2. INTRODUCTION
“Problem solving must not be treated by
teachers as a dish on the buffet table but rather
a sauce in every dish that is on the buffet table
serving a feast of mathematical dishes.”
3. OBJECTIVES
Teacher candidates will be able to:
Identify the role of the teacher in problem
solving
Explain types of problem solving lessons
Explain the stages in a problem solving
lesson
Assess problem solving lesson
4. THE ROLE OF THE TEACHER
The novice (first level)
A beginner (second level)
An enthusiast (third level)
An expert (fourth level)
5. THE NOVICE
Students have little or no understanding of
what problem solving is, of the meaning of
strategy, or of mathematical structure of the
problem. Most students at this level do not
know where to begin to solve a non-routine
problem.
Teacher assumes the role of a model.
6. A BEGINNER
Student understand the meaning of problem
solving, of strategy, and the mathematical
structure of a problem. They are able to
follow someone’s solution and can often
suggest strategies to be tried for problems
similar to those they have seen before.
Although they will participate actively in
group, many feel insecure about independent
problem solving
Teacher acts a prosthesis or crutch.
7. AN ENTHUSIAST
Student begin to feel comfortable with
problems. They suggest strategies different
from those who have seen used. They
understand and appreciate that problems
may have multiple solutions or in present
state of the problem may have “no solution”
unless an assumption is made.
Teacher becomes problem provider.
8. AN EXPERT
Student are able to select appropriate strategy
for most problems encountered and are
successful in finding solutions much of the time.
They show an interest in elegance and
efficiency of a solution and in finding an
alternative solutions to the same problem. They
suggest variation of old problem and are
consistently searching for problems to challenge
themselves.
Teacher serves a facilitator.
10. TYPES OF PROBLEM-SOLVING LESSONS
Teaching through problem solving;
Teaching about problem solving; and
Teaching for problem solving
11. STAGES IN PROBLEM SOLVING LESSONS
Modeling;
Coaching;
Explaining; and
Providing.
12. HOW TO ASSESS PROBLEM SOLVING?
Observing and questioning
Observation checklists
Scoring scales
Analytic scoring
Focused holistic scoring
Problem solving portfolios
Editor's Notes
It usually takes times to develop proficiency in problem solving. Students must be given problems to solve time and again throughout their entire mathematics at school.
Kantowski (1980) suggest that the role of the teacher may differ according to the levels of their students’ problem-solving capabilities. He has outlined the following characteristics of students at each level of problem solving development and the role of the teacher at each level.
s/he demonstrate the use of strategies to solve problems often thinking aloud and taking the students through the four stages of Polya’s problem solving model.
s/he guides them in their problem solving process, suggest alternatives when they come to a dead end. Provides them with problems that are carefully structured so they might fruitfully use the strategy they have observed
s/he provides the with more and more challenging problems to reinforce the usefulness of the strategies and to help them collect a store of related problems for future use
s/he motivates students to pose problems and facilitates the exchange of problems posed by students amongst them. s/he encourages them to view the world mathematically and a source of problems.
According to the teaching model, there are three phases in teaching process. The first phase helps students develop an understanding of a concept or a skill. The second one provides students with opportunities to consolidate the concept or skill. The third one provides opportunities for students to transfer their understanding to new situations. Problem solving involves students transferring their understanding of a concept or skill to familiar situations.
Three conceptions of problem-solving lessons will be discussed. Three lessons are described to illustrate each of the concept.
