Improving Engineering Mathematics Education at Queensland University of Technology Peter F. Coutis, Troy W. Farrell & Graeme J. Pettet Queensland University of Technology Brisbane, Queensland
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In this paper we describe the integration of a number of new teaching and learning initiatives into rst year engineering mathematics at the Queensland University of Technology. The project is aimed at improving learning outcomes by o ering a range of interesting and accessible learning opportunities and is based on a combined approach encompassing computer-assisted learning, peer tutoring and exible delivery initiatives. Details of the implementation of the project are described herein. Outcomes will be reported in detail at the Symposium upon completion of the project at the end of Semester 2, 1999.
Introduction Among the perennial challenges in engineering mathematics education are the instruction of large student cohorts characterised by an extraordinary range of backgrounds, abilities and attitudes [1], and the low pass rates that often result [2]. Also especially dicult is the task of motivating students to see the applicability of mathematical unit content to their speci c engineering discipline and, most importantly, delivering the material in a way conducive to this aim. This challenge is made all the more dicult by student and, in some cases, engineering faculty perceptions of engineering mathematics service courses as non-essential components of a degree program in engineering. As a result, the stereotypical engineering mathematics student tends to assume the role of a passive, rather than active learner, and is seldom prepared to do more than is absolutely necessary to achieve a pass grade in a given unit. Such a mindset is quite clearly not conducive to e ective learning and is not likely to result in a heightened appreciation of the power of mathematics in engineering applications. It is this problem and, in particular, our e orts to address it in engineering mathematics units at the Queensland University of Technology that represent the main focus of this paper. In this paper we outline the implementation of a project designed to improve learning outcomes and student satisfaction in a rst year engineering unit taught by the School of Mathematical Sciences at Queensland University of Technology (QUT). The project is funded under the 1999 QUT Teaching and Learning Development Small Grants Scheme and is planned in two stages. The rst stage - the implementation stage - will take place during Semester 2 1999, during which time the rst year engineering mathematics unit MAB131 - Engineering Mathematics 1A will be presented in a new format to a cohort of approximately forty students. (First semester cohorts typically number 300.) The second stage of the project is the dissemination/embedding stage. During this stage, a detailed analysis of the impact of the project will be conducted and reported at appropriate forums. At this time, a feasibility study will also be undertaken to determine the appropriateness of the new mode of delivery for other rst and higher year engineering mathematics service units o ered at QUT. 69
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Aims and Objectives of the Project The primary objective of this project is to integrate a combination of computer-based learning, peer tutoring and exible delivery initiatives into a core rst year engineering mathematics unit at QUT. Speci c aims of the project are: (i) to improve problem solving skills and concept development through an integrated strategy of computer-assisted learning and exploration, and group participation in problem-based learning modules, (ii) to improve technical communication skills and deep learning by encouraging students to take a more active role in the learning process through their participation in a mandatory peer tutoring scheme, and (iii) to instigate a more student-centred and exible assessment structure. By providing a wide range of interesting and accessible learning opportunities, we also hope to improve communication skills and quality of learning among students from non-English speaking backgrounds - a group typically making up a large proportion of student cohorts in engineering and, generally speaking, lacking in technical communication and academic literacy skills. In fact, a broad range of learning opportunities and experiences has been identi ed as an important issue in engineering mathematics education in general, due to the diversity in levels of mathematics preparation and attitudes towards the learning of mathematics that are typically encountered [1]. It is anticipated that the bene ts associated with these initiatives will be both pragmatic and educational in nature. In a pragmatic sense, it is hoped that the approaches adopted will have a positive impact on pass rates in the unit which, in recent years, have been as low as 60%. By encouraging the students to take a more active and heuristic approach to the acquisition of skills and techniques, we also aim to heighten appreciation of the usefulness of mathematics in engineering applications and, in combination with the adroit use of assessment tools, improve con dence to apply the techniques learned to subsequent quantitative components of degree courses in engineering.
Implementation The project is presently being implemented as part of the Semester 2 strand of the rst year engineering mathematics unit MAB131 - Engineering Mathematics 1A. Speci c details of the implementation of each of the main components of the project are provided below.
Computer-Assisted Learning The use of technology in teaching is currently an area of active debate. There are a number of schools of thought, although the general consensus seems to be that intelligently used in moderation, computers can be a valuable aid to improving student learning. For example, it has been proposed that \use of the new technology should be layered atop a traditional foundation" [3]. The minimisation of mundane mathematical tasks that enables a focus on understanding and modelling [4], has also been highlighted as one potential bene t associated with the use of computers in engineering mathematics education. In a study designed to gauge the impact of computer-assisted learning on performance in an engineering mathematics course,
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signi cant improvements in examination performance were noted for students studying with the assistance of graphics and computer algebra packages [5]. The importance of elucidating the link between lecture content and computer-assisted learning material is also emphasised, and it is suggested that incorporating computers into the assessment process may be one way of achieving this. It is important also, for engineering mathematics courses to keep pace with technological developments. In particular, today's engineers need to be well versed in mathematical modelling and technical computing principles [6]. Of particular interest here are suggestions in [6] for improving the quality of engineering mathematics education. These include concentration on practical, rather than theoretical issues, assignments and problems that cannot be solved by \crank the handle" or \ nd the formula" methods, problems where either too little or too much information is given so that students need to think very carefully about solution development, and the use of computer visualisation to improve student understanding of dicult concepts. These suggestions essentially summarise our philosophy for implementing computer-assisted learning in MAB131. The computer-based learning component of MAB131 will employ the computer algebra system Maple and is modularised in nature. In the rst two weeks of semester, students will complete a thorough introductory Maple module in order that they be familiar with the basic syntax and general capabilities of the system. The second formal Maple module will be presented towards the end of semester and will be structured as a problem-based group learning activity. For this activity, students will utilise Maple to solve a number of mathematically-based technical problems with applications to engineering. Students will be encouraged, in particular, to utilise Maple's visualisation capabilities and to undertake independent analytical exploration of the concepts involved. Extension of mathematical communication skills will be encouraged through a report writing component. A similar project o ered to a combined group of mathematics and engineering students taking a core rst year mathematics unit is described in [7]. The authors identify a number of positive outcomes, which they attribute to the use of technology. These include a reduction in the number of incorrect responses and non-attempts when compared to traditional handwritten assignments, increased student interest in applications of mathematics to real problems, and a willingness to explore these problems at levels beyond the scope of the assignment. The report writing component, which allowed students the freedom to work in their preferred style (graphical, numerical, or analytical), encouraged exploration and also enabled common student misconceptions to be easily pinpointed. The importance of choosing problems that are both mathematically accessible and also amenable to technological investigation is another guiding factor [8]. The successful introduction of a computer algebra system into second year engineering mathematics is reported in [9]. Interestingly, the authors suggest that some of the technical aspects of the system serve to underpin and reinforce the mathematical concepts.
Peer Tutoring Scheme Peer-assisted learning is of course, by no means a new concept in higher education (see, for example, [10, 11]) and the bene ts from such schemes can be classi ed in a number of ways. Qualitative evidence for an overall positive student reaction to a peer tutoring scheme among mathematics education students at the Chichester Institute of Higher Education, U.K. is detailed in [12]. Interestingly, peer tutors tended to perform at an above-average level in examinations for those topics that they were charged with teaching [13].
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In cases where students are employed for tasks not exceeding their capabilities, bene ts of peer tutoring schemes can include increased enthusiasm and interest in the discipline of mathematics among those students employed as peer tutors, as well as scal bene ts associated with a reduced dependence on casual academic sta [14]. For the present study, the major focus of the scheme is on improving learning outcomes. Experienced mathematics teachers at all levels would testify to the fact that the preparation required to teach a mathematical concept and the preparation required to simply solve a problem in a particular area are vastly disparate endeavours. The process of preparing to teach requires not only a basic understanding of the concepts involved, but also engagement in some level of \constructivist-like" thought whereby the creation of intellectual metaphors personalises the knowledge and enables a deeper conceptual understanding and the communication of these concepts to others. Although not all students will be capable of such deep learning outcomes, we envisage that providing a \learning through teaching" opportunity will result in some kind of positive outcome to most, and this is the basic philosophy behind the peer tutoring scheme to be employed in this project. Given the relative mathematical immaturity of most students in MAB131 and their general unfamiliarity with non-traditional modes of teaching, it was decided that only a very structured approach to the peer tutoring scheme would be likely to realise any signi cant bene t. The basic idea is that every student taking the unit will be required to demonstrate mathematical \leadership" and communication skills by leading a tutorial group through one section of the syllabus. Peer tutors, working in groups of three or four, will be asked to prepare and deliver a one hour tutorial. The tutorial problems assigned to each group will be given at least one week in advance and academic sta support will be provided to students in the preparation stage. The tutorials themselves will be structured as follows. Each session will commence with a short oral exam component where peer tutors will be given a simple problem to solve and explain to the class. This introductory problem will be similar, but not identical to the pre-prepared problems. Peer tutors will spend the rest of the tutorial - under the supervision of an academic sta member - assisting classmates with the assigned problems. Assessment in the peer tutoring scheme will be based on three equally-weighted components: a mark awarded by the class based on the preparedness and overall performance of the peer tutoring group, a mark awarded by the academic sta member based on the quality of solutions and learning materials that each group is required to supply to the class, and a third mark to be awarded on the basis of participation/attendance.
Flexible Delivery Flexible delivery of the course content in MAB131 will be achieved through a combination of world wide web resources and an element of \choice" in assessment. The most interesting aspect here is the assessment procedure for the examination component. Students will be given the opportunity to sit for three examinations during the semester, each of which constitutes one third of the course content. Additionally, a nal examination is held in the formal examination period and is designed to cover all the course material. Students can attempt as much or as little of this nal examination as necessary, and are guided in this decision by their performance in the mid-semester examinations, which carry the same weight as the respective parts of the nal exam. In essence, students choose how and when they ful ll their examination requirements for the unit.
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Project Outcomes Formative and summative student assessment tools will be used to gauge the success of the project in terms of the perceived value of adopted strategies, and in terms of student performance in major assessment components. Examination performance and surveys administered in Semester 1, 1999 will serve as a \baseline" against which surveys and examination results collected during the project implementation period in Semester 2 can be compared and contrasted. Particular attention will be paid to evidence of changing attitudes between the two groups, since links between attitudes towards mathematics and performance in engineering mathematics courses at tertiary level have previously been delineated [1]. A recent study of the e ects of a computer algebra system on the learning of, and attitudes towards mathematics amongst engineering students in Papua New Guinea [15] noted that, while there was anecdotal evidence for increased understanding, enjoyment and motivation for a group of students studying engineering mathematics under an experimental program which integrated a computer algebra system, there was not a statistically signi cant di erence between the examination performance of these students and those studying the unit in a traditional mode. It is proposed that the reason for this nding is the fact that examinations in the unit under consideration were written with little emphasis on conceptual understanding, thus negating any possible advantage attained through the use of computer-assisted learning. In the present study, we aim to incorporate assessment elements that will exploit and delineate deep learning outcomes and improved conceptual understanding obtained as a result of the computer-assisted learning component.
Summary In this paper we have given details of new learning initiatives presently being introduced in a rst year engineering mathematics unit at the Queensland University of Technology. Analysis of the impact of these initiatives will be completed at the end of the current teaching semester. A detailed report of the outcomes, including data analysis and descriptions of problems encountered will be provided at the Symposium in November. As a nal word, we would like to reiterate the following criteria, taken from Engineering Criteria 2000, the course accreditation criteria set by the Board of Directors of the U.S. Accreditation Board for Engineering and Technology (ABET) in 1996 [16]. The criteria which, we believe, are directly relevant to the initiatives described in this paper are: \an ability to apply knowledge of mathematics, science and engineering; an ability to function on multi-disciplinary teams; an ability to identify, formulate and solve engineering problems; an ability to communicate e ectively."
References [1] Shaw, C. T. & Shaw, V. F., (1997). Attitudes of rst-year engineering students to mathematics - a case study, International Journal of Mathematical Education in Science and Technology, 28(2), 289 - 301. [2] Fuller, M.L., (1994) Mathematics, relevance and motivation: a project in rst year engineering mathematics, The Role of Mathematics in Modern Engineering - Proceedings of AEMC94, The Institution of Engineers, Australia, Barton, ACT.
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[3] Krantz, S. G., (1993). How to Teach Mathematics, American Mathematical Society, Philadelphia, USA. [4] Bull, D., (1994). Recent developments in rst year engineering mathematics at Monash University, Caul eld, The Role of Mathematics in Modern Engineering - Proceedings of AEMC94, The Institution of Engineers, Australia, Barton, ACT. [5] Lawson, D.A., (1995). The e ectiveness of a computer-assisted learning programme in engineering mathematics, International Journal of Mathematical Education in Science and Technology, 26(4), 567 - 574. [6] Jenvey, S., (1994). Mathematics in electrical engineering courses - the past, the present and the future, The Role of Mathematics in Modern Engineering - Proceedings of AEMC94, The Institution of Engineers, Australia, Barton, ACT. [7] Coutis, P.F. & Wood, L.N., (1998). Utilising technology to maximise student learning: a case study, International Conference on the Teaching of Mathematics, John Wiley & Sons, New York, 80-82. [8] Coutis, P.F., (1999). Modeling the projectile motion of a cricket ball, International Journal of Mathematical Education in Science and Technology, 29(6), 789 - 798. [9] Lucy, D., Easton, A. & Li, N., (1998). The use of Mathematica at Swinburne, Proceedings of the Third Biennial Engineering Mathematics and Applications Conference, Adelaide, Australia. [10] Houston, K. & Lazenbatt, A., (1996). The introduction and evaluation of peer tutoring in undergraduate courses, Journal of Further and Higher Education, 20, 39 - 50. [11] Gold nch, J. & Summers, R., (1996). Peer- and self-assessment after group work, Mathematics Learning and Assessment, 5, 10-20. [12] Evans, W., Flower, J., & Holton, D., (1998). Peer tutoring in rst year undergraduate mathematics, International Conference on the Teaching of Mathematics, John Wiley & Sons, New York, 101-103. [13] Evans, W. [Personal communication] [14] Wood, L. N. [Personal communication] [15] Raj, L.L., (1998). The e ects of a computer algebra system on the learning of, and attitudes towards, mathematics amongst engineering students in Papua, New Guinea, International Conference on the Teaching of Mathematics, John Wiley & Sons, New York, 251-253. [16] Phillips, W.M., (1998). Engineering opportunity in the information age and the in uence on engineering education, Proceedings of the Third Biennial Engineering Mathematics and Applications Conference, Adelaide, Australia.
