Problem Solving
This Lecture Note was compiled by Dr. John Mubenwafor from various sources, including
Richard Mayer, Merlin Wittrock, Emma Wells, D. Elkind and A. C. Burris.
A major goal of education is to help students learn in ways that enable them to use what they
have learned to solve problems in new situations. In short, problem solving is fundamental to
education because educators are interested in improving students' ability to solve problems. This
entry defines key terms, types of problems, and processes in problem solving and then examines
theories of problem solving, ways of teaching for problem solving transfer, and ways of teaching
of problem solving skill.
The Definition of a Problem
What is a Problem?
A problem exists when a problem solver has a goal but does not know how to accomplish it.
Specifically, a problem occurs when a situation is in a given state, a problem solver wants the
situation to be in a goal state, and the problem solver is not aware of an obvious way to transform
the situation from the given state to the goal state. In his classic monograph, On Problem
Solving, the Gestalt psychologist Karl Duncker defined a problem as follows:
A problem arises when a living creature has a goal but does not know how this goal is to be
reached. Whenever one cannot go from the given situation to the desired situation simply by
action, then there has to be recourse to thinking. Such thinking has the task of devising some
action, which may mediate between the existing and desired situations. (1945, p. 1)
This definition includes high-level academic tasks for a typical middle school student such as
writing a convincing essay, solving an unfamiliar algebra word problem, or figuring out how an
electric motor works, but does not include low-level academic tasks such as pronouncing the
sound of the printed word “cat,” stating the answer to “2 2 =___,” or changing a word from
singular to plural form.
To truly be considered a problem, a mathematical quest must contain some effort or thought on
the part of the solver (Brownell, 1942; Polya, 1945).
What is Problem Solving?
"Problem solving refers to the systematic approach used to conceptualize a problem situation and
identify needs, analyze factors contributing to the problem situation, design strategies to meet
those needs, and implement and evaluate the strategies."
-Allen & Graden, 2002
"Recent research indicates that 85% of students served by problem-solving teams do not need
further evaluation for special education!"
-Hartmann & Faye, 1996
"It is the responsibility of teachers in regular classroom to engage in multiple educational
interventions and to note the effects of such interventions on a child experiencing academic
failure before referring the child for special education assessment."
-National Academy of Sciences, 1982
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"Recent surveys show that 3 to 5 percent of the students in public schools are referred each year
for psychological and educational assessment. About 92 percent of those students who are
referred are tested, and about 73 percent of those tested are placed in special education."
-Algozzine, Christianson and Ysseldyke, 1982
The previous quotes illustrate the fact that one of the most important occurrences for any child is
the decision by someone in the child's life to refer him or her for special education eligibility.
What becomes frustrating to those who make referrals is the length of time it takes to conduct a
traditional special education evaluation. There is often a time lag between referral and when
service or intervention is provided. Many have asked, "Why not get right to problem solving and
intervention so students get help faster and teachers/referrers are more satisfied?"
Problem solving occurs within the school setting at various levels and becomes more complex as
the resources needed to resolve a problem increases due to the significance of the problem. The
intent of problem solving is to resolve the problem using the necessary resources. The vast
majority of problem solving activities occur within the general education system. Often problem
solving begins with a teacher and parent communication together about how to support a
learning goal for a student. Teams of teachers at each grade level (elementary), or
core/department (secondary) meet regularly to support one another in finding solutions for
individual student concerns as well as to monitor and adjust instruction for groups of students.
Building level teams are available to offer additional expertise as needed for more challenging
needs. One outcome of problem solving for a small percentage of students may be referral for a
special education entitlement evaluation. Buildings are committed to providing special education
services to all children who meet state criteria for disability and demonstrate educational need,
though the primary goal of the problem solving process is enabled learning.
According to Mayer and Wittrock, problem solving is “cognitive processing directed at
achieving a goal when no solution method is obvious to the problem solver” (2006, p. 287). This
definition consists of four parts:
1. problem solving is cognitive, that is, problem solving occurs within the problem solver's
cognitive system and can only be inferred from the problem solver's behavior,
2. problem solving is a process, that is, problem solving involves applying cognitive
processes to cognitive representations in the problem solver's cognitive system,
3. problem solving is directed, that is, problem solving is guided by the problem solver's
goals, and
4. problem solving is personal, that is, problem solving depends on the knowledge and skill
of the problem solver.
Problem solving is the application of ideas, skills, or factual information to achieve the solution
to a problem or to reach a desired outcome.
In sum, problem solving is cognitive processing directed at transforming a problem from the
given state to the goal state when the problem solver is not immediately aware of a solution
method.
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TYPES OF PROBLEMS
Educational psychology has broken down problems in two different ways. The first way is to
make a distinction between well-defined and poorly (or ill) -defined problems.
Well-defined: This has a clear goal or solution, and problem solving strategies are easily
developed. A well-defined problem has a clearly specified given state, a clearly specified goal
state, and a clearly specified set of allowable operations. For example, “Solve for x: 2x + 11 =
33” is a well-defined problem because there is clear given state (i.e., 2 x 11 = 33), a clear goal
state (i.e., x = ___) and a clear set of operations (i.e., the rules of algebra and arithmetic).
Consider another, imagine that you are in school. If your teacher gives you a quiz that asks you
to list the first ten U.S. Presidents in order and name one important historical fact about each;
that would be a well-defined problem. The instructions and expected outcome is clear, and you
can use a simple memory recall strategy to come up with the correct answer.
Poorly (or ill) -defined: This is the opposite of well-defined problem. It is unclear, abstract, or
confusing, and does not have a clear problem solving strategy. An ill-defined problem lacks a
clearly specified given state, goal state, and/or set of allowable operators. For example, “develop
a research plan for a senior honors thesis” is an ill-defined problem for most students because the
goal state is not clear (e.g., the requirements for the plan) and the allowable operators are not
clear (e.g., the places where students may find information). Another example is, if your teacher
gives you a quiz that instead asks you, 'think about some history, then draw a picture, and be sure
to wash your hands,' you're not really sure what to do. What does the teacher expect of you? This
is a poorly-defined problem, because you don't know how to reach a solution or answer.
What makes a problem well-defined or ill-defined depends on the characteristics of the problem.
Although most important and challenging problems in life are ill-defined, most problem solving
in schools involves well-defined problems.
The second way that educational psychology has broken down different types of problems is by
making a distinction between routine and non-routine problems.
Routine: Just like the name indicates, a routine problem is one that is typical and has a simple
solution. When a problem solver knows how to go about solving a problem, the problem is
routine. For example, two-column multiplication problems, such as 25 x 12 = ___, are routine for
most high school students because they know the procedure. Routine problems are what most
people do in school: memorizing simple facts, how to do addition and subtraction, how to spell
words, and so on. However, in more advanced years or in more advanced subjects in school,
teachers might present students with non-routine problems that require critical thinking skills and
subjective solutions. For example, the ethics of social issues such as the death penalty, or the role
of civil rights in laws, or themes in famous literature, might be considered non-routine problems.
Non-routine:
When a problem solver does not initially know how to go about solving a problem, the problem
is non-routine. For example, the following problem is non-routine for most high-school students:
“If the area covered by water lilies in a lake doubles every 24 hours, and the entire lake is
covered in 60 days, how long does it take to cover half the lake?” Robert Sternberg and Janet
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Davidson (1995) refer to this kind of problem as an insight problem because problem solvers
need to invent a solution method (e.g., in this case the answer is 59 days). Non-routine
problems are more abstract or subjective and require more complicated or creative problem
solving strategies.
What makes problems either routine or non-routine depends on the knowledge of the problem
solver because the same problem can be routine for one person and non-routine for another.
Although the goal of education is to prepare students for solving non-routine problems, most of
the problems that students are asked to solve in school are routine.
THEORIES OF PROBLEM SOLVING
Many current views of problem solving, such as described in Keith Holyoak and Robert
Morrison's Cambridge Handbook of Thinking and Reasoning (2005) or Marsha Lovett's 2002
review of research on problem solving, have their roots in Gestalt theory or information
processing theory.
Gestalt Theory. The Gestalt theory of problem solving, described by Karl Duncker (1945) and
Max Wertheimer (1959), holds that problem solving occurs with a flash of insight. Richard
Mayer (1995) noted that insight occurs when a problem solver moves from a state of not
knowing how to solve a problem to knowing how to solve a problem. During insight, problem
solvers devise a way of representing the problem that enables solution. Gestalt psychologists
offered several ways of conceptualizing what happens during insight: insight involves building a
schema in which all the parts fit together, insight involves suddenly reorganizing the visual
information so it fits together to solve the problem, insight involves restating a problem's givens
or problem goal in a new way that makes the problem easier to solve, insight involves removing
mental blocks, and insight involves finding a problem analog (i.e., a similar problem that the
problem solver already knows how to solve). Gestalt theory informs educational programs aimed
at teaching students how to represent problems.
Information Processing Theory. The information processing theory of problem solving, as
described by Allen Newell and Herbert Simon (1972), is based on a human computer metaphor
in which problem solving involves carrying out a series of mental computations on mental
representations. The key components in the theory are as follows: the idea that a problem can be
represented as a problem space—a representation of the initial state, goal state, and all possible
intervening states—and search heuristics—a strategy for moving through the problem space
from one state of the problem to the next. The problem begins in the given state, the problem
solver applies an operator that generates a new state, and so on until the goal state is reached. For
example, a common search heuristic is means-ends analysis, in which the problem solver seeks
to apply an operator that will satisfy the problem-solver's current goal; if there is a constraint that
blocks the application of the operator, then a goal is set to remove the constraint, and so on.
Information processing theory informs educational programs aimed at teaching strategies for
solving problems.
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PROBLEM SOLVING AS A KIND OF THINKING
Problem solving is related to other terms such as thinking, reasoning, decision making, critical
thinking, and creative thinking.
Thinking refers to a problem solver's cognitive processing, but it includes both directed thinking
(which is problem solving) and undirected thinking (such as daydreaming). Thus, thinking is a
broader term that includes problem solving as a subset of thinking (i.e., a kind of thinking, i.e.,
directed thinking).
Reasoning, decision making, critical thinking, and creative thinking are subsets of problem
solving, that is, kinds of problem solving.
Reasoning refers to problem solving with a specific task in which the goal is to draw a
conclusion from premises using logical rules based on deduction or induction. For example, if
students know that all four-sided figures are quadrilaterals and that all squares have four sides,
then by using deduction they can conclude that all squares are quadrilaterals. If they are given
the sequence 2–4–6–8, then by induction they can conclude that the next number should be 10.
Decision making refers to problem solving with a specific task in which the goal is to choose
one of two or more alternatives based on some criteria. For example, a decision making task is to
decide whether someone would rather have $100 for sure or a 1% chance of getting $100,000.
Thus, both reasoning and decision-making are kinds of problem solving that are characterized by
specific kinds of tasks.
Creative thinking and critical thinking refer to specific aspects of problem solving, respectively.
Creative thinking involves generating alternatives that meet some criteria, such as listing all the
possible uses for a brick, whereas
Critical thinking involves evaluating how well various alternatives meet some criteria, such as
determining which are the best answers for the brick problem. For example, in scientific problem
solving situations, creative thinking is involved in generating hypotheses and critical thinking is
involved in testing them. Creative thinking and critical thinking can be involved in reasoning and
decision making.
COGNITIVE PROCESSES IN PROBLEM SOLVING
Mayer and Wittrock (2006) distinguished among four major cognitive processes in problem
solving: representing, in which the problem solver constructs a cognitive representation of the
problem; planning, in which the problem solver devises a plan for solving the problem;
executing, in which the problem solver carries out the plan; and self-regulating, in which the
problem solver evaluates the effectiveness of cognitive processing during problem solving and
adjusts accordingly. During representing, the problem solver seeks to understand the problem,
including the given state, goal state, and allowable operators, and the problem solver may build a
situation model—that is, a concrete representation of the situation being described in the
problem. Although solution execution is often emphasized in mathematics textbooks and in
mathematics classrooms, successful mathematical problem solving also depends on representing,
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planning, and self-regulating. In a 2001 review, Jeremy Kilpatrick, Jane Swafford, and Bradford
Findell concluded that mathematical proficiency depends on intertwining of procedural fluency
(for executing) with conceptual understanding (for representing), strategic competence (for
planning), adaptive reasoning, and productive disposition (for self-regulating).
According to Mayer and Wittrock (2006), students need to have five kinds of knowledge in order
to be successful problem solvers:
Facts: knowledge about characteristics of elements or events, such as “there are 100 cents in a
dollar”;
Concepts: knowledge of a categories, principles, or models, such as knowing what place value
means in arithmetic or how hot air rises in science;
Strategies: knowledge of general methods, such as how to break a problem into parts or how to
find a related problem;
Procedures: knowledge of specific procedures, such as how to carry out long division or how to
change words from singular to plural form; and
Beliefs: cognitions about one's problem-solving competence (such as “I am not good in math”)
or about the nature of problem solving (e.g., “If someone can't solve a problem right away, the
person never will be able to solve it”).
Facts and concepts are useful for representing a problem, strategies are needed for planning a
solution, procedures are needed for carrying out the plan, and beliefs can influence the process of
self-regulating.
TEACHING FOR PROBLEM SOLVING
Max Wertheimer (1959) made the classic distinction between learning by rote and learning by
understanding. For example, in teaching students how to compute the area of a parallelogram by
a rote method, students are shown how to measure the height, how to measure the base, and how
to multiply height times base using the formula, area = height x base. According to Wertheimer,
this rote method of instruction leads to good performance on retention tests (i.e., solving similar
problems) and poor performance on transfer tests (i.e., solving new problems). In contrast,
learning by understanding involves helping students see that if they can cut off the triangle from
one end of the parallelogram and place it on the other side to form a rectangle; then, they can put
1 x 1 squares over the surface of the rectangle to determine how many squares form the area.
According to Wertheimer, this meaningful method of instruction leads to good retention and
good transfer performance. Wertheimer claimed that rote instruction creates reproductive
thinking—applying already learned procedures to a problem—whereas meaningful instruction
leads to productive thinking—adapting what was learned to new kinds of problems.
Mayer and Wittrock (2006) identified instructional methods that are intended to promote
meaningful learning, such as providing advance organizers that prime appropriate prior
knowledge during learning, asking learners to explain aloud a text they are reading, presenting
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worked out examples along with commentary, or providing hints and guidance as students work
on an example problem. A major goal of meaningful methods of instruction is to promote
problem-solving transfer, that is, the ability to use what was learned in new situations. Wittrock
(1974) referred to meaningful learning as a generative process because it requires active
cognitive processing during learning.
TEACHING OF PROBLEM SOLVING
A more direct approach is to teach people the knowledge and skills they need to be better
problem solvers. Mayer (2008) identified four issues that are involved in designing a problemsolving course.
What to Teach. Should problem-solving courses attempt to teach problem solving as a single,
monolithic skill (e.g., a mental muscle that needs to be strengthened) or as a collection of
smaller, component skills? Although conventional wisdom is that problem solving involves a
single skill, research in cognitive science suggests that problem solving ability is a collection of
small component skills.
How to Teach. Should problem-solving courses focus on the product of problem solving (i.e.,
getting the right answer) or the process of problem solving (i.e., figuring out how to solve the
problem)? While it makes sense that students need practice in getting the right answer (i.e., the
product of problem solving), research in cognitive science suggests that students benefit from
training in describing and evaluating the methods used to solve problems (i.e., the process of
problem solving). For example, one technique that emphasizes the process of problem solving is
modeling, in which teachers and students demonstrate their problem-solving methods.
Where to Teach. Should problem solving be taught as a general, stand-alone course or within
specific domains (such as problem solving in history, in science, in mathematics, ETC.)?
Although conventional wisdom is that students should be taught general skills in stand-alone
courses, there is sufficient cognitive science research to propose that it would be effective to
teach problem solving within the context of specific subject domains.
When to Teach. Should problem solving be taught before or after students have mastered
corresponding lower-levels? Although it seems to make sense that higher-order thinking skills
should be taught only after lower-level skills have been mastered, there is sufficient cognitive
science research to propose that it would be effective to teach higher-order skills before lowerlevel skills are mastered.
Problem Solving Skills in Education
In school and in everyday life, we all have to solve a wide variety of problems. In school, these
problems might be how to complete an algebraic equation. In everyday life, problems might be
how to pay bills on a limited income. Either way, in order to be successful, we must have the
ability to solve different types of problems using different types of solution strategies.
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Depending on the type of problem, you have a lot of options regarding possible solution
strategies. The following are solutions that are common strategies for well-defined or routine
problems:
1. Algorithm: Algorithms are step-by-step strategies or processes for how to solve a problem
or achieve a goal. The most common example you might think of for using algorithms might be
in math class. When presented with an algebraic equation, you might have learned about how
to solve for x using certain well-defined steps. But algorithms can be used in other subjects as
well. For example, when you are learning about how to take apart and clean a car engine, you
will want to approach this problem using a set series of steps, making sure that you don't
misplace or mix up any of the parts.
2. Heuristics: Heuristics is another solution that many people use to solve problems. They are
general strategies used to make quick, short-cut solutions to problems that sometimes lead to
solutions but sometimes lead to errors. Heuristics are sometimes referred to as mental short-cuts,
and we often form them based on past experiences. You have probably used heuristics all the
time in your daily life, maybe without knowing what they are called. For example, when you go
to the store to buy a product, there will probably be several options on the shelf. When trying to
decide the quality of the different choices many people use the heuristic rule, 'you get what you
pay for,' meaning more expensive items will be of higher quality. While this might be true in
many cases, it's not necessarily always true. So, using this strategy does make for a quick
decision but it could backfire.
Another example of a heuristic is 'shorter lines will move faster.' Once you select the item you
want to purchase at the store, you head over to the cash registers. You pick the line with the
fewest number of people, assuming it will move the fastest. While normally this strategy would
work, you might get behind someone who needs to do a price check or a cashier who is new to
the job and doesn't know how to use the register. So, in summary, heuristics are a common
problem solving strategy for everyday life types of problems, and usually they lead to good
decisions but they can sometimes lead to mistakes.
3. Graphic representations: You might have used in the past. Graphic representations are
visual-based illustrations of a problem that might lead to clarification of a problem or creative
solutions. Examples of graphic representations are flow charts, diagrams, outlines or mind
maps. With any of these options, you can draw the problem out, and this might help you see the
problem in a new way.
4. The IDEAL Strategy: This is another problem solving strategy that comes from educational
psychology. ‘IDEAL’ is an acronym, meaning each letter stands for one idea. This strategy
includes five steps, each corresponding to one letter.
 The 'I' stands for identify, which is the first step in the IDEAL strategy. Here, you
identify what the problem is, as clearly as possible.
 The 'D' stands for define. Here, you define what the possible final goal or solution
might be.
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Next, the 'E' stands for explore. This is the step where you explore possible ways to
reach the solution or goal. You could use other strategies we've discussed here, such as
a graphic representation, heuristics, or brainstorming.
The fourth letter is 'A,' which stands for anticipate. In this step, you look forward to
possible outcomes of different solutions you've created, and try to see which solution
will work the best. You then choose one of the solutions and act on it. While the 'A' in
IDEAL stands for anticipate, not act, you can remember that the 'A' could include both
anticipation and action if it's helpful.
The final step in the IDEAL solution is the 'L,' which stands for look. After you have
chosen one solution and acted on it, you now look back and learn from what you did. So
again, you could think of the 'L' as standing for both look and learn.
Did your solution lead to a positive outcome? If not, go back to the beginning and try to find a
different solution. While the IDEAL solution may seem complicated with many steps to it, it's
actually a fairly intuitive way to solve problems. You identify the problem, define what you want
to achieve, explore possible solutions, pick one, and see what happened. Most of us use this
general strategy when problem solving; we might just not label it with the IDEAL name.
Problem solving involves a variety of skills and calls upon children to retrieve previously learned
information and apply it in new or varying situations. Knowing the basic arithmetic skills,
knowing when to incorporate them into new contexts, and then being able to do so are three
distinct skills. Having all three skills makes problem solving easier, but inability in one does not
mean that a student does not understand a problem. It may mean that the student’s learning style
has not been addressed. Similarly, because students can carry out the operations in isolation does
not mean they know when to apply them or how to interpret the numbers involved (Bley &
Thornton, 2001, p. 37).
Good problems include modifications that may be made for students with varying skills,
abilities, and learning styles. Multiple solutions and multiple methods of solution are encouraged
in a classroom that fosters a problem-solving atmosphere.
Let us now focus on more process and then on strategies of problem solving.
Polya’s Four-Step Process
Probably the most famous approach to problem solving is Polya’s four-step process described
below (Polya, 1945). Polya identifies the four principles as follows:
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back
The problem-solving process is merely a general guide of how to proceed in solving problems. In
many cases, steps of the process will overlap, thus it may not be possible to perform each step of
the process in the order given above. These four principles appear in many elementary-level
textbook series as early as the kindergarten grade level.
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Use of the problem-solving process and specific strategies may be mentioned in a state or district
model or framework.
Below we describing each of the four steps of Polya’s problem-solving process in more detail
together with a fifth step. This is not suggesting that Polya’s process is incomplete. In fact, the
fifth step, extend the problem, is mentioned in Polya’s manuscript as part of the fourth step—
look back. These ideas are separated so that the process of extending the problem, especially
relevant for teachers, does not become lost in the process of verifying the solution
Step 1: Understand the problem
To correctly solve a problem, you must first understand the problem. Below are some questions
that may help lead you (or a student) to an understanding of a given problem. Not every question
will be appropriate for every problem.
What are you asked to find or show?
What type of answer do you expect?
What units will be used in the answer?
Can you give an estimate?
What information is given? Do you understand all the terms and conditions?
Are there any assumptions that need to be made or special conditions to be met?
Is there enough information given? If not, what information is needed?
Is there any extra information given? If so, what information is not needed?
Can you restate the problem in your own words?
Can you act out the problem?
Can you draw a picture, a diagram, or an illustration?
Can you calculate specific numerical instances that illustrate the problem?
Step 2: Devise a plan
Use problem-solving strategies. The “plan” used to solve a problem is often called a problemsolving strategy. For some problems, you may begin using one strategy and then realize that the
strategy does not fit the given information or is not leading toward the desired solution; in this
case, you must choose another strategy. In other cases you may need to use a combination of
strategies. Several problem-solving strategies are described below (in no particular order). All of
these strategies can be found in elementary-level mathematics textbooks at around the third- or
fourth-grade level; many of these strategies are introduced as early as the prekindergarten level.
 Use guess and check. When a problem calls for a numerical answer, a student may make
a random guess and then check the guess with the facts and information given within the
problem. If the guess is incorrect, the student may make and check a new guess. Each
subsequent guess should provide more insight into the problem and lead to a more
appropriate guess. In some instances the guess and check strategy may also be used with
problems for which the answer is non-numerical.
 Draw a picture or a diagram/use a graph or number line. A picture or graph may
illustrate relationships between given facts and information that are not as easily seen in
word or numerical form.
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Use manipulatives or a model/act it out. When a problem requires that elements be
moved or rearranged, a physical model can be used to illustrate the solution.
Make a list or table. A list or table may be helpful to organize the given information. It
may be possible to make an orderly list or table of all possible solutions and then to
choose the solution that best fits the given facts and information from this list. In some
problems, the answer to the problem is a list or table of all possible solutions.
Eliminate possibilities. When there is more than one possible solution to a problem,
each possibility must be examined. Potential solutions that do not work are discarded
from the list of possible solutions until an appropriate answer is determined.
Use cases. It is possible to divide some problems into cases. Each case may be separately
considered.
Solve an equivalent problem. In some instances it is easier to solve a related or
equivalent problem than it is to solve a given problem.
Solve a simpler problem. It may be possible to formulate and solve a simpler problem
than the given problem. The process used in the solution of the simpler problem can give
insight into the more complex given problem.
Look for a pattern. Patterns are useful in many problem-solving situations. This strategy
will be especially useful in solving many real-world problems. “Patterns are a way for
young students to recognize order and to organize their world” (NCTM, 2000, p. 91).
Choose the operation/write a formula or number sentence. Some problems are easily
solved with the application of a known formula or number sentence. The difficulty often
lies in choosing the appropriate formula or operation.
Make a prediction/use estimation. One must closely consider all elements of a problem
in order to make a prediction or use estimation. This careful consideration may provide
useful insight into the problem solution.
Work the problem backward. If the problem involves a sequence of steps that can be
reversed, it may be useful to work the problem backward. Children at the early childhood
level may already have some experience in working backward. In solving many mazes
and puzzles, it is sometimes easier to begin at the end than to begin at the beginning.
Use logical reasoning. Mathematics can and should make sense. Logical reasoning and
careful consideration are sometimes all that is required to solve a mathematics problem
Step 3: Carry out the plan
Once a problem has been carefully analyzed and a plan is devised, if the plan is a suitable one for
the given problem, it is usually a relatively simple process to carry out the plan. However, in
some cases the original plan does not succeed and another plan must be devised. The original
strategy may need to be modified, or a new strategy may be selected. Students must realize that
not every problem will be solved within the first attempt. A failed attempt can be viewed as a
learning experience. Try to help students avoid getting frustrated or discouraged. Cooperative
learning teams can be used to encourage and engage students. Computers, calculators, or other
manipulatives may be useful tools when routine tasks are involved.
Step 4: Look back
Once an answer or solution is found, it is important to check that solution. Check all steps and
calculations within the solution process. Below are some questions that you (or your students)
may find useful in the looking back process.
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Is the answer reasonable?
Is there another method of solution that will easily verify the answer?
Does the answer fit the problem data?
Does the answer fulfill all conditions or requirements of the problem?
Is there more than one answer?
Will the solution process used be valuable in solving similar or related problems?
Step 5: Extend the problem
For a classroom teacher, an important part of the problem-solving process should involve trying
to create similar or related problems. A given problem may need to be simplified in order to be
used at a specific classroom level or with students that have special needs. A teacher may wish to
make a problem more complicated or to create similar related problems that are more difficult.
Elementary school students often extend the problem as part of a journal writing exercise as they
write their own story problems for a given situation.
It may be possible to generalize specific instances of a given problem. Teachers must be on the
lookout for opportunities to have students generalize and make conjectures. Teachers should
look for connections that can be made between given mathematics problems and solutions and
real-life situations. Teachers should also look for connections between given mathematics
problems and their solutions and other subject areas.
In summary, problem solving is the application of ideas, skills, or factual information to achieve
the solution to a problem or to reach a desired outcome. Before we solve a problem, we should
know what kind of problem it is. There are well-defined problems and poorly-defined
problems, as well as routine and non-routine problems. There are many possible problem
solving solutions or strategies: algorithms, heuristics, graphic representations, and the
IDEAL strategy
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