3.1
Unit 3 Problem solving: basic skills
Consider the action of making a cup of instant coffee. If you analyse the processes you need to go through, they are quite complicated. Just the list of items you need is quite long: a cup, a teaspoon, a jar of coffee, a kettle, water, and milk and sugar if you take them. Having found all these items, you fill the kettle and boil it;
use the teaspoon to put coffee into the cup;
pour the boiling water into the cup, just to the right level; stir; add milk and sugar; then put all the things you used away again. In fact one could break this down even more: we didn’t really go into very great detail on, for example, how you boil the kettle.
Although this is complicated, it is an everyday task that you do without thinking.
However, if you encounter something new, which may be no more complicated, the processes required to achieve the task may need considerable thought and planning.
Most of such planning is a matter of
proceeding in a logical manner, but it can also require mathematical tasks, often very simple, such as choosing which stamps to put on a letter. This thought and planning is what constitutes problem solving.
Solving most problems requires some sort of strategy – a method of proceeding from the beginning which may be systematic or may involve trial and error. This development of strategies is the heart of problem solving.
Imagine, for example, trying to fit a number of rectangular packages into a large box. There are two ways of starting. You can measure the large box and the small packages, and calculate the best way of fitting them in. You may make some initial assumptions about the
best orientation for the packages, which may turn out later to be wrong. Alternatively, you may do it by trial and error. If you have some left over at the end that are the wrong shape to fit into the spaces left, you may have to start again with a different arrangement. Either way, you will have to be systematic and need some sort of strategy.
With some problems the method of finding an answer might be quite clear. With others there may be no systematic method and you might have to use trial and error from the start.
Some will require a combination of both methods or can be solved in more than one way.
The words ‘problem solving’ are also used in a mathematical sense, where the solution sought is the proof of a proposition. ‘Problem-solving’ as tested in thinking skills
examinations does not ask for formal proofs, but rather asks for a solution, which may be a calculated value or a way of doing something.
Although many of the problems we shall look at here use numbers and require numerical solutions, the mathematics is usually very simple – much of it is normally learned in elementary education. Many problems do not use numbers at all.
As we saw in Chapter 1.3, there are three clearly defined processes that we may use when solving problems:
• identifying which pieces of data are relevant when faced with a mass of data, most of which is irrelevant
• combining pieces of information that may not appear to be related to give new information
• relating one set of information to another in a different form – this involves using experience: relating new problems to ones we have previously solved.
When solving problems, either in the real world or in examinations, you are given, or have, or can find, information in various forms – text, numbers, graphs or pictures – and need to use these to come up with a further piece of information which will be the solution to the problem.
The processes described above are the fundamental building blocks of problem-solving and can be expanded into areas of skill that may be brought together to solve more complex problems. The chapters in this unit divide these into smaller identifiable skill areas which can be tested using multiple-choice questions. Examples of such sub-skills are searching for solutions and spatial reasoning (dealing with shapes and patterns).
Later units deal with more complex problems, which can only be solved using several of these sub-skills in combination, and are closer to the sort of problem solving encountered in the real world.
The activity below gives an example of a simple problem; you can give either a simple answer or a more complicated one, depending on the degree of detail you consider necessary.
Luke has a meeting in a town 50 miles away at 3 p.m. tomorrow. He is planning to travel from the town where he lives to the town where the meeting is by train, walking to and from the station at both ends.
List the pieces of information Luke needs in order to decide what time he must leave home.
Then work out how you would proceed to plan his journey from these pieces of information.
Activity
Commentary
The chances are that you missed some vital things. You may have thought that all he needed was a railway timetable. Unless you approached the problem systematically, you may not have thought of everything.
Let us start by thinking of everything he does from leaving his house to arriving at the meeting.
1 He leaves his house.
2 He walks to the station.
3 He buys a train ticket.
4 He goes to the platform.
5 He boards the train when it arrives.
6 He sits on the train until it reaches the destination.
7 He leaves the train.
8 He walks to where his meeting is being held.
You can construct the pieces of information he needs from this list. They are:
1 The time taken to walk from his house to the station.
2 The time needed to buy a ticket.
(Remember to allow for queues!) 3 The time to walk to the platform.
4 The train timetable.
5 The time taken to walk from the station to where the meeting is being held.
Did you find them all? Perhaps you thought of some that I missed. For example, I didn’t think of allowing for the train being late. You could estimate this by experience and allow some extra time.
Now, to find out when he should leave home we need to work backwards. If his meeting is at 3 p.m., you can work out when he must leave the destination station to walk to the meeting.
You can then look at the timetable to see what is the latest train he can catch (allowing extra for the train to be late if appropriate). Then see from the timetable when this train leaves his home town. Continuing, you can determine when he should have bought his ticket, and when he should leave home.
Of course, you could do the whole thing by guesswork, but you might get it all wrong and, more to the point, you cannot be confident that you will have got it right.
In the sense we are using the word in this book, a ‘problem’ means a situation where we need to find a solution from a set of initial conditions. In the following chapters we shall look at different sorts of problem, different kinds of information, and how we can put them together to find solutions to the
problems. These chapters will lead you through the types of problem-solving exercises you will encounter in thinking skills examinations and give some indications about how you might
4 The following questions are based on a very simple situation, but require clear thinking to solve. Some are easier than others.
A drawer contains eight blue socks and eight black socks. It is dark and you cannot tell the difference between the two colours.
a What is the smallest number you will have to take out to ensure that you have a matching pair?
b What is the largest number you can take out and still not have a matching pair?
c What is the smallest number you can take out to be sure that you have one of each colour?
d What is the largest number you can take out and still have all of one colour?
e What is the smallest number you can take out to be sure you have a blue pair?
Answers and comments are on pages 315–16.
1 Imagine you are going to book tickets for a concert. List the pieces of information you need and the processes you need to go through in order to book the tickets and get to the concert. In what order should you do them? First list the main things, then try to break each down into smaller parts.
2 Consider something you might want to buy, such as a car, mobile phone or computer.
Make a list of the pieces of information you would need in order to make a decision on which make or model to buy.
3 Find a mileage chart that gives the distances between various towns (these can be found in most road atlases or on the internet). Pick a base town and four other towns. Consider making a journey that starts at the base town, takes in the other four and ends at the base town. In what order should you visit the towns to minimise the journey?
End-of-chapter assignments
approach such problems. However, learning to solve problems is a generally useful life skill and also, we hope, fun!
Summary
• In this chapter we have looked at what a problem is and how the word can be used in different ways.
• We have seen how information is used to contribute to the solution of a problem.
• We have looked at how various methods of using information can lead to effective solutions.