1. A DEVELOPMENTAL STUDY OF PLANNING: MEANS-ENDS ANALYSIS Move the discs from the right side to the left side as shown in the 1 st display Cannot place a larger disc on a smaller disc Move only one disc at a time

2. 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

3. 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

4. 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

5. DEVELOPMENTAL FINDINGS CAUSAL INFERENCE Contiguity infants in their 1 st 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

6. 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.

7. 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.”

8. A Child Study Of Analogical Reasoning 3, 4 and 5 year old children are presented with a three dimensional display and 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 tires that have been delivered to his garage and put them up on a shelf. But the shelf is too high and he doesn’t have a ladder so he can’t reach the shelf by himself. 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 of his tractor so he can take them to the market. But Bill isn’t tall enough to reach the top of the tractor by himself. How can he solve his problem? Solution: Stack two tires and stand on top of them.

9. Helping Children Engage in Analogical Reasoning “Seeing past superficial differences” Verbally prompt child to describe problem solutions across different problems Use multiple examples of problems with the same solution Use three dimensional displays and have the children “enact” the solutions

10. 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

11. 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 Problem Classic Problem Jean Piaget (1896-1980) Children’s Cognitive Development University of Geneva 1234 X

12. 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

13. Jean Piaget: Rods Task Classic Problem SteelCopperBrass

14. 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

15. 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

16. 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.

17. 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 4 th graders judge the deductive problem as true

18. 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 1 st problem was certain