May the best person win
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The Paralympics
The Paralympic Games include athletes with mobility impairments, visual impairments, brain damage and other disabilities. They started in the UK in 1948 with a small group of men who had been injured in the Second World War. The first post-war Olympic Games took place in London that year, and on the opening day Stoke Mandeville Hospital, which has a world-famous spinal injuries centre, organised a sports competition for some of the patients. The first official Paralympic Games, no longer restricted to war veterans, were held in Rome in 1960. The Paralympic Games now operate in parallel with both the summer and winter Olympics - and are called the 'Paralympic' Games from the Greek word 'para' meaning beside or alongside.
Team events and some individual events are for specific categories of athletes, such as wheelchair basketball, goalball and judo for the visually impaired, boccia for people with cerebral palsy (brain injuries), and rowing for people who only have use of their arms. In these events, people with other disabilities are excluded, and any variations in disability within the category are ignored.
In most individual events, athletes are divided into six broad categories - amputees, people with cerebral palsy, people with intellectual disabilities, wheelchair users, visually impaired people, and Les Autres. 'Les Autres' is the French term for 'others', and includes people with dwarfism, multiple sclerosis and congenital deformities. Within each of these categories, athletes are then further sub-divided into classes according to the degree to which their disability impacts on their performance.
Classifying athletes
It isn't only Paralympic sports where people are classified in some way. Golfers have their handicap as do race horses. Boxers and weight-lifters are classified according to their body weight, and many amateur sports classify people on the basis of what they have achieved so far.
Ideally each class should be as homogeneous as possible but different from other classes on the basis of certain criteria. In the case of the Paralympics, the criteria include sex, the nature of the event, the disability category and the degree to which the athlete's individual level of disability affects their performance. The classification needs to be seen to be fair, as far as possible. The Paralympics are a high stake competition, so the process also needs to be rigorous.
There are, however, many problems in grouping athletes into classes. It is difficult to apply handicapping in a consistent way across different classes. Racing conditions and changes in technology mean that classification frequently has to be revised. Whether a class is small or large may give athletes an unfair advantage or disadvantage. There have also been instances of cheating, with athletes claiming to be more disabled than they actually are in the hopes of being put into a class in which they would have an unfair advantage.
One way of avoiding such problems is to use a mathematical algorithm which focuses on athletes' performances, rather than the degree of their disability. This also avoids the need for committees to meet to debate which class an athlete should be put into and how classification should be updated as equipment or other factors change.
One proposed method involves calculating an overall median for an event, and also calculating median performances for each class within that event. The ratio of the median for a class compared to the overall median for the event tells us how good a particular class is relative to the others in the event. Competitors' performances are then scaled using this ratio to get their score. This process is illustrated at the end of this article.
Setting students
The classification of athletes for the Paralympics is a very similar process to that of classifying students for mathematics and other lessons. We might use criteria such as exam performance yet we are aware that this is a 'one off' snapshot of performance and therefore may not be entirely representative of a student's ability. To try and get a broader picture of performance we often add teacher assessment. Our attention may then turn to student confidence as we try and aim for conditions that will maximise a particular student's performance. Does the discussion of classification for the Paralympics above help further refine or inform our setting of students?
In the classroom - the school sports days
So what is the relevance here to school sports days and helping students to develop their mathematical thinking? The challenges we face when planning a sports day include:
- what events to choose
- how to allocate students to each event
- whether we need some sort of handicapping system or whether it is simply a 'flat playing field'?
The first challenge is taken up in one of this month's stage 1 problems, Our Sports. This encourages students to look at what events they might want to include in their sports day. This could be developed at stages 2 and 3, as a sampling problem. For instance, how many students should be asked to get a fair impression of the whole school view? Does it matter how good at sport those asked are? What would a representative sample look like?
The second challenge depends on the purpose of the sports day. For the Paralympics it is about peak performance and winning your event to gain the coveted gold medal. It is entirely competitive. For a school sports day it could be that everyone should take part, which raises the question of whether the most athletic should take all the 'medals' or not. There may be other issues which are as important as individual performances. The purpose will inform how students are allocated to each event. How important is it that we are modeling our school sports day on the real world of sport?
Students could be allocated to a particular event according to:
- previous performance
- their preferences
- ensuring everyone takes part in one event
- ensuring events are of equal size
- ??
By choosing one event - such as the 100m sprint - students could investigate the impact of different classification strategies. They could select participants using the their own classification strategies, try out the race with the different groups and then compare outcomes.
But how will they represent the results in order to make meaningful comparisons? For stage 2 and 3 students this could lead to the idea of an average as in some way typifying a particular group, with the range giving an indication of the variability within that group. Stage 3 and 4 students could think further about why the median might be preferable to the arithmetic mean.
The third challenge is informed by what we mean by 'fair'. This could lead to an investigation of different handicapping systems, such as that for golf or horse-racing. Students could explore how the system works and what it could be like in the school sports day. In school we may want the events to be fair yet perhaps not go as far as a handicapping system.
Questions to consider
How might the size of class affect your chances of winning?
- In a school setting, how might the size of the set/group affect a student's attainment outcomes?
- On what grounds do we/might we make a correlation between group size and ability?
What advantages are there in using median performances rather than mean, minimum or maximum performances for each class?
- How does this apply in a school setting in the way we may choose to group students?
Stage 3 students could also look at How Would You Score It?
Example of classification using median performances
Thanks to Professor David Percy, University of Salford, for his help with this section.
Here is a set of 50 random scores, between 0 and 100:
| 66 | 1 | 44 | 5 | 29 | 92 | 76 | 76 | 23 | 60 |
| 53 | 84 | 50 | 55 | 44 | 64 | 25 | 77 | 80 | 51 |
| 54 | 92 | 10 | 67 | 72 | 38 | 79 | 10 | 28 | 53 |
| 74 | 17 | 79 | 95 | 68 | 23 | 17 | 42 | 78 | 80 |
| 4 | 62 | 74 | 2 | 32 | 26 | 83 | 17 | 20 | 6 |
Suppose we want to divide the 'athletes', whose scores these are, into three equivalent groups so that we can judge who the overall medal winners should be. We could base our classification on the median, so that we make the median of each group the same, then having scaled all the scores, see whose score is best.
This process is illustrated below.
I start by subdividing the scores into three classes, according to whether the score is in the range 0 - 33 (Class A), 34 - 66 (Class B), and 67 - 100 (Class C). If a high score is best, this might indicate people who are severely affected by their impairment, those moderately affected, and those only slightly affected.
Class A: 1, 5, 29, 23, 25, 10, 10, 28, 17, 23, 17, 4, 32, 2, 26, 17, 20, 6 (18 members)
Class B: 66, 44, 60, 53, 50, 55, 44, 64, 51, 54, 38, 53, 42, 62 (14 members)
Class C: 92, 76, 76, 84, 77, 80, 92, 67, 72, 79, 74, 79, 95, 68, 78, 80, 74, 83 (18 members)
The overall median for the 50 scores is 53, and the three class medians are:
mA = 17, mB = 53, mC = 78.5
You could ask the question here whether the median of the central group is necessarily the same as the overall median (or is it simply a consequence of the members of the groups being symmetrically distributed?).
A simple scaling factor for each group can be calculated by dividing the minimum median by each class median, giving:
fA = 1, fB = 0.321, fC = 0.217
(17 x 1 = 17, 53 x 0.321 = 17, 78.5 x 17 = 17)
The raw scores are then scaled by these factors to give adjusted scores from which medal winners can be determined.
It turns out that all the medal winners would be in Class A, if a high score was best. These are the adjusted scores:
Class A: 1, 5, 29, 23, 25, 10, 10, 28, 17, 23, 17, 4, 32, 2, 26, 17, 20, 6
Class B: 21.2, 14.1, 19.3, 17.0, 16.0, 17.6, 14.1, 20.5, 16.4, 17.3, 12.3, 17.0, 13.5, 19.9
Class C: 19.9, 16.5, 16.5, 18.2, 16.7, 17.3, 19.9, 14.5, 15.6, 17.1, 16.0, 17.1, 20.6, 14.7, 16.9, 17.3, 16.0, 18.0
The ranges of classes with higher medians are reduced by the scaling process. Whereas for Class A 0 scales to 0, and 33 scales to 33, for Class B the minimum (34) and maximum (66) scale to 10.9 and 21.2 respectively, and for the minimum (67) and maximum (100) scale to 14.5 and 20 respectively. This favours the class with the lowest median, as we have seen.
This results from choosing data by randomly selecting integers from a range, which implies that, eg. 50 is twice as good as 25. Sporting data is not random and minimum scores are unlikely to be close to the bottom of the class, so such extreme results as this would be much less likely.
What difference would it make if class maxima were used, instead of class medians? If we assume that a high score is good, might this make for a fairer system of score adjustment?