I teach "Engineering Mathematics". What does "Engineering Mathematics" mean ? How is engineering mathematics different from any other sort of mathematics ? Does it has an underlying philosophy or approach ?
To address these questions, we need to firstly need acknowledge that most people who teach engineering mathematics don't generally see themselves as engineering mathematicians but normally identify themselves as either a mathematican who teaches engineers, an applied mathematician or an engineer who teaches mathematics. Not surprisingly, ideas about "Engineering Mathematics" reflect these different backgrounds and perspectives, so a generally agreed defintion and overall philisophy is unlikely. I think qustions about the nature of engineering mathematics point to a related question - what is engineering ?
On that question, many books, essays and papers have been written, and hundreds of defintions provided. Here are three:
"the profession in which the knowledge of the mathematical and physical sciences gained by study, experience and practice is applied with judgement to develop ways to utilise economics, materials and forces of nature for the progressive well being of human kind"
Engineering Council for Professional Development quoted in Johnston et al., "Engineering and Society", Prentice Hall, London, 2000, p. 533.
" the art of directing the great sources of power in nature for the use and convenience of man"
British Institute of Civil Engineering 1828 quoted in Ferguson, "Engineering and the Mind's Eye", MIT Press, Cambridge (USA) 2001.
"Engineers will translate the action the dreams of humanity, traditional knowledge and the concepts of science to achieve sustainable management of the planet through the creative application of technology"
Inst. of Professional Engineers New Zealand 1993 quoted in Johnston et al., "Engineering and Societry", Prentice Hall, London, 2000, p. 533.
The issues relating to the role of mathematics in engineering are quickly apparent in these definitions. The first defintion specifically mentions the role of mathematics, the second emphasizes "art" and third has "science" and "traditional knowledge" underpinning the actions of engineers. These differences are not just reflections of people's different preferences in defining their profession but reflect much deeper divisions in underlying philosophies about engineering. For example, Eugene Ferguson argues in his book "Engineering in the Mind's Eye" that a mathematical approach in engineering design at universities has been over emphsised at the expense of visualisation and drawing. Sharon Beder suggested in her book "The New Engineer" (MacMillian, Melbourne, 1998) that the amount of mathematics in traditional engineering courses reflected the desire of early engineering academics to impress other academics in their institutions of the intellectual rigor of their programs, rather than an analysis of what level of mathematics engineers really need. Of course, there are many more who would passionality argue that mathematics is a key aspect of engineering and central to its development.
For the purposed of this discussions, I will go with the defintion provided by the Engineering Council. The key terms in the definition for me are "knowledge", "study", "experience", "practice" and "judgement". Their use in the defintion suggest that mathematics and science, combined with practical experience, help the engineer to makes judgements and choices about how to utilise resources for the benefit of humanity. This implies to me that "Engineering Mathematics" must empasize the role of mathematics in making sound choices. As I labour toward some sort of coherent defintion (please be patient !!), a few aspects of the problem are becoming clear to me:
A) Engineers need to understand how mathematical principles can be applied to practical problems.
B) Engineers need to be confortable with using mathematics as a tool to inform judgements and choices.
C) Engineers need education in fundamental aspects of mathematics, in so far as a means of developing the skills associated the applications and forming judgements on technical matters.
I think the first two points are uncontroversial (though I am often surprised what some people would like to argue with !) but the third point very much reflects a judgement I have formed from personal experience. Some people argue that education in fundamental aspects of mathematics for engineers is more about "developing thinking and intellect", others would see as a simple educational necessity (i.e. don't run before you can walk). My view is somewhat different from both these positions, that is that educating engineers in fundamantal aspects of mathematics should always be done in the context of application and techical judgements. For example, when I teach techniques for solving differential equations, I emphasise right from the start the practical implications of these techniques and the strategies that engineers use to form and solve these types of problems. I would also agree with the proposition that pure intellectual appreciation of methematics should also encouraged among engineering students (the developing thinking argument) and in fact I think that both approaches (developing thinking/learning in context) can be complementary.
These deliberations don't lead me any closer to a clean defintion of "Engineering Mathematics" but rather simple emphasize how ideas about this area of knowledge are invarably interwinned with ideas about the nature of "Engineering" - a concept itself that is subject to debate and constantly evolving. I personally find this lack of defintion and evolutionary nature invigorating.