A DEVELOPMENTAL STUDY OF PLANNING: MEANS-ENDS ANALYSIS

Move the discs from the right side to the left side as shown in the 1st display

Cannot place a larger disc on a smaller disc

Move only one disc at a time

DEVELOPMENTAL FINDINGS FROM THE TOWER OF HANOI PROBLEM

Performance success from 3 to 6 yrs

older children can solve problems with more moves

What happens when a child can’t move a disc directly toward the goal

younger children break the rules

older children start to plan moves in advance

Case Study

Helping Children Solve The Tower of Hanoi Problem:Aiding Representation and Lessening the Cognitive Load

Represent Goal State so the child doesn’t have to keep it

in mind.

Incorporate the Key Rule about placing larger discs

on smaller ones in the actual task materials

Provide the child a Meaningful Story so the

task becomes more comprehensible

IMPORTANT PROBLM-SOLVING PROCESSES – CAUSAL INFERENCE

Contiguity

events occur close together in time and space

Precedence

event labeled “cause” precedes event labeled “effect”

Covariation

cause and effect consistently occur together

DEVELOPMENTAL FINDINGS CAUSAL INFERENCE

Contiguity

infants in their 1st year already use both temporal & spatial contiguity to infer causality

Precedence

By age 5 children consistently use the order of events (A-B-C) to infer cause-effect

Covariation

By age 8 children can use consistent co- occurrence to infer causality even with a time delay

The Role of Analogy in Problem Solving (Gick & Holyoak, 1983)

Problem #1: How to irradiate a tumor without harming surrounding tissue.

Solution: Attack tumor from many different angles with weak x-rays; At the point of intersect (tumor) x-rays are full strength.

Problem #2: How to attack a fortress without losing men on the roads into the fortress which are mined and prevent a large army from approaching the fortress.

Solution: Break up army and approach fortress in small groups from many of the roads; when the small forces meet at the fortress, the army will be at full strength.

Results Gick & Holyoak (1983)

Percentage of subjects who “see” the relation across problems

One story analog before the “radiation” problem = 29%

Two story analogs before the “radiation” problem = 45%

Two story analogs plus a “principle” = 62%

Principle:

“The general attributed his success to an important principle: If you need a large force to accomplish some purpose, but are pre-vented from applying such a force directly, many smaller forces applied simultaneously from different directions may work just as well.”

A Child Study Of Analogical Reasoning

3, 4 and 5 year old children are presented with a three dimensional displayand given a simple problem. They are asked to enact the solution.

Problem #1:John, the garage mechanic, has a problem. He needs to take all of the tiresthat have been delivered to his garage and put them up on a shelf. But theshelf is too high and he doesn’t have a ladder so he can’t reach the shelf byhimself. How can he solve his problem?

Solution: Stack two tires and stand on top of them.

Problem #2:Bill, the farmer, has a problem. He needs to put his bales of hay on top ofhis tractor so he can take them to the market. But Bill isn’t tall enough toreach the top of the tractor by himself. How can he solve his problem?

Solution: Stack two tires and stand on top of them.

Helping Children Engage in Analogical Reasoning

“Seeing past superficial differences”

Verbally prompt child to describe problemsolutions across different problems

Use multiple examples of problems with the same solution

Use three dimensional displays and have the children “enact” the

solutions

IMPORTANT PROBLM-SOLVING PROCESSES – SCIENTIFIC & LOGICAL

REASONING

Do children understand the logic of experimentation?

Not until they approach formal operations

Do children understand the logic of deductive reasoning?

Not fully until they approach formal operations and receive explicit instruction

• 1,2,3,and 4 contain colorless, odorless liquids.

• X contains an “activating solution”.

• Some combination of liquids (always including X) will give a YELLOW color.

• How can you find the combination that makes YELLOW?

Jean Piaget: Mixing Colors ProblemClassic Problem

Jean Piaget (1896-1980)Children’s Cognitive Development

University of Geneva 1 2 3 4

X

Jean Piaget: Mixing Colors Problem

Classic Problem

Jean Piaget (1896-1980)Children’s Cognitive Development

University of Geneva

1+x

2+x

3+x

4+x

1+2+x

1+3+x

1+4+x

2+3+x

2+4+x

3+4+x

1+2+3+x

1+2+4+x

1+3+4+x

2+3+4+x

1+2+3+4+x

Jean Piaget: Rods TaskClassicProblem

Steel Copper Brass

CONTROL OF VARIABLES: CANAL PROBLEM

You are asked to determine how canals should be designed to optimize boat speed. Working with an actual canal system and timing the boats from start to finish, you can conduct experiments to identify factors that influence speed. (boats are towed with a string and pulley system)

Variables:

large and small boats

square, circular, and diamond shaped boats

canal can be shallow or deep

you can make a boat heavier by adding a barrel

Counterintuitive: boats are faster in deeper canal, shallow canal due to greater turbulence

CONTROL OF VARIABLES: CANAL PROBLEM

A typical 11 year old child’s experimentation:

Trial 1: small, circular, light boat in a deep canal

Trial 2: large, square, heavy boat in a shallow canal

After Trial 2, the child concluded that weight makes a difference, but when asked to justify the conclusion, he simple said that if the boat in Trial 2 had been light it would have gone faster.

Trial 3: small, diamond-shaped, light boat in a shallow canal

Child predicts that the boat in Trial 3 would go faster than the boat in Trial 2 because “it depends on how much edging is on the thing” (a hypothesis about the shape)

Note: child fails to systematically test hypotheses, only notices confirmatory evidence

CONTROL OF VARIABLES: CANAL PROBLEM

A college student’s experimentation:

After numerous trials the student summarizes what she has accomplished so far.

Well, so far we worked with small boats. First, light, and then we added the weight to each of them, and we found that without the weight they would go faster. We also found out that the diamond shape was the fastest, with the circle being next. And the slowest was the square. Let’s take the bigger boats in the deeper water. We’ll start with the square and go in order.

Student notices the counterintuitive result with depth of canal. She immediately searches for a plausible explanation.

“My God! It does have an effect! It takes longer in shallow water! The only thing I can figure out is that the depth of water would have something to do with the buoyancy. The added water, adds more buoyancy, making the boat sit up higher in the water.

UNDERSTANDING THE DIFFERENCE BETWEEN DEDUCTIVE & INDUCTIVE REASONING

Deductive Problem

All poggops wear blue boots Tombor is a poggop Does Tombor wear blue boots?

Inductive Problem

Tombor is a poggop Tombor wears blue boots Do all poggops wear blue boots?

Developmental Findings

Kindergarteners see both conclusions as true

4th graders judge the deductive problem as true

UNDERSTANDING THE DIFFERENCE BETWEEN DEDUCTIVE REASONING & GUESSING

Two toys are hidden in two different boxes Puppet looks in one container and sees a “red” toy, so concludes the “blue” toy is in the other box

How certain is the puppet?

Two toys are hidden in two different boxes Puppet announces that the “red” toy is in box 1

How certain is the puppet?

Developmental Findings

4 yr olds don’t see any difference in the problems, by 9 yrs all children knew the 1st problem was certain