School of Mathematics and Statistics

Curtin University of Technology

- The main challenges facing a tertiary mathematics department include: how to cater for students with varying backgrounds and how to cater for the changing mathematical competencies required by employer groups.
- identification of deficiencies in the mathematical background of new students through development and use of appropriate diagnostic tests;
- development of appropriate supplementary material to enhance course content and related applications;
- utilisation of technology, including computer generated presentations, software application programs and graphics calculators;
- development of computer mediated learning/self-learning materials;
- utilisation of appropriate videos to enhance course content, presentation and support students' motivation and development of understanding;
- development of teaching and assessment practices that meet the needs of the diverse range of students studying undergraduate mathematics, including the needs due to gender, cultural background, language background, mathematics capabilities, and learning style.

The School of Mathematics and Statistics at Curtin has some 3,500 students enrolled in its 27 First Year Mathematics units. Recently, the School has undertaken a project aimed at developing effective modes of mathematical instruction utilising all available technology. This project focuses on first year mathematics units and the main components are:

In this paper we shall briefly describe the project, providing details on such aspects as: the computer administered diagnostic test that was implemented in 1996; and our experiences with computer learning packages such as CALMAT.

A major task facing a Tertiary Mathematics Department, with respect to Teaching and Learning, is how to cater for students with varying backgrounds. The current trend, which is expected to continue, shows an increasing number of students seeking courses in Science and Engineering without the required mathematical skills. Our analysis of entering students with advanced level TEE mathematics shows that a high percentage are not confident with many of the Year 12 topics (for example, complex numbers) that are further developed in first year units. The analysis of first year 1995 mainstream science and engineering students showed that among students with a TEE aggregate of 340 or less 28% passed, 43% failed and 29% did not complete their first semester mathematics unit. To address this problem, the School introduced two new bridging units in 1996. Further, an alternative unit is available for students without TEE calculus.

The problems encountered in teaching first year mathematics are not new and are not unique to Australia. Universities world wide are utilising computer assisted learning to help overcome the problems and at Curtin we aim to make maximum use of computers where appropriate.

The problems encountered in the teaching of mathematics to first year scientists and engineers are now generally held to be more widespread than in previous years. ... there are acute financial pressures upon departments, ... providing mathematically weak or disadvantaged students with an alternative to textbooks or an expensive private tutor seems to be a much needed addition to the general higher education teaching strategy (Kelly, Maunder & Cheng, August 1994, Maths & Stats).Computer technology can be used both for diagnostics and for supplementary or revision material for a unit. In our project we intend to make effective use of computers in both areas. Computer assisted learning in mathematics is not a new phenomenon. Statements such as "Computer Algebra Systems (CAS) are already efficient teaching aids in mathematics courses" (Calmet, 1987) show that computers have been used in mathematics teaching for many years. Symbolic packages were in fact first used in WA tertiary institutions in the late seventies. Lawson (1995) acknowledged that it had not been conclusively proved that students actually derived any benefit from using computers in learning mathematics although his results showed a significantly better performance for the group of students using computer assisted learning (CAL). In the past the way computers were used was possibly not always appropriate and hence positive results were not achieved.

The School has developed and implemented a number of strategies for assisting the transition from Secondary to Tertiary Mathematics. This is part of a major project developed in 1995. The project aims at developing effective modes of instruction, presentation and assessment of first year mathematics utilising all available technology. The main achievements to date are summarised below.

The task to find or write a suitable diagnostic test was not trivial. Time constraints dictated that an existing package had to be used in 1996. The CTI book Maths & Stats Guide to Software for Teaching was used as a starting point. This book lists over a thousand software packages covering a wide range of applications and gives a brief description of each package along with the cost and a contact address. The package that we chose was MCtest that was developed by Dr S Hibbard at Nottingham, England. The questions in this test of course assumed an English A level background which is different from the Australian TEE. Some of the questions were removed and new ones were added and the test was used in February 1996 by students about to start a degree course in engineering or science.

The test was not entirely satisfactory and our own test has been developed for use in 1997. One of the problems was that we attempted to test and give feedback on too many areas. The areas that were identified as important were: algebra; equations and inequalities; trigonometry; logarithmic and exponential functions; matrices; derivatives; chain rule; integration; integration by substitution; and applications of derivatives. The software randomly selected two questions from each topic to give a total of twenty questions. With only two questions on each topic the diagnostic profile was not appropriate. It indicated remedial, revision or competent which was not necessarily a true reflection of the students' ability.

In the test to be used in 1997 the number of areas has been reduced to seven and the number of questions increased to twenty five. A larger bank of questions has also been created with at least fifteen on each topic and these have been graded as: easy, medium and hard. The diagnostic profile will be more specific and not only tell the students the areas in which they are deficient but also recommend the appropriate units in which to enrol. This will be based on their mathematics TEE subjects and on the diagnostic test. That is, students may be advised to complete a bridging unit before enrolling in the mathematics unit needed for their course.

The CALMAT software is being used in an effort to improve the students' background knowledge. Too frequently we hear students say "I understood all that you did in the lecture but when I tried to do some questions ...". The reason that they cannot complete questions often comes down to the lack of appropriate algebraic skills. It may well be argued that today these skills are becoming less essential as a package such as Maple can do all these tasks but this argument, in our opinion, is not a valid one. It is essential for students to have basic arithmetic skills even though calculators have removed the drudgery of long tedious calculations. Similarly, they need the basic algebraic skills to use hand in hand with any software package.

CALMAT has fifty topics split into twelve different sections. This paper considers some of the details from the Algebra section. Topics from the algebra section include: introduction to algebraic expressions; polynomials; factoring polynomials; algebraic fractions; exponents; radicals; and further algebra. Students can access the required topic and before commencing it they are told with which topics they should already be proficient. The topic is then split into between three and fifteen subsections. Factoring polynomials, for example, has eleven subsections which includes four sections with exercises. The sections typically give some theory and worked examples. For example, Difference of Two Squares is a subsection of factoring polynomials. There is one page only of explanation and the information is built up slowly in fifteen parts with the student needing to press enter to view the next line. The following shows the completed screen.

Perhaps the most important factoring rule is the Difference of Two Squares

Here are some examples of its use

Before commencing the first exercise the student is reminded of the required syntax for entering an expression and is led through one example. Once in an exercise set the program will not let you out until you have four consecutive correct answers! This can be frustrating but students do need to be able to complete basic algebraic manipulations without making careless mistakes and they should therefore be encouraged to thoroughly understand the techniques before attempting the exercises. After entering an incorrect answer there is the option to try again or view the solution.

The use of symbolic computations started more than 20 years ago. One of the first symbolic packages that came to WA tertiary institutions in the late seventies was REDUCE, a lisp-based package that provides the user with the ability to keep track of long chains of algebraic symbols. Although this gives a hint of the way symbolic computations will influence mathematical teaching and research, it took more than a decade and the appearance of many other symbolic packages before their use in teaching became a reality. Initially, the problem was the amount of memory required to run a symbolic package, which limited its implementation to mainframes. The advances in microcomputer technologies as evidenced by the ubiquitous Macintosh and the Personal Computer has meant that nowadays, many of the more popular packages are available in a form suitable for implementation in small local networks or on stand-alone workstations.

The impact of the symbolic computational tool on our curriculum was noted in the recent report on the strategic review on the Mathematical Sciences that was prepared by the Working Party appointed by the National Committee for Mathematics of the Australian Academy of Sciences.

The packages that are most used in universities in Australia are Maple, Mathematica and Derive. Part of their popularity is the fact that they have reasonably priced student versions. Macsyma is another package, but it has not had a great impact in Australia just yet. The above is by no means an exhaustive list.

One of the main concerns in using a symbolic package in teaching is that it should give accurate results. The result of a review was published in 1995. The review looked at seven such packages (Axiom, Derive, Macsyma, Maple, Mathematica, MuPad and Reduce), and graded each of them according to the number of problems it can solve correctly without too much prompting from the user. The three top packages were found to be Macsyma, Maple and Mathematica, in that order. Since then, the new version of Maple has resolved most of the shortcomings identified in the review. In 1996, Curtin opted for a campus wide licence for Maple, which is now technically available on all platforms (although we are still waiting for some platform versions to arrive from Canada). The choice of Maple is influenced by the fact that it has already been in use in teaching in some units taught by our School.

The user friendly environment makes Maple a useful tool for teaching and learning. It is not necessary to limit our examples to artificial problems where all quantities come out as whole numbers or nice fractions of 1/2 or 1/3, etc. One need not be limited by the size of matrices that can be used to model a realistic problem, nor by the need to use nice round numbers as entries of the matrix. New examples in linear algebra as well as calculus can be generated on the spot. The ability of the student to easily reproduce class examples and consider new ones allows for better understanding of concepts which may have eluded the students in the classroom. Calculations are reproducible, and the computations can be encoded using the Maple programming language. The graphics capability allows one to illustrate properties of functions as they are defined.

It is undeniable that the computer development in recent years has great consequences for Mathematics. To keep up with the technological development we will have to rethink on how some concepts in mathematics should be taught, as well as on how the mathematical curriculum should be made more relevant. One such area is in the teaching of differential equations. Solution in series provides an example of a relatively long chain of symbols that can be manipulated. The ability to represent complicated functions and to identify new patterns, also influence the way we approach a research problem. It should be noted that the use of symbolic packages does not obviate the need for a thorough grounding in the conceptual understanding of the manipulations that are being carried out. Students, when first introduced to the package will be a bit apprehensive if they do not know what is expected of them. The speed at which Maple can provide the answer in graphical or analytical form makes it an invaluable aid in teaching. As a learning tool, it is important that the students learn how to reproduce the illustrative examples as well as construct new examples for themselves. It is also important to stress to the students what part of the use of the package is examinable. They have a dislike of the use of the package in a lecture to illustrate solutions, if they are then told that they are expected in a examination to do equivalent examples by hand. If they have to learn the use of the package, they need to be rewarded for doing so. This means assignment or tutorial work should be structured so that it will lead to their mastery of one or more concepts or techniques, the mastery of which will earn them a portion of the assessment for the unit. The examination of the same techniques should not then be repeated using "conventional" method. It is easy to assess whether students can use the package to do a problem. It is not so easy to assess whether they can use the package to discover new facts. The latter can only come from long exposure to usage and is perhaps not a short term goal.

Calmet Jacques, The role of computer-based symbolic manipulation packages in mathematics teaching for engineers. *International Journal of Mathematical Education in Science and Technology*, 18(5), 1987.

Lawson D. A., The effectiveness of a computer-assisted learning programme in engineering mathematics. *International Journal of Mathematical Education in Science and Technology*, 26(4), 1995.

Please cite as: Caccetta, L., Hollis, P., Siew, P. F. and White, B. (1997). Strategies for assisting the transition from secondary to tertiary mathematics. In Pospisil, R. and Willcoxson, L. (Eds), Learning Through Teaching, p62-67. Proceedings of the 6th Annual Teaching Learning Forum, Murdoch University, February 1997. Perth: Murdoch University.http://lsn.curtin.edu.au/tlf/tlf1997/caccetta.html |

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