How to solve math problems

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Updated: 22 October, 2007

How do you teach a child to solve math problems?

Observations

• Some teachers (including parents and tutors) teaches problem solving by showing and explaining the solution to the pupils. They may break the solution into steps and check that the pupil understand each step before going on to the next. Typically, what happens when a child asks his father to help solve a math problem is that the father mentally tries various methods or alternatives until he arrives at a solution. This may take a few seconds or much longer (fumbling around initially, making mistakes, etc). He then explains the solution to his child.
• Sometimes teachers work backwards to get the solution. The father mentioned above may have worked out the answers instantly without much difficulty or he may have used calculators or other skills such as algebra. He then uses model diagrams to explain the solution to his child. Because he already knows the answer, he draws the model diagram in such a way that it explains what he already knows. For example, he may start off by dividing a bar into 16 sub-units or by grouping the bars in a particular way when there is no indication from the information given in the question that this is required. (He should have started the model with information that the child, not knowing the answer, could reasonably extract from the question.)
• When a pupil is unable to solve a math problem from an assessment book, some teachers refer to the answer key and explain the given solution to the pupil.

The teachers described above were explaining or teaching the solution to the pupil. The pupil may understand the solution to the given question and be able to apply the procedure or solution method to solve other similar questions. However, he may not be able to solve a related but unfamiliar problem.

Teachers focused on explaining the solutions to problems misses something important. The pupil sees the solution without seeing the teacher thinking (or maybe struggling) to arrive at the solution. What is missing from the teacher's instruction is how to find the solution. How did the teacher pull the rabbit out of the hat? The pupil knows the various mathematical concepts, heuristics and algorithms. But how does the teacher know which method or combination of methods to use for the different problems? (Unfortunately, some pupil erroneously conclude at this stage that they do not have the aptitude for math.) The pupil understands the plan of action to solve the math problem and is able to carry it out. However, he is not trained to prepare the plan himself. The strategies and thinking that brings forth the solution (the skills the child needs most for problem solving) are not taught.

"There are two aims which the teacher may have in view when addressing to his students a question or a suggestion...: First, to help the student to solve the problem at hand. Second, to develop the student's ability so that he may solve future problems by himself."
G. Polya

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