QCA consultation on level 3 mathematics: MEI position paper Summary This paper is being circulated in April 2009 to coincide with a consultation that QCA is conducting into possible changes to level 3 mathematics, mostly for first teaching in 2012. Its target audience is everyone with an interest in mathematics at this level. The issues involved are really important and we hope that there will be a large and representative response to the consultation. Overall This consultation covers possible changes to AS/A Levels in Mathematics and Further Mathematics; we recommend rejection of the QCA proposals, in favour of no change at this time. It also covers other level 3 mathematics courses; these might benefit large numbers of students and merit serious consideration. Timing There is to be a general review of qualifications in 2013. Any changes made at this stage should be designed to provide information for that review rather than to pre-empt its findings. New GCSEs will be first examined in 2012 so the proposed new specifications, with first teaching also scheduled in 2012, cannot be informed by their impact. AS/A Level Mathematics and Further Mathematics The consultation offers, as its preferred option, a model for AS and A Levels in Mathematics and Further Mathematics which is fundamentally flawed. This is referred to as Model A in the overarching issues questionnaire. While existing syllabuses are manifestly successful, the proposal is to replace them with others that are new and completely untried. It is by no means certain that it is possible to write satisfactory syllabuses in line with Model A. This would be a very high risk gamble, much greater than that of Curriculum 2000, with students, universities and, indeed, the nation itself as potential losers. There would be a reduction in the quality of mathematics learnt; the applied mathematics would be seriously fragmented and its underlying principles lost. The proposed changes to Further Mathematics would reduce its uptake. The proposed loss of flexibility would narrow participation in AS and A Level Mathematics, setting the government’s target of 80 000 A Level students at risk. The proposals would compromise the government’s whole STEM agenda. Use of Mathematics and Use of Statistics The “Use of …” syllabuses could attract more students to continue mathematics and statistics post-GCSE and this would be in the national interest. There is a real danger that smaller institutions offer only Use of Mathematics, making it impossible for their students to take A Level Mathematics. The untested design of Use of Statistics could cause a reduction in numbers studying statistics at this level if it replaces AS and A Level Statistics. The 2013 Review should consider the place of such syllabuses in the national framework. QCA consultation on level 3 mathematics MEI position paper, April 2009 Rationale This paper is written by MEI but we hope that it will be widely read by those with an interest in mathematics at this level, including everyone teaching AS and A Level Mathematics and Further Mathematics, no matter which specification they are following. The commentary reflects our views on the issues and we have not set out to present different opinions; indeed we doubt our ability to make the case for options that we regard as misguided. We are, however, aware that not everyone will share our views; if you are one of those, we would still urge you to respond to the questionnaire so that the outcome from the consultation is as representative as possible. General points about AS and A Level Mathematics and Further Mathematics Reasons not to change in 2012 There are a number of reasons why 2012, or indeed a year or two either side of it, is a bad time to be introducing change. The 2013 review In 2013 there will be a general review of 14-19 qualifications. This will look at the overall provision including, among other things, the relationship between A Levels and Diplomas. Presumably the review will begin fairly early in 2013, say March or April. By then students would be part way through the first year of a two-year A Level on the new syllabus. The 2012 syllabus could be out-of-date before any student had even gone half-way through it. New mathematics specifications from 2012 could prejudice the outcome for mathematics. There would be reluctance to change the new mathematics specifications so soon and so there is a danger that mathematics would not embrace the opportunities available to other subjects and, consequently, be out of step with them. If it ain’t broke, don’t fix it … unless you can make it better The existing A Level Mathematics and Further Mathematics syllabus is working remarkably well, with A Level numbers now at their highest since 1991 and set to rise again in 2009 and in 2010. This is the most successful syllabus we have had for a long time, so it would be foolish to change it without good reason. It should always be remembered that AS and A Level Mathematics serve a wide range of users, many of whom have particular needs. A good syllabus balances those needs but in doing so, it is always open to criticism from particular groups that it does not quite meet their requirements. The success of the present syllabus is partly due to its inbuilt flexibility, allowing most of the particular needs of different groups to be met. This includes the very successful arrangements for Further Mathematics. Any change to meet the needs of a particular group runs the risk of acting against those of others, and this is particularly true of any loss of flexibility. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 2 An untested design Two mathematics A Levels are currently being piloted but the criteria for consultation are not the same as either of them. So the proposal is that, in 2012, a syllabus that is working well and is based on 20 years of experience and development should be replaced by one that is completely untested. Even if there were no other causes for concern, this would be taking a very large gamble, arguably greater than that which led to the Curriculum 2000 fiasco. At stake are the futures of whole cohorts of students and, with them, the future of mathematics in this country. Such a course of action would be the height of irresponsibility. There are, however, many reasons to doubt the success of syllabuses based on the proposed criteria. The most significant of these are set out later in this document. Reduction in uptake of A Levels in Mathematics and Further Mathematics 2008 saw the numbers taking A Level Mathematics exceed the government’s 2014 target of 56 000. There was also a significant increase in the numbers taking Further Mathematics. A revised target of 80 000 taking A Level Mathematics has now been set. In our view, the proposed changes to A Level Mathematics and the complete separation of Mathematics and Further Mathematics specifications would make the attainment of this target highly unlikely. These points are explained further later in this document. Following on from GCSE The proposed schedules for changes to GCSE and A Level would have two drawbacks. First, the new A Level would have been designed without any evidence of the effects of the new GCSE(s). It is not just a case of it making no sense at all to design a new A Level until the new GCSE has settled down and it is possible to assess the strengths and weaknesses of students who have taken it. It is much worse than that; a syllabus would be put in place in the knowledge that it would not be fit for purpose The second point about this schedule is that it would create a cohort of guinea pig students. Current year 8 students will start the new mathematics GCSE in 2010; the same students would then start on the new mathematics A Level in 2012. Arguments put forward in favour of change It is perhaps worth considering the arguments that are put forward in favour of change. In our view, these arguments are, at best, weak and, in several cases, false and dangerous. Mathematics should have 4 units (“reducing the burden of assessment”) It is argued that most other subjects have moved to four units; leaving A Level Mathematics with six units will mean that students think it is harder than other A Levels and so they will not choose it. (There are in fact several subjects that have retained 6 units.) Most subjects started teaching four unit A Levels in 2008. Mathematics will continue with six units until at least 2012. If the larger number of units in A Level Mathematics was going to cause a reduction in uptake, we would be seeing the effect now. However, the opposite is the case; there is an increase in the uptake of mathematics in the current Year 12. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 3 The reduction from six units to four units is sometimes described as “reducing the burden of assessment”. Such a move might make it easier to administer examinations but is it to the advantage of learners? If there are fewer examinations, then the stakes for each one are higher and there is less opportunity for weaker students to be encouraged by, and to build on, their successes. For many students such a change would increase the pressure of assessment. Ensuring a balance of AS and A2 units in A Level Mathematics The Mathematics A Level differs from that of other subjects in that it may consist of 4 AS and 2 A2 units or 3 of each. This is sometimes presented as a problem. However, in a recent report the independent regulator, Ofqual, dismissed this suggestion. “While the demand of an A level made up of three AS plus three A2 units appears to be higher than an A level comprising four AS plus two A2 units, reviewers judged that the matter of breadth versus depth had to be taken into account. They considered that studying two different AS applications units requiring understanding of significantly different mathematical concepts was of equivalent demand to studying two units (one AS and one A2) in one application, and that this increased demand in terms of breadth offset the reduction caused by doing two AS units instead of one AS and one A2 unit.” (Review of standards in GCE mathematics in 2004 and 2007, Ofqual, 2009) Removing the need for complex grading arrangements It is claimed that many people do not understand the structure of A Level Mathematics and Further Mathematics and so it is necessary to separate the qualifications and remove modules which could be used for either. This is a spurious argument. The current structure of A Level Mathematics and Further Mathematics is well understood by mathematics teachers and students and it gives rise to qualifications that are understood by end-users. The rules of aggregation are inevitably more complicated than those for subjects that have only a single award, but they have long been dealt with by the awarding bodies’ systems. Further details, in the form of examples or answers to frequently asked questions, could easily be made available via the internet. Administrative convenience should not be allowed to undermine the quality of mathematics learnt in our schools and colleges. Making the arrangements for awarding A* in mathematics the same as in other subjects Introducing a major syllabus change in order to bring the criteria for awarding A* into line with those for other subjects would be a case of the tail wagging the dog. We should not even contemplate subordinating the needs of the large majority of candidates to administrative arrangements relating to the most talented. For other subjects, A* is awarded on the basis of performance in all three A2 modules but, for mathematics, where some candidates do 4 AS units and 2 A2 units, it is to be awarded on performance in the 2 compulsory A2 units, C3 and C4; they are pure mathematics units covering the subject core. However, the way that assessment works in mathematics, with differentiation by task rather than by outcome, means that fundamental differences from other subjects are already inevitable. What matters for mathematics, or indeed any subject, is that the method used should be fit for purpose. This should be the case for an award that will be based on the compulsory A2 core. However, mathematicians see an optional extra paper with built-in stretch and challenge, for example the Advanced Extension Award, as preferable. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 4 Introducing “stretch and challenge” Providing students with stretch and challenge cannot be done in the same way in all subjects. In some subjects, it is possible to differentiate by outcome on the same question so that the best candidates score more highly but most candidates can access the question and write something. This is not the case in mathematics. A question which some students find stretching and challenging will be considered pedestrian by some students and completely impossible by others. Great care must be taken when introducing stretch and challenge into Mathematics A Level examinations. Mathematics is already seen by some students as being a subject for the cleverest only. If the examination papers become less accessible, this will reduce participation just when we want to increase it. The wording of the overarching issues questionnaire implies that new A Levels will be so challenging as to render the Advanced Extension Award (AEA) unnecessary. If this turns out to be the case, we will have a disaster on our hands with plummeting uptake. Choice is a bad thing (providing students with “equality of opportunity”) It is also claimed that the choice available in A Level Mathematics is undesirable and so should be removed. Two reasons are given for this. The first is that some options are easier than others, or that some people think they are. The substantive point is dismissed in the recent Ofqual report: “Overall, reviewers concluded that the demands caused by the range of options available remained constant over time, and the converging practice across awarding bodies was helpful to teachers and candidates. Reviewers also judged that there was no significant difference in demand between the optional routes available through the syllabuses in 2007.” (Review of standards in GCE mathematics in 2004 and 2007, Ofqual, 2009) Perceptions of difficulty should not form the basis for decision making; they are subjective and inevitably differ from one person to another. It is also argued that, since not all schools can offer students a choice of units at A Level, the choice should not be available to anyone. This is a very dangerous argument, reducing everyone to a lowest common level rather than setting standards to which people can aspire. School mathematics should be a rich experience. Removing options would be a body blow to the many schools and colleges that currently offer them successfully and see it as important to do so. A recent survey carried out by AQA found that over 85% of teachers were in favour of retaining choice. It is also argued that removing choice would benefit Higher Education with all students arriving with the same background experience. However, most university courses recruit students with a wide variety of previous qualifications from within the UK and overseas, and, even among those who have taken A Level, there are big differences in what they actually know. So the idea that removing choice will make life much easier for universities may well be no more than a myth. However, this argument misses the point that the primary purpose of education is to benefit students. Studying mathematics at A Level gives a wide range of students a greater insight into the mathematical aspects of their various university courses. This enables them to gain deeper understanding of the subjects they have chosen for their degrees. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 5 Increasing the emphasis on proof It has recently been suggested that proof is not examined in A Level Mathematics and so there needs to be a change. This is untrue. The current subject criteria for A Level Mathematics explicitly include proof and so awarding bodies are obliged to test it. If some examinations are not addressing proof, this is a matter for the regulators to act on. If the regulators are unable to do so then it is difficult to see how changing the structure of A Level Mathematics will improve matters. Moreover, the draft criteria for Mathematics A Level do not include proof explicitly in the assessment objectives, whereas it is in the current assessment objectives for A Level Mathematics. The draft criteria given for Model A remove disproof by contradiction from Mathematics and place it in Further Mathematics. It is not clear how reducing the proof requirements for A Level Mathematics will result in an increase in its assessment. Making links between pure and applied content We are deeply suspicious of the argument that the proposals would make better links between pure and applied mathematics. Our fear is that the applied mathematics will be seen as little more than an excuse to provide additional practice for the techniques learnt in the pure: kinematics would provide practice of calculus, the binomial distribution of the binomial theorem, and so on. The principles of applied mathematics would be lost. The applied content in the mathematics criteria is described as “modelling”. We are concerned that this word is being used to cover the low quality of the proposed applied mathematics provision and the lack of a sound underlying philosophy. This would seem to be a case of the emperor’s new clothes, an invalid attempt to cover the inadequacy of these proposals with a coating of respectability. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 6 The proposals for A Level Mathematics and Further Mathematics: Model A Outline of Model A Model A is clearly QCA’s preferred model. For the reasons given in this paper, we believe it would be a national disaster. A Level Mathematics would consist of 4 units. No choice of units would be available. The pure content would be virtually the same as the present core content. In this model, the core for A Level would be extended to the applied mathematics. The expected way of meeting the draft criteria would be to have 2 pure units, each twice the size of the current core units, and 2 applied units, each consisting of a mixture of topics drawn from mechanics, statistics and decision mathematics. Further Mathematics would consist of six equally weighted units. The common core would be extended to cover 2 compulsory pure Further Mathematics units, one AS and the other A2. The 4 other units could be chosen from a range of options but there would be no units which could be used in either Mathematics or Further Mathematics. The criteria for Mathematics would also allow 4 equally weighted units, each consisting of a mixture of pure and applied content, but this model has not been discussed in any of the meetings preceding this consultation. It would produce all of the problems described below, and a few more of its own as well. Detailed comments on Model A The applied mathematics design In the present syllabus, each applied strand has its own set of units, for example Mechanics 1 to 4, in which concepts are built up in a coherent, sequential framework. A student may include Mechanics 1 in either AS Mathematics or in AS Further Mathematics (but obviously not in both). This would not be possible in Model A which does not allow any unit to be available within either Mathematics or Further Mathematics. By contrast, Model A requires two separate starts to mechanics (etc), one in Mathematics and another in Further Mathematics. That raises the question of what topics should be placed in the AS Further Mathematics applied units and there is no simple answer, because of the following restrictions. No Further Mathematics material may duplicate the AS or A2 Mathematics applied content. The content of the AS Further Mathematics unit in any strand cannot be dependent on the A2 Mathematics applied content, which will usually be studied in Year 13. Each unit must be worthwhile in its own right and should not include unimportant topics as space-fillers. The units should present learners with a clear, logical route through the strand building up the concepts and the principles underlying their application. We do not believe that it is possible to design coherent sets of units for the applied strands under these conditions. Consequently, any implementation of Model A would involve low quality applied mathematics provision. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 7 Fragmentation In the draft criteria for Model A, the proposed applied content in AS Mathematics includes little bits of five discrete topics. Each of these would attract its own examination question and students would be taught how to answer those questions. There is no connection between the topics so they would learn no applied mathematics, just how to answer a few types of examination questions. The only two sections of the AS unit that would be at all close are the kinematics and the dynamics but the connection between them, Newton’s 2nd Law, cannot be made as it is in the A2 content. This AS unit would give students a completely false introduction to the nature of applied mathematics. Instead of meeting the various strands as coherent bodies of knowledge, each with its own underlying principles and methodology, they would learn them as no more than isolated techniques. Just the same problems would continue into A2. These units would be positively damaging to the mathematical education of young people, who would learn lessons that they would have to unlearn if they were to make subsequent progress. Lack of interchange between Mathematics and Further Mathematics units A fundamental feature of the design of the present syllabus is that some applied units can be allocated to either Mathematics or Further Mathematics. Model A removes this facility. It is seen as a valuable feature of the present syllabus by many students and teachers, but apparently as a problem by QCA. The present system makes it viable for some schools that could not otherwise do so to offer Further Mathematics; they can share the teaching of some modules with Mathematics. This possibility would be removed by taking away the overlap between the two syllabuses. Some students’ initial motivation for taking Further Mathematics is the belief that taking extra modules will improve their grades in the single Mathematics because they can be interchanged. Once they have started the course, their motivation changes as they enjoy the course and become more mathematically fluent. However, removing the overlap between Mathematics and Further Mathematics could remove their initial motivation for starting the course. Thus removing the overlap between Mathematics and Further Mathematics is likely to have two negative effects: reducing the number of schools offering Further Mathematics and reducing the number of students opting to do it in those schools where it is available. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 8 Uptake of Mathematics It seems very likely that the introduction of Model A would have a negative impact on the uptake of AS and A Level Mathematics. This prediction is based on three reasons. The first is that many schools and colleges enter students for their first module in January of Year 12. This is usually C1. For many students this is formative assessment at an early stage in their sixth form course, telling them how they are getting on and if they have to work harder; there is, of course, the possibility of re-sitting the following June. Combining C1 and C2 into a single double weight unit would make this impossible. Two effects are predictable; some students will not start, or drop out early, because the initial step is too large and others will continue through to June, flounder in the examination and give up mathematics altogether. Model A offers no choice in the applied mathematics and so some students would see what is on offer as being less relevant to them than is the case at the moment. This is particularly likely to be the case with those who want to take statistics units to support their other A Levels. The third reason is that there is a growing belief, to which we subscribe, that the recent increase in uptake of mathematics has been accentuated by the high profile of Further Mathematics. There is evidence that offering Further Mathematics improves the status of mathematics throughout a school or college, and attitudes towards it. The implication of this is that the damage, as described above, that Model A would do to Further Mathematics would also reduce the uptake of Mathematics. For all these reasons, we believe that Model A represents a real and serious threat to the government’s target of 80 000 A Level Mathematics students by 2014. Loss of choice Model A would remove all choice from the single A Level Mathematics. At present, many teachers offer their students a choice of which applied units to take and guidance as to what will best support their intended careers. This practice is consistent with the diversity of university courses which involve mathematics, nearly all of them as a service subject. This is not a chance phenomenon; it is a manifestation of the fact that in one form or another mathematics is very widely used in work and life. It is entirely appropriate that this should be recognised in what is available in schools, allowing individual students to be as well prepared as possible for the next stages of their lives. Some teachers are not comfortable with all of mechanics, statistics and decision mathematics. The proposal would force them into teaching material they do not know well. This is not a good way to raise standards. Additional Further Mathematics Model A recommends removing Additional Further Mathematics qualifications. These have been run successfully by MEI for 18 years and, more recently, by Edexcel. They reward those taking 15 (AS) and 18 (full A Level) modules and are taken by very talented individuals who love mathematics. (Typically these students take at least three other subjects as well.) Although Additional Further Mathematics is not taken by many students, those who do so are enthusiastic about mathematics; such an attitude should be encouraged. We should be glorying in the achievements of these students, not trying to cut them back. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 9 New qualifications There is a need to widen participation in mathematics post-16 and to have a curriculum and qualifications that support this. We believe that a variety of such qualifications should be developed now so as to inform the 2013 review. So we welcome, in principle, the Use of Mathematics and the Use of Statistics syllabuses for teaching from 2011. However, the focus of their development should be to inform the 2013 review and so they should be monitored to see what effects these new qualifications have. A Level Use of Mathematics The A Level in Use of Mathematics is based on the current AS in Use of Mathematics. It would consist of six units; three of these are core units, including one Free Standing Mathematics Qualification (FSMQ), and the other three are FSMQs. Although we welcome the likelihood of wider participation that will probably accompany the Use of Mathematics, we would urge the introduction of safeguards to ensure that it does not have unintended consequences. At present, over 50% of schools and colleges offering A Level Mathematics do so with cohorts of fewer than 15 students. Such schools will not be able to run both Mathematics and Use of Mathematics A Levels. A possible, even likely, scenario is this. Some schools teach A Level Use of Mathematics instead of Mathematics. Their students then offer it for university entrance, telling admissions tutors that Mathematics was not available to them. Some admissions tutors feel obliged to accept such students. Word gets out and more schools teach Use of Mathematics instead of Mathematics. Mathematics and Further Mathematics are effectively replaced by Use of Mathematics. There is a consequent reduction in the standard of mathematics in our schools and colleges. These days the standard response to this possibility is that it would not happen. However it was taken very seriously in the recent past when the extension of AS Use of Mathematics to a full A Level was disallowed by QCA because of precisely these fears. We recommend that the following measures be taken in the period leading up to the 2013 review. Only schools and colleges that also enter candidates for A Level Mathematics should be allowed to offer A Level Use of Mathematics. Careful consideration must be given to ensuring that students and teachers understand the pathways available from A Level Use of Mathematics. Research should be conducted to establish the positive features of this syllabus, leading to advice on how they could best be incorporated into a new standard A Level Mathematics. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 10 A Level Use of Statistics AS Use of Statistics would replace the current AS Statistics offered by OCR (on behalf of MEI) and AQA. Similarly A Level Use of Statistics would replace A Level Statistics. However the design is quite different from that of the current Statistics specifications. At AS there would be 2 compulsory units, both of them FSMQs. The third unit would consist of two controlled assessment tasks based on the content of the other two. The same pattern would apply at A2. There would be no choice in content; it would necessarily be the same for all awarding bodies. On paper it would seem possible that the Use of Statistics syllabuses would open up opportunities to the large number of students who will need statistics when they go on to university a year or two later. However, it is by no means certain that this will happen. Those same students could now be taking AS and A Level Statistics but the uptake of these courses could, and should, be very much higher. It is not obvious that re-naming them, or the proposed design changes, will alter that situation. The proposed move away from the Statistics syllabuses would seem to be based on hope rather than evidence. There is no reason to believe that substantial numbers of schools and colleges that do not offer AS or A Level Statistics will be attracted to the new course. There is, however, a danger that institutions that currently offer Statistics would not offer Use of Statistics because of the substantial differences between the courses. The controlled assessment could provide an opportunity for students to solve substantial and meaningful problems using statistics. However, it would need to be well organised. Professional development and support for teachers would be essential to enable them to implement the course effectively. We see no virtue in the content being prescribed, particularly at this developmental stage of a new qualification. The only possible justification for the proposed level of prescription would be if students wanted to aggregate units from different awarding bodies into the same AS or A Level qualification. There is no evidence that this is likely to happen or that it would be realistic to expect the awarding bodies to handle the extra administration that would be involved. We have four major recommendations concerning the Use of Statistics and how it should be used to inform the 2013 review. Research should be carried out to discover why such a small proportion of students who will need statistics at university study it in Years 12 and 13. At this stage it should run alongside AS and A Level Statistics rather than replace those specifications. Awarding bodies should be free to determine the content, subject to the usual approval procedures. A pilot qualification involving 3 Use of Mathematics units and 3 Use of Statistics units should be developed and field-tested. This qualification could be very useful to a wide range of students. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 11 Links Reading through this position paper, it will be clear that we feel huge concern about many of QCA’s proposals. In this we are not alone. The Royal Statistical Society’s position statement can be found at http://www.rss.org.uk/curriculum . The paper you are reading can be found at: http://www.mei.org.uk/qcaconsultation.shtml. A number of related documents and links may be found at the same location including the following: “Finding your way round the QCA consultation on level 3 mathematics”, a guidance document written by MEI; a copy of the submission of the Greater Manchester Sixth Form Colleges Mathematics Group who wrote recently to QCA to express their concern; a link to the relevant page of the QCA website and direct links to the consultation documents. A final word Mathematics is not only a beautiful and exciting subject in its own right but also one that underpins many other branches of learning and is consequently fundamental to the success of a modern economy. However, we are in no doubt that, if the main proposals in this consultation were to go ahead, many young people in our schools and colleges would have an impoverished experience of mathematics, meeting less of its richness and diversity, and that they would carry this deficit with them in adult life. The last few years have seen great successes for level 3 mathematics, with large increases in uptake at A Level and even larger increases in Further Mathematics. A change of culture is taking place in which the subject is coming to be much more highly valued. Let us decline QCA’s invitation to snatch defeat out of the jaws of victory. QCA consultation on level 3 mathematics MEI position paper, April 2009 Page 12