This document discusses the importance of teaching problem solving in mathematics classrooms. It argues that most problems posed in classrooms are actually exercises, where students practice specific algorithms, rather than true problems that require strategy and creative thinking. The document advocates for teaching problem solving using heuristics like understanding the problem, devising a plan, carrying out the plan, and looking back. It also discusses problem solving strategies and the role of the teacher in facilitating problem solving lessons.

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Problem Solving
Dr. Mary Pat Sjostrom

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Why teach problem solving?

“Learning to solve problems is the principal reason for studying
mathematics.” (NCSM, 1977)

“Problem solving is not only a goal of learning mathematics but
also a major means for doing so.” (NCTM, 2000)

“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.” (NCTM, 2000)

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Problem or Exercise?
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A problemis a task for which one has no ready strategy for
solving; there are no memorized or prescribed methods.
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Most of the “problems” posed in U.S. mathematics classrooms
are really exercises. The purpose of an exercise (like the
purpose of exercises done in a physical workout) is to develop
skill with a particular method or algorithm.

Exercise (like working out) is important, but it should not be the
primary goal of the curriculum. Some research has found that
as much as 90-95% of time spent in mathematics classroom is
devoted to exercises – learning specific algorithms and
practicing them through exercise.

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“Students learn from the kind of
work they do in class.”
(Hiebert, 1997)

Practicing paper-and-pencil
skills on worksheet exercises
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faster at doing these types of
exercises.

Watching the teacher
demonstrate methods for
solving problems

better at imitating methods on
similar problems. (Also may
learn that they do not have the
capability to think on their own.)
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build new relationships. In other
words, construct understanding.

Reflecting on how things
work, how certain ideas or
procedures are
similar/different, how what
they know relates to new
situations

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Problem Solving can be taught
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Many in the United States think people are born problem
solvers or they are not – they “got the math gene”!
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Schoenfeld examined expert and novice problem solvers and
found that experts carry out Polya’s steps, reflect on and
monitor their thinking, and move flexibly between the steps.
Novices tended to jump to step 3; they grab some numbers and
just try to work the problem.

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Problem Solving can be taught

Japanese teachers routinely begin a lesson by posing a
problem, allow students time to work on it, lead a discussion of
student ideas, then have students go back to work.
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The lesson concludes with student presentations of their
solutions.
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Then the teacher helps the students to summarize the
mathematics and if possible generalize (e.g., develop a formula
or articulate a concept).

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Role of the Teacher
Teaching through problem solving requires a change in the role of
the teacher, a shift in how the teacher thinks about teaching and
learning.

Students do most of the mathematics – it may seem like the
teacher “isn’t teaching.”
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The teacher must select quality mathematical tasks that allow
students to learn mathematics content as they figure out strategies
and solutions.
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The teacher must plan and ask questions that encourage students
to verify their solutions and reflect on the strategies they use.
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Listening is critical – the teacher listens to students! (Who talks
most in your class? Who listens?)
(Van de Walle, 2010)

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Polya, 1945
Devise
a
Plan
Look Back
Problem Solving
Heuristics
Understand
the
Problem
Carry out
the
Plan

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Problem Solving Heuristics
Understand
What
the Problem
is known?
What are the data?
What are the conditions?
What are you trying to find out?

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Problem Solving Heuristics
Devise
Have
a Plan
you solved a problem like this
before?
What strategies might work in this
problem?

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Problem Solving Heuristics
Carry
out the Plan
Check
each step.
Can you see that each step is
correct? Can you prove that it is
correct?
If you aren’t making progress, feel
free to return to steps 1 and/or 2.

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Problem Solving Heuristics
Look
Can
Back
you check the result?
Does the answer make sense?
Can you get the same answer a
different way?
Can you use the result, or the
method, to solve a different problem?
Is there more than one answer?

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Problem Solving Strategies
Working
Finding
backwards
a pattern
Adopting
Solving
a different point of view
a simpler problem
Considering
extreme cases
Posamentier et al., 2010

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Problem Solving Strategies
Making
a drawing or diagram
Intelligent
guessing and testing
Accounting
for all possibilities
Organizing
data
Logical
reasoning
Posamentier et al., 2010

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References
Hiebert, J., et al. (1997). Making Sense: Teaching and Learning Mathematics with Understanding.
Portsmouth, NH: Heinemann.
National Council of Supervisors of Mathematics. (1997). Position paper on basic mathematical skills.
Retrieved Feb. 9, 2011, from
http://www.mathedleadership.org/docs/resources/positionpapers/NCSMPositionPaper01_1977.pdf
National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics.
Reston, VA: NCTM.
Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.
Van De Walle, J. A, Karp, K, & Bay-Williams, J. M.. (2010). Elementary and middle school mathematics (7th
edition). Boston: Pearson Education.