Mathematical Challenge 1999-2000

Edward M Patterson

Here are the figures for the total numbers of entries in Mathematical Challenge for each session since the competition began:

Table 1: Numbers of entries for Mathematical Challenge

These are the numbers of entries, which must not be confused with numbers of entrants. If, in one session, 3 sets of problems are circulated, then one entrant can submit 1, 2 or 3 entries.

The total number of entries over all the 24 sessions is 124517. The figures suggest that the yearly total peaked at 14316 in 1996-97, after which there was a decline. However, this rudimentary analysis of the figures conceals more than it reveals. The figures should be looked at in the light of various substantial changes which have been introduced from time to time. The following brief history of the competition since it began in 1976 might be helpful.

1976-77. Although this initial year was exciting and enjoyable, it was very hard work for many people. Five sets of five problems each proved to be too much for most contestants, as well as for teachers and markers, so it was agreed that in 1977-78 there would be four sets of four problems each.

1977-85. During the early years, the competition was regarded as being suitable for students in S5 or S6. Indeed in an information leaflet it was stated that 'Nothing outside the O grade Mathematics syllabus will be assumed', which might of course be translated (incorrectly!) into 'Everything inside the O grade mathematics syllabus will be assumed'. A few very bright younger members of the schools did take part and some were outstandingly good. The organisers were pleased to see them, but hoped also to hear of students who had not previously shown much interest in Mathematics, but had been converted to the subject as a result of their experiences in responding to Mathematical Challenge. Year by year the competition followed the same style, with some changes in the numbers of entries, but nothing substantial. Shortages of teachers may have had an influence here and there, but sometimes the problems themselves were very ambitious.

1985-91. In session 1985-86, a major change took place. It was suggested by some of the markers in Section II that Mathematical Challenge should encourage junior students by making more of the problems accessible to them. It was decided that there would be four sets of problems, with five problems in each set. The first three problems would be suitable for students in S1 or S2 and these students would be judged on their entries for these problems. Students in S3, S4, S5 or S6 would be judged on their attempts at all five problems in each set.

That new system was in force for six sessions. The numbers of entries from students in S1 and S2 increased, but at the same time the numbers of entries from S5 and S6 diminished. Of the 3500 entries in 1985-86, 1348 were from S1 or S2. However, of the 4422 entries in 1990-91, 3146 were from S1 or S2. Thus the numbers from S1 and S2 changed from about 38% of all entries in 1985-86 to about 71% in 1990-91. This was a worrying development and at the same time there was concern about students in S3 and S4, for whom there seemed to be no satisfactory arrangements within the system.

1991-95. At a special meeting of the National Committee in May 1991, it was agreed that a revised system would operate from session 1991-92 onwards. The competition would be split into three 'Divisions': Junior for S1 and S2, Middle for S3 and S4, and Senior for S5 and S6. Separate sets of problems would be used in the three divisions, although some overlap would be allowed. There would be three sets of problems, with four problems in each set. Contestants could enter a higher division than that indicated by their year, but not a lower division. Having chosen such a division, they were expected to remain in it. From the outset the new system seemed to work well, even though for the first two sessions the total number of entries did not show any sign of a big increase, being at much the same level as in 1990-91. Then there were two significant changes. One was a rapid increase in the numbers of entries in the Junior and Middle Divisions and the other was the emergence of the Primary Division. The possibility of extending the competition to primary schools had been aired earlier, but there had been doubts about its viability.

1. Would it be possible to establish and retain contact with all the primary schools in Scotland?

2. The selection of problems suitable for pupils in primary school would be difficult, since few (if any) members of the National Committee had experience of teaching in a primary school.

3. If large numbers of entries were received from primary schools, it might prove to be difficult to find an adequate number of experienced markers.

However, following a definite request from Section 3 (Lothian, Borders and Dumfries & Galloway), an experiment was tried in session 1994-95, when entries for two specified problems in each set in the Junior Division were invited.

1995-99. The new Primary Division was formally introduced at the beginning of session 1995-96. Contestants, mainly from P7, established the status of the new Division quickly and enthusiastically. [Entries from primary schools are included in Table 1.]

The numbers of entries in the Primary Division so far are shown in Table 2, along with the numbers of entries in the Secondary Divisions ( Junior, Middle and Senior) since the inception of the new Primary Division.

Table 2: numbers of entries from primary schools and secondary schools

^* trial run, mainly in Section 3

It is clear from the tables that the organisers of the four Sections have had to cope with exceptionally high numbers of entries, partly because of a rapid increase in the entries from secondary schools and partly because of the enthusiasm of pupils and teachers in primary schools. Section 4 (covering the same area as the former Strathclyde Region) was under particular strain because of the size of its population. Two things were done at the beginning of session 1998-99 to try to help.

(1) Section 4 was split into two parts, one of which would be the new Section 4.

This would consist of Glasgow, Argyll and Bute, North and South Lanarkshire, and East and West Dunbartonshire. The other part of the old Section 4 would be part of the new Section 5. This would consist of Ayrshire, Renfrewshire and Inverclyde (formerly in the old Section 4) and Dumfries & Galloway (formerly in Section 3).

(2) The organiser of the new Section 4 would look after the secondary schools in his or her area. The primary schools in the area would be covered by a new 'sub-section', to be run by the Faculty of Education of the University of Glasgow.

1999-2000. Two further changes have taken place.

(1) The organiser of Section 3 will look after the secondary schools in his or her area. The primary schools in the area will be covered by a new sub-section, to be run by the Moray House Institute of Education in the University of Edinburgh.

(2) For the Secondary Divisions (i.e. Junior, Middle and Senior) there are to be two sets of five problems each instead of three sets of four problems each. Such a change makes comparison of the numbers of entries difficult. The figure 4898 (at the foot of the third column in Table 2) gives the total number of entries for two sets of problems. The figure immediately above, 7819, gives the total number of entries for three sets of problems. Figures in earlier years suggest that the third set of problems would bring about 30% of the sum of the first and second, so that if there had been a third set, then there would have been roughly 0.3 times 4898 entries, which is approximately 1469. Therefore the total for the year could have been about 4898 + 1469 = 6367. This dubious estimate gives a figure which is still considerably less than 7819.

Thus there appears to have been a genuine reduction of numbers of entries from Secondary Schools, whilst the number from Primary Schools has continued to increase. To some extent the numbers of entries from Secondary Schools have been restricted by the requirement that when more than 35 entries are submitted for a particular set of problems then at most 35 will be accepted; however, a local organiser may decide to modify this requirement, for example by requesting that a school should provide one or more teachers who are willing to act as markers. The requirement was introduced in session 1998-99. What effect it has had on the number of entries is not clear. Further tentative deductions might be made from the figures for the numbers of entries in the Junior, Middle, Senior and Primary Divisions.

Numbers of entries or of entrants do not tell us much about the standards achieved by the entrants, but the numbers of Gold, Silver and Bronze Awards are at least measures of quality. In the early years of the competition, there were two grades of award : Prizes (consisting of money) and Certificates of Honourable Mention. In 1986-87 there were additional competitions in which contestants had to design (i) a Mathematical Challenge T-shirt or (ii) a Mathematical Challenge Mug. Both competitions attracted a number of contestants. T-shirts were given away to award winners in 1987 and mugs in 1988. The winning T-shirt was very good, but there was only one size, which created another problem for the smallest award winners. The winning mug was an immediate success and has featured on every award day since 1988. There are now 13 different mugs in a complete set. Other items such as pens and rulers have also been given away. Prizes of money were not given after session 1994-95. Anyone who was afraid of the terrible effect that this withdrawal might have on the number of entries should study the figures for 1995-96 and 1996-97.

The way in which the competition has developed is both interesting and worrying. We still need to encourage young people to take an interest in Mathematics to the extent that they study the subject at a university, taking an honours course if they have the ability and commitment. Several young award winners in Mathematical Challenge with evident talent in problem solving and a good appreciation of

Mathematics and its applications, have disappointed me in recent years by choosing to study subjects such as Law or Medicine because they have been advised that qualifications in these subjects will guarantee a post with security of tenure, whereas (allegedly) a degree in Mathematics will not. I wonder how many people have wasted their talents by allowing themselves to be diverted by well-intentioned but not necessarily well-informed financial advisers.

There are five more tables, which show the numbers of entries and numbers of awards in sessions 1998-99 and 1999-2000 for each of the sections. Remember that in the secondary divisions there were 3 sets of 4 problems each in 1998-99 but 2 sets of 5 problems each in 1999-2000.

Mathematics and its applications, have disappointed me in recent years by choosing to study subjects such as Law or Medicine because they have been advised that qualifications in these subjects will guarantee a post with security of tenure, whereas (allegedly) a degree in Mathematics will not. I wonder how many people have wasted their talents by allowing themselves to be diverted by well-intentioned but not necessarily well-informed financial advisers.

There are five more tables, which show the numbers of entries and numbers of awards in sessions 1998-99 and 1999-2000 for each of the sections. Remember that in the secondary divisions there were 3 sets of 4 problems each in 1998-99 but 2 sets of 5 problems each in 1999-2000.

Table 3 : Section 1 (North)

Table 4 : Section 2 (East and Central)

Table 5 : Section 3 (Lothian & Borders)

Table 6 : Section 4 (West)

Table 7 : Section 5 (South West)

Over the years, many commercial, educational and industrial concerns have contributed to Mathematical Challenge. Current and recent sponsors are: BP Amoco, the Bank of Scotland, Digital Equipment Co Ltd., The Edinburgh Mathematical Society, IBM UK Ltd, The London Mathematical Society, Marks and Spencer, NCR Ltd., Royal Mail, The Royal Society of Edinburgh, The Scottish International Education Trust and the Society of Petroleum Engineers. These organisations, and many before them, have contributed either in cash or in kind. Some of them, such as The Scottish International Education Trust, have no strong connection with Mathematics, but have given substantial financial support for a number of years. Without the backing of all the bodies the survival of Mathematical Challenge would have been doubtful to say the least. And we have even had contributions from individuals, in particular from Professor L E Fraenkel of the University of Bath; the discerning reader will have noticed that his name has appeared in every issue of the Journal since no. 24, except that in no. 29 his name was misspelled (apologies to Professor Fraenkel for this). The interest he has shown in Mathematical Challenge and his major contributions to it are greatly appreciated.

As far as the day to day working of Mathematical Challenge is concerned we have relied on the goodwill of a whole army of setters (in some cases inventors) of new problems, solvers, markers and advisers - all essential to the competition if it is going to work smoothly. Local organisers have undoubtedly been extremely busy, but much appreciated help has come from administrative and secretarial staff in universities and other institutions. The backing which teachers and parents have given to the competition is most important. Where it has been possible to invite teachers and parents to presentation ceremonies they have come in force and with evident approval.

As someone whose time to go has come, I thank you all and wish you well for the future.

Edward M Patterson

22 November 2000