Analogical Problem Solving

Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate of the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumor will also be destroyed. At lower intensities, the rays are harmless to healthy tissue, but they will not affect the tumor, either. What type of procedure might be used to destroy the tumor with the rays and at the same time avoid destroying the healthy tissue?

A small country was ruled from a strong fortress by a dictator. The fortress was situated in the middle of the country, surrounded by farms and villages. Many roads led to the fortress through the countryside. A rebel general vowed to capture the fortress. The general knew that an attack by his entire army would capture the fortress. He gathered his army at the head of one of the roads, ready to launch a full-scale direct attack. However, the general then learned that the dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely, since the dictator needed to move his troops and workers to and from the fortress. However, any large force would detonate the mines. It therefore seemed impossible to capture the fortress. However, the general devised a simple plan. He divided his armies into small groups and dispatched each group to the head of a different road. When all was ready, he gave the signal and each group marched down a different road. Each group continued down its road so that the entire army arrived together at the fortress at the same time. In this way, the general captured the fortress and overthrew the dictator.

Analogies involve reasoning about relations, in particular about relational similarity, so that a correspondence is established between one set of relations and another.

In principle, reasoning may involve at least three types of correspondences:

"property correspondence" - e.g., is a 3 inch disk more like a quarter or a pizza?

"concept correspondence" - e.g., is a turkey a bird?

"system correspondence" - e.g., is the internal structure of an atom like the solar system?

Analogical reasoning is primarily concerned with systemic correspondences, as in problem solving, where a solution to a known problem may be applied to solving a structurally similar problem.

How is Analogical Reasoning Measured?

'Classical' Analogies

Traditional measure is the a : b : : c : d task, defined by Aristotle as "an equality of proportions -- involving at least four terms-- when the second is related to the first as the fourth is to the third." An example might be

bicycle : handlebars : : ship : rudder.

Young children perform poorly on the classical analogy task. Piaget gives an example of a 5-year-old who has arranged four pictures into the sequence bird : feathers : : dog : hair. Asked to explain why the pairs of pictures go together in this fashion, the child replies, "the dog eats the bird, those are the feathers!"

Early failure on these tasks is usually attributed to the late appearance of the ability to represent or reason about "higher-order" relations.

Reasoning by analogy in problem solving

This depends on the similarity of relational structure between a known solved problem and a novel problem. For example, it has been shown that solving the Duncker's radiation problem can be facilitated by presenting a (solved) problem concerning an army and mined roads.

Young children can perform well on problem solving by analogy. Anne Brown gives an example of a 3-year-old who has reasoned that the hawkmoth caterpillar and the porcupine fish have evolved analogous solutions to warding off predators:

Adult: are they the same kind of stories?

Child: Yes, they are the same Both of them have a mean guy that wants to eat them all up They both get mean and scary so they [the predators] run away!

Adult: They're pretty smart, huh?

Child: Just like me!

What Develops?

1. The structural ability to reason analogically, including the ability to resist false counter-suggestions that would destroy the analogy. On this view, children's early success at analogical problem solving is either accidental or an artifact of the experimental methods that are used.

2. The knowledge to apply particular analogies. On this view, children's early failure on classical analogy problems is due to use of problems involving facts or relations with which the children are not sufficiently familiar.

3. The metacognitive ability to seek analogical relations between known solved problems and novel problems.

Developmental Theories of Analogical Reasoning

Not surprisingly, Piaget offered a stage theory to explain development of analogical reasoning (with regard to classical 4-term analogies). His theory was based on the following set of tasks:

1. Children were given sets of pictures featuring objects and parts of objects and asked to sort them into "pairs that go together." This tested ability to relate the a and b (or c and d) terms. These 'lower-order' relations were mostly associative in nature.

2. Children were then asked to sort the pairs of pictures into "sets of four that go together." This tested their ability to form analogies. These 'higher-order' relations were primarily causal.

3. If children formed analogies successfully, their ability to resist counter-suggestions was tested. For example, for the analogy

bicycle : handlebars : : ship : rudder

children might be asked, "Would a bell also go with the bicycle? Then what would you have to choose for a ship?"

Piaget's Stages

Stage I (Preoperational) - egocentric responses using idiosyncratic relations.

Stage II (concrete operational) - occasional, limited success

Stage III (formal operational) Success on all aspects of the tasks.

Counterevidence to Piaget

Goswami & Brown reasoned that children might fail Piaget's tasks because they lack knowledge of the relations entailed. On this view, children ought to be able to solve classical analogy problems provided the analogies are based on familiar relations.

The task: children were presented with a horizontal array of 3 pictures (the a, b, and c) terms and were given 5 pictures from which to chose one to complete the sequence.

Analogies were based on familiar, physical, causal changes, e.g.,

playdoh : cut playdoh : : apple : ?

Distractors included an appearance match (e.g., a ball), an associative match (e.g., a banana), an alternative object with the correct transformation (e.g., cut bread), and the correct object with an incorrect transformation (e.g., a bruised apple).