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Why is problem solving important?

Cognitively Guided Instruction (CGI)Achievement increases when teachers know more about the problem solving strategies of their students

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What is problem solving?Using heuristics? Memorized rules of thumb, and procedures Using algorithms? A complete procedure.

• Gestalt: Reasoning about the whole?

• What you do when you don’t know what to do?

• Representing the problem until you can solve it. (Polya, Mayer)

•“Coming up with the answer” (4th grader)

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Types of problems

Well defined- “What is four times six?” Small problem space Clear operation Clear result

Ill defined- “What could we do to improve our quality of life?” Vast problem space Arguable solution

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Story Problem

Floor tiles are sold in squares 30 centimeters on each side. How much would it cost to tile a rectangular room 7.2 meters long and 5.4 meters wide if the tiles cost $0.72 each?

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A Problem Solving Model(Mayer, 1987)

Problem Representation

Problem Solution

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Problem Representation

Translation

Linguistic: knowledge of language used in the problem. For example…•“Floor tiles” and “tiles” refer to the same objects. •The room is a rectangle.

Semantic: facts about the world. One meter equals 100 centimeters.

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Problem Representation (Cont.)

Translation

Integration: Finding the right problem schemas to use on this item, for example: this is an area problem so…Area= Length x Width, and then you figure you

can…• Multiply the two dimensions of the room to

get its area,• Multiply sides of a tile to get its area, • Divide the total room area by the area of one tile to find number of tiles that will fit the room.

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Translation

IntegrationProblemRepresentation

ProblemSolution

Planning and Monitoring

Strategic information: Knowledge of howTo plan a solution once a representation has been created for the problem.•Solving an equation•Evaluating your progress

Execution

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Translation

IntegrationProblemRepresentation

Planning & monitoring

Execution ProblemSolutionProcedural: Carry out the steps

7.2 x 5.4= 38.88.3 x .3= .09 tile area in meters38.88/.09= 432 total tiles 432 x .72= $311.04

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The Answer is…..

$311.04

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TranslationIntegration

ProblemRepresentation

Can I correctly interpret the words in the problem? (linguistic & semantic)

Can I putsome structureto the givens?(schematic) Plan & Monitor

ExecutionCan I workwith the problemnow that I have represented it?(strategic)

Can I carry out the remaining steps my plan calls for to get an answer?(procedural)

ProblemSolution

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What Hinders Problem Solving?

Functional fixedness: failing to see a tool in a new or unorthodox way.Response set: Problems bias you toward a particular type of strategy. What percent is 4?

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What helps Problem Solving?

Draw a pictureUse multiple representations of the contentBreaking the problem down to identify subgoals.Working backwards and forwardsRemaining calm and relaxed: Regulated adrenaline levels.

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What kind of problem solving do we emphasize?

Most textbooks emphasize the problem solution phase of problem solving.Our problem solving is influenced by our representations. Placing values on a number line Asking students to draw a picture of a problem

Research has indicated that we can improve our representation skills (Brenner, et al, 1999)

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Discovery and Guided Discovery

Discovery: Open problem solving with little teacher involvement.

Guided Discovery: Teacher Structures the problem solving task and provides guidance at key points

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Direct or Expository Instruction

Highly organized presentation: Teacher organizes material and sequences itPresentation of procedures firstEmphasizes deductive logic

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Guided Discovery v.s. Direct or Expository

Instruction

Should we give concepts or procedures first?Can you think of a way that discovery has worked or not worked for you?How would you use discovery learning in a lesson?

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OutlineWhat is transfer?Why is studying transfer important? A brief transfer task.What are the components of transfer?Are there different types of transfer?What do we know about how transfer works?What kind of new research is being done?

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What is Transfer?

Transfer is taking knowledge acquired from one set of tasks and applying that knowledge to a new task.Fundamental question: “Are people able to see commonalties in related problems and make use of them in their attempts to find solutions?”

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Importance of Transfer I

Transfer has become an important part of the definition of conceptual understanding.Part of NCTM and NSES curriculum standards.A means of testing or demonstrating understanding.

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Importance of Transfer II

Crucial difference between experts and novices. Ex. Feynman: O-rings in cold

temp.

Teachers and schools aspire to be teaching critical thinking skills.Do you transfer knowledge between problems?

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In a far away land there was a fortress with an evil dictator who mistreated the local villagers.At that time there was also a general who commanded a rebel army, and he wanted to seize control of the fortress and free the villagers.There were lots of roads to the dictator’s castle but the dictator had them booby-trapped with bombs so that only a small number of persons could travel safely on each road at a time.

If the general had sent sufficient forces down one of the roads to the castle, the explosives would have blown-up his troops.

The Problem...

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The general decided to spread out his forces and send small numbers of troops down each of the roads to the fortress. The general’s troops were then able to avoid the bombs and converged on the fortress at the same time, capturing it from the dictator, and liberating the villagers.

-Adapted from Gick & Holyoak, 1980.

The Solution…..

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Given a human being with an inoperable stomach tumor, and rays which destroy organic tissue at sufficient intensity, by what procedure can one free the patient of the tumor by using these rays and at the same time avoid destroying the healthy tissue which surrounds it?

Consider a New Problem...

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The Dictator Problem...

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The Tumor Problem...

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The Two Problems are Analogous

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Important Points About This Transfer Task

This problem is one of the classic problems in cognitive psychology.The problem is a good example of how the cognitive study of transfer began.Today, the definition of transfer has widened: Research now studies the application of different areas of knowledge to new problems.

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What are the Components of Transfer?

Surface cues. Problem structure.

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An Example from School Math...

Billy has three marbles and Alley has five marbles. How many do they have altogether?Surface cues: Marbles, Billy and Alley.Structure: 3 + 5= 8.How do we help students to look past the surface features to the problem structure?

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Different Types of Transfer I

Positive Transfer: Previous learning facilitates performance. Moving from one column addition to

two column addition.

Negative Transfer: Previous learning hinders performance. Drawing familiar objects.

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Different Types of Transfer II

Near Transfer- The problems share many of the same features. Example: One column addition to two

column addition.

Far Transfer - Different surface features, but the problems share some underlying similarities. Example: Using addition to take an

inventory.

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Tumor ProblemAn example of far positive transfer.The two problems are analogous.Both use the divide and conquer strategy.Without hints 30% success rate on second problem. 10% success rate working only the second problem.

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How Does Transfer Work?

Schema theory.Reasoning by analogy.

Schemata at work: Blind Date

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Schema Theory

Schemata are organized networks of knowledge that you use to make sense of the world.Smallest unit of meaning in thinking: Analogous to a concept.Give coherence to information by influencing your: Ability to pick useful strategies. Ability to remember relevant facts that go

with the current problem.

dog

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has...

Ears

Legs

Mammal

Is a has...

Cute Annoying Dangerous

Big teethis...Small

warm blooded

is...

Bark

cold nose

mouth

produces..

are

A Schema Fit for a Dog..

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has...

Ears

Legs

Mammal

Is a has...

Cute Annoying Dangerous

Big teethis...Small

warm blooded

is...

Bark

cold nose

mouth

produces..

are

Activated Dog Schema

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has...

Ears

Legs

Mammal

Is a has...

Cute Annoying Dangerous

Big teethis...Small

warm blooded

is...

Bark

cold nose

mouth

produces..

are

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The Schema of an Expert

Larger schemas: More knowledge.More integrated Knowledge: More connections within and between schemata.Perhaps learning that is integrated with prior knowledge transfers more frequently.

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Thinking by Analogy

Schema activation: “It’s like a puppy but it never gets any bigger?” “It goes yip, yip, and never shuts up.”One schema of knowledge (puppy) is mapped on to Chihuahua schema.

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Dictator (base) Tumor (target)

Problem features:FortressMined roadsStreams of attacking troops that vary in size.

Solution:Small groups of troops move along each road to converge on the fortress

Problem features:TumorSurrounding tissueRays that can vary in intensity

Solution:Weak rays are all focused on the tumor

Analogous Problem Structures

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Positive Transfer Remains Elusive

Studies finding low rates of transfer.In everyday life it appears that we transfer knowledge all the time.Typical experiments may be a difficult way to study successful transfer.

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Transfer in the Classroom...

Recent research in classrooms has shown some evidence of transfer (Brenner, et al., 1997).Connecting tables, graphs, and equations: multiple representations of content.

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2x+1Multiply the number by 2 then

add 1 to it.We ordered… We received…

2 5

5 11

? 21

3 ?

02468

101214161820

1 2 3 4 5 6 7 8 9 10

Bags of Flour

Y=2x+1

-Bags of flour-We ordered… We received…1 3

4 9

6 13

3 7

Can you write a rule that explains the shipping errors?

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A Classroom Based Study of Transfer.

The study involved two groups of students: A treatment group receiving our experimental

unit. A comparison group that received their regular

instruction.

The students were local middle school students taking their first pre-algebra class. After 20 days of instruction the two groups were tested.

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Transfer Results...

Students were able to transfer their understandings of tables, graphs and equations to a function word problem more frequently than the comparison group.Interviews with the students over a year later indicated that they still remembered content from the lessons.

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Conclusion: Will this Lecture Transfer?

You now... Know the definition of transfer. Know that it is recognized an

important part of the national math and science standards.

Have worked some problems designed to demonstrate transfer.

Have considered new ideas about how problems might be related. (near / far)

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Conclusion (cont...)

Can think about whether previous learning ultimately helps or hurts your efforts to solve a new problem. (positive/negative)

Have an understanding about the basic cognitive principles that underlie transfer. (schemata, analogical reasoning)

Have seen an example of some of the latest research in transfer.

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Can you connect all 9 dots using only four straight lines with no gaps between the four lines?

Puzzle Problem

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Quiz

5 points multiple choice, 5 short answerKnow the information processing model in depth.Know the types of transferKnow about experts/novicesDifferent types of memory: schema, script, procedural, declarative….What helps/hurts problem solvingGeneral features of Mayer’s problem solving model

Our memory experiment: serial position curve