2. Using and applying mathematics

"How use doth breed a habit in a man"
Shakespeare, 'Two Gentlemen of Verona'

2.1 The applications of mathematics are so wide-spread and the techniques used so varied that it is difficult to envisage any coherent scheme of study that could credibly be described simply as 'the use and application of mathematics'. Clearly a meaningful synthesis can only be extracted from consideration of the details included in the Program of Study and Attainment Targets and, as implied in the preamble, this amounts to little more than learning correct nomenclature and typical applications of the individual topics covered in the following sections.

2.2 A general point which perhaps merits more attention is that words in mathematics are often used with a specialised meaning, different or more restricted than in general language. Where this arises it is important that a definition of the word in the context is given and that pupils are clear that implications arising from any more general usage or other contexts cannot be imported into the mathematical reasoning.

2.3 The section on selecting and using mathematics is likely to create practical difficulties with mixed ability groups, particularly the suggestion that pupils should be taught to 'try different mathematical approaches to problems' and 'develop their own mathematical strategies'. With the more aware pupils the idea that there are alternative methods of solution which may be quicker or easier can stir the imagination but for the less advanced the prospect is much more likely to be inhibiting.

2.4 Perhaps the most cogent points in this section are the need for pupils to present information and results clearly, and to explain their reasoning. In the early stages of learning mathematics students tend to become unwittingly conditioned to the situation that problems presented to them (a) have a solution and (b) yield to the methods currently being studied. They need to appreciate that eventually, in real life, this will not always be so and others will need to follow their reasoning and working. Another important issue is that they acquire the habit of checking their results and considering whether they are reasonable.

2.5 The dangers of the suggestion that pupils should be taught 'to make conjectures of their own based on the evidence they have produced' have been dealt with in the Introduction. The essential need is rather that they develop an ability to justify and validate their conjectures reliably: it is important for their future development that they are not left with any misconceptions as a result of their conjectures.