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# Why Do Problem Solving?

Using a problem solving approach to teaching and learning maths is of value to all students and especially to those who are high achieving. The reasons for using problem solving are summarised below.

• Problem solving places the focus on the student making sense of mathematical ideas. When solving problems students are exploring the mathematics within a probem context rather than as an abstract.
• Problem solving encourages students to believe in their ability to think mathematically.  They will see that they can apply the maths that they are learning to find the solution to a problem.
• Problem solving provides ongoing assessment information that can help teachers make instructional decisions.  The discussions and recording involved in problem solving provide a rich source of information about students' mathematical knowledge and understanding.
• Good problem solving activities provide an entry point that allows all students to be working on the same problem.  The open-ended nature of problem solving allows high achieving students to extend the ideas involved to challenge their greater knowledge and understanding.
• Problem solving develops mathematical power.  It gives students the tools to apply their mathematical knowledge to solve hypothetical and real world problems.
• Problem solving is enjoyable. It allows students to work at their own pace and make decisions about the way they explore the problem.  Because the focus is not limited to a specific answer students at different ability levels can experience both challenges and sucesses on the same problem.
• Problem solving better represents the nature of mathematics. Research mathematicians apply this exact approach in their work on a daily basis.

There are several reasons why it is important to do problem solving or to take a problem solving approach to mathematics, especially with more able students. We discuss these under three headings below: to benefit the student, to better represent the subject and to benefit the teacher.

## To benefit the student

Probably the first way that a student’s exceptional ability comes to the fore is because they solve sums accurately, efficiently, and speedily. Certainly in primary school, students stand out because they learn new concepts quickly, manipulate numbers intuitively and easily remember number facts.

Learning and applying algorithms in standard situations can quickly become not very interesting for them. Introducing a problem solving approach to mathematics can present bright students with a challenge. It gives them an opportunity to think outside the square and develop their confidence in themselves.

Not all bright students’ ability thrusts them to the fore. Some gifted children may, for whatever reason, quietly sit in a class and be content with the regular programme. Problem solving activities can give these students more motivation and challenge and provide them with an interest in the subject and renewed confidence in themselves.

## To better represent the subject

Some observers see mathematics as a subject where success depends on learning rules that are to be followed without thought. We have tried in the Bright Sparks section to show the research aspect of mathematics. This aspect is the creative side of mathematics and is what research mathematicians do on a day to day basis. The highs and lows of their creative activity are felt by mathematicians in the same way that they are by artists, musicians and sports people.

In the Bright Sparks section we have tried to show that solving problems parallels very closely the creation of mathematics. By encouraging problem solving in mathematics classrooms we are helping show that there is another aspect to mathematics that has a much more human face and is more interesting than simply following rules.

## To benefit the teacher

There are also some advantages to implementing a problem solving approach for teachers. Being the teacher of a really exceptional student can be very challenging. It may be in the best interests of all if a mentor can be found to extend the students and relieve the teacher of some pressure. However, the Bright Sparks and problem solving sections of this website may be of some help to a teacher ‘in need’. Once an able student is familiar with using a problem solving approach they should be able to work independently on a mathematical problem for an extended period of time, choosing and exploring generalisations and extensions to the problem that interest them. We hope that it works for you and that you can see how to use it to develop the next bright spark that comes into your class.