Imagine for a moment that you are a parent and that you normally drop your daughter off at school on your way to work. One day you both pile into the car and discover that the car won’t start. What do you do?
Let’s suppose, for the sake of argument, that it is important you get to work and your daughter gets to school as soon as possible. However, you have to get your car fixed. You decide to call a taxi and phone a garage from work to ask them to go out and look at the car. You then realise the garage will need a key for the car. While you are waiting for the taxi you call the garage and explain the situation. You say you will drop the keys off at the garage on your way to work. Okay so far. The next problem is how to get home and pick your daughter up from school in the evening. You decide that the simplest thing would be to take a bus from work that stops close to the garage and pick up the car (assuming there’s nothing seriously wrong with it). You can then go and pick up your daughter in the car. She may have to wait in school for a while but, with a bit of luck, she shouldn’t have to wait longer than about quarter of an hour.
Five minutes later, the taxi arrives.
The point of this little story is that all you have done to solve the problem is to stand beside the telephone and think. It is exceedingly useful to be able to imagine— to think about—the results of an action or series of actions before you actually perform them. But what exactly does it mean to think? Thinking involves reasoning about a situation, and to do that we must have some kind of dynamic “model” of the situation in our heads. Any changes we make to this mental model of the world should ideally mirror changes in the real world. So where does this mental model or mental representation come from?
Before engaging in any kind of conscious problem-solving activity the solver needs to understand the problem in the first place. Understanding a problem means building some kind of representation of the problem in one’s mind, based on what the situation is or what the problem statement says and on one’s prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving.
What kind of representation do we form of a problem? When we talk of a mental representation we are referring to the way that information is encoded. The word “rabbit” can be represented visually as a visual code, and by what it sounds like as a phonological code. We also know what “rabbit” means, so there must be a semantic code (a “meaning code”). The way we encode information from a problem situation is often based on what we are told or what we read. When we read a piece of text, for example, we not only encode the information that is explicitly stated, but we also have to make inferences as we read to make sense of the text. Most of these inferences are so automatic that we are often unaware that we have made any inferences at all. Bransford, Barclay, and Franks (1972) presented people with the following sentence:
They later gave a recognition test to some of the subjects that included the sentence: 2. Three turtles rested on a floating log and a fish swam beneath it.
Bransford et al. had hypothesised that subjects would draw the inference that the fish swam beneath the log (notice that this is not stated in the original sentence). Indeed, the subjects who were presented with the second sentence on the recognition task were as confident that it had been presented originally as those subjects who had been given the original sentence on the recognition task. The point here is that one’s memory of a situation, based on a reading of a text, may include the inferences that were drawn at the time the representation of the text was constructed or retrieved.
Problem solving, then, involves building a mental representation of a problem situation, including any inferences you make on reading the problem, that will allow you to carry out some action. In other words, for the mental representation to be of any use it has to include some idea of what you can do that will allow you to move from the initial problem situation to the goal. Now it follows that if you don’t know much about the domain, or you have never attempted this kind of problem before, then your understanding of the problem is unlikely to be all that good. Glaser (1984, p. 93) explains why:
At the initial stage of problem analysis, the problem solver attempts to ‘understand’ the problem by construing an initial problem representation. The quality, completeness, and coherence of this internal representation determine the efficiency and accuracy of further thinking. And these characteristics of the problem representation are determined by the knowledge available to the problem solver and the way the knowledge is organised.
This does not mean that the representation has to be “complete” before any problem solving can take place. If you had a “complete” representation of a problem then you wouldn’t have a problem, as you would know exactly how to get from where you are now to where you want to be. A problem only exists when it is not immediately obvious how to get from where you are now to your goal. An adequate representation should at least allow you to see what moves you can possibly make and allow you to start heading towards your goal. Chapter 2 deals with how we build mental representations of problems and how we use them to work towards a solution. (The topic is revisited in Chapter 7.)
However, there is also the case of those problems where our initial representation gets us nowhere. The difficulty here lies in finding a different way of representing the problem (this is sometimes referred to as
“lateral thinking”, although the concept of re-representing problems in order to find a solution goes back
long before the term was invented). Unfortunately, knowing that you should find a new way of representing the problem does not in fact help you very much—you still have to find this “new way of representing” it. Nevertheless when a new representation comes to mind a solution is often immediately obvious; or, at least, you often know what to do so that a solution can be reached very quickly. The kind of problem solving that involves being stuck for a while until you suddenly see what to do is called insight, and is the subject of
Chapter 3.
Insight is not confined to so-called “insight problems”. The same phenomenon may occur when solving typical textbook algebra problems, for example, or in solving simple everyday problems. Here again the initial representation may be extremely unhelpful, and only when a new representation is found can the poor student or DIY enthusiast apply the solution procedure that is made obvious by the new representation. The study of insight problems can therefore tell us something about everyday problems and textbook problems.