Dr. Mary Pat Sjostrom
Bear with me…
this is relevant to this module.
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)
Problem or Exercise?
A problemis a task for which one has no ready strategy for
solving; there are no memorized or prescribed methods.
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.
We can approach and solve teaching as we approach
and solve a mathematics problem:
heuristically, rather than algorithmically.
How do Polya’s heuristics apply to teaching?
Understand the Problem
What are your goals for students?
What concept do you want students to understand?
What is the Big Idea of the mathematics?
What are the real world connections?
What do you expect students to already know about
What misconceptions might students have?
Where would you like student thinking to be at the
end of this lesson?
Devise a plan
How do you usually teach this lesson?
What problem (or what coherent sequence of
problems) can you pose that will engage students
in developing this mathematical concept?
What resources are available to you? For the
How do you expect students to respond to
How will you know students understand?
Carry out the plan
Teach the lesson.
Listen to student thinking.
Students should do the mathematics. (You
don’t need the practice, they do!)
Monitor and reflect on your teaching.
You may have to modify the plan as the
How do you think the lesson went?
What worked? What didn’t work?
Did students learn what you wanted them to
learn? How do you know?
What would you do differently if you could
teach the lesson again?
Based on what happened today, what will you
do when the class meets again?
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
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.
Posamentier, A. S., Smith, B., &Stepelman, J. (2010). Teaching secondary mathematics: Techniques and
enrichment units (8th Edition). Upper Saddle River, NJ: Merrill Prentice Hall.
Van De Walle, J. A, Karp, K, & Bay-Williams, J. M.. (2010). Elementary and middle school mathematics (7th
edition). Boston: Pearson Education.