Consideration of a range of A and AS statistics syllabuses, both free-standing and contained within mathematics courses.

Background

The Review Group comprised Anne Hawkins, Neville Hunt, David Lennox and David Cassell. Between them, they have considered a range of A and AS statistics syllabuses, whether free-standing or contained within mathematics courses. (The group has not looked in depth at the statistics content of other disciplines such as geography, psychology, etc., although comments from some teachers of these subjects were solicited.)

The Group was aware that a number of changes are anticipated:

• The centralisation of assessment of 3 Boards, EdExcel, AQA and OCEAC.
• The likely reduction in the number of syllabuses, in response to government pressure. The move to modular packages at A and AS level facilitates this, but will probably mean great variation in content under a broad umbrella of a particular award title.
• SCAA has determined that what little statistics and probability there was in the core mathematics syllabus(es) will be removed. This could result in some students qualifying in A or AS level mathematics with no study of statistics or probability beyond GCSE.

The Review Group looked at syllabuses, examination papers and examiners’ reports, coursework requirements, and teachers’ preparedness to teach statistics and probability. Syllabuses considered were drawn from OCEAC, AEB, WJEC, EdExcel and NEAB. All Boards were offering a number of different syllabus combinations, and terminal or (linear) and modular structures. The review covered Mathematics, Further Mathematics, Statistics, various combinations including Statistics, and MEI Structured Mathematics.

Syllabuses

Despite the wide range of syllabuses, there was evidence of considerable common ground. The emphasis was on mathematical modelling, and mathematical analysis. The stated aim was to develop a sense of coherence and progression in mathematics. The statistical syllabuses, however, did not generally reflect this aim. For most of these, the content was illogically ordered. There were inappropriate emphases on theoretical/algebraic aspects, under-estimation of the intellectual demands in more applied, conceptual aspects and over-emphasis on ‘more techniques’ being synonymous with ‘more advanced’. Project work, where required (or optional), is hampered by lack of preparatory teaching in the necessary applied skills, e.g. experimental design.

In general, the contents of the syllabuses were very similar. MEI and London included some extra (applied) subject matter, but not necessarily in the right place, nor in an integrated fashion. The Cambridge AS Statistics tried for a more applied, more ‘statistical’ syllabus, but the result was rather mundane, with too little support for teachers.

The NEAB proved to be an exception. Its statistics syllabuses were accompanied by a completely separate set of aims and objectives, confirming the Review Group’s view that the ‘standard’ aims and objectives of the mathematics syllabuses are inadequate and inappropriate for the purposes of statistical education. The content and approach of the NEAB syllabuses were clearly oriented towards the development of more applied skills and understanding, not merely the accumulation of analytical techniques. There was more support for teachers in interpreting the requirements of the syllabus. It was felt that the NEAB’s AS- and A-level Statistics syllabuses were more promising as support for students’ other disciplines or preparation for further study in statistics than those of the other Boards.

Assessment

In the assessment process of most of the Boards, there is a strong emphasis on calculation and plotting charts and graphs. This appeared to be a repetition of skills that should have been adequately tested at GCSE level. There are minimal demands on candidates to interpret or comment on the results of their calculations, and evidence from examiners’ reports that these skills are not being developed. (The OCEAC AS-level Statistics was the exception in that it had a compulsory comprehension question concerning a given set of data. NEAB statistics papers and examiners’ reports were not available for consideration.) In general, the weighting of marks for different parts of answers, however, makes it clear that the examiners themselves are sending the wrong signals to teachers and students. The AEB and OCEAC boards tended to employ more imaginative and plausible contexts, but with so little emphasis on application and interpretation it makes little difference to the candidates whether the context is real or not. Lack of choice of questions in the shorter examinations associated with modular courses should at least encourage candidates to revise the whole syllabus. The Boards are slow to respond to new technologies. They do not ban the use of calculators, which turns the questions that they set into trivial exercises. Rather than using the opportunity to examine other more relevant aspects of statistics, however, they insist on candidates showing the steps in their working (i.e. ‘method’ marks go for hand calculation).

Format

In the majority of syllabuses where coursework was included, its assessment was based upon general mathematical criteria that were not appropriate for the needs of statistics. The most comprehensive guidance was given in NEAB syllabuses, but the suggested choice of topics for A-level work were too similar to those given at GCSE level. Expecting teachers to interpret vague terms such as ‘trivial’, ‘poor’, ‘routine’, ‘satisfactory’, etc., creates problems in moderating the coursework assessments. The exceptions to this were found in the NEAB, MEI and London Modular syllabuses, where reasonable descriptors were given, along with better guidance for teachers. The real problem seemed to be related to the philosophy that statistics is part of mathematics, and hence its coursework is wrongly presumed to be mainly to do with techniques, rather than with data and its interpretation. Very few of the syllabuses distinguished between the different assessment aims that could, and should, apply.

Teachers

More recently qualified mathematics teachers had generally been taught statistics in their degrees. This was not necessarily the case for those who had qualified ten or more years ago. All reported having taught statistics topics that they had not met in their own training. In general, the mathematicians felt that statistics lacked the rigour of mathematics, and although they became more comfortable with teaching it, there was a feeling that ‘it lowered the tone’. There was some evidence that statistics was being used for less able students, allowing them to study ‘mathematics’ when they would find a full course in pure mathematics too demanding. There was a suggestion that this was all part of the general decline in standards of A-level mathematics.

The teachers of other disciplines (psychology, biology, etc.) seemed to lack confidence in their own understanding of the underlying principles of statistics. A number of them reported that their own training had tended to take the form of ‘cook-book’ techniques. Students' difficulties with statistics in these contexts seemed to be reflected in less emphasis on their being assessed on this aspect of the course and hence in the marginalisation, rather than integration, of statistical methodology. If user-discipline teachers wanted their students to acquire their statistical training within mathematics or statistics courses there were problems because particular techniques were not taught early enough to meet the needs of the user-discipline. There were also clear mismatches between what the user-discipline contexts needed by way of content and approach, and what constituted mathematics and/or statistics syllabuses.

Recommendations

Particularly at this time of change in the organisation of assessment, and anxieties about the decline in standards of mathematics, there would seem to be renewed opportunities for the RSS to speak out on the subject of statistical education. It appears that what is currently offered as statistical education at A- and AS-level, as well as its style of delivery, serves neither those who need it for their user-discipline studies, nor those who will specialise in statistics, nor those who wish to specialise in mathematics. Within the membership of the RSS, there is considerable expertise in the field of statistical education. Pressure by such individuals’ for change, though, cannot match the level of influence that a united professional body could, and should, exert. The following would appear to be immediate priorities for the RSS to propose.

The removal of statistics and probability from the core mathematics syllabus(es) is a matter for concern. However, it is also an opportunity to argue that it was not appropriate or adequate in the first place, being essentially a somewhat trivial re-visit of GCSE content.

The mathematical community needs to be persuaded to re-consider the nature of ‘mathematics’ at A-level. It is not only the statistical content of these syllabuses that is inappropriate. The emphasis should not merely be on students being able to carry out the steps of a memorised (or given) algorithm with accuracy. Rather, it should it be on students being able to use appropriate tools to shed light on (and thereby help to solve) realistic problems.
Real coherence and progression within the subject of statistics can only be achieved by a rethink of the content and approaches of most of the available A- and AS-level syllabuses. Progression is not achieved by the acquisition of more techniques, but rather by the development of skills and conceptual understanding.

Depending on the government’s response to Dearing, the recommendation that the Committee made for broader-based patterns of post-19 education (including GNVQS) could be helpful for promoting the AS-level study of statistics. The reality is that this would be likely to be within a mathematics context. This is less than ideal, but with appropriate changes in content and more emphasis on principles of statistical methods and concepts, this could work. NEAB have already shown something of what can be achieved. The other Boards should be encouraged to follow their example.
Changing the way statistics is taught within the context of mathematics would not only be better for statistics. It would also benefit mathematics (by introducing more meaningful, in intellectual terms, work), and user-disciplines (by focusing on relevant, and usable, content).

Examinations should reflect available technology. Other forms of assessment of statistics are more appropriate than those currently used. At present, the real intellectual demands of the discipline of statistics are lost, because the available technology makes the objects of the assessment process trivial exercises.
To implement changes effectively, support for teachers needs to be integrated into the syllabus documentation, the assessment guidelines, and the recommended teaching materials. More particularly, appropriate initial and in-service training courses need to be developed. Funding will be needed for this.