TEACHING MATHEMATICAL PROBLEM SOLVING

In USA :

1. Philosophy of mathematics education for the 21 st century :
   The goal of teaching mathematics is to help all students develop mathematical power
   ie to produce effective problem solvers and powerful mathematical thinkers.

2. Mathematics must be seen as an integrated whole ( not as a separate and unrelated topics),
    as a part of human experience, emerging from everyday experience, interaction with science
    and technology and other fields.
   

3. Spend more time on developing broad- based mathematical problem solving skills
    ( general problem solving techniques ie heuristics) and less time on perfecting routine
     computations.

4. Teaching mathematics as problem solving

  - problem solving as a means as well as a goal of instruction
  - apply problem solving skills to solve problems in new contexts with emphasis on multi-steps
    and non-routine problems.
  - recognize and formulate (posing) problems from real word situations/phenomena
  - mathematics is problem-centered and application- based
  - subject to be investigated, discovered, explored and created

5. Problem solving is seen as the most important means to develop powerful mathematical thinkers.

TEACHING AND LEARNING APPROACHES IN MATHEMATICS

There are two main approaches in teaching and learning mathematics based on the psychological theories in education:

 Behaviorist approach :

1. drill -practice (practice makes perfect)
2. mastery of skills (lower order thinking skills- knowledge, comprehension and application)
3. performance- based  (how to do) - suitable for routine/familiar problems
4. focus on algorithm (procedures/steps of calculation)
5. mistakes and errors should be avoided/minimized
6. teacher- centered (focus on teaching)

Cognitive approach :

1. construction of meaning (searching for meaning)
2. conceptual understanding (higher order thinking skills - analysis, synthesis and evaluation)
3. thinking- based (emphasis on why) - suitable for non- routine/ unfamiliar problems
4. focus on heuristic (general methods of solving problems) - Polya's Model
5. mistakes and errors is good indicators of misconceptions and difficulties
6. student- centered (focus on learning)

Discuss with suitable examples on how to use behaviorist and cognitive approaches in teaching and learning mathematics in the classroom. Why teachers need to master both approaches?