University of London Mathematics 1 Study Guide
The study of mathematics can be very rewarding. It is particularly satisfying to solve a problem and know that it is solved. Unlike many of the other subjects you will study, there is always a right answer in mathematics problems. Of course, part of the
excitement of the social sciences arises from the fact that there may be no single ‘right answer’ to a problem: it is stimulating to participate in debate and discussion, to defend or re-think (and possibly change) your position.
It would be wrong to think that, in contrast, mathematics is very dry and mechanical. It can be as much of an art as a science. Although there may be only one right (final) answer, there could be a number of different ways of obtaining that answer, some more complex than others. Thus, a given problem will have only one ‘answer’, but many ‘solutions’ (by which we mean routes to finding the answer). Generally, a mathematician likes to find the simplest solution possible to a given problem, but that does not mean that any other solution is wrong. (There may be different, equally simple, solutions.)
With mathematical questions, you first have to work out precisely what it is that the question is asking, and then try to find a method (hopefully a nice, simple one) which will solve the problem. This second step involves some degree of creativity, especially at an advanced level. You must realise that you can hardly be expected to look at every mathematics problem and write down a beautiful and concise solution, leading to the correct answer, straight away. Of course, some problems are like this (for example, ‘Calculate 2 + 2’ !), but for other types of problem you should not be afraid to try various different techniques, some of which may fail. In this sense, there is a certain amount of ‘trial and error’ in solving some mathematical problems. This really is the way a lot of mathematics is done. For obvious reasons, teachers, lecturers and textbooks rarely give that impression: they present the solution right there on the page or the blackboard, with no indication of the time a student might be expected to spend thinking — or of the dead-end paths he or she might understandably follow — before a solution can be found. It is a good idea to have scrap paper to work with so that you can try out various methods of solution. (It is very inhibiting only to have in front of you the crisp sheet of paper on which you want to write your final, elegant, solutions. Mathematics is not done that way.) You must not get frustrated if you can’t solve a problem immediately. As you proceed through the subject, gathering more experience, you will develop a feel for which techniques are likely to be useful for particular
problems. You should not be afraid to try different techniques, some of which may not work, if you cannot immediately recognise which technique to use.