Apply Polya's heuristics for problem solving to the "problem" of teaching.

1. +
Problem Solving
Dr. Mary Pat Sjostrom
Bear with me…
this is relevant to this module.

2. +
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)

3. +
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.

4. +
Devise
a
Plan
Look Back
Problem Solving
Heuristics
Understand
the
Problem
Carry out
the
Plan
Polya, 1945
Remember these?

5. Can we view
teaching
as
+ problem solving?

6. +
Devise
a
Plan
Look Back
Teaching
as
problem solving
Understand
the
Problem
Carry out
the
Plan
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?

7. +
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
this concept?

What misconceptions might students have?

Where would you like student thinking to be at the
end of this lesson?

8. +
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
students?

How do you expect students to respond to
__________?

How will you know students understand?

9. +
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
lesson progresses.

10. +
Look Back

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?

11. +
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.
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.