Importance of Mathematics

4 April 2017

The main reason for studying mathematics to an advanced level is that it is interesting and enjoyable. People like its challenge, its clarity, and the fact that you know when you are right. The solution of a problem has an excitement and a satisfaction. You will find all these aspects in a university degree course. You should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures.

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When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the wealth of the country. The importance of mathematics The everyday use of arithmetic and the display of information by means of graphs, are an everyday commonplace. These are the elementary aspects of mathematics. Advanced mathematics is widely used, but often in an unseen and unadvertised way. The mathematics of error-correcting codes is applied to CD players and to computers. •The stunning pictures of far away planets sent by Voyager II could not have had their crispness and quality without such mathematics. •Voyager’s journey to the planets could not have been calculated without the mathematics of differential equations. •Whenever it is said that advances are made with supercomputers, there has to be a mathematical theory which instructs the computer what is to be done, so allowing it to apply its capacity for speed and accuracy.

The development of computers was initiated in this country by mathematicians and logicians, who continue to make important contributions to the theory of computer science. •The next generation of software requires the latest methods from what is called category theory, a theory of mathematical structures which has given new perspectives on the foundations of mathematics and on logic. •The physical sciences (chemistry, physics, oceanography, astronomy) require mathematics for the development of their theories. •In ecology, mathematics is used when studying the laws of population change. Statistics provides the theory and methodology for the analysis of wide varieties of data. •Statistics is also essential in medicine, for analysing data on the causes of illness and on the utility of new drugs. . •Travel by aeroplane would not be possible without the mathematics of airflow and of control systems. •Body scanners are the expression of subtle mathematics, discovered in the 19th century, which makes it possible to construct an image of the inside of an object from information on a number of single X-ray views of it.

Thus mathematics is often involved in matters of life and death. These applications have often developed from the study of general ideas for their own sake: numbers, symmetry, area and volume, rate of change, shape, dimension, randomness and many others. Mathematics makes an especial contribution to the study of these ideas, namely the methods of •precise definitions; •careful and rigorous argument; representation of ideas by many methods, including symbols and formulae, pictures and graphics; •means of calculation; and the obtaining of precise solutions to clearly stated problems, or clear statements of the limits of knowledge. These features allow mathematics to provide a solid foundation to many aspects of daily life, and to give a comprehension of the complexities inherent in apparently quite simple situations. For these reasons, mathematics and calculation have been associated from earliest times. In modern times, the need to perform rapid mathematical calculations in war time, particularly in ballistics, and in decoding, was a strong stimulus to the development of the electronic computer.

The existence of high speed computers has now helped mathematicians to calculate and to make situations visual as never before. Also this calculation has developed from numerical calculation, to symbolic calculation, and currently to calculation with the mathematical structures themselves. This last is very recent, and is likely to lead to a major transformation. These capacities change, not the nature of mathematics, but the power of the mathematican, which increases perhaps a millionfold the possibility to comprehend, to argue, to explore.

There is also a reverse interaction. The notion of computing would not have made sense without Mathematics, and it was the analysis of the methods of Mathematics by mathematicians, philosophers, logicians and engineers which led to the concept of a programmable computer . Indeed, two mathematicians, von Neumann in the USA and Turing in the UK, are known as the fathers of the modern computers. Analysis of computing, and attempts to make it as reliable as possible, needs deep Mathematics, and this need is likely to grow.

A computer, unless it is programmed, is just a box made of metal, glass, silicon, etc. Programming expresses algorithms in a form suitable for the computer. Mathematics is needed as a language for specification, for determining what is to be done, how and when, and for the verification that the programs and algorithms work correctly. Mathematics is essential for the correct use of computers in most of their applications and the mathematical needs of computing have sparked off many new and exciting questions.

Thus computers, while they have, fortunately, done away with the need for humans to carry out routine calculations, have also required from mathematicians a deeper analysis of the process and logic of computation, and its representation in a machine. The imagination of mathematicians is also stirred by its rigorous nature, which forces them to follow through the logic of their ideas. There are many examples of mathematicians producing apparently strange and inapplicable theories, noting simply that this is the way the mathematics seems to go, only to find these vindicated perhaps decades later by surprising applications.

A recent example is the theory of knots, which was developed as a part of pure mathematics since 1870. A wonderful advance in 1985 showed how the theory could be applied in physics in relation to quantum theory, and in biology in relation to the way DNA unknots itself before dividing. Similarly, modern notions of chaos and fractals were pioneered by mathematicians in the early years of this century. Now fractals are a practical tool for compressing data on computer discs. The study of mathematics can satisfy a wide range of interests and abilities. It develops the imagination.

It trains in clear and logical thought. It is a challenge, with varieties of difficult ideas and unsolved problems, because it deals with the questions arising from complicated structures. Yet it also has a continuing drive to simplification, to finding the right concepts and methods to make difficult things easy, to explaining why a situation must be as it is. In so doing, it develops a range of language and insights, which may then be applied to make a crucial contribution to our understanding and appreciation of the world, and our ability to find and make our way in it. Mathematics as a career

Those who qualify in mathematics are in the fortunate position of having a wide range of career choices. The abilities •to use logical thought, •to formulate a problem in a way which allows for computation and decision, •to make deductions from assumption, •to use advanced concepts, are all enhanced by a mathematics degree course. It is for this reason that mathematician are increasingly in demand. With a mathematics degree, you should be able to turn your hand to finance, statistics, engineering, computers, teaching or accountancy with a success not possible to other graduates.

This flexibility is even more important nowadays, with the considerable uncertainty as to which areas will be the best for employment in future years. [The most recent surveys show graduates in mathematicians and computer science at the top of the earning lists six years after graduation. ] Computer science has a considerable mathematical component, which is becoming more important as the designers of software are required to prove that the software meets its specification. This kind of rigour is one of the basic techniques of mathematics, and can be learned only through a mathematics course.

What to look for in a University Degree course The first decision you need to make when choosing a degree course concerns the subject matter. Do you want to make mathematics alone, or do you want to mix it with another subject in a joint honours degree or in some modular structure? There are arguments both ways. The advantage of a single honours course, particularly when your main interest is mathematics, is that it offers the opportunity to study a wide range of topics and methods within your subject. The advantage of a mix is the wider range of skills which you may attain.

This, coupled with a broader education, may widen the types of job opportunities available. For example, employers are looking for communication skills, and are often happier if students have done some assessed essay or report writing during their degree. Learning these skills can also be an aspect of a single honours course which involves history of mathematics, problem solving, and other forms of project work. Teaching and assessment One of the things you should look for is the range of tutorial provision and of personal contact.

Mathematics is a subject where you learn to do things, and you need feedback on whether your own approaches are the correct ones, in relation to both detail and overall plan. The use of continuous assessment is often an advantage, since in real life you are tested on the quality of the work you can produce, as well as the ability to do that work quickly and under pressure. It is becoming more important for you to show to a prospective employer that you can produce well thought-out work to a high standard, and that you can communicate what you know, both in writing and orally.

Mathematics is a wonderful and exciting subject, but how it works, and why it is so successful, are not matters which are easy to understand, even by its practitioners. Perhaps full explanations would require deeper analyses of language, psychology and thought than are at present available. Nonetheless, it is desirable in a degree course for there to be some discussion and analysis of the processes used in mathematics (symbols, abstraction, generalisation, formalisation, proof, problem solving, etc. , to help you grasp the underlying ideas, set the subject in context, and to relate the learning of technique to an understanding of how these techniques fit into a general scheme. You must see how the apparently abstract nature of mathematics is one of the reason for its power, since it enables the exploitation of analogies between apparently diverse situations. A Bangor graduate now employed in industry wrote that “a mathematics graduate who can see which of a range of techniques are applicable to a problem in hand is worth his weight in gold”. The new M Math degree

This year sees many more universities offering a 4 year degree called M. Math, or something of a similar character, following its introduction in 1994. What is this degree? Why has it been introduced? Is it for you? The main reason for its introduction is the continuing progress in Mathematics which is hinted at in the section “The importance of mathematics” in the previous article “Why choose Mathematics”. The extraordinary advances that have occured in the subject over the last ten years or so have led a recent book to be called ” Mathematics, the New Golden Age” (Keith Devlin, Penguin).

OUTPUT from the “system” has changed. What about the INPUT, that’s you? In schools, Mathematics has changed. G. C. S. E. has introduced young mathematicians to a much more interactive view of their subject. Investigations, when they work well, allow one to see Mathematics being created by oneself. The subject is not proclaimed from on high by some omnipotent being. It is a human activity, it is challenging, often hard work, often fun and very rewarding intellectually. There are, however, still only 24 hours in each day, and 7 days in each week. Something has gone.

The average ability of young mathematicians to manipulate algebra has slipped. This has meant that sixth formers have been finding the hill to climb from G. C. S. E. to A-level has become even steeper. It has also meant that Universities have had to slow down the presentation of algebra in degree courses. Algebraic manipulation is important. It feeds into calculus and analysis, into areas of applied Mathematics, probability and statistics, so all material has had to be slowed down to some extent. On the positive side, the experience of investigations at G. C. S. E. as meant that students are often less passive and want to interact with the material at a more personal level than was previously possible. There is yet another influence changing the way Universities present their Mathematics. The future employers of mathematicians demand the same expertise as before in the subject, but they also expect that their future employees will be able to write reports, to communicate, and present the conclusions of their investigations, to formulate questions and interpret the solutions, qualities that were not always there in the past.

For this, like all graduates, mathematicians need “transferable skills”. They need to be able to explain, to talk to experts in other disciplines, to question, and to investigate. All this has put a big strain on the 3 year degree courses. We need to slow down the presentation of material, to increase the depth and breadth of the material with the new advances in Mathematics and its applications, to include the training in “transferable mathematical skills”. What should be done? The answer put forward by a working group from the main mathematical learned societies was a 4 year degree option.

Students would have the choice. If they wanted to become professional mathematicians then they would probably take the 4 year degree. If they did not want to use their mathematical knowledge in their future careers, their choice might be a standard 3 year B. Sc. This would still give them an extremely valuable broadly based, highly “numerate” degree but the additional technical depth of the MMath type degree would not be demanded. The BSc student would get a training in logical thought, planning, formulation etc. but would not meet the more specialist parts of Mathematics in such quantity or depth.

The aspiring “professional mathematician” would need that extra year in the MMath course in order to come to grips with that extra depth and breadth. The MMath is thus somewhat different in conception from the old MSc, and is only an initial degree. It does ressemble other Master’s programmes in other European countries in its basic conception. So, after all that, an MMath is a 4 year first degree for those hoping to become professional mathematicians. Is it for you? I cannot answer that, but probably neither can you at the moment. The decision has not got to be made immediately.

You can swop from the standard BSc course to the MMath in nearly all cases provided your grades are good enough, or you can swop the other way. A final decision has to be made before entering the third year, for various administrative reasons relating to grants. If you find Maths at University fun and challenging and after 2 years you think that being, say, a research mathematician, statistician or operational researcher appeals to you, try the 4 year degree. Remember though that the two degrees have a different aim: neither is “better” than the other.

If you opt for the BSc and afterwards find a taste for deeper Mathematics, it seems likely that the conventional MSc route will still be available, even probably thriving, so your decision is not a condemnation to a life without maths!! The new degree schemes are a reaction to the changing face of Mathematics, in which new insights are gained, new concepts emerge from the darkness, new questions are formulated. As you consider if a degree in Maths is for you, remember that while Mathematics is one of the oldest subjects, it is constantly being rejuvenated and it is very much alive as it heads into the 21st century.

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