Analogical Reasoning

Types of reasoning

• Content free reasoning:– Deductive reasoning

• Reasoning by similarity– Inductive reasoning– Analogical reasoning:

• analogical transfer: solving problem in one domain based on solution in another domain

• Analogical inference: generalizing properties/relations from one domain to another

Analogical transfer

Reason from base problem (previously solved) to target problem

• Recognition: identify a potential analog• Abstraction: abstract general principle from base

problem• Mapping: apply principle to target

Radiation Problem

• Problem – Patient has an abdominal tumor (in center of

body) – Radiation will kill tumor – But rays strong enough to kill tumor will also

destroy healthy tissue that rays pass through on way to tumor

• Solution?

Solution

INCORRECT CORRECT

Gick & Holyoak (1980)

Only 20% solved it

Participants were more likely to solve the problem if they were given an analogy.

Fortress Problem

• Army general wants to attack a fortress in center of forest.

• Many roads lead to fortress like spokes of a wagon wheel.

• All the roads have land mines that are triggered by heavy traffic.

• If entire army travels down a road, land mines will go off.

Solution?

Solution

The general sends small groups of soldiers down each of the roads, co-ordinating their movements so that they all arrive at the fortress at the same time.

Structural Similarities

Fortress Problem Radiation Problem

Fortress TumorMined Roads Surrounding tissueAttacking troops Rays varying in

intensity

Small groups of troopsConverging on fortress Weak rays focused

on tumor

Recognition Process

• Radiation problem hard problem:– Preceded by Fortress problem, 20% solved it– If given hint that problems are related, 92%

solved it• Subjects do not produce corresponding solutions

with any great frequency unless given a hint• These results demonstrate the difficulty of

recognizing the relevance of one problem and its solution to another

Gick & Holyoak (1980)

Transfer aspects of fortress problemto radiation problem

• Fortress problem was given with three different solutions: – Dispersion: the general’s army approaching the

fort from several roads at once.– Tunnel: a general digging a tunnel to the fort– Open-route: a general attacking the fort via a

supply route

Problem Solutions

The DISPERSION solution, in which small doses of radiation are aimed at the tumor from different directions so that the radiation accumulates and destroys the tumorous tissue;

Solutions to the Radiation problem fell into three main categories:

The OPEN-ROUTE solution, in which radiation is delivered to the tumor through the esophagus (tube leading from the throat to the stomach);

The TUNNEL solution, in which an incision is made which leaves the tumor in place but which allows direct delivery of radiation.

Distribution (%) of Solutions

What do the results show?

Abstraction Process

– Does performance improve if principle of base problem is stated? No

– Showing diagrams helps? No

– Showing multiple problems with related solutions helps? Yes

Analogical Inference

“An atom is like the solar system”

TARGET BASE

Knowledge about the base domain can be used to reason about the target domain. Structure mapping theory is a theory for how this could work.

Structure Mapping Theory(Dedre Gentner, ’83, ’89)

Theory on how to interpret an analogy

1) set up correspondences

2) focus on matches of relations, not attributes

3) focus on higher-order relations

4) extend knowledge about target by mapping relations from source to target

ALIGN RELATIONS

BASED ON ALIGNED

RELATIONS,ALIGN OBJECTS

“the nucleus is like the

sun”

Mental Models• Knowledge structure used to understand how

objects and physical processes work• Sometimes, mental models are based on poor

analogies and can lead to incorrect predictons• Example: thermostats. How to heat up a room

quickly?

| | 50 | | 60 | | 70 | | 80 | |

Current temp. Desired temp.

OR?Kempton, ‘86

if you said “80”, what might your mental model be based on?

Children’s Mental Models

• Children were told facts:– Columbus

sailed around the earth

– Shown space photographs

• What is their Conception of earth given these facts?

Nussbaum ‘79

“there are two earths, one is flat, the other is a sphere out in space”

Intuitive Physics

• Intuitive physics is based on naive mental models of physical processes

•The diagram shows a curved tube, lying flat on a table. A ball is shot into the opening and out the other end. What is the ball’s path after it shoots out the tube?

Kaiser, McClosky & Profitt ‘86

OR?

ResultsKaiser, McClosky & Profitt ‘86

(The C-curve problem is the same but shows just the C part of the spiral)

Explanation by Kaiser et al.

• Grade school children start constructing theories of motion

• Theories of persistence of motion: “an object persists in the way it is moving, unless there is friction”. E.g., bouncing ball continues to bounce

• Theory is wrongly applied to spiral problem: overgeneralization

Intuitive Physics with a set of gears

Schwartz & Black, 1999

Suppose we swing this gear in this

direction

What is the direction of this gear?

Schwartz & Black, 1999

Suppose we swing this gear in this

direction

?

Schwartz & Black, 1999

Suppose we swing this gear in this

direction

?HINT:

Time A Time B

Path XPath Y

Path Z

From McCloskey, 1983

Mental models might be initially be based on perceptual simulations (e.g., turning the gears in your mind)

After training, these models might shift to more abstract rules to solve problems