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Teaching mathematical modelling toundergraduate engineering studentsD. N. P. Murthy a & N. W. Page aa Department of Mechanical Engineering , University ofQueensland , St. Lucia, Queensland, AustraliaPublished online: 09 Jul 2006.
To cite this article: D. N. P. Murthy & N. W. Page (1981) Teaching mathematical modellingto undergraduate engineering students, International Journal of Mathematical Education inScience and Technology, 12:2, 235-243, DOI: 10.1080/0020739810120217
To link to this article: http://dx.doi.org/10.1080/0020739810120217
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INT. J. MATH. EDUC. SCI. TECHNOL., 1981, VOL. 12, NO. 2, 235-243
Teaching mathematical modelling toundergraduate engineering students
by D. N. P. MURTHY and N. W. PAGE
Department of Mechanical Engineering, University of Queensland,St. Lucia, Queensland, Australia
(Received 13 May 1980)
In this paper we present the structure and details of a subject to teachmathematical modelling to undergraduate engineering students.
1. IntroductionTraditional methods used by engineers when studying both old and new
engineering systems rely heavily on experimentation involving either the systemitself or a scaled physical model. The main drawbacks of this approach are that theyare costly and lack flexibility. Till a few decades back, the high cost and limited powerof computational methods prevented the use of mathematical models as analternative method in the study of engineering systems. However, over the last twodecades, the cost and power of computational methods have improved dramatically,thus offering scope for the wide use of mathematical models in the study (i.e. analysisand design) of complex engineering systems.
Unfortunately, most undergraduate engineering programmes have fallen shortin providing adequate training in the building of mathematical models. In a typicalundergraduate engineering programme students are taught basic sciences andmathematical techniques in the first and second years. In subsequent years, mostteaching effort is devoted to the analysis of standard models of various engineeringsystems using the mathematical techniques learnt in the earlier part of theprogramme. The student has very limited exposure to the building of mathematicalmodels. As a consequence, his ability to use modern analytical and computationalmethods to solve new and challenging problems is limited.
Since engineers are being called upon to tackle more and more complex systems',it is essential to equip new engineers with the tools most useful in obtaining new andinnovative solutions. We believe that mathematical modelling with its intrinsiceconomy and flexibility is one such tool. With this in mind a subject, 'MathematicalModelling', has recently been prepared and introduced into the undergraduateprogramme in Mechanical Engineering at the University of Queensland. In thispaper we present the details of the subject, the approach taken and discuss studentresponse to the subject and its success in teaching mathematical model building.
2. Problems in teaching mathematical modellingMathematical modelling is a relatively difficult subject to teach, as modelling is
both an art as well as a science. The different stages in the evolution of a mathematicalmodel, from the simplest and somewhat unrealistic to the most complex and trulyrepresentative, requires not only an understanding and appreciation of a largenumber of concepts but also the development of certain innovative skills and
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236 D. N. P. Murthy and N. W. Page
attitudes. It requires a high sense of creativity, intuition and foresight, skills whichare difficult to teach formally. Thus, effective teaching of model building requiresboth a formal lecture format as well as alternative formatsthe latter to develop this'art' aspect of modelling. In this context, the studio approach (used very effectivelyin the teaching of design) is highly suited for it allows the student to learn anddevelop the needed skills and attitudes by his own mistakes and constructivecriticism from his fellow students and lecturers. This type of an approach combinedwith other activitiese.g. brain-storming sessionsoffer scope for developingattitudes of innovation and self-discovery, both of which are qualities needed tobecome a good model builder. Thus a course to teach mathematical modelling needsto be carefully structured so as to develop in the student both the science as well asthe art aspect of modelling.
Another difficulty in the teaching of mathematical modelling is that it requires avery strong interaction between two totally different worldsan abstract world ofmathematical concepts on the one hand and a real world of the physical orengineering system being modelled on the other. In general, the real physical systemis complex and the mathematical model must be a simplified representation of it, sothat it is amenable to mathematical analysis. The task of building an adequatemathematical model requires decision-making at various stages to resolve thevarious conflicting requirements (e.g. complexity of model and hence adequacyversus the solvability of the model). This can be done satisfactorily only when themodel builder has a good grasp of mathematical concepts and formulations and adeep understanding of the physical aspects of the system being modelled. The skill isto select a suitable mathematical formulation (a formulation which is abstract andmakes no sense outside mathematics) as a dummy and clothe it in terms of thephysical description of the system being modelled so that it becomes a mathematicalmodel (hopefully adequate!) of the system.
Since the types of mathematical formulations encountered by - engineeringstudents in the earlier years are limited, an important task of the modelling subject isto expose them to many more new mathematical formulations and develop theirconfidence to learn more about these on their own. This aspect-is important for it notonly introduces self study but allows a bigger selection of formulations to choosefrom in model building. The self study activity is very desirable for each modellingexercise involves something new which the builder has often to learn by himself.
A serious problem encountered in teaching mathematical modelling to engineer-ing students is the change in mental attitude required on the part of the student. Thisarises because in earlier years, engineering students are given only well-definedproblems, formulations which have a unique solution or answer. Although theyrecognize towards the very end of their undergraduate programme that problemformulation and solutions are not unique, they in general lack confidence and areinsecure in tackling ill-defined or imprecise problems. In modelling, a variety ofmodels can be built for a particular system depending on the modeller's degree ofunderstanding of the system.
3. Structure of the subjectDue to the reasons discussed in the previous section, teaching mathematical
modelling requires a non-standard approach. The students not only need to learn avariety of concepts and techniques, but also need to develop their skills in aspectssuch as self initiative, creativity, critical thinking, self learning and decision-making.
Application of chasing to differential equations 237
In order to achieve this difficult goal, the subject described here was structured insuch a fashion that lectures,