ACTIVITY 3.7 Rate problems

1. The Mad Bird problem

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start towards each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead at 100 mph to the front of the second train. When the bird reaches the second train it turns back and flies towards the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour, how many miles will the bird have flown before the trains meet? (Posner, 1973, pp. 150–151)

Figure 3.14. Two variants of the Nine Dots problem presented in MacGregor et al. (in press).

2. The River Current problem

You are standing by the side of a river which is flowing past you at the rate of 5 mph. You spot a raft 1 mile upstream on which there are two boys helplessly adrift. Then you spot the boys’ parents 1 mile downstream paddling upstream to save them. You know that in still water the parents can paddle at the rate of 4 mph. How long will it be before the parents reach the boys? (Hayes, 1989a, p. 25)

(problems involving some kind of cover story rather than a simple sum) require the solver to represent the problem appropriately before a way of solving it (a solution procedure) becomes obvious. The study of insight problems therefore has important consequences for our understanding of how we solve all kinds of problems. Activity 3.7 contains two examples of distance=rate×time problems and ways of representing them.

There are two features of these kinds of algebra problems that are important. The first is that you often have to throw common sense out of the window (Birds don’t fly at 100 mph and anyway why would it fly back and forth like that? How do the parents know their sons are adrift on a raft two miles away?). Common sense, or general world knowledge, can often interfere with such problem solving. Second, both problems are written in such a way that certain features are more salient than others—that is, they stand out. One effect of this is that you may be led into representing the problems in a certain way. In the first case you may be led to represent the problem from the point of view of the bird. In the second case you may be led to represent the problem from the point of view of the person on the bank. Both these representations of the problems make the solutions harder.

A representation of the Mad Bird problem from the bird’s point of view, as in Figure 3.15, involves trying to add up how far the bird travels each time it flies from one train to the other. This makes the solution rather difficult. However, the problem is readily solved if we ignore the bird for the moment and concentrate on the trains as in Figure 3.16. All you need to find out to begin with is how long it takes the trains to meet. Both trains travel at a very sluggish 25 miles an hour and meet after travelling 25 miles so

they take an hour to meet. The problem now becomes how far does the bird travel in an hour at a speed of 100 miles an hour. The answer is obviously 100 miles.

The River Current problem can be represented in two ways as in Figures 3.17 and 3.18. In Figure 3.17 the parents are paddling at 4 mph but the river current is 5 mph so they are actually heading away from the observer at 1 mph. The boys are travelling in the same direction at 5 mph so the difference between the two speeds is 4 mph. That is, the boys are approaching their parents at 4 mph. If they travel 4 miles in one hour then they will travel 2 miles in half an hour.

In Figure 3.18 we can forget about the observer and take the point of view of the boys on the raft. We can also forget about the river current, as it affects the parents and the boys equally (similarly, if you are sitting on a train travelling at 80 mph you don’t normally think of a ticket inspector as moving at 82 mph—you can ignore the train’s speed as it affects you both equally and regard the inspector as moving at a much more sedate 2 mph). The only relevant figure therefore is the speed at which the parents are paddling, i.e., 4 mph. In which case they will reach the boys in half an hour.

Figure 3.15. The Mad Bird problem from the bird’s point of view.

Figure 3.16. The Mad Bird problem from the trains’ point of view.

Figure 3.17. The River Current problem from the observer’s point of view.

INFLUENCING PROBLEM REPRESENTATIONS: THE EFFECT OF INSTRUCTIONS

The two examples just quoted illustrate how we are often “persuaded” to form a particular representation of a problem by the way a problem is worded. The Mad Bird problem is likely to generate a mental representation involving a bird flying between the two trains. The River Current problem invites you to represent the problem in terms of the speed of the river past an observer, so that’s what you do. You are given information, some of which appears to be particularly salient by the way the problem is worded, and so the representation you form is based on that salient information.

The effect of instructions on the representations people form of problems was investigated by Hayes and Simon (1974; Simon & Hayes, 1976) using two variants of the Tower of Hanoi problem. When two problems have an identical underlying structure (as revealed, for example, by state-space analysis) they are said to be isomorphic (see also Chapter 4). The two isomorphs of the Tower of Hanoi problem used by Simon and Hayes are shown in Table 3.1. In the move isomorph, globes are transferred from one monster to another in much the same way that disks in the Tower of Hanoi are moved from one peg to another and with the same constraints—if a peg (monster) has more than one disk (globe) only the smallest can be transferred to another peg (monster); a large disk (globe) cannot be placed on a peg (moved to a monster) that has a smaller disk

TABLE 3.1

The Monster problem

A move isomorph

Three five-handed extraterrestrial monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighbourhood, both monsters and globes come in exactly three sizes with no others permitted: small, medium, and large. The medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-sized globe. As this situation offended their keenly developed sense of symmetry, they proceeded to transfer globes from one monster to another so that each monster would have a globe proportionate to his own size. Monster etiquette complicated the solution of the problem as it requires:

1. that only one globe may be transferred at a time,

2. that if a monster is holding two globes, only the larger of the two may be transferred; 3. that a globe may not be transferred to a monster who is holding a larger globe.

By what sequence of transfers could the monsters have solved the problem?

Figure 3.18. The River Current problem ignoring the observer.

A change isomorph

Three five-handed extraterrestrial monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighbourhood, both monsters and globes come in exactly three sizes with no others permitted: small, medium, and large. The medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-sized globe. As this situation offended their keenly developed sense of symmetry, they proceeded to shrink and expand the globes so that each monster would have a globe proportionate to his own size. Monster etiquette complicated the solution of the problem as it requires:

1. that only one globe may be changed at a time;

2. that if two globes are of the same size, only the globe held by the larger monster can be changed; 3. that a globe may not be changed by a monster who is holding a larger globe.

By what sequence of changes could the monsters have solved the problem? (Adapted from Simon & Hayes, 1976, pp. 168–169)

(globe). The change isomorph is trickier. It involves changing the sizes of globes, and the globes stay with a single monster. Actually what has happened is that the equivalent of pegs and disks have swapped round in the two representations. The equivalent of a disk in the move isomorph is actually a peg in the change isomorph.

Simon and Hayes developed a computer program called UNDERSTAND that would take as input the problem instructions and identify the goal, states, and operators that applied to the problem. This information was then fed to a general-purpose program called the General Problem Solver. The UNDERSTAND program predicted that the representations solvers would generate (in terms of states and operators) would be entirely determined by the problem instructions. For example, a move in the change isomorph would be represented as a particular monster changing the size of its globe. When Simon and Hayes (1976) analysed the verbal protocols of people attempting to solve the two variants, they found that their subjects were powerfully influenced by the version of the story they were given, as UNDERSTAND predicted.

This chapter has shown that our initial representation of a problem can often determine its difficulty, or even whether it is possible to solve it at all. In insight problems in particular we often hit an impasse early on that prevents further search for a solution. Whereas in well-defined problems a solution may be found by searching through a problem space, in insight problems a solution can only be found by searching for a useful problem space. Recently theorists have shown that the processes involved in trying to solve ill- defined or abstractly defined insight problems are essentially similar to those used in solving well-defined problems. We are now in a position to say much more precisely why such problems are difficult.

SUMMARY

1. The way we represent problems when we encounter them has a powerful influence on our ability to solve them. Occasionally, our representation of a problem is so poor that we get stuck—we reach an impasse.

2. It is sometimes possible to get out of an impasse by such means as: • focusing on a different aspect of the problem;

• looking at extreme conditions; • trying to find an analogy;

• re-encoding aspects of the problem—trying to see them in different ways; • relaxing constraints that we have inadvertently placed on the problem.

3. Gestalt psychologists were interested in how we represent and “restructure” problems. They viewed thinking as often either reproductive, where we use previously learned procedures without taking too much account of the problem structure, or productive, where thinking is based on a deep understanding of a problem’s structure and is not structurally blind.

4. They were also interested in the failures of thinking due to:

• functional fixedness, when we fail to notice that an object can have more than one use;

• the effects of set, when we apply previously learned procedures when a simpler procedure would work.

5. Insight problems would appear to pose problems for information-processing theories of problem solving because they do not appear to involve sequential, conscious, heuristic search. Consequently some researchers have viewed insight as a special case of problem solving. Others have tried to fit insight into traditional information-processing accounts:

• Kaplan and Simon saw insight as a search for a representation rather than a search in a representation.

• Keane pointed out that functional fixedness could be understood in terms of the salient properties of objects. That is, functional fixedness is due to the organisation of semantic memory.

• Ohlsson argued that we access one operator at a time based on our initial interpretation of a problem. Retrieving operators is an unconscious process involving spreading activation (and hence also depends on the organisation of semantic memory). When we cannot retrieve an operator based on our initial representation we reach an impasse and have to change the representation before we can access a relevant operator.

6. MacGregor et al. have produced a detailed model of the processes involved in solving abstractly defined insight problems such as the Nine Dots problem that incorporate heuristics such as means-ends analysis, and show how the conditions for an insightful solution might arise.

7. The study of insight problems is important because the processes involved are often the same as those involved in establishing an appropriate representation of a situation or word problem. Without an appropriate representation, no relevant operators can be accessed and problem solving reaches an impasse or becomes more difficult.

8. The way a problem is phrased can have a powerful effect on the representation that we form of it. Simon and Hayes showed how an initial representation can be formed from problem instructions and how it can be modelled.