Part 1

How to study

University Study
Pattern of work
Lectures
Tutorials
Cooperation
Books and libraries
Vacation work

University Maths
Introduction
Studying the theory
Problem solving
Writing mathematics

Applied Maths
Pure vs applied
Applied problems
Writing out solutions

 

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Since one cannot be a mathematician without solving problems, you should do some at every occasion, between lectures, and at frequent intervals while reading. They may be exercises from a book, or problems from a sheet which you have been given. Where you get the problems from is not so important, provided that they span a range of difficulty, so that there are some which you expect to solve, and some which will really stretch you.

In solving a mathematical problem, there are usually three stages:

(1)

understanding the problem;

(2)

experimentation (to find a method which looks likely to work);

(3)

verifying and writing out the solution.

These stages will be examined more closely in the section on Applied Maths and in Part 2: Proofs, but for the moment we will merely make outline some general principles.

Suppose then that you are faced with a taxing problem. Here are some guidelines as to how you might proceed if you find the problem difficult. These suggestions are listed in roughly the order in which they should be tried, but not all of them will be suitable for any given problem.

(i)

While searching for a method, do not worry about details; fill them in later, if you can.

(ii)

Try working backwards from the answer (if this is given); this may show you the connection between the assumptions and the conclusion; eventually, you will have to turn the argument around so that it runs in the right direction.

(iii)

Draw a diagram, if appropriate (many students are very reluctant to do this). Although diagrams rarely constitute a solution in themselves, they often show you something which you would not otherwise have noticed.

(iv)

If the problem is a general one, try special cases, or simpler versions of the problem.

(v)

Make sure that you have read as much as possible of the relevant theory. Re-read your lecture notes and books; write down on a single sheet of paper all the information which appears to be relevant.

(vi)

Look in your notes and books for similar worked examples.

(vii)

If you are still stuck, put the problem aside and come back to it later (e.g. the next day). Surprisingly often, the solution will then stare you in the face.

(viii)

If you are still stuck, ask your fellow-students (see Cooperation).

(ix)

If the worst comes to the worst, ask your tutor.

Let's be cheerful, and suppose that somehow or other you have found a method which you think may work. Inevitably, there will be some details which you have skipped over, or which you are not completely confident about, so there remains the task of writing out a complete and accurate solution. For example, you may have to reverse the direction of your argument, or you may have to set out a formal proof by induction. This stage also serves to verify your solution, in the sense that it shows you whether the method actually works. It may be that when you try to write the solution out, you find there is some unexpected difficulty which you had overlooked, in which case you will have to return to experimentation. We shall describe some of the general principles of writing mathematics in Writing mathematics, and some particular aspects will be discussed in detail in Part 2.

Even if you think that you have solved a problem correctly and arrived at the correct answer, pay attention to any solution given in the back of the book or in tutorials. If the other solution looks different from yours, think about whether yours is equally valid; if your solution now looks dubious, make sure that you understand what is wrong.