Problem-Solving in Mathematics Education: Going Beyond Basics

Problem solving refers to the process of tackling a problem to try and solve it. In mathematics problem solving makes use of mathematical processes which enable pupils to develop new insights, and sometimes new procedures. It involves exploration, discovery and analysis. Problem solving begins with a task which the pupils understand and are willing to engage in, but for which they have no immediate solution. It is associated with developing and learning ways to tackle and solve problems. According to Broomes, Cumberbatch, James and Petty (1995) problem solving should no longer be viewed as an activity in which pupils engage after they have acquired certain mathematical concepts and skills.

Problem solving should be viewed both as means of acquiring new mathematical knowledge and as a process for applying what has been previously learned.

George Polya has also propose a four-step process for problem solving. These four steps are understand the problem, devise a strategy/plan for solving the problem, carry out the strategy/ plan and check for results or look back and check.

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These four steps are important in helping to do problem solving. The first stage of Polya’s problem solving is understanding the problem. In order for the students to solve a problem in a mathematics classroom they must first understand the problem or understand what they are asked to find or do. The problem must be read carefully then analyzed. Polya taught teachers to ask students questions such as:Do you understand all the words used in stating the problem?What are you asked to do or show?Can you restate the problem in your own words?Can you think of a picture or diagram that might help you understand the problem?Is there enough information to enable you to nd a solution?

Polya's Second Principle: Devise a plan

Polya mentions that there are many reasonable ways to solve problems.

The skill at choosing an appropriate strategy is best learned by solving many problems. You will nd choosing a strategy increasingly easy.

A partial list of strategies is included: Guess and check Look for a pattern

Make an orderly list Draw a picture
Eliminate possibilities Solve a simpler problem
Use symmetry Use a model
Consider special cases Work backwards
Use direct reasoning Use a formula
Solve an equation Be ingenious

1Polya's Third Principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals.

Polya's Fourth Principle: Look back

Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn't. Doing this will enable you to predict what strategy to use to solve future problems. So starting on the next page, here is a summary, in the master's own words, on strategies for attacking problems in mathematics class.

This is taken from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957,