Broad- based mathematical problem solving strategies

...in USA (cont..)

NCTM (National Council of Teachers of Mathematics) - focus on concept development and problem solving.

By learning and acquiring a variety of broad-based ( general ) mathematical problem-solving
strategies (heuristics) students are equip to be a better problem solvers across the topics in mathematics or transferring those skills to a variety of problems.  Some of the general problem solving strategies are :

1. Characterize the problem : What is given? What is needed?What is missing? etc
2. Have you seen this before? : or different form ?
3. Look for pattern : eg Gauss recognized a pattern 1+2...+100 = ?
    1+100=2+99=...101 (50 pairs)
     50 @ 101 = 5050
4. Simplification/reduction :  can the problem be broken up into smaller or manageable
    sub-problems?
5. Work backwards : when trying to prove a theorem, it may begin from the conclusion and back track logically
6. Modeling/simulation : a mathematical model may be developed that simplify some complicated process/phenomena in the real word (representing/translating into a mathematical forms eg table, diagram, chart, graph, equation, relationship, function, inequalities, matrices, etc)
7. Logical reasoning/arguments - inductive and deductive reasoning,
8. guess and check/improve - develop a sense of estimation
9. make and test conjectures
10. formulate/pose problems from situations within and outside mathematics


     

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?




 

PERSPECTIVES OF MATHEMATICS ANT ITS RELATION TO TEACHING AND LEARNING

Perspectives of mathematics :

Mathematics as a dynamic and continually expanding field of human creation, involving the process of inquiry, thinking, reasoning (with its intellectual rigour), discovery and invention as well as a cultural product of various civilizations.

1. Mathematics is a practical and problem- driven and problem solving knowledge (ie mainly arise from practical or real- life situations).

2. Mathematics is a science of numbers, shape and space and relationships.

3. Mathematics as a language

4. Mathematics as a way of thinking


Assignment 1 :

Discuss in what ways do the perspectives of mathematics influence the curriculum design in mathematics at the secondary school level ?

Assignment 1

Assignment 1 (20%) with presentation

Problem solving is a fundamental process and an integral part of mathematics.
Critically discuss with suitable examples the epistemology (origin/sources/discovery/grows/development/potential/validity/limitation (if any) of one mathematical concept/topic/branch in the contexts of problem solving (refer to at least one mathematics textbook at the primary or secondary school).

Sinopsis MPS 1813

MPS 1813 -  PENYELESAIAN MASALAH DALAM PENDIDIKAN MATEMATIK
                      (PROBLEM SOLVING IN MATHEMATICS EDUCATION)

                      Key concepts/questions :

                      1. Mathematics - brief history and philosophy of mathematics related to problem 
                          identification, formulation and solution.

                     2.  Mathematics as problem solving - models (strategies, methods, heuristics
                          and techniques).

                     3.  Mathematics education (teaching, learning and assessment) from the perspective of
                          problem solving.

                     4.  Research on problem solving in mathematics education.

                     5.  Issues and trends of problem solving in mathematics education.

                     



                    

MATHEMATICS - THE SCIENCE OF PATTERNS AND ITS IMPLICATIONS TO TEACHING AND LEARNING

Mathematics is one way of understanding the world and universe.
Patterns are everywhere. Patterns occur in geometry (eg wallpaper/tiles patterns using various geometrical shapes/figures), in music,  in human behavior (eg voting patterns )

What the mathematician does is mainly looking for a  pattern or to examine abstract patterns which give rise to different branches of mathematics - numerical patterns/patterns of numbers and counting  (arithmetic and number theory), patterns of shape, symmetry and regularity (geometry; the mathematics of beauty- transformation), patterns of motion and change (mathematics in motion- calculus), patterns of reasoning and communicating (logic; logical arguments/connections), patterns of chance (probability theory - making prediction etc), patterns of closeness and position (topology), and so on.

Those patterns can either real or imagined, visual or mental, static or dynamic, qualitative or quantitative,
purely theoretical or utilitarian.

Those patterns can arise from the world or phenomena around us, from the depths of space and time
( geometry of the universe; three- dimensional physical universe etc), or from the inner workings of the human mind.

Discuss the implications of maths as a science of patterns to the teaching and learning of mathematics in secondary schools.