A2 Evidence from national mathematics competitions:

The questions below come from two national events taken in 1995. Each is aimed at the top 35% of the relevant age group. Thus the population is reasonably representative of those who might expect at a later date to go on to higher education. Our information suggests that schools do indeed restrict their entries to something like the top 35%, and this should be borne in mind when assessing the data. All indications point to the conclusion that the schools that take part tend to be above average in their aspirations and staffing.

The first group of questions comes from the UK Intermediate Mathematical Challenge for pupils in English School Years 9--11 (10--12 in Northern Ireland, S3 or S4 in Scotland), which had an entry of 115,000 from 1,500 schools. The second group of questions comes from the UK Junior Mathematical Challenge for pupils in English School Tears 7 and 8 (8 and 9 in Northern Ireland , S1 and S2 in Scotland), which had an entry of 105,000 from 1,400 schools.

Both events are one hour papers with twenty five multiple-choice questions of graded difficulty. Candidates are advised to concentrate their efforts, in the first instance, on Questions 1--15, and to attack later questions selectively. While the problems are technically elementary, they all require pupils to select and coordinate two or more simple steps. We have excluded only those problems which are sufficiently unusual, off-beat, or hard to make the interpretation of statistics problematic. The questions remaining are those which test standard curriculum material in a relatively straightforward way. (To allow for differing backgrounds, in the case of the UK JMC we have further restricted the selection to concentrate on problems on `number'.) After each option we give the percentage of candidates choosing that answer. The balance consists of those who failed to answer the question. The percentage corresponding to the correct answer is underlined.

Our central professional concern is with the quantity and quality of those proceeding to university. This has led us to consider such students' preparation at 16--19, and hence to comment on weaknesses in the curriculum at 5--16. Of the two sets of problems given here, those from the Intermediate Challenge are of most immediate concern; the Junior Challenge questions have been included mainly to complete the overall picture.

Among pupils' responses to the Intermediate Challenge questions we draw particular attention to the following. The success rate on Q1 is very low (given that candidates come roughly from the top third of the ability range), and the numbers choosing option C suggests that many think division by single digit numbers depends only on the last two digits. The response to Q7 reflects our concern 4C (Section 4): instead of using the given information to analyse the diagram mathematically, pupils have simply assumed that a triangle in the position shown necessarily has a right angle at A. The response to Q12 is most striking: where previous generations would be expected to calculate the percentage increase, we only ask how the percentage increase could be calculated. Finally, the responses to Q15 and Q17 illustrate basic weaknesses which hinder the subsequent development of algebraic technique.