The United Kingdom Mathematics Trust and its Challenges
(based on a talk given at the BSRLM conference at Brighton University on 12th November 2016)
The latest issue of Maths Challenges News celebrates the 20 years of its existence. In 1995, there were two hundred and fifty thousand students participating in nine or ten Mathematics competitions around the country. Since the various bodies responsible came together and formed the UKMT, the number participating in the competitions at Junior, Intermediate and Senior levels has increased to 670,000 in 2016. The UKMT is a very successful charity. The whole enterprise depends on a large community of volunteers who do the work of setting problems, marking scripts and running the many events and competitions for those who do well in the JMC, IMC and SMC.
The Challenges give students an opportunity to spend 60 minutes, or 90 in the case of the Senior Challenge, grappling with the non-routine problems set by the UKMT.
In many cases, the solutions rely on the students being able to bring together different ‘schema’ or organised bodies of learned knowledge. For instance, what is the acute angle between the hands of a clock at 1610? The student would need to understand the analogue clock, the division of a full turn into 360 degrees and the four rules with fractions. As these ‘schema’ are likely to have been learned in different contexts, a special mental effort is required to bring them together.
Or a question from the Senior Challenge of 2005:
(1+1/2)(1+1/3)(1+1/4)(1+1/5)……………(1+1/2004)(1+1/2005)
This requires knowledge of the different ways of representing mixed numbers, conventions of using brackets, and rules for cancelling down fractions. The student may be tempted down several pathways such as writing 1+1/2 as 1.5, or trying to multiply out brackets (all 2004 of them!) because they have been habituated to respond to those particular visual stimuli in that way. But if they manage to write it as
(3/2)(4/3)(5/4)(6/5)………..(2005/2004)(2006/2005),
the solution falls out easily by cancellation. I have seen students smiling at that one.
Looking at the big picture:
The range of these averages for the JMC, IMC and SMC are, respectively, to the nearest unit are 18, 11 and 13.
The Junior and Intermediate Challenges are marked out of total of 135. The SMC (recently at least) has been marked out of 100, but the students starts with 25 marks.
The average student is scoring nearer one third than half the marks. From what is written in the Year Books, it may be inferred that the UKMT would like to increase the average scores. It would presumably be easy to increase the average by setting routine problems, but that would defeat the whole purpose. So there is a considerable challenge in composing suitable problems. The Year Books discuss each Challenge and often express dismay:
JMC 12: What is the smallest four-digit positive integer which has four different digits? A 1032 B 2012 C 1021 D 1234 E 1023
The comment on this question ran ‘You might think that it would hardly be possible to set an easier question but only just over half the candidates gave the right answer. Why was the option C – 1021 – so popular? Was it carelessness that resulted in a failure to notice that the question asked for the smallest four digit positive integer which has four different digits. Or did many pupils not understand the meaning of different digits?
A practising teacher has to walk past a thousand mistakes every day, and so becomes somewhat inured to error. Students’ errors cease to dismay! However, it may be worth pointing out that the noun ‘integer’ here has two qualifying adjectives and one qualifying adjectival phrase, and that the word four and integer appear twice in slightly different contexts. As students seldom come across sentences like this, it is not surprising that so many misunderstand. The question has a high mind-boggle coefficient!
But how can Mathematical problems be set without using the language of Mathematics precisely?
Ten years of diachronic data from one particular school
Skews of distributions of marks
Working out the skews for each of the challenges, in one particular school, between 2005 and 2016 only seems to tell us that the skews are generally positive. Otherwise there is little in the way of a pattern. The graphs of National score distributions published in the Year books also tend to have positive skews.
The kurtoses for the challenges make little sense. Perhaps our year groups are too small for any pattern to emerge, or perhaps the stepped distribution resulting from the marking scheme is the reason for the random appearance.
The means for each Challenge do seem to tell a story. This particular school selects from the top half of the ability range and its average scores tend to be close to the national average.We can usually see a particular Year Group doing better on the Junior Challenge in Year 8 than they did in year 7, and gradually increasing their average scores on the Intermediate Challenge in Years 9, 10 and 11.
But as the National means, as we have seen, jump about in various range of roughly 10 marks, it makes more sense to work with the differences of our means from the national mean.
If we do this and ask Excel to give us a bar graph, it does look as though students are, on average, getting one or two more questions correct over the years. (Questions could be asked about the methodology of lumping together the JMC and IMC means here, and leaving out the SMC data – which has not yet been tabulated.)
A measure of the number of Golds, Silvers and Bronzes achieved by the School each year appears to confirm the trend. (3 points for each Gold, 2 for Silver, 1 for Bronze, summed, divided by the number of entrants, multiplied by 100 and rounded to the nearest unit. A score of 100 would indicate that the average student achieved a Bronze.)
Interventions have taken place as and when they have occurred to the Head of Department, and as time and energy permit.
- Encouragement of Junior School to enter students for the Primary Maths Challenge.
- Encouragement of Junior School to enter Year 6 students for the Junior Mathematics Challenge. This began in 2009.
- Provision of packs of past papers and solutions to students at the Junior and Intermediate levels.
- Allocation of lesson time to practise of challenge questions with class discussion.
- Provision of video solutions of challenge questions.
- A new resource intended to support students’ preparation for the Challenges.
- From 2015 , provision of ‘prompt’ videos. The rationale behind these videos is that simply watching a video solution does not engage the students directly in mathematical activity. Therefore the ‘prompt’ videos first of all read the question out loud to the listener, and then suggest lines of attack – or approach – to solving the problem. The student is left to complete the work.
No doubt the best and most professional maths teachers promote a problem-solving approach in their classes, as a matter of course. But, ideally, teaching problem-solving in small groups, or one to one, is best. Mentoring students is very time-consuming. It is a reasonable guess that the average Challenge entrant appends little time preparing for the Challenges.
Therefore there may be some value in ‘prompt’ videos – or something similar – which engage willing students more closely with the questions and give them a better chance of answering them. For what it is worth, the writer’s personal observations of his own students’ engagement with these videos in the Mathematics Computer Room, and data from the Prep School suggest measurable benefit.
References
UKMT Year Books
Skemp R., The Psychology of Learning Mathematics
Watson A., Jones K. and Pratt D., Key Ideas in Teaching Mathematics: Research-Based Guidance For Ages 9-19
Samson I., Demathtifying – Demystifying Mathematics
