEB site will burden them with extra work which is time consuming.
When we generalise the inequalities to two the variable case, it is not feasible to employ the tabular approach illustrated in Table 1. However, it is not a hard task to sketch a function with two variables with today's technology. Most software packages provide an opportunity to examine the graph from different perspectives by rotating and translating axes. In fact, such a property can be built as an animation program which automatically presents the view of the graph from several different perspectives. Using colour graphing, as in the one variable case, will further enhance an understanding of the graph of the function.
For example, let us identify the region defined by
xy > 1 for x and y > 0 (2)

First of all, the domain restriction, x and y > 0, can be identified by simply examining the coordinate axes and using colouring approach, Figure 3.
Figure 3. Identifying domain restriction for Equation (2).
The next step could be the examining of the inequality, xy - 1 > 0, as a graph of y = 1/x, Figure 4, and identify the regions satisfying y > 1/x. Then, by combining Figure 3 and Figure 4 we can explain why the lower shaded region will be dropped from the solution by colouring or animation. Some software such as the Graphic-Calculator can be easily employed for this purpose.
Figure 4. Regions identifying xy - 1 > 0 using Graphic Calculator.
Perhaps, the last step, as a complementary step to Figure 4, is to sketch the inequality, xy - 1 > 0, as a graph of the function, f(x, y) = xy - 1, in Figure 5. Then, the same region can be identified from Figure 5 by referring to Figure 4.
Figure 5. The graph of (x, y) = xy - 1 using Graphic Calculator
Obviously, it will not be an easy task to explain such a detailed solution to this problem for both external and internal students without having help from technology such as establishing a hypermedia WWW site for mathematics.

5. Conclusions
We believe that there should be more contact between the university and high school maths teachers if the use of technology in teaching is to be a success. If high school teachers prepare the students during their education for the technology environment, then lecturers at the university level could employ these environments more efficiently in their teaching. We should also stress here that the introduction of a WWW site/mathematics laboratory with a reformed calculus textbook such as the Harvard Consortium text rather than a traditional textbook, is more beneficial in this approach.
We also believe that the reformed calculus approach in a computer laboratory environment should be designed as a coherent set of materials to use with students who want to learn in a context that encourages a deeper level of thinking, reasoning, reading, talking, writing and problem solving even though this demands more work from students and lecturers.
We should also stress here that using the technology does not imply giving up some traditional teaching/learning tools which have already proved effective over the years. As David Glertner [GLE94], who is a computer scientist at Yale University, pointed out in an educational discussion forum, that:
While we bemoan the decline of literacy, computers discount words in favour of pictures and pictures in favour of video. While we fret about the decreasing cogency of public debate, computers dismiss linear argument and promote fast, shallow romps across the information landscape.
Hence, as we are enter the computer age in education, we must be aware of the side effects. In order to get best out of technology in teaching/learning it should be complementary to the skills of students:
Solve the problem with technology, confirm the solution with your skills; solve the problem with your skills, confirm the solution by using technology.
The use of the on-line mathematics teaching methods with WWW can not only balance the distance education and on-campus approach, but also provide a link between traditional classroom mathematics and the computerised classroom of Maple or Mathematica. This approach may open up an exploration for which one did not have the motivation when covered without the technological tools.

References

[BAT94]
Bates, A. W. (1994). Educational Multi-Media in a Networked Society. Proceedings of ED-MEDIA 94 - World Conference on Educational Multimedia and Hypermedia, Vancouver, Canada; June 25-30, 1994.

[BOG95]
Bogacki, P. (1995). Computer-based Calculus Project Group at Old Dominion University. [Online]. Available WWW: http://claymore.math.odu.edu/~web/cbii/calculus.html (e-mail: bogacki@math.odu.edu).

[BRU93]
Bruce, B. C., Peyton, J. K., and Batson, T. (eds).(1993). Networked-based classrooms-Promises and realities. Cambridge University Press, New York, 1993.

[EUS94]
Eustace, K. (1994) Interactive Education: Distributed hypermedia and the networking of students on the World-Wide-Web. CEGV Conference: The Information Superhighway: The implications for Education, Geelong, Australia, September 19-20, 1994.

[FEL95]
Fellows, G. (1995). AussieMOO Home Page. [Online]. Available WWW: http://silo.riv.csu.edu.au/ (e-mail: gfellows@csu.edu.au).

[GEI94]
Geissinger, H. (1994) CSU Distance Students' Attitude toward Computer Use. Occasional Papers in Open and Distance Learning. No.15, Charles Sturt University.

[GLE94]
Glertner, D. (1994, November 9). EDUPAGE. [Online]. Available e-mail: listproc@educom.edu.

[HAL93]
Hall, N. and Elliott, A. (1993). A metacognitive approach to teaching mathematics through computer supported environments, 343-349. In Contexts in Mathematics Education, Proceedings of the 16th Annual Conference of the Mathematics Education Research Group of Australasia, (MERGA), Brisbane.

[HUG95]
Hughes-Hallett, D. and Gleason, A. M. (1995). Harvard University, Core Calculus Consortium [Online]. Available WWW: http://archives.math.utk.edu/calculus/crol.html (e-mail: calculus@math.harvard.edu).

[KAP92]
Kaput, J. (1992). Technology and Mathematics Education. In D. A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, 515-556, New York, Macmillan.

[KER92]
Kearsley, G. (1992). Multimedia Projects: Issues and Applications. Journal of Educational Multimedia and Hypermedia. Vol.1, 103-110.

[LOO93]
Looms, P. (1993). Interactive Multimedia in Education. In Latchem, C, Williamson, J and Henderson-Lancett, L. (Eds) Interactive Multimedia. Kogan Page: London.

[MOR95]
Moore, L. and Smith, D. (1995). Project CALC at Duke University (Calculus As a Laboratory Course). [Online]. Available WWW: http://archives.math.utk.edu/calculus/crol.html; Available gopher: poincare.math.upenn.edu Directory: /00/InteractiveMathematicsTextProject/ File: AboutProjectCALC

[NAT95]
National Teaching Development Grant Projects. (1995). Improving University Teaching 1995, , Union Offset Co. Pty Ltd, Canberra.

[ROM93]
Romiszowski, A. F.(1993). Developing interactive multimedia courseware and networks: Some current issues in interactive multimedia. In Latchem, C, Williamson, J and Henderson-Lancett, L. (Eds.) Interactive Multimedia. Kogan Page: London.

[TTI95]
Teaching Technology Implementation Committee. (1995). Second Interim Report: A Teaching Strategy for Charles Sturt University, April 1995. [Online]. Available WWW: /

[TSA95]
Tsang, P. and Wright, M. (1995). Developing a WWW Multimedia Bulletin Board. [Online]. Available e-mail: ptsang@csu.edu.au.

[UHL95]
Uhl, J. and Woods, D. (1995) The Calculus & Mathematica Distance Education Program (C&M DEP). [Online]. Available WWW: http://www-cm.math.uiuc.edu/dep/

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