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