Chapter 7 Problem Solving and Theorem Proving

Geometric representation of Pythagoras Theorem, with semi circles and squares.

Figure 7.1: Geometric representation of Pythagoras Theorem, with semi circles and squares.

One learns, and loves, mathematics by doing it, not just by reading it - Mathematics is not a spectator sport! Therefore tackling exercises and problems is an essential part of learning mathematics and by far the most efficient way of consolidating and checking your understanding of the course. Books and notes should always be studied with pen and paper to hand.

Using Worked Examples

Worked examples will be part of your lectures and text books. Some will be presented with the full written explanations however many will leave part (or all) of the working for you to complete; an example is presented in the Appendix which shows a partial and full solution.

When reading examples you should ensure you can follow each step by:

  • Working through line by line, filling in any gaps.

  • Make a note of steps you do not understand so you can work on them later. Ask a friend or tutor.

  • Annotate the processes; your understanding; moments of enlightenment.

Exercises

Exercises are questions intended to train you to become skilled in the straightforward application of techniques and/or to give you examples of concepts which have been introduced in lectures. If possible, you should try them before going to any class based on them, so that you can raise your difficulties. It is hardly sensible to spend your time in the class working through questions which you can do, only to find out later that you have difficulties with some of the others. You should try to complete all the exercises set, not just those which have to be handed in.

Note that unless a specific numerical value is asked for it is normally preferable to leave mathematical constants and functions, such as \(e\), \(\ln\) or \(\cos\), in your answers. If you are asked to give your answers to a prescribed degree of accuracy do so (and not to a greater degree as given by your calculator or computer). For example writing \(\frac{\sqrt{3}}{2}\) is usually better than an approximation such as \(0.866\).

Problems

Problems (or more complicated examples) are not always straightforward: they usually require some insight and creative use of the mathematics concerned. You may find that you are unable to do them all and that some of them you cannot even start! There is no recipe for solving problems but here are some of the things you can try. The following sequence indicates how you might go about tackling more involved problems/exercises.

Getting started:

  • Have lecture notes, relevant textbooks and calculator at hand.

  • Read the entire question carefully and identify the key concepts involved, (e.g. by comparison with the key word list from your lecture notes).

  • Identify the mathematics likely to be involved and make sure that you know and understand it. Re-read the relevant section of your notes if necessary. Look for similar problems.

  • Draw a picture if the problem allows it. In applied mathematics diagrams are almost always essential, (e.g. to show the forces acting on a body), but they can also be very helpful in other areas of your work. Experiment with pictures, diagrams, patterns – they often provide a real breakthrough. Make use of any software packages which may be available to help you experiment more freely with different diagrams. They can also help by, for example, solving equations or evaluating integrals that you may not be familiar with.

  • If the problem is a general one, look at special cases or formulate and solve a simpler problem first.

  • If the problem is abstract, look at concrete examples, (e.g. put numbers in place of letters), take specific examples of sequences or sets and so on.

  • Think around the problem and write down your thoughts, (why not try producing a ‘patterned note’ or ‘flow diagram’ of the problem?). What you write down often gives you other ideas and helps to build up a solution step-by-step. What you don’t write down often vanishes into thin air and is not remembered at the vital moment.

  • Don’t be afraid to experiment and perhaps go up blind alleys. Although the final solution should be logically ordered you may find it helpful in developing it to work backwards from the result to be obtained (if you are given it) as well as forward from the data of the problem. Be prepared to gloss over details to start with in an attempt to obtain the broad outline of a solution. You may be able to fill in the details later.

  • When you are writing a ‘final’ solution, try to maintain good writing standards in both English and ‘maths’ by using sentences to explain your working and appropriate mathematical symbols to produce coherent arguments. (This will take time and practice.)

What if you get stuck?

It may not feel like it, but getting stuck (and finding a way out) is the key to learning. Be prepared to struggle with the harder problems and to try different approaches until you find the way to the solution. If, after several attempts, you remain stuck ask for help.

  • Re-read the question and make sure that you know the mathematical meaning of all the technical words and that you can give examples of their use.

  • Notice that key words like ‘deduce’, ‘hence’ and ‘otherwise’ often suggest ways of doing problems.

  • Make a list of key concepts involved in the statement and re-read the relevant theory.

  • See if you can do easier problems of the same type.

  • If you are still in difficulty then look for a fresh approach (e.g. from textbooks containing similar examples).

  • Discussion with other students often produces new insight. Explaining ideas to others will help to clarify your own thinking. Don’t be afraid to share ideas; you will find that others share your difficulties but each can contribute new ideas or understandings. At the same time, try to develop your own thinking: there is little value in simply copying someone else’s work and any work to be handed in for assessment should be your own, of course. If you feel you have reached a dead-end, then don’t give up – in the course of solving a problem new key concepts may appear and the suggestions above may need to be retraced.

  • If you get really stuck with a problem, don’t sit at it for too long. Try something else, have a break, or sleep on it. It is important to know when to stop trying: and note your difficulties for discussion at the next problems class or tutorial.

  • Do not be afraid to ask your tutor, lecturer or other relevant member of staff for help.

Reflecting on your work.

  • Even if you haven’t completed your work, hand in what you have done. This will enable your lecturer to advise you on how best to proceed. It is better to do this than to hand in a full answer copied from another student.

  • Try to place the problem in context relative to the whole course; look back at what you have done, check the arguments and look for links between the result and the method with the rest of the course.

  • In writing out the final solution, pay careful attention to the logical flow of the argument. Read it back to yourself to see that it makes sense. Check your working, with numbers if possible. Keep the question and your solution filed for future reference at revision time.

  • Handing in work is most important, you will recieve feedback which allows you to find out whether your ideas are right, whether you’ve understood the problem and whether your solutions are correct (even if the final answers look right). You may have to do this anyway for continuously assessed courses.

  • If model solutions are provided then make use of them. Compare your solutions with those given. Sometimes you may learn more from a model solution to a problem for which you have found a correct solution than you will from solutions to problems which have baffled you.

  • If in the course of attempting problems which are set you feel that there are shortcomings in either your lecture notes or your understanding, then make a note of these and bring them up at the next tutorial. You may also find it helpful to attempt further problems and exercises from relevant textbooks or to transcribe worked examples.

  • If several of you experience considerable difficulties with a certain part of your work, then approach your lecturers and tutors collectively. If the problems persist then approach your staff-student committee (perhaps through your class representative).

Problems involving Proofs

Understanding theorems and their proofs is often a source of difficulty. When tackling a new theorem, find out first of all what it says and how it is used in the subject. Next try to identify the three or four main steps in the proof to get a broad outline of how it works.

Two of the more difficult things in learning mathematics are the construction of proofs and the appreciation of theorems. These are inseparable from thinking mathematically. Practice and experience have no substitutes but you will find the following useful:

  • Use diagrams and pictures to explore the truth or falsity of statements through appropriate illustrations.

  • Study lecture notes and set textbooks in order consciously to identify the assumptions and the conclusions of theorems.

  • Try to locate when each assumption is used in a proof. It may help to highlight such places in a particular colour. You can sometimes appreciate why such an assumption is needed by investigating a counter-example to the theorem in which the assumption does not hold.

  • Identify definitions you need to know (find out what the definitions are if you are unsure).

  • Identify the notation used (and find out what it means if you don’t already)

  • Attempt proofs of ‘parallel’ theorems mentioned in lectures or left as exercises in the lecture notes or in the set textbook. Look for simple special cases. Familiarity with the simple makes the complex easier.

  • Gain practice in the various methods of proof. Identify the ideas in proofs and note their re-appearance.

Summary

Get started:

  • Identify and key concepts/notation/words.
  • Use any similar, worked examples to help guide you.
  • Draw a picture (this can help clarify the problem).
  • Get stuck in and try different things; making mistakes is the only way to learn what works (and what doesn’t). You will get better at ‘spotting’ the way to go as you gain experience.

If you get stuck:

  • Take a short break, then re-read the question and ensure you understand each part.
  • Discuss with your peers.
  • Try a similar (perhaps slightly easier) problem.
  • Ask your tutor/lecturer for some advice.