Brief Introduction to Problem
Solving
"If you cannot solve a problem, then
there is an easier problem you cannot solve: find
it."
(George Polya: How to Solve It.
Princeton, 1945).
Last revised Spring 2002. This is a chapter-length introduction problem solving and roles of computers in problem solving. Click here for a short book on the same topic
Introduction
Problem and Task
Team
Domain Specificity
What is a Formal
Problem?
Representations
of a Problem
Representing
Problems Using Computers
Problem Posing
and Clarification
Some
Problem-Solving Strategies
A General
Strategy for Problem Solving
Working Toward
Increased Expertise
Transfer of
Learning
Project-Based Learning
Summary of
Important Ideas
References
Introduction
This document gives a brief overview of the "subject" of problem solving and of roles of Information and Communications Technology (ICT) in problem solving. It is targeted specifically toward preservice and inservice teachers. The ideas from this document can be woven into instruction in almost any curriculum area or methods course.
This document focuses both on solving problems and
on accomplishing tasks. We will use the term problem
solving to refer to both solving problems and
accomplishing tasks. Our goal is to help improve the
quality of education that students receive in our
educational system.
Our educational system attempts to differentiate between
lower-order cognitive (thinking) skills and higher-order
cognitive (thinking) skills. In recent years there has
been increased emphasis on higher-order skills. In very
brief summary, we want students to learn some facts, but we
also want them to learn to think and solve problems using
the facts.
Often the "thinking" that we want students to do is to
recognize, pose, and solve complex, challenging problems.
Thus, one of the goals of education is to help students to
get better at posing, representing, and solving problem. A
few schools actually offer specific courses on problem
solving. For the most part, however, students learn about
problem solving through instruction in courses that have a
strong focus on a specific content area. Every teacher
teaches problem solving within the specific subject matter
areas of their curriculum.
Many people have observed that the "every teacher teaches
problem solving" is a haphazard approach, and that the
result is that students do not get a coherent introduction
to problem solving. When a student reaches a specified grade
level, can the teacher be assured that the student has
learned certain fundamental ideas about posing,
representing, and solving problems? Can the teacher assume
that a student knows the meaning of the terms problem,
problem posing, and problem solving? Can the teacher be
assured that the students know a variety of general purpose
strategies for attacking problems? In our school system at
the current time, the answer to these questions is "no."
Thus, each teacher is left with the task of helping their
students master both the fundamentals of problems solving
and then the new problem-solving topics that the teacher
wants to cover. This document covers the basics
(fundamentals) of problem solving. It is designed as a
general aid to teachers who need to cover the fundamentals
with their students.
Of course, the fundamentals need to be interpreted and
presented at a grade-appropriate level. This document does
not try to do that. It is left to the individual reader to
understand the fundamental ideas and then present them in a
manner that is appropriate to their students.
This document places particular emphasis on several
important problem-solving ideas:
- Posing, representing, and solving problems are
intrinsic to every academic discipline or domain. Indeed,
each discipline is defined by the specific nature of the
types of problems that it addresses and the methodologies
that it uses in trying to solve problems.
- There are some tools (for example, reading and
writing) that are useful in addressing the problems in
all disciplines. Information and Communications
Technology (ICT) provides us with some new and powerful
tools that are useful aids to problem solving in every
discipline.
- Much of the knowledge, techniques, and strategies for
posing, representing, and solving problems in a specific
domain requires a lot of knowledge of that domain and may
be quite specific to that domain. However, there are also
a number of aspects of posing, representing, and solving
problems that cut across many or all domains, and so
there can be considerable transfer of learning among
domains. Our educational system should help all students
gain a reasonable level of knowledge and skill in these
broadly applicable approaches to problem solving.
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Problem and Task
Team
Donald Norman is a
cognitive scientist who has written extensively in the area
of human-machine interfaces. Norman (1993) begins with a
discussion of how tools (physical and mental artifacts) make
us smart. David
Perkins (1992) uses the term "Person Plus" to refer to a
person making use of physical and mental tools. In many
situations, a person with appropriate training, experience,
and tools can far outperform a person who lacks these
aids.
In this document we use the term Problem or Task Team
(P/T Team) to refer to a person or a group of people and
their physical and mental tools. Figure 1 illustrates the
P/T Team. These concepts are explained in subsequent
paragraphs.
Figure 1. People aided by physical and mental
tools.
Figure 1 shows a person or a group of people at the
center of a triangle of three major categories of aids to
solving problems and accomplishing tasks:
- Mental aids. Even before the invention of
reading and writing, people made use of notches on bones
and other aids to counting and to keeping track of
important events. Reading, writing, and arithmetic are
mental aids. These have led to the development of books,
math tables, libraries, calculators, computers, and many
other mental aids. Mental aids supplement and extent
capabilities of a person's mind.
- Physical aids. The steam engine provided the
power that led to the beginning of the industrial
revolution. Well before that time, however, humans had
developed the flint knife, stone ax, spear, bow and
arrow, plow, hoe, telescope, and many other aids to
extend the physical capabilities of the human body. Now
we have cars, airplanes, and scanning electron
microscopes. We have a telecommunications system that
includes fiber optics, communications satellites, and
cellular telephones
- Education. Education is the glue that holds it
all together. Our formal and informal educational systems
helps help people learn to use the mental and physical
tools as well as their own minds and bodies.
ICT is a combination of both mental and physical aids.
One way to think about this is the use of computers to
automate factory machinery. Such machinery stores/contains a
certain type of knowledge, and the machinery can use that
knowledge to carry out certain manufacturing tasks. An
artificially intelligent, computerized
robot provides another example of a combination of
mental and physical tools.
The mental aids and physical aids components of a P/T
Team are dynamic, with significant changes occurring over
relatively short periods of time. The pace of change of ICT
seems breath taking to most people.
On the other hand, our formal educational system has a
relatively slow pace of change. This has led to the
interesting situation of many preschool children growing up
with routine access to mental and physical aids, learning
their use through our informal educational system, and then
encountering formal education that is woefully inadequate in
dealing with such aids. For example, many elementary school
students have more ICT knowledge and skill than do their
teachers.
People who are skilled at functioning well in a P/T Team
environment have a distinct advantage over those who lack
the knowledge, skills, and access to the facilities. Such
analysis leads to the recommendation that the P/T Team and
problem solving should be a central themes in education.
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Domain Specificity
Each academic discipline focuses on a category of
problems that help to define the discipline and
methodologies for solving these problems. Chemistry,
history, and mathematics are different disciplines because
they address quite different types of problems and have
developed quite different methodologies for addressing
problems.
Moreover, each academic discipline has a huge amount of
accumulated knowledge. A mathematician can spend a lifetime
studying a specific subdomain such as algebra, geometry, or
statistics, and not fully master just one specific
subdomain. A musician cannot hope to gain a high level of
expertise in each musical instrument and type of music.
Similarly, an artist cannot hope to gain a high level of
expertise in each art medium.
Research into problem solving has indicated that one needs considerable domain-specific knowledge and skills to solve pose, represent, and solve problems within that domain. People use the term "domain specificity" when discussing this idea. Thus, it is not surprising that formal education is usually broken up into specific courses that focus on specific components of specific domains. This approach allows courses to be designed and taught by people who are relatively competent in the subject matter domain of the course.
Domain specificity is a major challenge to our
educational system. For the most part, "real world" problems
cut across different domains. Thus, we teach students in a
domain-specific manner and environment, and expect that they
will transfer their knowledge and skills to the
interdisciplinary problems they encounter outside of school.
This is a huge leap of faith, and what actually happens is
that relatively few students make such transfers of
learning.
Fortunately, the situation is not quite as bleak as it
sounds. While many aspects of problem solving are specific
to the academic area (domain) of the problem, there are also
many ideas about problem solving that cut across all
domains. Thus, with appropriate education and experience, a
person can gain some general expertise in problem solving
that is useful in addressing any new problem that they might
encounter.
Moreover, there is a gradual increasing understanding of
how to teach for transfer. That is, progress in the domain
of transfer of learning is beginning to provide teachers
with specific information about how to teach for
transfer.
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What is a Formal
Problem?
There is a substantial amount of research literature as
well as many practitioner books on problem solving
(Polya,
1957); Frensch and Funke, 1995; Moursund,
1996).
Problem solving consists of moving from a given initial
situation to a desired goal situation. That is, problem
solving is the process of designing and carrying out a set
of steps to reach a goal. Usually the term problem is
used to refer to a situation where it is not immediately
obvious how to reach the goal. The exact same situation can
be a problem for one person and not a problem (perhaps just
a simple activity or routine exercise) for another
person.
Figure 2. Problem-solving process--how to achieve final
goal?
Here is a formal definition of the term problem. You
(personally) have a problem if the following four conditions
are satisfied:
- You have a clearly defined given initial
situation.
- You have a clearly defined goal (a desired end
situation). (Some writers talk about having multiple
goals in a problem. However, such a multiple goal
situation can be broken down into a number of single goal
situations.)
- You have a clearly defined set of resources that may
be applicable in helping you move from the given initial
situation to the desired goal situation. There may be
specified limitations on resources, such as rules,
regulations, and guidelines for what you are allowed to
do in attempting to solve a particular problem.
- You have some ownership--you are committed to using
some of your own resources, such as your knowledge,
skills, and energies, to achieve the desired final
goal.
These four components of a well-defined problem are
summarized by the four words: givens, goal, resources, and
ownership. If one or more of these components are missing,
we call this a problem situation. An important aspect
of problem solving is realizing when one is dealing with a
problem situation and working to transform that into a
well-defined problem.
People often get confused by the resources (part 3) of the definition. Resources do not tell you how to solve a problem. Resources merely tell you what you are allowed to do and/or use in solving the problem. For example, you want to create a nationwide ad campaign to increase the sales by at least 20% of a set of products that your company produces. The campaign is to be completed in three months, and not to exceed $40,000 in cost. Three months is a time resource and $40,000 is a money resource. You can use the resources in solving the problem, but the resources do not tell you how to solve the problem. Indeed, the problem might not be solvable. (Imagine an automobile manufacturer trying to produce a 20% increase in sales in three months, for $40,000!)
Problems do not exist in the abstract. They exist only
when there is ownership. The owner might be a person, a
group of people such as the students in a class, or it might
be an organization, or a country. A person may have
ownership "assigned" by his/her supervisor in a company.
That is, the company, or the supervisor has ownership, and
assigns it to an employee or group of employees.
The idea of ownership is particularly important in
teaching. If a student creates or helps create the problems
to be solved, there is increased chance that the student
will have ownership. Such ownership contributes to intrinsic
motivation--a willingness to commit one's time and energies
to solving the problem.
The type of ownership that comes from a student
developing a problem that he/she really wants to solve is
quite a bit different from the type of ownership that often
occurs in school settings. When faced by a problem
presented/assigned by the teacher or the textbook, a student
may well translate this into, "My problem is to do the
assignment and get a good grade. I don't have any interest
in the problem presented by the teacher or the textbook." A
skilled teacher will help students to develop projects that
contain challenging problems, and the problems are ones that
the students really care about.
Many teachers make use of project-based
learning within their repertoire of instructional
techniques. Within PBL, students often have a choice on the
project to be done (the problems to be addressed, the tasks
to be accomplished), subject to general guidelines
established by the teacher. Thus, students have the
opportunity to have an increased level of ownership of the
project they are working on. Research on PBL indicates that
this ownership environment can increase the intrinsic
motivation of students.
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Representations
of a Problem
There are many different ways to represent a problem. A
problem can be represented mentally (in your own mind),
orally, in writing, on a computer, and so on. Each type of
representation has certain advantages and disadvantages.
From a personal or ownership point of view, you first
become aware of a problem situation in your mind and body.
You sense or feel that something is not the way that you
want it to be. You form a mental representation, a mental
model, of the problem situation. This mental model may
include images, sounds, or feelings. You can carry on a
conversation with yourself--inside your head--about the
problem situation. You begin to (mentally) transform the
problem situation into a well-defined problem.
Mental representations of problems are essential. You
create and use them whenever you work on a problem. But,
problems can be represented in other ways. For example, you
might represent a problem with spoken words and gestures.
This could be useful if you are seeking the help of another
person in dealing with a problem. The spoken words and
gestures are an oral and body language representation or
model of the problem.
You might represent a problem using pencil and paper. You
could do this to communicate with another person or with
yourself. Writing and drawing are powerful aids to memory.
You probably keep an address book or address list of the
names, addresses, and phone numbers of your friends. Perhaps
it contains additional information, such as email addresses,
birthdays, names of your friends' children, and so on. You
have learned that an address book is more reliable than your
memory.
There are still other ways to represent problems. For
example, the language and notation of mathematics are useful
for representing and solving certain types of problems. For
example: A particular type of carpet costs $17.45 per square
yard--how much will the carpeting cost for two connecting
rooms? One room is 16 feet by 24 feet, and the other room is
12 feet by 14 feet.
Figure 3. Two rooms to be carpeted.
Conceptually, the problem is not too difficult. You can
form a mental model of the two rooms. Each room will be
covered with carpet costing $17.45 per square yard. So, you
need to figure out how many square yards are needed for each
room. Multiplying the number of square yards in a room by
$17.45 gives the cost of the carpet for the room. Add the
costs for the two rooms, and you are done.
Note that this is only one of the many possible ways to
conceptualize this problem. You may well think of it in a
different way.
The field of mathematics has produced the formula A = LW
(Area equals Length times Width). It works for all
rectangular shapes. Making use of the fact that there are
three feet in a yard, the computation needed to solve this
problem is:
Answer = $17.45 (16/3 x 24/3) + $17.45 (12/3 + 14/3)
Perhaps you can carry out this computation in your head.
More likely, however, you will use pencil and paper, a
calculator, or a computer.
There are two key ideas here. First, some problems that
people want to solve can be represented mathematically.
Second, once a problem is represented as a math problem, it
still remains to be solved.
Over the past few thousand years, mathematicians have
accumulated a great deal of knowledge about mathematics.
Thus, if you can represent a problem as a math problem, you
may be able to take advantage of the work that
mathematicians have previously done. Mental artifacts, such
as paper-and-pencil arithmetic, calculators, and computers,
may be useful. Indeed, ICT-based computational
mathematics is now an important approach in representing
and attempting to solve a wide range of mathematics
problems.
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Representing
Problems Using Computers
One particularly important feature of a mental model is
that it is easily changed. You can "think" a change. This
allows you to quickly consider a number of different
alternatives, both in how you might solve a problem and in
identifying what problem you really want to solve. You can
quickly pose and answer "What if?" types of questions about
possible alternative actions you might take.
Other representations, such as through writing and
mathematics, are useful because they are a supplement to
your brain. Written representations of problems facilitate
sharing with yourself and others over time and distance.
However, a written model is not as easily changed as a
mental model. The written word has a permanency that is
desirable in some situations, but is a difficulty in others.
You cannot merely "think" a change. Erasing is messy. And,
if you happen to be writing with a ball-point pen, erasing
is nearly impossible.
When a problem is represented with a computer, we call
this a computer model or a computer representation of the
problem. For some problems, a computer model has some of the
same characteristics as a mental model. Some computer models
are easy to change and allow easy exploration of
alternatives.
For example, suppose the problem that you face (that is,
the task you want to accomplish) is writing a report on some
work that you have done. You write using a word processor.
Thus, you produce a computer model of the report. You know,
of course, that a key to high quality writing is "revise,
revise, revise." This is much more easily done with a
computer model of a report than it is with a paper and
pencil model of a report. In addition, a computer can assist
in spell checking and can be used to produce a nicely
formatted final product.
In the representation of problems, computers are useful
in some cases and not at all useful in others. For example,
a computer can easily present data in a variety of graphical
formats, such as line graph, bar graph, or in the form of
graphs of two- and three-dimensional mathematical
functions.
But a computer may not be a good substitute for the
doodling and similar types of graphical memory-mapping
activities that many people use when attacking problems.
Suppose that one's mental representation of a problem is in
terms of analogy, metaphor, mental pictures, smells, and so
on. Research that delved into the inner workings of the
minds of successful researchers and inventors suggests this
is common and perhaps necessary. A computer may be of little
use in manipulating such a mental representation.
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Problem
Posing and Clarification
Many of the things that people call problems are actually
poorly defined problem situations. In this case, one or more
of the four components of a clearly defined problem are
missing. For example, you turn on a television set and you
view a brief news item about the homeless people in a large
city and the starving children in a foreign nation. The
announcer continues with a news item about students in our
schools scoring poorly on an international test, relative to
those from some other countries. The announcer presents each
news item as a major problem. But, are these really clearly
defined problems?
You can ask yourself four questions:
- Is there a clearly defined given initial situation?
(Do I really know the facts? Can I check out the facts
through alternative sources that I feel are
reliable?)
- Is there a clearly defined goal? (Is it really clear
to me how I would like things to be? Are there a number
of possible goals? Which goal or goals seem most feasible
and viable?)
- Do I know what resources are available to me that I
could use to help achieve the goal? In addition, are
there rules, regulations, and guidelines that I need to
know about as I work to solve this problem?
- Do I have ownership--do I care enough to devote some
of my own resources? (Am I willing to spend some of my
own time and money on achieving the goal?)
If you can answer "yes" to each of these questions, then
you (personally) have a formal, clearly defined problem.
Often, your answer to one or more of the questions will
be "no." Then, the last question is crucial. If you have
ownership--if you really care about the situation--you may
begin to think about it. You may decide on what you feel are
appropriate statements of the givens and the goal. You may
seek resources from others and make a commitment of your own
resources. You may then proceed to attempt to solve the
problem.
The process of creating a clearly defined problem is
called problem posing or problem clarification. It usually
proceeds in two phases. First, your mind/body senses or is
made aware of a problem situation. You decide that the
problem situation interests you--you have some ownership.
Second, you begin to work on clarifying the givens, goal,
and resources. Perhaps you consider alternative goals and
sense which would contribute most to your ownership of the
problem situation.
Identifying and posing problem situations, and then
transforming them into well-defined problems, are
higher-order thinking tasks. These tasks are not adequately
addressed in our educational system. To a very large extent,
students are asked to work on problems that are posed by the
teacher and/or the curriculum materials. The problems tend
to be quite limited in scope and typically lack a "real
world" quality. Typically students are not asked to explore
problem situations such as hunger, homelessness, prejudice,
terrorism, and so on. They tend to (incorrectly) "learn"
that all problems have solutions, and that they are "dumb"
or not working hard enough if they do not find "the
solution" to a problem that has been assigned.
The result of the problem-posing process is a problem
that is sufficiently defined so that you can begin to work
on solving it. As you work on the problem, you will likely
develop a still better understanding of it. You may redefine
the goal and/or come to understand the goal better. You may
come to understand the given initial situation better;
indeed, you may decide to do some research to gain more
information about it. Problem posing is an ongoing process
as you work to understand and solve a problem.
Problem posing is a higher-order thinking skill that is
an integral component of every domain. Moreover, it is a
component of problem solving that cuts across all discipline
areas. Some additional general purposed problem-solving
ideas are given in the next two sections.
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Some
Problem-Solving Strategies
A strategy can be thought of as a plan, a heuristic, a
rule of thumb, a possible way to approach the solving of
some type of problem. For example, perhaps one of the
problems that you have to deal with is finding a parking
place at work or at school. If so, probably you have
developed a strategy--for example, a particular time of day
when you look for a parking place or a particular search
pattern. Your strategy may not always be successful, but you
find it useful.
Every problem-solving domain has its own strategies.
Research suggests:
- There are relatively few strategies that are powerful
and applicable across all domains. (Breaking a big
problem into smaller problems is one of these
general-purpose strategies. Doing library research is
another general-purpose strategy.) Because each subject
matter (each domain) has its own set of domain-specific
problem-solving strategies, one needs to know a great
deal about a particular domain to be good at solving
problems within that domain.
- The typical person has few explicit strategies in any
particular domain. This suggests that if we help a person
gain a few more domain-specific strategies, it might make
a significant difference in overall problem-solving
performance in that domain. It also suggests the value of
helping students to learn strategies that cut across many
different domains.
The idea of breaking big problems into smaller problems
is called the top-down strategy. The idea is that it may be
far easier to deal with a number of small problems than it
is to deal with one large problem. For example, the task of
writing a long document may be approached by developing an
outline, and then writing small pieces that fill in details
on the outline.
Library research is a type of "ask an expert" strategy. A
large library contains the accumulated expertise of
thousands of experts. The Web is a rapidly expanding global
library. It is not easy to become skilled at searching the
Web. For example, are you skilled in using the Web to find
information that will help you in dealing with Language Arts
problems, Math problems, Science problems, Social Science
problems, personal problems, health problems, entertainment
problems, and so on? Each domain presents its own
information retrieval challenges.
An alternative to using a library in an "ask an expert"
approach is to actually ask a human expert. Many people make
their livings by being consultants. They consider themselves
to be experts within their own specific domains, and they
get paid for answering questions and solving problems within
their areas of expertise. The "Ask
ERIC" system provides a human interface to the ERIC
information retrieval system.
The various fields of science share a common strategy
called Scientific Method. It consists of posing and testing
hypotheses. This is a form of problem posing and problem
solving. Scientists work to carefully define a problem or
problem area that they are exploring. They want to be able
to communicate the problem to others, both now and in the
future. They want to do work that others can build upon.
Well done scientific research (that is, well done problem
solving in science) contributes to the accumulated knowledge
in the field.
You have lots of domain-specific strategies. Think about
some of the strategies you have for making friends, for
learning, for getting to work on time, for finding things
that you have misplaced, and so on. Many of your strategies
are so ingrained that you use them automatically--without
conscious thought. You may even use them when they are
ineffective.
The use of ineffective strategies is common. For example,
how do you memorize a set of materials? Do you just read the
materials over and over again? This is not a very effective
strategy. There are many memorization strategies that are
better. A useful and simple strategy is pausing to review.
Other strategies include finding familiar chunks,
identifying patterns, and building associations between what
you are memorizing and things that are familiar to you.
Some learners are good at inventing strategies that are
effective for themselves. Most learners can benefit greatly
from some help in identifying and learning appropriate
strategies. In general, a person who is a good teacher in a
particular domain is good at helping students recognize,
learn, and fully internalize effective strategies in that
domain. Often this requires that a student unlearn
previously acquired strategies or habits.
Problem-solving strategies can be a lesson topic within
any subject that you teach. Individually and collectively
your students can develop and study the strategies that they
and others use in learning the subject content area and
learning to solve the problems in the subject area. A
whole-class project in a course might be to develop a book
of strategies that will be useful to students who will take
the course in the future.
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A General
Strategy for Problem Solving
Here is a general six-step strategy that you can follow
in attempting to solve almost any problem. This six-step
strategy is a modification of ideas discussed in Polya
(1957). Note that there is no guarantee of success. Keep in
mind that not every problem is solvable. Also, you may lack
the knowledge, skills, time, and other resources needed to
solve a particular problem. However, this six-step strategy
might get you started on a pathway to success.
- Understand the problem. Among other things, this
includes working toward having a clearly defined problem.
You need an initial understanding of the Givens,
Resources, and Goal. This requires knowledge of the
domain(s) of the problem, which could well be
interdisciplinary.
- Determine a plan of action. This is a thinking
activity. What strategies will you apply? What resources
will you use, how will you use them, in what order will
you use them? Are the resources adequate to the
task?
- Think carefully about possible consequences of
carrying out your plan of action. Place major emphasis on
trying to anticipate undesirable outcomes. What new
problems will be created? You may decide to stop working
on the problem or return to step 1 as a consequence of
this thinking.
- Carry out your plan of action. Do so in a thoughtful
manner. This thinking may lead you to the conclusion that
you need to return to one of the earlier steps. Note that
this reflective thinking leads to increased
expertise.
- Check to see if the desired goal has been achieved by
carrying out your plan of action. Then do one of the
following:
- If the problem has been solved, go to step 6.
- If the problem has not been solved and you are
willing to devote more time and energy to it, make use
of the knowledge and experience you have gained as you
return to step 1 or step 2.
- Make a decision to stop working on the problem.
This might be a temporary or a permanent decision.
Keep in mind that the problem you are working on may
not be solvable, or it may be beyond your current
capabilities and resources.
- Do a careful analysis of the steps you have carried
out and the results you have achieved to see if you have
created new, additional problems that need to be
addressed. Reflect on what you have learned by solving
the problem. Think about how your increased knowledge and
skills can be used in other problem-solving situations.
(Work to increase your reflective
intelligence!)
Many people have found that this six-step strategy for
problem solving is worth memorizing. As a teacher, you might
decide that one of your goals in teaching problem solving is
to have all your students memorize this strategy and
practice it so that it becomes second nature. This will help
to increase your students' expertise in solving
problems.
Many of the steps in this six-step strategy require
careful thinking. However, there are a steadily growing
number of situations in which much of the work of step 4 can
be carried out by a computer. The person who is skilled at
using a computer for this purpose may gain a significant
advantage in problem solving, as compared to a person who
lacks computer knowledge and skill.
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Working
Toward Increased Expertise
One of the goals of instruction in any subject area is to
help students increase their expertise at posing,
representing, and solving problems in the subject area.
People can get better at whatever they do. A person can get
better at a sport, at a hobby or craft, or in an academic
field. A person's level of expertise can increase through
learning and practice. A person who is really good at
something relative to his/her peers is considered to be an
expert.
It is important to distinguish between having some level
of expertise and being an expert. The word expertise does
not mean any particular level of ability. For anything that
you can do, you can imagine a scale of performance that runs
from very low expertise to very high expertise. When a
person has a high level of expertise in some particular
area, we call this person an expert. Bereiter and
Scardamalia (1993) contains an excellent summary of research
about expertise.
Figure 4. An "expertise" scale.
Research on expertise indicates that it takes many years
of study, practice, and hard work for a person to achieve
their full potential in any particular area of expertise.
For example, consider any one of the eight areas of
intelligence identified by Howard
Gardner. If a person is naturally talented in one of
these areas and works really hard for 10 to 15 years within
that specific area, they are apt to achieve world class
status in that area. It is a combination of talent and hard
work over many years that allows a person to achieve a high
level of expertise in an area.
Because it takes so much time, and effort to achieve a
high level of expertise in just one narrow field, few people
achieve a high level of expertise in multiple fields. For
example, consider how few professional athletes perform at a
world class level in two different sports. Or, consider the
general practitioner versus the specialists in medicine.
One of the successes in the field of Artificial
Intelligence has been the development of Expert
Systems--computer programs that exhibit a considerable
level of expertise in solving problems within a specific
domain. In a number of narrowly defined domains, Expert
Systems or humans working together with an expert system can
perform at a quite high level of expertise. This, of course,
has profound implications in education. Suppose a computer
program (an Expert System) exists with a specific domain
that is being covered in a school curriculum. Now, what do
we want students to learn about solving problems in that
domain? Do we want students to learn to compete with the
Expert System, or learn to work with the Expert System?
You may think that such questions do not pertain to our
ordinary curriculum. But, one can think of a handheld
calculator as having some Artificial Intelligence. More
sophisticated calculators can solve a wide range of math
problems. A spelling checker in a word processor has a
certain level of expertise, as does a grammar checker. The
point is, progress in Artificial Intelligence is providing
us with powerful aids to problem solving (that is,
resources) in many different domains.
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Transfer of
Learning
Transfer of learning deals with transferring one's
knowledge and skills from one problem-solving situation to
another. You need to know about transfer of learning in
order to help increase the transfer of learning that your
students achieve.
Transfer of learning is commonplace and often done
without conscious thought. For example, suppose that when
you were a child and learning to tie your shoes, all of your
shoes had brown, cotton shoe laces. You mastered tying
brown, cotton shoe laces. The next year you got new shoes.
The new shoes were a little bigger, and they had white,
nylon shoe laces. The chances are that you had no trouble
whatsoever in transferring your shoe-tying skills to the new
larger shoes with the different shoe laces.
However, there are many transfer of learning situations
that are far more difficult. For example, a secondary school
math class might teach the metric system of units. From the
math class, students go to a science class. Frequently the
science teacher reports that the students claim a complete
lack of knowledge about the metric system. Essentially no
transfer of learning has occurred from the math class to the
science class.
The goal of gaining general skills in the transfer of
your learning is easier said than done. Researchers have
worked to develop a general theory of transfer of
learning--a theory that could help students get better at
transfer. This has proven to be a difficult research
problem.
At one time, it was common to talk about transfer of
learning in terms of near and far transfer. This theory of
transfer suggested that some problems and tasks are so
nearly alike that transfer of learning occurs easily and
naturally. This is called near transfer. Other problems and
tasks required more concentrated effort and thinking for
transfer to occur. This is called far transfer.
The theory of near and far transfer does not help us much
in our teaching. We know that near and far transfer occur.
But, what is "near" or "far" varies with the person
attempting to do the transfer. We know that far transfer
does not readily occur for most students. The difficulty
with this theory of near and far transfer is that it does
not provide a foundation or a plan for helping a person to
get better at transfer.
In recent years, the low-road/high-road theory on
transfer of learning, developed by Salomon & Perkins
(1988), has proven to be more fruitful. Low-road transfer
refers to developing some knowledge/skill to a high level of
automaticity. It usually requires a great deal of practice
in varying settings. Shoe tying, keyboarding, steering a
car, and memorized arithmetic facts are examples of areas in
which such automaticity can be achieved and is quite
useful.
On the other hand, high-road transfer involves: cognitive
understanding; purposeful and conscious analysis;
mindfulness; and application of strategies that cut across
disciplines. In high-road transfer, there is deliberate
mindful abstraction of the idea that can transfer, and then
conscious and deliberate application of the idea when faced
by a problem where the idea may be useful.
For example, consider the top-down strategy of breaking a
big problem into smaller components. You can learn the name
and concept of this strategy. You can practice this strategy
in many different domains. You can reflect on the strategy
and how it fits you and your way of dealing with the
problems you encounter. Similar comments hold for the
library research strategy.
Eventually, you can incorporate a strategy into your
repertoire of approaches to problem solving. When you
encounter a new problem that is not solved by low-road
transfer, you begin to mentally run through your list of
strategies useful in high-road transfer. You may decide that
breaking the problem into smaller pieces would be an
effective strategy to apply. Or, you may decide that library
research (a Web search) is a good starting point.
Two keys to high-road transfer are mindfulness and
reflectiveness. View every problem-solving situation as an
opportunity to learn. After solving a problem, reflect about
what you have learned. Be mindful of ideas that are of
potential use in solving other problems.
Of course, there are a wide range of problems that lie
between those easily handled by low-road transfer and those
that require the careful, conscious, well-reasoned, mindful
approaches suggested by high-road transfer. Earlier in this
document we discussed the many years of hard work required
to gain a high level of expertise in a domain. To a large
extent, this work results in moving many problems from the
middle ground in the domain toward the low-road transfer end
of the scale. More and more of the problems that you
encounter in the domain are quickly and easily solved,
almost without conscious thought and effort.
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Project-Based Learning (PBL)
PBL is an individual or group activity that goes on over a period of time, resulting in a product, presentation, or performance. PBL typically has a timeline, milestones, and other aspects of formative evaluation as the project proceeds.
Doing a project and solving a problem have much in
common. For example, in PBL a student or team of students
typically have considerable latitude in posing the details
of what will be accomplished in the project. There are
limited resources, such as time. There is a clear goal of a
product, presentation, or performance. The student or team
of students may well develop a high level of ownership as
they work on the project. Thus, any PBL environment is a
good environment for teaching problem solving.
PBL shares much in common with Process
Writing. In the United States, the roots of Process
Writing are often traced back to the Bay
Area Writing Project circa 1975. A six step version of
Process Writing is:
- brainstorming
- organizing the brainstormed ideas
- developing a draft
- obtaining feedback
- revising, which may involve going bask to earlier
steps
- publishing
This list can be viewed as a strategy for accomplishing
the task (solving the problem) of doing a writing
project.
Summary of
Important Ideas
Each classroom teaching situation provides an environment
that can be used to help students improve their
problem-solving and higher-order thinking skills. Students
will make significant progress if:
- They have ownership of the problems to be solved and
the tasks to be accomplished. They are intrinsically
motivated.
- The problems to be solve and the tasks to be
accomplished are challenging--they stretch the
capabilities of the students.
- There is explicit instruction on key ideas such as:
- Problem posing. Working to achieve a
clearly defined problem. As you work to solve a
problem, continue to spend time working to define the
problem.
- Problem representation (Yarlus and Sloutsky,
2000).
- Building on the previous work of yourself and
others.
- Transfer of learning.
- Viewing each problem/project as a learning
opportunity. As you work on solving a problem, work to
learn things that will help you in the future Do
metacognition. Do a conscious, considered analysis of
the components and the overall process in each
challenging problem that you address. This will help
you to get better at solving problems.
- Roles of Information and Communications Technology
in problem solving.
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References
Bereiter, C., & Scardamalia, M. (1993). Surpassing
ourselves: An inquiry into the nature and implications of
expertise. Chicago and La Salle, IL: Open Court.
Frensch, P. & Funke, J., (Eds.). (1995). Complex
problem solving: The European perspective. Hillsdale, NJ:
Lawrence Erlbaum Associates.
Moursund, D.G. (1996). Increasing your expertise as a
problem solver: Some roles of computers. Eugene, OR: ISTE.
Some
chapters are available online.
Norman, D. (1993). Things that make us smart: Defending
human attributes in the age of machines. Reading, MA:
Addison-Wesley.
Perkins, D. (1992). Smart schools: Better thinking and
learning for every child. NY: Free Press.
Polya, G. (1957). How to solve it: A new aspect of
mathematical method (2nd ed.). Princeton, NJ: Princeton
University Press.
Problem Posing and Problem Solving: The BioQUEST Library
[Online]. Accessed 10/23/01: http://bioquest.org/indexlib.html.
Problem Posing in Academic Writing [Online].
Accessed 10/23/01: http://www.temple.edu/writingctr/cw06001.htm.
Salomon, G., & Perkins, D. (1988, September).
Teaching for transfer. Educational Leadership, 22-32.
Yarlus, A.S. and Sloutsky, V.M. (2000). Problem
Representation in Experts and Novices: Part 1. Differences
in the Content of Representation; Part 2. Underlying
Processing Mechanisms.
Part 1 [Online] Accessed 11/2/01:
http://www.cis.upenn.edu/~ircs/cogsci2000/
PRCDNGS/SPRCDNGS/posters/yar_slo.pdf.
Part 2 [Online] Accessed 11/2/01: http://www.cis.upenn.edu/~ircs/cogsci2000/
PRCDNGS/SPRCDNGS/PAPERS/SLO-YAR.PDF
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