PROBLEM SOLVING
If I ask you, ‘What is 6 + 6?’, unless you are a young schoolchild, you will be able to retrieve the answer 12 straight from memory. On the other hand, if I ask you, ‘What is 37 + 568?’, you have to do some problem solving. Being numerate means that you know how to solve this problem: it calls for a standard application of arithmetic procedures, and these procedures can be drawn from memory. This kind of problem-solving is called simply routine problem solving. In contrast, creative problem solving cannot be done according to a formula because there are no standard procedures in memory. As we experience the same problem type over and over again, what was at first creative may become routine, of course.
Search space
Consider this anagram problem:
What two-word phrase can be made out of these letters:
SPROGOLIBVELM?
What strategies would you employ to solve it? The simplest is blind search, in which you just move the letters around blindly until a phrase appears. The possibilities here are enormous, so blind search is clearly not a very smart way to proceed. But how do we constrain the search? There are some sequences of letters in English that are legal and commonplace (like ‘pro’), some that are rare (like ‘goli’), and some that are downright impossible (like ‘blvm’). So a smarter strategy is to try constructing fragments from common grammatically legal combinations, then trying sequences that are more and more rare. Fragments will serve to cue word possibilities that you know, which will help speed up the search. With practice, people who like anagrams in crosswords develop a number of ways to constrain the search space. All problems can be construed in terms of search spaces, though this is more obvious with some problems than with others. In their classic book Human Problem Solving, Newell and Simon (1972) illustrat d the problems of search space more thoroughly than anyone had before. One problem they studied in some detail is the following (cover the solution and try the problem first):
For each letter, substitute one digit, such that the whole thing fits the laws of (base 10) arithmetic; in the example below D = 5:
DONALD
+ GERALD
ROBERT
Solution:
526485
+197485
723970
You will notice that your perception of what is involved in the problem increases as you work on it. For instance, to begin with, you may not have noticed the problem of carrying. That is, you will need to add 1 to a column left of the one you are working on if the sum exceeds 9.
Newell and Simon (1972) collected speak-aloud protocols – they required people to say aloud what they were doing while they were attempting problems like this. This helped them to analyse in detail the steps people go through in problem solving. There were two main findings: 1. People set up initial representations of problems, which influence the search space. 2. They employ general purpose techniques, called heuristics, which help constrain the search space. So, with the problem above, Newell and Simon found several possible representations. For instance, some people saw it as being one based on word meaning. Suppose the puzzle was:
BILL + WAS = KING
A person might reason that BILL = William the conqueror → 1066, therefore B = 1, I = 0, L = 6. This kind of reasoning turns out to be inappropriate for our particular problem given above. Other examples might be described as typographic – E looks a bit like 3, etc. – and cryptographic – using some sort of systematic code, like A = 1, B = 2, etc. Neither applies to our particular example, but the important point here is that our initial conception of the problem can alter the way in which we attempt to solve it. Understanding how people develop a problem space – the representation of a problem in the head of an individual – is a major aspect of work on problem solving. (The more general idea of a Mental Model is discussed later in this chapter.) For instance, when we learn how to problem solve, we must first recognize when seemingly different problems have a common logical structure.
Looking for a common structure
A classic study of how underlying common structure might be spotted was carried out by Gick and Holyoak (1980; 1983). They examined how experience with a puzzle called the ‘military problem’
(Holyoak, 1984) affected performance on a second problem,
called the ‘radiation problem’ (Duncker, 1945):
The military problem
A general wishes to capture a fortress located at the center of a country. There are many roads radiating out from the fortress. All have been mined, so that although small groups of men can pass over the roads safely, any large force will detonate the mines. A full-scale direct attack is therefore impossible. What should the general do? (Holyoak, 1984, p. 205) The radiation problem Imagine that you are a doctor treating a patient with a malignant stomach tumor. You cannot operate because of the severity of the cancer, but you must destroy the cancer. You could use highintensity X-rays. However, the intensity needed is such that the beam would destroy the healthy tissue that the rays must pass through. A lower intensity beam would not harm the healthy tissue, but would also not destroy the cancer. How can you use X-rays to destroy the tumor without destroying the healthy tissue? (adapted from Duncker, 1945) The solution to the two problems is very similar. In the case of the radiation problem, the solution is o direct weak X-rays from a number of different points outside of the body, and to set the sources up so that the beams converge at the site of the tumor. That way, no single beam is strong enough to cause damage to healthy tissue, but the combined effect on the tumour is enough to destroy it. The military problem has a solution based on the same principle: small groups of soldiers are sent along different roads at the same time, converging as one big army at the fortress. Gick and Holyoak had participants do the military problem first. One group of participants simply read the problem in the belief that they were just to recall the wording. Under those circumstances, only 30 per cent derived the correct solution to the radiation problem. However, if the participants were given two similar problems before the radiation problem, then there was more transfer. In general, though, the more superficially similar problems are, the better the transfer (Holyoak, 1990). So spotting the similarity of problems is far f om automatic.
If I ask you, ‘What is 6 + 6?’, unless you are a young schoolchild, you will be able to retrieve the answer 12 straight from memory. On the other hand, if I ask you, ‘What is 37 + 568?’, you have to do some problem solving. Being numerate means that you know how to solve this problem: it calls for a standard application of arithmetic procedures, and these procedures can be drawn from memory. This kind of problem-solving is called simply routine problem solving. In contrast, creative problem solving cannot be done according to a formula because there are no standard procedures in memory. As we experience the same problem type over and over again, what was at first creative may become routine, of course.
Search space
Consider this anagram problem:
What two-word phrase can be made out of these letters:
SPROGOLIBVELM?
What strategies would you employ to solve it? The simplest is blind search, in which you just move the letters around blindly until a phrase appears. The possibilities here are enormous, so blind search is clearly not a very smart way to proceed. But how do we constrain the search? There are some sequences of letters in English that are legal and commonplace (like ‘pro’), some that are rare (like ‘goli’), and some that are downright impossible (like ‘blvm’). So a smarter strategy is to try constructing fragments from common grammatically legal combinations, then trying sequences that are more and more rare. Fragments will serve to cue word possibilities that you know, which will help speed up the search. With practice, people who like anagrams in crosswords develop a number of ways to constrain the search space. All problems can be construed in terms of search spaces, though this is more obvious with some problems than with others. In their classic book Human Problem Solving, Newell and Simon (1972) illustrat d the problems of search space more thoroughly than anyone had before. One problem they studied in some detail is the following (cover the solution and try the problem first):
For each letter, substitute one digit, such that the whole thing fits the laws of (base 10) arithmetic; in the example below D = 5:
DONALD
+ GERALD
ROBERT
Solution:
526485
+197485
723970
You will notice that your perception of what is involved in the problem increases as you work on it. For instance, to begin with, you may not have noticed the problem of carrying. That is, you will need to add 1 to a column left of the one you are working on if the sum exceeds 9.
Newell and Simon (1972) collected speak-aloud protocols – they required people to say aloud what they were doing while they were attempting problems like this. This helped them to analyse in detail the steps people go through in problem solving. There were two main findings: 1. People set up initial representations of problems, which influence the search space. 2. They employ general purpose techniques, called heuristics, which help constrain the search space. So, with the problem above, Newell and Simon found several possible representations. For instance, some people saw it as being one based on word meaning. Suppose the puzzle was:
BILL + WAS = KING
A person might reason that BILL = William the conqueror → 1066, therefore B = 1, I = 0, L = 6. This kind of reasoning turns out to be inappropriate for our particular problem given above. Other examples might be described as typographic – E looks a bit like 3, etc. – and cryptographic – using some sort of systematic code, like A = 1, B = 2, etc. Neither applies to our particular example, but the important point here is that our initial conception of the problem can alter the way in which we attempt to solve it. Understanding how people develop a problem space – the representation of a problem in the head of an individual – is a major aspect of work on problem solving. (The more general idea of a Mental Model is discussed later in this chapter.) For instance, when we learn how to problem solve, we must first recognize when seemingly different problems have a common logical structure.
Looking for a common structure
A classic study of how underlying common structure might be spotted was carried out by Gick and Holyoak (1980; 1983). They examined how experience with a puzzle called the ‘military problem’
(Holyoak, 1984) affected performance on a second problem,
called the ‘radiation problem’ (Duncker, 1945):
The military problem
A general wishes to capture a fortress located at the center of a country. There are many roads radiating out from the fortress. All have been mined, so that although small groups of men can pass over the roads safely, any large force will detonate the mines. A full-scale direct attack is therefore impossible. What should the general do? (Holyoak, 1984, p. 205) The radiation problem Imagine that you are a doctor treating a patient with a malignant stomach tumor. You cannot operate because of the severity of the cancer, but you must destroy the cancer. You could use highintensity X-rays. However, the intensity needed is such that the beam would destroy the healthy tissue that the rays must pass through. A lower intensity beam would not harm the healthy tissue, but would also not destroy the cancer. How can you use X-rays to destroy the tumor without destroying the healthy tissue? (adapted from Duncker, 1945) The solution to the two problems is very similar. In the case of the radiation problem, the solution is o direct weak X-rays from a number of different points outside of the body, and to set the sources up so that the beams converge at the site of the tumor. That way, no single beam is strong enough to cause damage to healthy tissue, but the combined effect on the tumour is enough to destroy it. The military problem has a solution based on the same principle: small groups of soldiers are sent along different roads at the same time, converging as one big army at the fortress. Gick and Holyoak had participants do the military problem first. One group of participants simply read the problem in the belief that they were just to recall the wording. Under those circumstances, only 30 per cent derived the correct solution to the radiation problem. However, if the participants were given two similar problems before the radiation problem, then there was more transfer. In general, though, the more superficially similar problems are, the better the transfer (Holyoak, 1990). So spotting the similarity of problems is far f om automatic.
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