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Saturday, 29 September 2012

Trick or Treat?

In this article we discuss the role of 'tricks' in mathematics, that they can be either simply misleading or content or process driven and how they can provide positive or negative experiences for the learner. The more we have thought about this the more we are aware that we have only scratched the surface of this issue, but we are sharing our early thoughts and hope it will elicit further discussion and refinement. Throughout we shall refer to mathematical content and mathematical process. You can think of the former as knowledge and the latter as skill.

In short, we want to explore the question: what is a mathematical trick and when are tricks good and when are they bad?

Focusing attention on tricks

Here are seven reasonably quick questions chosen to highlight various aspects of the word 'trick' in mathematics. As you try to solve them, do you see any part of their solutions as 'tricks'? Do you feel that you need to use a 'trick' to solve them? Why?

The solutions are available at the end of the article. It will be better to try the questions before looking at the solutions, as the way in which a trick is received by the learner is an important aspect of this article!

1. As I was going to St Ives, I met a man with seven wives. Every wife had seven sacks; And every sack had seven cats; And every cat had seven kits. Kits, cats, sacks, wives: How many were going to St Ives?

2. Find the missing term in this sequence: 110, 20, 12, 11, ... , 6, 6

3. Continue the sequence J, F, M, A, M, ..

4. Evaluate 22(xcos(4x2)+x2sin(4x))dx

5. Without using a calculator, find the height h in the diagram below

6. Find x if x6=272x

7. Find a pair of natural numbers the cubes of which differ by a million

Assessment of the problems

As we shall see, the word 'trick' might be applied to these problems in various ways. Although for each one we focus on a major point which arises, the range of issues uncovered are not mutually exclusive. In each case, we provide an example of 'positive' classroom use and an example of 'negative' classroom use. Positive uses might lead to improved understanding or enjoyment, whereas negative used might lead to frustration, mistrust or confusion.

Question 1 - Deliberately misleading the solver

This is the classic trick question: the question is specially designed to catch out and fool the solver. If you have seen this question before, recall or imagine seeing it for the first time. Imagine a youngster trying to solve the question by attempting to work out 1+7+7×7+7×7×7+7×7×7×7 only to be told 'No. The answer is 1. The others were coming away in the opposite direction'. How would such a learner feel? Despite looking mathematical, there is no mathematical content in the answer.

Positive use: Posed in a mathematically honest way to try to get students to convert a question into a written sum: the focus would be on the mathematical process rather than the answer itself. The 'trick' would then be a fun twist.

Negative use: Phrasing the question in a way which emphasises the combinatorial aspect or in a way in which people might start a long calculation, where the aim is to focus on catching out the solvers.

Question 2 - Could be a trick, depending on the context in which the question is received

To solve problem this you need to try to interpret the digits in different number bases. Is this a trick? It depends on the context in which the question is received. Consider it as part of a series of questions involving arithmetic sequences: in this context it would be a trick question, given out of context and designed to mislead. If it were given in the context of work on number bases it would take on a very different, positive character. Furthermore, if presented as a pure puzzle to challenge your thinking then it could also have positive benefits (what mathematics could I bring to bear that will help me solve this problem?). This problem immediately raises the issue that context is important in determining whether a question is a trick question or not. In short, were this question presented as a year 7 question or in a set of exercises on algebraic sequences this would be very misleading. Were it presented in the context of a set of A-level exercises on modular arithmetic, the problem would take on a very different flavour.

Positive use: To encourage creative engagement with number bases, or with a suggestion that the standard mathematical content for sequences will not work.

Negative use: With no hints that 'standard' methods of sequences will fail. Given to students who might understand number bases, but are unlikely themselves to think of number bases as a method of solution.

Question 3 - A question immediately solved only by spotting the answer

The answer to this sequence requires you to spot the pattern of months of the year. Is this pattern-spotting process a mathematical exercise? It is interesting to consider how it differs from the previous question. In question 3 there is some sense that there is virtually no calculation needed to solve the problem once the solver has made the connection with the months of the year, whereas in the question 2 a logical process needs to be undertaken to apply the idea of number bases. We make the point that this is a non-mathematical puzzle: the 'trick' can be applied instantly without calculation or recourse to mathematical reasoning of any sort.

Positive use: To encourage students that letters don't always need to stand for numbers

Negative use: Given in any situation in which solvers are expecting a numerical answer

Question 4 - Quick shortcut to a solution using an advanced mathematical procedure

This is the first question to sensibly allow multiple methods of solution. The answer could be derived at length by using integration by parts multiple times, but the mathematical trick of spotting that the integrand is odd allows you almost immediately to arrive at the answer. This is a mathematical process trick. This question could be posed negatively as a trick to mislead the solver into wasting time on a lengthy algebraic calculation, but the 'trick' to arrive immediately at the solution has a reusable quality to it: it is a trick that might well become a tool in future problem solving situations. We assume in this case that the problem would be used in a context designed to elecit the development of a useful mathematical tool rather than in circumstances which are misleading.

Positive use: The question could be used to provide some integration by parts practice with the interesting answer zero leading naturally onto the discussion of the clever trick which might be employed next time.

Negative use: Given to a mixed group of students for whom some will immediately spot the trick, whereas others will start working on integration by parts. Do the trick spotters sit around waiting, or do the integrators have their work interrupted by finding out the answer before they have finished their computation?

Question 5 - Neat shortcut using a simple mathematical process trick

This problem allows for multiple methods of solution. One possibility is trigonometric which requires a calculator; another approach involves spotting firstly that the base triangle is a 6-8-10 Pythagorean triangle and secondly that a cunningly placed pair of lines allows you to use similar triangles to solve the problem. Using these lines feels like a trick because once they are in place the problem becomes much simpler. Placing additional lines on a diagram is a useful trick which could be applied to a wide variety of problems: it is a mathematical process trick. Whilst this trick does not immediately give rise to the answer, it does open up alternative avenues and raises the issue of elegance in a solution.

Positive use: Given to students who are likely, with enough necessary scaffolding, to be able to think of the trick of applying additional lines to the diagram.

Negative use: Given in any context in which students will become completely stuck.

Question 6 - Tricking the solver by using a misleading context

This question was placed towards the end to make you suspicious. Were we, the question setters, trying to trick you into giving the wrong answer? No. But hopefully this reinforces the point about the context in which a question is set leading to the attitude of the solver towards the problem. It also raises an addition point. To solve this equation you most probably mentally rearranged to get 3x=33 and then divided to get x=11. But rearrangement of terms and then division are both in themselves tricks. Anyone who has ever taught algebra to a class of year 7 pupils will understand that, initially, such rearrangements and divisions are far from obvious for most pupils. They become mathematical content tricks, such as the integration trick in question 4 eventually becomes to more experiences mathematicians: to solve equations like these you can follow these steps

Positive use: Misleading contexts might be used to encourage learners to think outside the box.

Negative use: Used in a situation where solvers doubt their knowledge or ability to do the question, so that they feel suspicious of the question.

Question 7 - Quick solution using an advanced mathematical content trick

This is a misleading question because there are, in fact, no natural numbers the cubes of which differ by a million. A very quick way to see this is to note that Fermat's last theorem says that there are no whole number solutions to a3+b3=c3. Since 1 million is the cube of 100, we know that there is no possible whole number solution to the question. The trick is to use a theorem: without knowledge of the theorem the question is very difficult; with the theorem it is trivial. We can consider the theorem to be a mathematical content trick.

Positive use: Provided in a time-constrained way as an interesting introduction to a fascinating mathematical context.

Negative use: As an unsupervised homework task: students might, believing that there is an answer, spend a great deal of time working on the problem and failing to make progress.

Motivated by the discussion of the previous questions, we are now in a good position to try to categorise tricks in mathematics.

Categorising tricks in mathematics

Clearly, whether a question is received as involving in some way a trick very much depends on the way in which it is presented and the mathematical skill and knowledge of the solver. In fact, when writing this article we disagreed many times on whether a given activity constituted a 'trick', and came to realise that a trick very much depends on the learner and context. We concluded, however, that it is possible to provide a simple breakdown of tricks into three simple categories:

1. Trick questions
2. Questions which can be more quickly solved using mathematical tricks
a. Questions using mathematical process tricks
b. Questions using mathematical content tricks

Whether a question falls into category 1 depends very much on the context in which the question is presented. Whether a questions falls into category 2a or 2b very much depends on the mathematical level of the problem solver.

Discussing these categories more generally, away from specific examples, will allow us better to understand the purpose of such questions in mathematics learning. We do not attempt to provide a complete list or categorisation in this article; this might be the focus of further, more detailed investigation.

Trick questions

These are questions specifically designed to mislead the solver in some way, such as:

1. You are deliberately misled into creating a difficult answer or following a difficult path when there is a simple, immediate answer or route to the answer
2. You are misled into creating a simple answer when the actual answer is more difficult.

Trick questions can be made (intentionally or otherwise) by

1. Choice of misleading wording or notation
3. Choice of underlying mathematics that the solver is unlikely to know or think of

A trick question is likely to result in the solver immediately seeing the solution upon being told the trick.

Why might you use a trick question? A main reason is to reinforce a learning objective (such as always read the rubric) or to make a lesson more memorable. However, be aware that tricking the solver can lead to either positive or negative emotional responses:

Positive: Intrigued, amused, motivated, full of wonder.

Negative :Embarrassed, foolish, irritated, angry

It is important to be aware that the right sort of trick at the right time can lead to a sense of delight, but the wrong sort of trick can confuse or embarrass the learner, create a sense of mistrust or defensive environment in the classroom.

Mathematical tricks

Clearly, a trick question can be created in any subject, including mathematics. However, many of our examples use 'mathematical tricks'

The best way to understand the use of a mathematical trick is to imagine that you are stuck on a problem and can find no way forward. You cannot start and then, suddenly, there is a light bulb moment and you can suddenly see how to proceed. You might on occasions be able to see your way immediately and obviously to the answer, or you might still need to do some work to process or apply the trick ' however, there is a definite 'ah!' as you realise how to proceed. At a most basic level, the light bulb is the trick and it jumps you from not knowing how to start to the problem to being confident as to how to proceed. Sometimes the trick will be a piece of firm knowledge, such as a theorem; at other times it will be a process of sorts, such as cleverly splitting up a diagram

In the example questions, the mathematical light bulb moments were

1. No mathematical trick
2. Thinking of applying different number bases
3. Thinking of the sequence of the months of the year
4. Realising that the integral was odd
5. Thinking to apply lines to the diagram
6. Realising that the equation can be rearranged
7. Spotting the connection to Fermat's last theorem

This list highlights the fact that mathematical tricks can be broken down into tricks of content and tricks of process. To be clear, mathematical 'content' is a piece of factual, mathematical knowledge, such as the definition of trig ratios; mathematical 'process' is the means and skill by which you use and apply your general mathematical content knowledge. In almost every mathematics lesson, students use, are made aware of, or even subjected to a range of mathematical tricks. Whether these are viewed as tricks largely depends on the familiarity of the learner to the mathematical ideas. Consider, for example, the familiar knowledge that there is a formula for the solution of any quadratic equation. Why might it be thought of as a trick? Imagine asking a non-mathematician to find a number which equals a quarter of its square added to 35/36. In a couple of moments, the mathematician can spot the quadratic equation and will soon find the exact answer. It is no surprise that ancient cultures held such knowledge in almost magical esteem! On a more day to day level, there is often an element of magic in the learning of new mathematics. Although this magic might fade over time teachers might do well to remember the mystery that new mathematics holds for their learners.

Conclusion

We have explored some issues surrounding tricks in mathematics. Some questions might be received as tricks and some questions might be solved using tricks.

Questions might be crafted to be received as tricks in various ways, such as wrapping simple content in a new and novel way or making questions using pieces of content which are unfamiliar to the solver. Whether a question is received as a trick often very much depends on the context in which the question is given

A mathematical trick involves a 'light bulb' moment or 'ah' which moves the solver forward. Whether or not a question involves an overt mathematical trick very much depends on the experience of the solver: on first encounter, almost any piece of new mathematical content beyond mere definition might be viewed as a trick. It is only with practice that they are integrated into your repertoire of knowledge and become tools. As you accumulate more content tricks you become a more knowledgable mathematician; as you develop a repertoire of more mathematical process tricks you become a more skilled mathematician. Acquisition of a toolkit of tricks is a valuable and necessary part of the development of mathematical skill.

Footnote
Just as we finished this article we received notification of a new WIKI-style site called 'The Tricki'? (http://www.tricki.org/). This site was recently started by colleagues in the mathematics department here in Cambridge. They say:

'It [The Tricki] can be briefly described as a wiki for mathematical tricks -- hence its name. However, the word "trick" should be interpreted very broadly, and it might be better to use a phrase such as "mathematical technique". It is interesting to reflect how much this description, and the site itself, complements the views expressed in this article.

1. One. I was going to St. Ives. The others were coming in the opposite direction.

2. 10 (in base 6). The sequence is the same number 6 starting in base 2, then base 3, then base 4 etc.

3. The next letter is J for June. The letters are the first letters of the months of the year

4. Zero. The integrand is an odd function evaluated over the range [-2, 2]. Thus, the positive area will cancel out the negative area.

5. The length of the hypotenuse of the base triangle is 10 (because it is a 6-8-10 triangle. By dropping a perpendicular from the highest point and a line intersecting this at right angles (see diagram below) we see that triangles AXY and ABC are similar. Since the hypotenuse of the larger triangle is double that of the smaller triangle, we see that XY must be half of 8, which is 4. Thus, the total height is 10 .

6. Rearranging we see that 3x=33, so x=11.

7. Note that a million equals (100)3, so we are trying to set the sum of two cubes equal to a cube. This is impossible because of Fermat's last theorem.

Engaging Students, Developing Confidence, Promoting Independence

How do we develop positive attitudes towards mathematics and learning mathematics?

- Use a wide range of tasks and resources
- Enthusiastic teachers, with a 'can do' positive attitude
- Plenty of opportunities for students to experience success
- Hands-on approaches to learning
- Use real life examples and explore links with other subjects
- Offer positive role models of mathematicians
- Maths Clubs - e.g. older students mentoring younger students
- Posters publicising maths
- Share learning with parents (e.g. maths evenings to encourage positive attitudes amongst parents)
- 'Make it enjoyable': Maths challenges, competitions, puzzles of the month, celebrate achievements

How do we develop confident learners who are able to work independently and willing to take risks?

- Acknowledge all contributions positively, encourage learning from mistakes, welcome wrong answers as the springboard to new understanding
- Use positive language
- Encourage independent and small group research
- Value different approaches to solving problems

How do we develop good communicators - good at listening, speaking and working purposefully in groups?

- Plan lessons which focus on group work
- Set a rule that groups are not 'allowed' to move on until all the students understand
- Allow time for presentation of findings
- Set the rule: "Don't ask the teacher - ask at least three other students first"
- Teachers take a step back and ask students to explain to the class their methods and reasoning
- Mix up groups - expect students to take on a variety of roles and work with a variety of people
- Ask students to prepare tests and answers for younger age group
- Ask students to make a podcast or film on a given topic

How do we develop students who have appropriate strategies when they get stuck?

- Offer higher-order, open ended tasks to get students used to being 'stuck'
- Encourage students to explain their difficulty to the rest of the class - vocalise the problem, "say it out loud". Follow-up with an open discussion of the options available
- Offer tasks in which students have to identify and correct errors and encourage similar reflection on their own work
- Create a culture in which 'thinking outside the box' is valued

How do we develop lessons that maintain the complexity whilst making the mathematics accessible?

- Give plenty of time to engage in and 'solve' problems - the process is more important than the answer
- Use investigational tasks which can be accessed by everyone but can have different levels of outcome - low threshold, high ceiling tasks
- Be positive about any steps students take towards solving the problem, however small
- Present tasks in different formats
- Encourage a supportive environment in which students work together, discuss ideas and turn to each other for help

How do we develop students' ability to make connections (e.g. see/utilise different aspects of mathematics in one context, see applications in other areas)?

- "Where have we seen this before?"
- Present problems that can use many areas of maths
- Present open problems which allow students to ask their own questions and develop the need to learn something new
- Present problems based on real life and cross curricular contexts
- Invite outside speakers and professionals to discuss the use of maths in their jobs

How do we develop critical learners who value and utilise differences (e.g. different approaches/ routes to solution)?

- Encourage group work, peer assessment, rotation feedback, discussion
- Change the composition of groups regularly
What are the strengths and weaknesses of this method?
When might you use this method?
- Encourage contributions from all the students
- Require students to explain their solution
- Emphasise method rather than outcome
- Bring students together for mini-plenaries to share and compare approaches
- Set problems which can be solved in a variety of ways

May the Best Person Win

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?

Example of classification using median performances

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 each class median by the minimum 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?