82 Unit 3 Problem solving: basic skills
3.2 How do we solve problems?
We have seen that a problem consists of a set
of information and a question to answer. In
order to solve the problem we must use the
information in a certain way. The way in
which we use it may be quite straightforward –
it may for example be simply a matter of
searching a table for a piece of data that
matches given conditions. In other cases,
instead of searching for a piece of data, we may
have to search for a method of solution. The
important thing in either case will be to have a
strategy that will lead to the solution.
Many publications give (in various forms)
the procedure:Data Process Solution
This is all well and good, and indeed
represents a way problems can be solved. It
says nothing about what the words and, in
particular, the arrows mean. It is in this detail
that the key to problem solving is found. In
simple terms, we are concerned with
identifying the necessary pieces of data and
finding a suitable process. There are no hard
and fast rules; different problems must be
approached in different ways. This is why
problem solving appears in thinking skills
examinations; it tests the ability of candidates
to look at situations in different ways and to be
able to use many different strategies to find
one that works. Whilst a knowledge of the
different categories of problem, as identified
by the syllabuses and the various chapters of
this unit, will help, you will always need to
have an open mind and be prepared to try
different approaches.
There are several ways problems may be
approached. A term that is used a lot is
‘heuristic’ (see for example How to Solve Itby G. Polya [Penguin, 1990] – a book on
mathematical problem solving). This word
comes from the Greek ‘to find’ and refers to
what we might call ‘trial and error’ methods.
Alternative methods depend on being
systematic: for example, an exhaustive search
may lead to an answer. Previous experience of
solving similar types of questions will always
be a help.
Imagine you are going out and can’t find
your house keys. Finding them is a problem in
the sense meant by this section of the book.
The heuristic method (and sometimes the
quickest) is to run around all the likely places
to see if they are there. After the likely places,
you start looking at the less likely places, and
so on until they turn up or you have to resort
to more systematic methods. There are two
systematic ways of searching. The first (using
experience) involves thinking carefully about
when you last came into the house and what
you did; this can be the quickest method. The
other (which in mathematical terms is often
known as the ‘brute force’ method) involves
searching every room of the house thoroughly
until they are found. This is often the most
reliable method but can take a very long time
and most people will use it as a last resort.
When people are solving problems, they
may use all of these methods, often in the
order given above. This is quite logical, as the
heuristic method can lead to a very rapid
solution whilst the systematic search is
slowest. One of the prime skills you need in
tackling problem-solving questions in
examinations is to make a good judgement of
which method is the most appropriate one to
use in any set of circumstances.