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Reflections on teachingmathematics to engineers andscientistsM.J. O'carroll aa Department of Mathematics and Statistics ,Teesside Polytechnic, Borough Road , Middlesbrough,Cleveland TS1 3BA, EnglandPublished online: 09 Jul 2006.

To cite this article: M.J. O'carroll (1985) Reflections on teaching mathematics toengineers and scientists, International Journal of Mathematical Education in Scienceand Technology, 16:2, 323-325, DOI: 10.1080/0020739850160232

To link to this article: http://dx.doi.org/10.1080/0020739850160232

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INT. J. MATH. KDUC. SCI. TKCHXOI.., 1985, VOL. 16, NO. 2, 323-325

Reflections on teaching mathematics toengineers and scientists

by M. J. O'CARROLLDepartment of Mathematics and Statistics, Teesside Polytechnic,

Borough Road, Middlesbrough, Cleveland TS1 3BA, England

(Received 2 July 1984)

As well as giving a participant's reflection on the conference, the paperpresents some personal views on possible directions for mathematics teaching.

1. IntroductionThese are reflections both of the conference and of my own experiences. They do

not set out to give an objective summary of the conference, not even from the Britishpoint of view. In our final group meeting it was acknowledged that one of the benefitsof this cordially active conference was that we would each reflect on it later, in ourdifferent ways, and that it would be sure to influence our thoughts about, and maybe,our approach to, our work. In my case this brings a timely stimulation of enthusiasmin reviewing the organization, methods and costs of my department's 'service'teaching.

Set on an island in large still-frozen waters, and complete with gymnasium andforest running track through the fresh smelling pines, the conference centrepromised a special opportunity for clarity of thought and vision. And so it was, butthe event could not be characterized as leisurely so much as intense and busy.

Conference organizers know that size, pace and schedule are importantparameters in meeting the objectives. At this one, it worked. The opening smallgroup modelling workshops provoked widespread active participation and thisfollowed on at the right level in the main conference sessions. There was a goodvariety both of contributed lectures and of modes of activity (lectures, discussions,group work, posters, video, etc). As if that were not enough, every coffee break andlate evening was filled with further discussions, embracing both technical and moregeneral mathematical modelling questions, such as the existence of God (demon-strating the non-uniqueness of modelling approaches) and the pursuit of happiness(clearly a question of optimal control), which were despatched with that broaddecisiveness which comes with the late hours of the night.

This then was the style of the conference, open and active, and truly Anglo-Swedish in that much of the discussion, experiences and ideas were exchangedbetween the United Kingdom and Swedish delegates rather than in disjoint groupsfrom each country. It was enormously helpful that the level of English languageamong the Swedish delegates was so high.

2. Concepts of modellingSince the conference started with delegates working on raw modelling projects,

thoughts about modelling and its teaching held a particularly prominent place.Contributions on case studies, simulation and modelling embraced several aspects of

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324 M.J.O'Carroll

this topic, which has been quite extensively developed and tested in courses in theUnited Kingdom in recent years. As a result, many of the Swedish delegates wereinterested to hear of these experiences, and in some cases, are planning developmentsthemselves. The conference will have been a helpful stimulant to that.

There is a mixture of concepts of modelling in both countries. To my mind thesefocus on two aspects: (a) the relation between mathematics and the real world, and(6) the nature of problem solving. The former is particularly concerned with theformulation and interpretation phases of the whole modelling process, and itinvolves confrontation with 'dirty data' which may not lend themselves to directmathematical formulation but are obscured by contingent matters (measurementconditions, fraud, prejudice, inaccessibility, substitution of unknown ingredients,etc). It also involves a struggle to decide what question to answer. We saw some ofthis in the modelling workshop. The latter embraces notions of scientific methodwhich can be taken in the large (including the real world) or sometimes in the small(solving problems of a mathematical nature to start with, perhaps by simplified orapproximate methods).

Relative to the training of scientists or engineers, my own view is that the former(real world) aspects are particularly essential, and that many of the fascinatingproblems of the latter type are too artificial to capture the essence of modelling.

Examples of this latter type of problem are:

(i) How good an estimate do you get for route length by counting the number ofkilometre squares a route enters on a map?

(ii) Of all U.S. cities, no two pairs being the same distance apart, let each beconnected to its nearest neighbour, what is the maximum number ofconnections possible for any one city?

The first of these involves some interpretations and question-defining, and we haveused it at Teesside, but both are more or less abstractions, not confrontations withreality. I am not sure this distinction was clear at the conference but I hope it will be.

3. Directions of engineering mathematics teachingThe twin pressures of financial stringency and academic quality are widespread

in the United Kingdom, and the same tendency occurs in most countries includingSweden. The conference heard of various developments with some bearing on this.

The modelling and other case study and creative work is naturally staff intensiveand consumes time and resources in selecting and preparing good material. Help canoften be drawn from industry in providing and enhancing project studies. This allhelps to make the mathematics active and relevant, and helps to overcome thecompartmentalization and inter-disciplinary barriers which tend to occur.

Self-learning materials, texts, slides and videos, are being used to reduce theteaching manpower in the classroom for both the basic delivery of information andfor remedial tutorial support. The more sophisticated materials however are verycostly to produce, and much interest was shown in simpler less ambitious materials,including simple video production. There is no Utopia here and somewhere along theline close attention of the lecturer in tutorial support for small groups of students isnecessary, for both mathematical and psychological reasons.

At the University of Warwick, what appeared at first to be a low manpower self-learning system, was in fact backed up by extensive tutorial support. At K T H a

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Reflections on teaching mathematics to engineers and scientists 325

special post was created for a lecturer consultant to run a clinic for students from9.00 a.m. to 3.00 p.m. each day. Other Swedish institutes had a rotating 'assistant onduty' arrangement. Student help, and small group tutorial support, are here to stay,regardless of computer tutorial systems.

Engineering is becoming more sophisticated too, and as always there is pressureof material on syllabuses. For these and financial reasons, the time available forteaching mathematics is not increasing and sometimes is diminishing. This hasrepercussions on the type of mathematics which can be effectively taught.Increasingly rigour and even mathematical structure may have to give way tomethods and techniques, and yet the engineer must know when confidence can beplaced in these techniques. The education must then provide a competence in using asuitable set of methods: a clear knowledge of when they work and memorabledemonstrations of when they fail; and ability to consult mathematical advice whenappropriate. Counter examples and costs of methods should be prominent.

Where is it leading us? I see the future bringing more self-learning for the "breadand butter" material, and this paying for the quality-enhancing developments ofmodelling project activity and collaborative teaching (with engineering features) ofthe context and reliability of the mathematics. The self-learning materials will needto be a mixture of higher quality products distributed and exchanged among manycentres, and low cost materials quickly and easily produced for specific courses.

4. Follow upThe stimulation from such events brings visions of all kinds of consequent

actions, most of which are overtaken by the pressure of more urgent and mundanematters back at 'home'. In our group discussion delegates recognized this but alsothe probable effect our own reflections would have on our future work. Moretangibly, there is a desire, and useful function, for further communications andexchanges of one sort or another. I have agreed to pursue possible staff exchanges andto send course information. The KTH has a definite policy of 'internationalization'and further exchanges would accord with this.

Many of us will have done some 'shopping' at the exhibitions and I am pleased tosay that various extra slides, posters, books, etc. are on their way to TeessidePolytechnic and I am setting up a simple video system such as demonstrated by JohnSearle. The meeting may have unwittingly prompted collaboration between U.K.delegates who already know each other well. Ed Robson and I hope to collaborateover acquiring and using a mathematical analysis computer package, and followinghis masterly performance at table tennis, Terry Wilkinson and I spent a little timedrafting a table tennis simulation program and are interested in a 'Robot Ping Pong'competition just announced.

In conclusion, the meeting was a timely and stimulating exchange of views andexperiences. There was a surprising yet reassuring commonality of difficulties andconsiderations facing delegates from both nations, and a great sense of fraternity inteaching mathematics to engineers and scientists. This was an extremely enjoyableand rewarding event, and I am pleased to express the gratitude and appreciation dueto the organizers and sponsors.

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