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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

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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 formats—the 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 activities—e.g. brain-storming sessions—offer 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 worlds—an 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 roles—each 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 assignment—predicting 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 purpose—the 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 discussion—Overall 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 characterization—basic 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 concepts—time 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 model—inparticular, 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 informulating descriptive models given a real physical system. The student began toappreciate the characterization of the system variables in terms of length and timescales; causal relationships, etc. Typical systems used in the past for this assignmenthave been road transportation in a city, setting up a new factory, and a thermal powerstation. This assignment was designed to stress the need for a deep understanding ofthe physical aspects of the system being modelled.

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Application of chasing to differential equations 239

Assignment 3 was similar to assignment 1 in that the student was asked to build aseries of increasingly complex models for a given system. With the experience of thefirst two assignments, the student had by this stage some exposure to the modelbuilding process. He was therefore able to tackle this assignment much moresatisfactorily than assignment 1. Also, by re-examining his solution to the firstassignment, the student could measure for himself the change and progressachieved. This critical examination is important in developing self-confidence andskills of self analysis. However, at this stage, the process of self learning and criticalthinking was still in its infancy. In the past, we have used a simple engineeringsystem for this assignment, e.g. a pneumatic pump excited by an electromagnet.This example was chosen because students have some familiarity with the variouscomponents and subsystems, and this assisted them in the modelling task.

For the remainder of the semester, there were no more lectures. Instead, the timewas spent working on two major assignments and attending seminars. The twoassignments were aimed at developing self learning and critical thinking. Inassignment 4 the student was asked to examine a particular modelling methodologyin depth. In general, it was a topic covered briefly in the lectures, but the student wasencouraged to choose from other topics also, e.g. bond-graphs or simulation(analogue and digital). In assignment 5, the student examined different models builtfor a specific engineering system. In the last four weeks, students gave seminars onthe material they had researched for assignment 5. The aim of the seminars was topresent the models critically, so as to evoke critical examination and questioningfrom fellow students and staff members.

Since these two assignments were of central importance to the subject they aredescribed in more detail below. The aims of the two assignments are best illustratedby the figure, which shows the paths taken by the student in completing theassignment. This illustrates and reinforces the main feature of modelling: stronginteraction between two different worlds, mathematical and physical.

In assignment 4, each student studied a specific topic (mathematic formulation ortechnique) from the list given below:

(i) Non-linear ordinary differential equations(ii) Parabolic partial differential equations

(iii) Hyperbolic partial differential equations(iv) Differential difference equations(v) Linear time varying equations

(vi) Linear time series formulation(vii) Non-linear time series formulation

(viii) Markov chains(ix) Regression (linear and non-linear) models(x) Parameter estimation—deterministic formulation

(xi) Parameter estimation—stochastic formulation(xii) Analogue simulation

(xiii) Digital simulation(xiv) Simulation languages(xv) Design of experiment

(xvi) Validation methods

The student was asked to read standard texts to understand the details of theformulation or technique selected. For example, in the case of parabolic equations,

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240 D. N. P. Murthy and N. W. Page

->- Different physical systems

IDifferentmathematicalformulationsS techniques

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sr

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A: Path of Assignment # 4

B: Path of Assignment »5

Paths of assignments 4 and 5.

this involved the understanding of the correct mathematical specification so that theformulation is well posed and an understanding of the various methods of solution.The student needed to recognize which formulations allow exact solution; thevarious approximate analytical methods of solution (e.g. regular and singularperturbation); the various computational methods of solutions (e.g. use of analogue,digital and hybrid computers fo'r obtaining solution). In addition, he was asked tomake a library search, using abstracts, to compile a list of physical systems for whichthe formulation has served as a useful dummy in building mathematical models.Thus, in this exercise, the student moved along a horizontal line on the grid shown inthe figure. The student was asked to present a 30 page report on the assignment.

In assignment 5, the student was asked to undertake a journey in which he movedon the grid along a vertical line—i.e. he was asked to select a physical system and do alibrary search to compile a list of different mathematical models as dummies thathave been applied to the system under study. In the report he was asked to give adetailed description of the physical aspects of the system and a critical examination ofdifferent models—their usefulness and shortcomings. This is important for differentmodels are suited for different purposes. The report (20 to 30 pages long) alsoexamined the student's ability to build new models in that he was asked to suggestimprovements to current models or to suggest a new formulation for the systemunder study.

This last assignment comes closest to the way an engineer would be expected toproceed when asked to study a system using mathematical models. This thereforebecomes a format which he can use throughout his life.

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Application of chasing to differential equations 241

Some of the specific systems which the students have examined in the past threeyears are

(i) Pollution in a river(ii) Atmospheric pollution

(iii) Thermal power station(iv) Traffic flow(v) Demand and technology forecasting

(vi) Models for production line(vii) Models in mining industry

(viii) Drug absorption in body(ix) Brand selection(x) Dynamics of advertising

(xi) Models of maintenance(xii) Models in sport . .

Several of these topics involved the study of non-engineering systems but wereincluded to illustrate to the student that mathematical modelling techniques havegeneral application and that many of these systems share with engineering systemscommon mathematical dummies.

4. Tutorials and seminarsThe tutorials (or group sessions) associated with the first three assignments were

run largely along the lines of critical examinations of the student effort and brain-storming sessions in which the students were asked to give further suggestions andideas relating to the particular assignment. The role of the lecturer in these tutorialswas primarily that of providing leadership when the sessions got bogged down orseriously side-tracked. These early tutorials fulfilled a number of importantfunctions in addition to covering the subject matter of the assignments themselves.The students developed confidence in speaking out, in exposing their ideas forcomment and criticism. Thus, they were prepared for an active participating role inthe seminars which replaced both tutorials and lectures later in the course.

As the course progressed, the lecturer played an increasingly minor role intutorials and seminars. It is worth stressing that active participation is essential fordeveloping self examination and giving and receiving constructive criticism.

5. Assessing student performanceThe student grades were based on the following distribution:

Per centAssignments 1-3 20Assignment 4 30Assignment 5 30Participation in seminars and tutorials 20

Total 100

As indicated, the final grade was based solely on assignments and participation inseminars and tutorials. The use of a formal examination was considered inap-propriate to a course of this type.

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242 D. N. P. Murthy and N. W. Page

In most of their undergraduate programme, engineering students assume apassive role, in tutorials—they work on specific problems alone and seek the lectureror tutor's assistance only when a difficulty is encountered. In modelling, it isabsolutely essential that the students assume an active role. In order to encourageand force the students into this role, 20 per cent of the grade for our course was givenfor participation; 10 per cent being allotted for presentation of his seminar onassignment 5 and 10 per cent for questioning and participation in other tutorials andseminars.

6. Student responseThis subject has been running for the last three years at the University of

Queensland and is an elective subject which students can take either in their third orfourth year of study. The average enrolment has been ten per year—an ideal numberfor any larger or smaller size would have made the structure of the subject lesseffective.

A much larger number showed initial interest in the subject but many droppedout after the first week or so when they had a clearer understanding of the course styleand requirements. The main reason students gave for this high attrition rate wastheir lack of confidence in coping with the unfamiliar structure and scope of thesubject. Those students remaining in the course tended to be drawn from the top halfof the class in terms of past performance.

The response of the students has been favourable. Most appreciated the subjectand saw the value of it. The only complaint that has been received concerned thework load. This problem was largely attributable to the students inefficiency incoping with the unfamiliar subject format—inefficient library research and lack ofappropriate consultation with the staff member.

In the initial stages the students were hesitant to participate in tutorials andseminars and in particular to criticize their fellow students. But towards the end ofthe semester, students developed the confidence to accept and give criticalcomments. Most students taking the subject enjoyed the stimulating experience ofself-learning and critical thinking, and on the whole we have been very pleased withthe student response.

7. Some c o m m e n t sThe main features which the students learned and appreciated were

(i) critical thinking and self criticism(ii) confidence to venture into self study

(iii) use of mathematical models in engineering (as well as in non-engineering)context

(iv) use of library facilities

The last point is important for traditionally undergraduate students do not makefull use of libraries in their programme. In order to assist students in the proper useof the library resources notes were handed out early in the semester. These gaveinformation about abstracting and indexing journals and various other journalswhere the student could find papers on models and modelling.

The students required a lot of personalized attention. When they were doing thelast two assignments, they required regular meetings with the staff member to helpthem understand points not clear or to steer them in the correct direction. The

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Teaching mathematical modelling 243

initiative was left to the student to contact the staff member. Some sought verylimited help (an hour or two) whilst others sought a lot (five to six hours). This one-to-one contact and interaction was a very important part of the student's learningprocess for it helped him to discuss confidentially his shortcomings and ways toovercome them.

Although there are many books on mathematical modelling, no single book wasUsed as a text. The available books on mathematical modelling were put in thereserve collection of the library and students were encouraged to read sections andchapters. The list of books on modelling put in the collection is given in references[2—10] at the end of the paper.

In addition to the books cited above students were asked to read chapters frommore general books. In this context, a chapter from [11] was useful for assignment 2.Reference [12] contains a collection of cross-disciplinary essays on models inscholarly thought which assisted the students to see the role of models in a very widecontext. Also, students were encouraged to read books on thinking (e.g. [13,14]) andon creativity (e.g. [15]). Students were also given a list of papers (from variousjournals) on the different aspects of mathematical modelling.

In the future we hope to provide them with detailed outlines of the topics coveredin the formal lectures, and with an exhaustive list of references which can be retainedby the student for use in later years.

8. ConclusionsIn this paper we have discussed teaching mathematical modelling to under-

graduate engineering students. We have presented the structure and organization ofthe subject which has proved to be fairly successful. We feel that the approach ishighly innovative and hope that this paper will result in the appearance of morepapers discussing alternative methods of teaching mathematical modelling toengineering students.

References[1] SUBRAMANIAN, R., MURTHY, D. N. P., and SMITH, T. C , 1976, Trans of Int. Assoc. for

Mathematics & Computer in Simulation, Vol. XVIII, pp. 239-44.[2] ANDREWS, J. G., and MCLONE, R. R., 1976, Mathematical Modelling (Butterworths).[3] HABERMAN, R., 1977, Mathematical Models (Prentice Hall).[4] ZIEGLER, B. P., 1976, Theory of Modelling and Simulation (John Wiley).[5] OSBORNE, M. R., and WATTS, R. O., 1977, Simulation and Modelling (University of

Queensland Press).[6] ROBERTS, F. S., 1976, Discrete Mathematical Models (Prentice Hall).[7] BENDER, E. A., 1978, An introduction to Mathematical Modelling (John Wiley).[8] MAKI, D. P., and THOMPSON, M., 1973, Mathematical Models and application (Prentice

Hall).[9] MIHRAM, G. A., 1971, Simulation Statistical Foundation and Methodology (Academic

Press).[10] KEMENEY, J. G., and SNELL, J. L., 1972, Mathematical Models in the Social Sciences

(MIT Press).[11] SAATY, T. L., 1959, Mathematical Methods of Operation Research (McGraw-Hill).[12] SHANIN, T., (ed), 1972, The rules of the Game (Tavistock).[13] DE BONO, E., 1969, The five day course in thinking (Penguin Books Ltd).[14] DE BONO, E., 1970, Lateral thinking (Penguin Books Ltd).[15] KOESTLER, A., 1969, The Act of Creation (Picador).

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