In 2003, candidates sitting a higher-tier GCSE maths paper were presented with a question about "Brian" and "George" and their ongoing golf battles. The first part of the question required pupils to interpret a cumulative frequency diagram to deduce the median and the interquartile range of Brian's scores. According to the examiner's report, almost every candidate scored full marks.
In the second part of the question, pupils were presented with a box and whisker diagram of George's scores and asked to decide, with reason, which player was more consistent in their scoring and then which was the better golf player. The examiners were less impressed here. "Part (b) was less successful. Candidates write lines of nonsense to justify an answer instead of giving a response in statistical terms," said the report.
A quick study of the examiner's report on any GCSE paper will reveal a similar story - one of the main areas where pupils lose marks time and time again is when they are asked to explain something, argue a point, or give a reason for their answer.
This apparent shortfall is not constrained to higher-level candidates. The first part of another GCSE paper aimed at pupils targeting a grade G or F challenged the candidates to work out the mean number of packets of crisps Anjum's 10 friends consumed. The majority of pupils had no problem with this, even though technically it should have been one of the most difficult questions on the paper. However, when candidates were shown the results of Alice's survey into the chocolate-eating habits of her friends and asked to make some comparisons, the wheels came off.
The examiner's report explains that "some candidates were unable to make any comment at all".
"Many simply restated facts already given. Many attempted to compare the means, but had difficulty expressing themselves clearly."
And it is not only GCSE candidates who display a lack of literacy skills. Before key stage 3 Sats were abolished, the questions asking pupils to "explain your answer" or "explain what these results show" tripped them up. Furthermore, the most descriptive of the core modules in A-level maths is C3, and this is the one in which candidates of all ability levels score lower.
Is the problem that pupils feel more comfortable and are more successful dealing with numbers than words? As my pupils like to point out whenever they are asked to write down more than one sentence, "Sir, this is maths, not English."
But pupils of all ages and abilities are losing important marks on questions for which they need literacy skills to communicate with the examiner.
From September 2010, for the first time, the quality of a candidates' written communication in maths is going to be formally assessed at GCSE level, so any shortfall in this area will grow in importance.
According to latest OCR GCSE specifications, for example, candidates will be expected to ensure that text is legible and that spelling, punctuation and grammar are accurate so that meaning is clear. Students must also present information in a form that suits its purpose, and use a suitable structure and style of writing. Furthermore, a Quality of Written Communication benchmark is being introduced at all levels by all GCSE boards to help them assess candidates' literacy skills.
But is it fair to punish pupils for poor literacy skills in maths? Shouldn't they be judged solely on their numeracy skills and let subjects such as English test their ability to construct sentences? After all, it is hard not to have sympathy for the pupil who hates writing but loves numbers, but whose success in the latter is being hampered by failure in the former.
However, this misses a crucial point. One of the most important aspects of maths is communication. What is the point of embarking upon a complicated, ground-breaking calculation if the mathematician cannot tell the world what the result means in a way everyone understands?
Without communication, maths is restricted to the number-bods and geeks who will happily do maths for fun without requiring any greater reason or motivation. Literacy in maths is important precisely because it gives maths a greater meaning and relevance. But this is difficult to explain to the average 15-year-old sitting at the back of your lesson on a Friday afternoon.
So, how should we deal with the problem of poor literacy in maths? The National Strategies website, run by the Department for Children, Schools and Families, has little say on the matter. On its "literacy across the curriculum" page, there is information on literacy in art and literacy in science but not in maths.
Pupils could be failing "explanation" questions for two reasons: first, they do not know the answer to the question or, second, they know the answer, but they are unable to communicate it.
For example, in the case of Anjum's friends, some candidates could calculate the mean number of crisps eaten, but were unable to compare this answer in a meaningful way with the answer to another survey. Likewise, for Brian and George's golf battles, the higher-tier GCSE candidates could work out a median golf score, but were unable to use that median to form an argument about who was the better golf player.
Ask a selection of pupils how to work out the mean or the median of a set of data and most will be able to tell you correctly. Ask the same group of pupils why we bother calculating the mean and the median, or what they tell us, and far fewer will give the correct answer. The same can be said for the answers to simultaneous equations, trigonometry questions, loci problems and a whole host of other mathematical topics.
Pupils are generally proficient at the skills required to carry out operations and solve problems, but are far less successful when it comes to knowing what their answers mean.
One way to deal with this may be to use real-life examples and data. The mean number of crisps may have little relevance or spark scant interest in pupils.
However, the mean expenses claim of a Labour MP, or the mean audience figures for The X Factor rounds containing John and Edward compared with those after they were voted off the show, may impart greater meaning.
There is a second simple strategy for the classroom: when a pupil volunteers an answer to a question, immediately challenge another pupil to explain exactly what that answer means.
What does it tell us, why have we worked it out, what is its relevance? These questions may cause pupils to think more about their answers and increase their understanding in the process.
Once pupils know the answer to those kinds of questions, they need to be able to communicate these answers clearly and appropriately.
One way of achieving this could be to give pupils a set of answers to exam questions and ask them to mark them. Include major misconceptions and examples of weak literacy in the answers in order to flag them up to pupils. They can then compare their marks with the official mark scheme and examiner's report.
You can also do "court case" lessons where pupils are presented with a set of data, given a hypothesis, divided up into small groups (prosecution or defence) and asked to make coherent arguments to support their point of view. The rest of the class then acts as a jury and votes on the argument they found most convincing.
Another option involves jigsaw activities where important mathematical words (or words important for arguing a point, such as "conversely" or "validated") are matched with their definitions.
If pupils can not only understand a concept, but also convey their understanding and make well-informed coherent arguments, it will not only aid their exam success but also improve their life chances in general.
Craig Barton is an advanced skills teacher in Merseyside and the creator of mrbartonmaths.com
- Match up "arguing" words with their definitions using the freely available Tarsia software. www.mmlsoft.com
- Questions and mark schemes can be found by visiting the website of your exam board, or by purchasing an exam bank such as Test Base. See www.testbase.co.uk for key stage 3, Exam Quest, www.examquest.co.uk for KS4, or Solomon Papers, www.solomon-press.commaths A.htm for KS5.
- Find statistics so you can discuss comparisons. The TSM site (www.tsm-resources.comuseful-files.html) has real-life data sets.