Stressing the relevance of mathematics to engineering/ technology anymore will only be like carrying coals to Newcastle. The redundancy is adduced by the view that a good part of all engineering can be called applied mathematics (and, therefore, many areas of mathematics can be termed as theoretical engineering!). Even areas once considered ‘pure’ in mathematics have found applications in applied sciences and engineering. Use of Ramanujan’s Sums Cq(n) in signal processing and computer vision, application of his Graphs in coding theory and of theory of partitions and q-series in statistical analysis and many of number theory concepts in discrete mathematics, communication engineering and modern and mathematical physics are only a few examples.
All knowledge is the result of quest for answers to problems and most problems in engineering are first to be modelled into mathematical ones. Subsequently, testing the formulated problem for solvability, modifying it, if necessary, into one in feasible form, with approximations or simplifications, and actually solving, which may require newer methods or extensions of known methods involving numerous numerical techniques, if closed form solutions are not available… are all in the domain of mathematics.
Interpretation of the solution in the engineering context, of course, is the engineering job. Thus a good engineer must also imbibe a good mathematical skill. A difference is that while a mathematician may design theoretically a new equipment, like a component of a space craft, an engineer would create and test the equipment or its prototype. It is said that mathematicians ‘write’ proofs while engineers ‘produce’ proofs. Mathematics, according to Aristotle, is ‘science of quantity’ and, to ingeniare’
(Latin for ‘engineer’) is to ‘contrive, devise’. Thus the two complement each other.
For this reason only, in most of the engineering branches/specialisations, engineering mathematics courses are offered mandatorily in at least the first three semesters of UG classes and the first semester of PG classes. For some UG branches, some additional mathematics-based courses are also offered, like Discrete Mathematics for CSE, and
Numerical Analysis, Operational Research and/or Reliability Theory as regular/elective subjects for mechanical engineering.
In some universities, even for biology-based branches like biotechnology, bioinformatics and biomedical engineering, the subject of mathematics is a must in the qualifying HSC examination; and the mathematics papers for them in BE/BTech are also the same as for the other branches. Interestingly, in some universities like Queen’s University, Canada, the subject of ‘Mathematical Engineering’ is offered as a
PG programme/specialisation for engineers. Queensland University of Technology, Brisbane, Australia, offers a ‘BE(Hons) / Bachelor of Mathematics’ dual degree programme of five years’ duration in which students can choose any engineering major like civil, computer and software systems, medical or process engineering and specialise in applied and computational mathematics, decision theory or statistical science.
The reasons quoted for such clubbing include the growing need for application of mathematics in engineering, scientific and industrial areas, like IT, management, and finance involving planning, designing, construction, manufacture, quality assessment and control. A seemingly sound engineering structure may fail because of a mathematical instability which was not detected earlier. Phenomena as diverse as chaos in semiconductor lasers, patterns in traffic jams and weather and devastating catastrophes like floods, tsunamis and global warming are subjects of mathematics which could be fully exploited for the better functioning of the engineer and to the good of the mankind.
No mathematics knowledge would be complete without a workable computer knowledge which in turn is a basic tool for an engineer. Accordingly, ‘Engineering Mathematics is at the heart of Engineering.’ But not everything seems to be all right with the mathematical preparedness of our students aspiring to be engineers. There are two types of inadequacies in this regard. The first is that of the student and the second is that of the teacher.
“The past decade has seen a serious decline in students’ basic mathematical skills and level of preparation on entry into higher education. Many students embark on engineering degree programmes without the necessary mathematics skills required for the course…” This quote is from the website of the Engineering Subject Centre, a national centre delivering support to learning and teaching in the U.K. higher education community. With respect to Tamil Nadu, we are riddled with a paradox: the ‘best’ students (many securing 200/200 marks in Mathematics paper in the State Board Class XII examination) of the state are the students of CEG, the ‘prime’ engineering college of the state; but percentage of complete passes in the first semester examinations touched as low as 25 per cent within the past few years, Mathematics being one major culprit.
A clear and major reason is that the skill they earned in that subject up to Class XII was not mathematical, but related only to memory testing. Clear grasping of concepts and readiness to apply them in slightly different contexts are lacking in many of them.
Further, aiming only at high scores, many would have omitted to study some crucial sections of the syllabus, like Integration and Differential Equations, which are important for engineers, and even the entire Class XI portion. Change in medium of instruction, higher pace of teaching and learning, diffused individual attention and the like may be some more reasons. Smart engineering (‘ingenium’ = cleverness) students should waste no time in identifying and rectifying such deficiencies in them.
Teachers influence the lives of children both actively and passively from their very tender age. Among them, teachers of mathematics have an added responsibility: in addition to the subject of mathematics, they have to teach children how to learn mathematics. Basics must be thoroughly explained before going to formulas. Though preparation for examination is important, there cannot be lacunas in the logical development on the subject lest the subject should become a myth and out of bounds for an otherwise intelligent student. Originality should be encouraged and discussions should be promoted.
A few guidelines for the young teachers of mathematics that will help make their teaching more effective and purposeful.
Instead of asking the students to ‘memorise’ a mathematical formula, make the student derive the formula at least once completely from the first principles. While trying to reach a new address, do not hire a driver for your car; you drive yourself to the address overcoming hurdles. Then you will remember the destination better, or reach it yourself quickly following the route you have taken.
Abstract and analytical statements can be better understood and remembered by referring to a geometrical interpretation, if available. For, writings are better than words and drawings are better than writings. Equations of curves and surfaces, their maxima/minima and two-variable linear programming problems are some examples.
To drive home that mathematics is not mystic or God-given, but only developed by humans like us, give suitable anecdotes related to mathematical concepts, involving their proponents and developers. The fact that Euler, to whom Ramanujan is often compared, was very earthly, fathering 13 children, must be an interesting piece. The never-ending feud between Newton and Leibnitz on the credit for discovery of differential calculus must be an absorbing intellectual episode. Einstein was a ‘hopeless idiot’ in his school days and Ramanujan failed in English in FA class.
In suitable contexts give famous quotes of related mathematicians/scientists which will enrich and enliven your lecture and the subject. Laplace would often appeal to his students: “Read Euler, read Euler; he is the master of us all!” Euler, when he lost his eye sight, did not sigh, but said, “Now I will have fewer distractions.”
Wherever possible, relate the lesson with suitable areas of application, preferably from the branch/specialisation of the class being handled. Transform techniques for electrical engineers and algebraic eigenvalue problems for civil engineers are some examples.
Instil confidence in your students. Prepare complete notes of what to teach, with references, typical examples, possible variations, problems for further exercise, etc. Check the class notes and home works of the students at random. Update your notes every year using your growing experience and knowledge and make it a valuable reference material which can even be published later.
Conduct tutorial classes seriously. Tutorials are sometimes more important than your lectures — in the sense that students ‘learn’ the concepts by ‘doing’ rather than by ‘hearing’. Thus backlog in learning is avoided, reducing the fear of examinations. For an engineering student, promptness is more important than even intelligence.
Make the class interactive. Invite questions and suggestions from the students during your lecture. Encourage originality explicitly. Such measures not only make teaching and learning more effective, but also help keep the class alive, alert and tuned. While evaluating students’ performance, particularly during central valuation, give credit to alternative methods of answering, instead of shunning.
Teaching faculty in mathematics and engineering departments can do collaborative research, benefiting both the groups. Very often, researchers in mathematics newly design or extend a mathematical technique, but illustrate its application over a suitable, but hypothetical problem without an explicit application. If, instead, the problem chosen is of interest to engineers, not yet solved or solved by a different method, the result will have its value doubled! M.Phil. candidates are not expected to do highly rigorous research like Ph.D. candidates. Their dissertations can be even surveys, or expository in nature. Each M.Phil. student in mathematics can be made to study thoroughly and prepare and submit their dissertation in an area of importance for researchers, like the matrix methods for eigenvalue problems, Chebyshev and other polynomial collocation methods, boundary value problems, finite element method, perturbation technique, integral equations, spline approximations, etc. (Of course, the dissertation may have to contain an application section also, for completeness.)
Within a few years of the implementation of this scheme the department will have a good collection of reference material of immense value for researchers in mathematics and engineering.
The above article is based on a lecture, delivered by the writer at Anna University, Chennai, a few years ago, for senior Mathematics faculty.
The writer was formerly professor of mathematics, College of Engineering, Guindy, also heading entrance examinations and admission.
