Strategies for solving arithmetical problems

Students develop and use a range of methods when solving problems. These strategies tend to become more sophisticated as better ways of determining the answer are developed.

The range of strategies that students use to solve problems in mathematics is best described through an example.

Children learn the forward sequence of number words initially in the same way as they learn the alphabet, as a continuous string. To find the answer to this question they need to know the correct sequence of number words, to match the count to the objects and to recognise that the last number stated signifies the total.

A child who counts them all but starts again from one is using a less sophisticated strategy than a child who counts on from nine. To be able to count on from nine, children need to be able to cope with the forward sequence of number words. If you ask children what number comes after nine, they will often initially count from one to find the answer. In developing the ability to count on, students need to know the sequence of number words well enough to continue counting from any number.

Typically, students move through developing knowledge of the sequence of number words, to combining and counting all the objects they can see, to counting on and eventually to using addition facts.

When approaching addition questions using a counting on strategy, a more efficient strategy is to always count on from the larger number. Similarly, in answering 3 + 9, many students will often "bridge to ten", saying: "1 + 9 is 10 and I have 2 more, making 12".

The purpose of this example is to demonstrate the range of solution strategies students use and to emphasise the need to help them to develop sophisticated strategies. One of the difficulties with inefficient strategies is that, although they are slower, they still work. This means that inefficient strategies can be very persistent. A student asked to find 8 + 3 could count out 8, then count out 3 and finally count all the objects to obtain an answer. If this strategy persists in later years, the amount of mental effort needed to obtain the answer makes it difficult to achieve further learning.

It is important to learn to use addition facts automatically, as it allows the student to attend to other features of problems. Basic strategies can persist even after students develop more sophisticated approaches. Competent adults will occasionally revert to using their fingers to count on at times, because this strategy achieves the correct answer and doesn't require as much thinking as using addition facts.