PROBLEM POSING, PROBLEM SOLVING AND PEDAGOGY

Assumption (epistemologically) -  mathematics as result of human problem posing and solving
                                                    -  mathematics as a mental construction (creation, invention or
                                                       discovery)  

In term of learning, social constructivism identifies all learners of mathematics as creators of mathematics involving problem posing and solving.

Therefore as a consequences of problem posing and solving pedagogy :

1. School maths for all should be centrally concerned with mathematical problem posing and solving
    (reduce content- oriented mathematics curriculum)

2. Inquiry, investigation, problem posing or formulation should occupy a central place in the school
    maths curriculum and precedes problem solving

3. The pedagogy (teaching, learning and assessment) should be process and inquiry(or investigation)
    focused (vs product)

4. learner-centered view of investigation as a learner directed activities (new questions posed, new
    situations are generated and explored- promotes active learning)

5.  Increase learner autonomy and self- regulation ( develop reflective and meta-cognitive skills)


Mathematical problem posing (formulation, investigation etc) is divergent (creative thinking and higher order thinking) as the process of mathematical problem solving (critical thinking and higher order thinking - eg by using Polya method)) is convergent.








                                             

PROBLEM-BASED LEARNING(PBL) IN MATHEMATICS

Paradigm shift from traditional teaching model (content-based, teacher directed and student as knowledge recipient) to problem- based learning model( problem motivated, teacher as facilitator and student as problem solver).

Main characteristic of PBL approach is that,
the problem (real-world problem : unstructured and authentic (vs simulated) is the starting point of learning. (discuss its implications to the current maths curriculum).

eg (solve the following problems)

1. The costs for two different kinds of heating systems for a three- bedroom home are given below

solar system - cost to install rm 29700 and operating cost/year is rm 150
electric system - cost to install rm 5000 and operating cost/year is rm 1100

After how many years will total costs for solar heating and electric heating be the same?
What will be the total costs for both systems at that time?

2. Two ordinary six-sided dice are rolled, what is the probability of getting a sum of 8?
3.  Working together, Ahmad and Ali can complete a job in 4 hours. Working alone, Ahmad requires 
     6 hrs more than Ali to do the job. How many hrs does it take Ali to do the job if he works alone?

Benefits of PBL:

1. creating meaningful learning(content/topic/concept) through inquiry (emphasis on critical, logical  creative thinking, deep reasoning and metacognition))

2. encourage the development broad -based mathematical problem solving strategies (heuristics)
 rather than content learning in a limited sense.

3. development of self-directed/regulated/independent learners - students assume major responsibility
    for the acquisition of knowledge.




 

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.