Teaching mathematical modelling to undergraduate engineering students

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  • This article was downloaded by: [Laurentian University]On: 11 October 2014, At: 18:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

    International Journal ofMathematical Education in Scienceand TechnologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmes20

    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

    0020-739x/81/1202 0235 $02.00 1981 Taylor & Francis Ltd

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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.

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  • 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, assignments and seminars all played significant roleseach to develop one or more of the qualities necessary in a good model builder. Toillustrate the approach to the teaching of mathematical modelling, we now discussdetails of the course that we have run over the last few years. The subject was of onesemester duration (13 weeks) and involved a work-load of 8 hours per week. Thisincluded attending lectures and seminars and self study. The course outline is shownin the table.

    In the first week there was no lecture and students were asked to work onassignment 1. In it, the student was asked to build a series of mathematical models ofincreasing complexity given a system or some information about a system. Sincestudents had no formal exposure to modelling, this assignment was aimed atassessing self initiative, critical thinking and preconceived ideas. For a more detaileddiscussion on the rationale for this assignment, the reader is referred to [1]. In thepast we have used a non-engineering example for this assignmentpredicting thefuture population given the census data for the last 100 years. To motivate thestudents, a short write-up was given to indicate the relevence of the problem toengineering and technology. The assignment was discussed in the second week withparticular reference to some of the concepts and attitudes needed. This made thestudent appreciate his shortcomings and thus started him on a journey of self-examination and learning. Also, by this time the students had a feel for what thesubject would demand of them for the next 12 weeks. This was very important as itenabled the student to review his decision to continue in the subject or not. Thisaspect is discussed below in 6.

    From the second week onwards, two formal lectures were delivered each week for6 weeks. In each lecture a specific topic was briefly discussed and its relevance tomathematical modelling highlighted. Only the salient points were presented. Thisapproach served a dual purposethe student could appreciate the totality ofmathematical modelling without getting bogged down in the various details (i.e.examine the forest as opposed to details of each tree) and more importantly, itestablished a base for the student to embark on a self study into the details of one or

    Course outline

    Week Lecturesf Assignmentf/Seminarsf

    1 - , 1 Assignment 1 { Completion2 LI, L2 J I Group discussionI W'Vt ) Assignment 2 { Completion4 L5,L6 j { Group discussion5 L7, L8 1 A s s i g n m e n t 3 j Completion6 L9, L10 J I Group discussion7 LI 1, LI 2 Assignments 4 and 5 Initial consultation

    Assignments 4 and 5:Further consultation9

    10111213 Group discussionOverall review

    t For details see the text.

    Assignment 4: DueAssignment 5:Individual presentation atseminars and group discussion

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  • 238 D. N. P. Murthy and N. W. Page

    more topics in a later assignment. The topics covered in the various lectures were asfollows:

    LI: System characterizationbasic conceptsL2: Models and model buildingL3: Translating system to descriptive modelL4: Classification of mathematical formulationL5: Ordinary differential equation formulationL6: Partial differential equation formulationL7: Markov chain formulationL8: Time series modelsL9: Parameter estimationL10: Structure identification and design of experimentLI 1: Validation of modelLI 2: Pitfalls in modelling

    In order to illustrate the level and scope of the course content, a brief descriptionis given below of the lecture material.

    In the first lecture, system characterization, the student was made to appreciatevarious basic conceptstime and length scales; causality and correlation; degree ofunderstanding. These were illustrated by citing simple systems which were familiarto the students. In the second lecture various types of models were introduced andthe modelling process discussed. The role of descriptive model which contains therelevant features of the system needed for modelling was highlighted. This isimportant, for mathematical modelling is viewed as a clothing of a dummymathematical formulation in terms of the variables of the descriptive model. Thethird lecture examined different ways of representing the descriptive modelinparticular, the use of graph theory formulation.

    In lecture four, various mathematical formulations that can be used as dummiesin obtaining mathematical models were discussed. In lectures five through eight,four different formulations were briefly discussed. In each, the precise mathematicalformulation was presented and methods for obtaining solutions briefly discussed.The student was made aware of the many other formulations not discussed, e.g.stochastic differential equations formulation and differential difference formulation.

    The last four lectures (lectures nine through twelve) examined various otheraspects of mathematical modelling. These required a basis in statistics and elementsof decision making. The last lecture dealing with pitfalls, was aimed at exposing thestudent to potential traps of various types which can occur in mathematicalmodelling.

    To assist the student in self study, a detailed list of references was given at thevery beginning of the course. Supplementary lists were handed out as and whenneeded.

    During the period when the formal lectures were being given, the students didassignments 2 and 3. Assignment 2 was aimed at giving the student practice informula...

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