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)
2
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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4
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?
5
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
Integration
Problem
Representation
Planning and Monitoring
Strategic information: Knowledge of how
To plan a solution once a representation has
been created for the problem.
•Solving an equation
•Evaluating your progress
Problem
Solution
Execution
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Translation
Integration
Problem
Representation
Planning & monitoring
Execution
Procedural: Carry out the steps
7.2 x 5.4= 38.88
.3 x .3= .09 tile area in meters
38.88/.09= 432 total tiles
432 x .72= $311.04
Problem
Solution
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The Answer is…..
$311.04
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Can I correctly interpret the words in the problem? (linguistic & semantic)
Can I put
some structure
to the givens?
(schematic)
Can I work
with the problem
now that I have
represented it?
(strategic)
Translation
Integration
Problem
Representation
Plan & Monitor
Problem
Solution
Execution
Can I carry out the
remaining steps my plan
calls for to get an answer?
(procedural)
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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?
13
What helps Problem Solving?
Draw a picture
Use multiple representations of the
content
Breaking the problem down to identify
subgoals.
Working backwards and forwards
Remaining calm and relaxed: Regulated
adrenaline levels.
14
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)
15
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
16
Direct or Expository
Instruction
Highly organized presentation: Teacher
organizes material and sequences it
Presentation of procedures first
Emphasizes deductive logic
17
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?
18
Outline
What 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?”
20
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.
21
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?
22
The Problem...
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.
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The Solution…..
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.
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Consider a New Problem...
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?
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The Dictator Problem...
26
The Tumor Problem...
27
The Two Problems are Analogous
28
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?
31
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.
33
34
Tumor Problem
An 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.
35
How Does Transfer Work?
Schema theory.
Reasoning by analogy.
Schemata at work: Blind Date
36
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.
37
dog
A Schema Fit for a Dog..
cold
nose
Mammal
has...
Is a
is...
warm
blooded
Ears
mouth
produces..
Bark
has...
.
is.
Small
Legs
has
..
is...
Big teeth
are
Cute
Annoying
Dangerous38
Activated Dog Schema
cold
nose
Mammal
has...
Is a
is...
warm
blooded
Ears
mouth
produces..
Bark
has...
.
is.
Small
Legs
has
..
is...
Big teeth
are
Cute
Annoying
Dangerous
39
40
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.
41
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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Analogous Problem Structures
Dictator (base)
Tumor (target)
Problem features:
Problem features:
Fortress
Tumor
Mined roads
Surrounding tissue
Streams of attacking troops
Rays that can vary
that vary in size.
in intensity
Solution:
Solution:
Small groups of troops
Weak rays are all
move along each road to
focused on the
converge on the fortress
tumor
43
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.
44
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.
45
2x+1
-Bags of flourWe ordered… We received…
1
3
Multiply the number by 2 then
add 1 to it.
We ordered… We rece ived…
4
2
5
5
11
?
21
3
?
6
3
9
13
7
Can you write a rule
that explains the shipping
errors?
20
18
16
14
12
10
8
6
4
2
0
Bags of Flour
Y=2x+1
1
2
3
4
5
6
7
8
9
10
46
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.
50
Puzzle Problem
Can you connect all 9 dots using only four straight lines with no
gaps between the four lines?
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Quiz
5 points multiple choice, 5 short answer
Know the information processing model in
depth.
Know the types of transfer
Know about experts/novices
Different types of memory: schema, script,
procedural, declarative….
What helps/hurts problem solving
General features of Mayer’s problem solving
model
Our memory experiment: serial position curve
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