Problem
Problem Solving
Solving
‘‘Mathematics is something that you do,
not something that you understand.’’
Arthur B. Powell
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Problem
Problem Solving
Solving
Problem Solving: As
Topic in Mathematics
or
Approach of Mathematics Teaching
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Problem
Problem Solving
Solving
“We believe that the primary goals of
mathematics learning are understanding and
problem solving, and that these goals are
inextricably related because learning
mathematics with understanding is best
supported by engaging in problem solving.”
Lester and Lambdin
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Problem
Problem Solving
Solving
KEY POINT TO CONSIDER
In order to develop understanding, we
should try and recognize and emphasize
in our teaching the process of problem
solving, rather than simply concentrating
on getting the answer right.
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Problem
Problem Solving
Solving
WHAT IS PROBLEM SOLVING?
The meaning of mathematical
‘‘problem solving’’
is neither unique nor universal.
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Problem
Problem Solving
Solving
Definition:
The National Council of Teachers of
Mathematics (NCTM, 2000):
Problem solving is ‘engaging in a task for
which the solution method is not known in
advance’.
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Problem
Problem Solving
Solving
Definition:
In order to find a solution, students must
draw on their knowledge, and through this
process, they will often develop new
mathematical understandings. Solving
problems is not only a goal of learning
mathematics but also a major means of doing
so.
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Problem
Problem Solving
Solving
Definition:
Bell and Burkhardt:
‘Problem solving is the activity called into
play when there is a demand to apply
knowledge, skill and experience to
unfamiliar situations’.
Again we have the unfamiliar or unknown method of approaching the problem.
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Problem
Problem Solving
Solving
Definition:
Mayer (1992):
Problem solving is cognitive processing
directed at achieving a goal when no
solution method is obvious to the problem
solver. According to this definition,
problem solving has four main
characteristics.
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Problem
Problem Solving
Solving
Definition:
Mayer (1992):
First, problem solving is cognitive—it
occurs within the problem solver’s cognitive
system and can be inferred indirectly from
changes in the problem solver’s behavior.
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Problem
Problem Solving
Solving
Definition:
Mayer (1992):
Second, problem solving is a process—it
involves representing and manipulating
knowledge in the problem solver’s cognitive
system.
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Problem
Problem Solving
Solving
Definition:
Mayer (1992):
Third, problem solving is directed—the
problem solver’s thoughts are motivated
by goals.
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Problem
Problem Solving
Solving
Definition:
Mayer (1992):
Fourth, problem solving is personal—the
individual knowledge and skills of the
problem solver help determine the difficulty
or ease with which obstacles to solutions can
be overcome.
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Problem
Problem Solving
Solving
Definition:
Hayes (1989):
‘‘Whenever there is a gap between where
you are now [an initial situation] and
where you want to be [an adequate
response], and you don’t know how to find
a way [a sequence of actions] to cross that
gap, you have a problem’’.
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Problem
Problem Solving
Solving
Definition:
Mayer and Wittrock (1996):
‘‘A problem occurs when a problem solver
wants to transform a problem situation
from the given state into the goal state but
lacks an obvious method for accomplishing
the transformation’’.
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Problem
Problem Solving
Solving
In the mathematics and mathematics
education literature, no universally
accepted definition exists for the
mathematical terms ‘‘task’’, ‘‘problem’’,
or ‘‘exercise’’.
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Problem
Problem Solving
Solving
Problem solving is an integral part of
all mathematics learning, and so
it should not be an isolated part of
the mathematics program.
Good problems will integrate
multiple topics and will involve
significant mathematics.
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Problem
Problem Solving
Solving
Standards: (NCTM)
Instructional programs from
prekindergarten through grade 12
should enable all students to—
• build new mathematical
knowledge through problem
solving;
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Problem
Problem Solving
Solving
Standards: (NCTM)
Instructional programs from
prekindergarten through grade 12
should enable all students to—
• solve problems that arise in
mathematics and in other
contexts;
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Problem
Problem Solving
Solving
Standards: (NCTM)
Instructional programs from
prekindergarten through grade 12
should enable all students to—
• apply and adapt a variety of
appropriate strategies to
solve problems;
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Problem
Problem Solving
Solving
Standards: (NCTM)
Instructional programs from
prekindergarten through grade 12
should enable all students to—
• monitor and reflect on the
process of mathematical
problem solving.
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Problem
Problem Solving
Solving
How can problem solving help students
learn mathematics?
Good problems give students the chance
to solidify and extend what they know
and, when well chosen, can stimulate
mathematics learning.
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Problem
Problem Solving
Solving
The teacher’s role in choosing worthwhile
problems and mathematical tasks is
crucial. By analyzing and adapting a
problem, anticipating the mathematical
ideas that can be brought out by working
on the problem, and anticipating students’
questions, teachers can decide if particular
problems will help to further their
mathematical goals for the class.
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Problem
Problem Solving
Solving
There are many, many problems that are
interesting and fun but that may not lead
to the development of the mathematical
ideas that are important for a class at a
particular time.
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Problem
Problem Solving
Solving
Choosing problems wisely, and using
and adapting problems from
instructional materials, is a difficult
part of teaching mathematics.
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Problem
Problem Solving
Solving
In the primary curriculum in England
and Wales, problem solving comes under
the strand of ‘Using and applying
mathematics’. Guidance notes provided
by the Department for Education and
Skills (DfES, 2006a) identifies five
themes within this strand:
Mathematics Across the Curriculum
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Problem
Problem Solving
Solving
•Solving problems.
•Representing – analyse, record, do, check,
confirm.
•Enquiring – plan, decide, organize,
interpret, reason, justify.
•Reasoning – create, deduce, apply, explore,
predict, hypothesize, test.
•Communicating – explain methods and
solutions, choices, decisions, reasoning.
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Problem
Problem Solving
Solving
Mayer (1985) identified the following
factors as contributing towards problem
solving performance:
• Practice in recognizing problem types.
• Practice in representing problems –
whether concretely, in pictures, in
symbols, or in words.
• Practice in selecting relevant and
irrelevant information in a problem.
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Problem
Problem Solving
Solving
Orton and Frobisher (1996) identified the
following processes within problem solving:
• Operational processes of collecting and
ordering data.
• Mathematical process of searching for
patterns.
• Reasoning processes of analysis and reflection.
• Communication processes in describing
methods.
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Problem
Problem Solving
Solving
The most famous ‘processes’ involved in
problem solving by Polya (1957):
• Understanding
unknowns or data.
the problem, e.g. looking for
• Devising a plan, e.g. looking for related problems,
restating the problem in a different way.
• Carrying
step.
out the plan, in particular checking each
• Looking back, e.g. checking the result, obtain the result
differently, use the result for other problems.
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Problem
Problem Solving
Solving
Standard for Grades 9-12
In high school, students’ repertoires of problemsolving strategies expand significantly because
students are capable of employing more complex
methods and their abilities to reflect on their
knowledge and act accordingly have grown.
Thus, students should emerge from high school
with the disposition, knowledge, and strategies
to deal with the new challenges they will
encounter.
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Problem
Problem Solving
Solving
Standard for Grades 9-12
As in the earlier grades, problems and
problem solving play an essential
role in students’ learning of
mathematical content and in helping
students make connections across
mathematical content areas.
?
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Problem
Problem Solving
Solving
Standard for Grades 9-12
What should problem solving
look like in grades 9 through 12?
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Problem
Problem Solving
Solving
Standard for Grades 9-12
Problem solving plays a dual role in the
high school curriculum. On the one hand,
solving problems that have been
strategically chosen and carefully
sequenced is a fundamental vehicle for
learning mathematical content.
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Problem
Problem Solving
Solving
Standard for Grades 9-12
Most mathematical concepts or
generalizations can be effectively
introduced using a problem situation
that helps students see important
aspects of the idea to be generalized.
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Problem
Problem Solving
Solving
Standard for Grades 9-12
On the other hand, a major goal of high
school mathematics is to equip students
with knowledge and tools that enable them
to formulate, approach, and solve problems
beyond those that they have studied.
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Problem
Problem Solving
Solving
Standard for Grades 9-12
High school students should have significant
opportunities to develop a broad repertoire of
problem-solving (or heuristic) strategies. They
should have opportunities to formulate and refine
problems because problems that occur in real
settings do not often arrive neatly packaged.
Students need experience in identifying problems
and articulating them clearly enough to determine
when they have arrived at solutions.
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Problem
Problem Solving
Solving
The following problem serves multiple
purposes. It gives students an opportunity
to build their content knowledge during the
problem solving process, to learn or practice
some heuristic strategies, and to make
connections among various ways of
thinking about the same mathematical
content.
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Problem
Problem Solving
Solving
Problem:
How many rectangles are there on a
standard 8 × 8 checkerboard?
Count only those rectangles (including squares)
whose sides lie on grid lines.
For example, there are nine rectangles on a 2 × 2
board, as shown in next figure.
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Problem
Problem Solving
Solving
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Problem
Problem Solving
Solving
This problem provides students with an
opportunity to review counting
techniques and show their power, as
well as to use their prior knowledge in
other approaches to solving the
problem.
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Problem
Problem Solving
Solving
A teacher and a class with a problemsolving disposition will be quick to
formulate interesting extensions such as,
What would an analogous threedimensional problem look like?
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Problem
Problem Solving
Solving
What should be the teacher’s role
in developing problem solving in
grades 9 through 12?
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Problem
Problem Solving
Solving
Successful problem solving requires
knowledge of mathematical content,
knowledge of problem-solving strategies,
effective self-monitoring, and a productive
disposition to pose and solve problems.
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Problem
Problem Solving
Solving
A significant part of a teacher’s
responsibility consists of planning problems
that will give students the opportunity to
learn important content through their
explorations of the problems and to learn
and practice a wide range of heuristic
strategies.
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Teaching
Teaching Problem
Problem Solving
Solving
Teaching is itself a
problem-solving
activity.
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References:
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