Problem-Solving and Mathematics

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By Remediana Dias

Teachers devote much time to work with numbers and the practice of four rules, and many children achieve a satisfactory level of competence in this field, but few have sufficient opportunity for learning how to apply the skills they acquire to the solving of problems. Too few schools make opportunities for the development and extension of mathematical understanding which arises in children’s play, in their interest and in work in other parts of the curriculum.

The process of problem-solving is not new. It has gone on since man first existed. The human race was learning and achieving through problem-solving long before anyone gave any thought to the processes involved. People learnt what they wanted to know without worrying about it. We have a natural desire to find out and to explore. If this desire is nurtured from the start it becomes an extremely powerful force, not only in mathematics, but in all education. It is part of our cultural heritage and is, or can be very stimulating.

The success of developing problem solving in the classroom depends heavily on how the teacher sees his/her role in delivering the curriculum. Those who see their role as having the sole responsibility for controlling and leading the maths lesson, by dictating rules and giving quick tips on how to remember them, often, unintentionally, impose their own thinking styles on their pupils. In an effort to remember those rules and tips, pupils’ own ideas are often marginalized or even precluded and the most powerful motivator of all, the pleasure of getting there by themselves is thwarted.

Socrates is quoted as having said, “The ideas should be born in the student’s mind and the teacher should act only as a midwife.’ Unfortunately because school is about accelerating and guiding the learning process, the teacher often has to sow a seed in the student’s mind before it can develop and be born. However, the best problems are still those that occur naturally.

DEVELOPING STRATEGIES

In all problem-solving, whether it be from a natural source or teacher initiated, children should be assisted to develop their own strategies and make them as efficient as possible. Other more direct approaches can always be illustrated, bearing in mind that it is better that children use a less efficient method which they understand than an efficient one that they don’t. The learner’s own understanding is a vital element in any problem-solving  and they should feel comfortable in using the mathematics they know. After all this is how mathematics works in real life.

We should allow children time and latitude to break down the problem or investigation, using their own techniques and inventing their own strategies. If we start breaking it down into small steps for them we are in danger of fragmenting the problem so much that they become unable to see links between different mathematics topics. If the problem is both suitable and interesting, it will motivate children to solve it.

There is always the dilemma of whether skills and facts should be taught before problem-solving is introduced, or whether they should be an outcome of problem-solving. If it is postponed until techniques are developed, some children who take a long time to learn those techniques and  a short time to forget them again, will expend a great deal of energy in learning and relearning to do neat rows of sums, often missing out on problem-solving altogether. If the facts and skills taught in the classroom are going to be of use in real life then pupils must be able to use them with confidence and apply them to real problems no matter how much or how little they know.

Problem-solving is for everyone. It is not exclusive for those who can work with numbers. The techniques of reasoning, discussion and logic which are the essence of all investigative work, should be in order before symbols are introduced. Symbols are not the concept but are representative of everything that has been understood before they are a shorthand way of communicating what is known. Therefore early problem-solving should be about making concepts meaningful rather than numeric symbols. It should flow from free choice of topics to those gathered together to encourage a particular outcome.

EARLY PROBLEM-SOLVING

There are two major stages in early problem-solving that precede any work with symbols. They integrate mathematical relationships and create flexibility with mathematical ideal and concepts. Pupils who have difficulty with symbols need not be excluded from problem-solving activities as they can become highly motivated by these two pre-symbolic stages.

Stage 1: Physical experience

Very young problem-solvers work in silence and usually alone, exploring and learning how to handle the environment at their own pace, using trial and improvement methods. The materials are real and therefore appealing to the senses and very enjoyable. Pupils’ involvement and ability to concentrate is quite remarkable. No one evaluates their investigations or suggests that they should working in another way.

Stage 2: Language to describe that experience

Once a child begins to acquire language, discussion can take place. He/she is now able to question ‘why’  and does so frequently. The development of speech brings with it the development of verbal reasoning and logic. Children are able to discuss their investigations and improve on their methods if they wish.

Getting involved, discussing and reason are important problem-solving states. During these early stages few children experience difficulty. It is when symbols are introduced that difficulties begin to occur. Techniques can be very hard to learn if they are not lined to previous experiences.  For example: take the sum of 7*3=21. Do you know what I am talking about? It could be the number of days in 3 weeks, the number of sides on seven triangles. It is when mathematics is about nothing at all that it is difficult for most children to understand and apply. The lack of understanding and meaning can lead to lack of motivation and then using and applying mathematics becomes a problem in itself.

Problem-solving and investigations should pervade all mathematics learning. They invite pupils to question, challenge, discuss, explore and record all areas of mathematics including numbers. Individual learning is dependent on individual experience and individual ability to process mathematics but we all learn better if that experience is enjoyable. Problem-solving is enjoyable, it allows us to use the methods in which we feel confident. The direction of the solution is up to the solver.

Remediana Dias is a Dyslexia Practitioner based in Dubai.

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