The Post-14 Mathematics Inquiry
Making Mathematics Count
The Report of Professor Adrian Smith's Inquiry into Post-14 Mathematics Education
Chapter 4 - Action on Current and Future
Mathematics Pathways
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The Working Group on 14-19 Curriculum and Qualifications
Reform
- Current concerns over GCSE Mathematics in England
- Concerns over key skills and Application of Number (AoN): FSMQs and AS Use of Mathematics
- Concerns relating to GCE Mathematics
- Concerns with Adult Numeracy
- Possible Future Pathway Models for Mathematics 14-19
- Current concerns over GCSE Mathematics in England
The Working Group on 14-19 Curriculum and Qualifications Reform |
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| 4.1 | The Post-14 Mathematics Inquiry has proceeded in parallel with the work of the Working Group on 14-19 Curriculum and qualifications reform in England, chaired by Mike Tomlinson. In its Interim Report in February 2004 (DfES/0013/2004), the Working Group outlined broad proposals for the phase of 14-19 learning, including the development of a new diploma framework that would cover the whole of the 14-19 learning programme. The Interim Report includes proposals to move away from the existing age-related qualifications to a system offering more opportunities for students to achieve qualifications in their own time and at their own ability and aptitude level, while offering coherent pathways of progression. Such a framework should provide candidates with opportunities to demonstrate and record specific mastery of skills and topics rather than recording overall levels of success or failure. A key feature of the Tomlinson proposals is a single 14-19 learning continuum in place of the current perception of 14-16 and 16-19 as two distinct phases. |
| 4.2 | The Working Group proposals will encourage more students to obtain level 3 qualifications. So far as mathematics is concerned, the proposals incorporate the possibility of more specialist study of mathematics beyond a mandatory core of foundational mathematics. Although the Mathematics Inquiry has proceeded independently from the Working Group on 14-19 Reform, the Inquiry has found there to be a strong consensus within the mathematics community in favour of a diploma type of approach to qualifications. |
| 4.3 | So far as mathematics is concerned, the Post-14 Mathematics Inquiry agrees with the 14-19 Working Group’s conclusion that the present qualifications framework is in need of a radical overhaul. The first part of this chapter will discuss in detail the concerns expressed to the Inquiry about the current framework. This will lead us to make specific short- and medium-term recommendations regarding the current framework. We see these not only as important steps towards improving the current structure, but also as contributing to a longer term direction of travel, compatible with the Tomlinson notion of progressive pathways, each with its own mathematical components. We see some version of the latter as the key to providing a structure whereby all students have access to a relevant mathematics pathway appropriate to their learning needs, and relevant to end destinations in the workplace, or continuing education, post-19. In the view of the Inquiry, mathematics should be seen as an integrated whole when designing 14-19 pathways. We begin with a review of concerns with the current structure. |
Current concerns over GCSE Mathematics in England |
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| 4.4 | The Inquiry has no doubt that, compared with the previous O-level/CSE structure, GCSE Mathematics has been beneficial to many more students and has provided them with an adequate background for further study in the subject. However, respondents to the Inquiry have raised a number of serious concerns, which the Inquiry believes to be well founded. |
| 4.5 | In GCSE Mathematics, only about 50 per cent of the candidature achieves the iconic ‘pass’ grades A*-C. Many repeat GCSE to try to improve their grade, having failed to reach at least a C grade first time round at age 16. There do not appear to be easily accessible data on resit performance, but respondents to the Inquiry report that a significant number of resit candidates do not achieve an improved grade. More generally, far too few young people in England achieve level 2 qualifications in mathematics and England seriously lags behind its European competitors in this respect. |
| 4.6 | Whilst accepting that the decision to have a three-tier arrangement for mathematics was made with the best of intentions, respondents to the Inquiry have overwhelmingly expressed grave concern that GCSE mathematics is now the only GCSE subject where a grade C is not accessible on all the tiers. In the light of concern expressed about the three-tier structure, the regulatory bodies have given further consideration to appropriate assessment mechanisms and is running a pilot of a two tier GCSE examination in mathematics with OCR. |
| 4.7 | The pilot scheme has three examination papers and all candidates sit a combination of two of these. Candidates studying the Foundation PoS are entered for Paper 1 (targeting grades E-G) and for Paper 2 (targeting grades C-D). Those studying the Higher PoS are entered for Paper 2 and for Paper 3 (targeting grades A*-B), but could be entered for Papers 1 and 2 if they are having difficulty with the course. Every student therefore has access to a grade C and there will be only one route to each grade. The pilot will run through two complete cycles. The first examinations took place in June 2003 and there will be a second round in June 2004. Ministers will be notified of outcomes and advised of any proposed changes in December 2004. Any modification of the current arrangements in England will require Ministerial approval. |
| 4.8 | The Inquiry has been informed that the QCA would wish to see this two-tier assessment structure become the standard examination structure for GCSE Mathematics within a few years. The assessment structure would then mirror the revised curriculum structure, a correspondence that many respondents clearly believe to be important. We believe that this was the original intention at the time of the 1999 revision of the curriculum, but was shelved on the grounds that such additional change to the system would be have been too disruptive at that time. The majority of respondents to the Inquiry seem to believe that most teachers would now welcome the shift to two-tier examining as fitting in more naturally with their curriculum planning and setting into cohorts for the Key Stage 4 Higher and Foundation Programmes of Study. |
| 4.9 | The three-tier assessment structure does not mirror the two-tier structure of the revised curriculum. Moreover, the existing three-tier arrangement for assessing GCSE Mathematics disbars about one third of candidates from having access to a grade C. Since grade D is the highest grade achievable on the Foundation Tier papers, respondents report that many students feel themselves to have been classed as “failures” by their teachers before they even start the course. The Inquiry shares this concern. Rightly or wrongly, public opinion – no doubt much influenced by school league tables – has come to regard a grade C at GCSE as a minimal acceptable level of attainment. It therefore seems to the Inquiry totally unacceptable to be entering some 30 per cent of the age cohort into a tier in which “externally perceived success” (ie grade C) is unattainable whatever the level of achievement. |
| 4.10 | The existing arrangements for assessing GCSE Mathematics allow raise issues regarding the interpretation of GCSE grade B. This grade can be awarded both on the Intermediate Tier papers and also on the Higher Tier papers. However, respondents are clear that the algebraic and geometric content associated with the Intermediate Tier is significantly less than the algebraic and geometric content associated with the Higher Tier and that this means that there cannot be an unambiguous interpretation of GCSE grade B in mathematics. In particular, there is concern in relation to preparedness for AS/A-level mathematics. Many clearly feel that, without some form of bridging course, candidates obtaining a grade B in mathematics on the Intermediate tier have an inadequate basis for moving on to AS and A2. They have had too little fluency in algebra and too little routine practice with reasoning about geometric properties and relations. |
| 4.11 | However, it has been put to the Inquiry that the tactical behaviour of schools and pupils is being influenced by the perception that it is easier to get a grade B for GCSE mathematics by being entered for the Intermediate Tier. We have been informed that when grade B was first introduced as a possible outcome on the Intermediate Tier, entries for the Higher Tier fell from nearly 30 per cent to about 15 per cent of the candidate cohort and have remained relatively stable since then. The Inquiry finds these consequences of the current arrangements to be worrying, both in terms of the interpretability of grades and the perverse incentives it provides for placing pupils on educationally inappropriate pathways. We suggest that those piloting the twotier system take on board these concerns. |
| 4.12 | The GCSE Mathematics examinations in summer 2003 were the first to assess the revised two-tier curriculum. Anecdotal evidence to the Inquiry indicates that the new coursework regulations have caused some problems to some teachers and pupils. It may also be the case that teachers may not have fully acquainted themselves with the content of the Higher PoS. The Inquiry is not in a position to properly assess the new two-tier initiative or the two-tier assessment, but nevertheless believes that serious consideration should be given to moving to a two-tier structure. |
| Recommendation 4.1
The Inquiry recommends that, subject to the present pilot being fully and successfully evaluated, immediate consideration be given by the QCA and its regulatory partners to moving as soon as is practicable to a two-tier system of overlapping papers for GCSE Mathematics in England, Wales and Northern Ireland. The Inquiry recommends that the regulatory authorities try to recruit more schools and colleges to take part in preimplementation piloting after summer 2004. |
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| 4.13 | Many respondents clearly feel that mathematics is not rewarded sufficiently at level 2 in comparison to English and science and this is also reflected in responses given in focus groups organised by QCA on behalf of the Inquiry. It is widely believed by pupils and teachers that the amount of effort required to achieve a single GCSE award in Mathematics is similar to the amount of effort required to gain the two awards in English Language and Literature or to gain a Double Award in Science. There is a widespread concern that this is adding yet further to the perception of mathematics as a disproportionately hard subject and may be adversely affecting pupils’ subsequent choices post- 16. The Inquiry believes that this to be a serious issue and supports the view that serious consideration should be given to making a double award available for mathematics for the higher tier route (either in the current structure, or in a revised two-tier structure). |
| 4.14 | We acknowledge that consideration needs to be given as to how to do this so as to ensure that such a double award is on a par with the double award for GCSE Science. The Inquiry has not had the time or resources to provide detailed practical recommendations regarding the necessary curriculum and/or assessment adjustments required (in either the two-tier or three-tier structures). However, we would wish to make the following clear recommendation. |
| Recommendation 4.2
The Inquiry recommends that, at the earliest possible opportunity, consideration should be given by the QCA and its regulatory partners to re-designating GCSE Mathematics, appropriately modified if necessary, to merit a double award at level 2. This re-designation should be considered in tandem with the possible move to a two-tier system (see Recommendation 4.1). |
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| 4.15 | Many respondents have expressed serious doubts about the value of GCSE mathematics coursework, in particular the data-handling component. There is concern that current requirements lead to a rather artificial approach to analysing and interpreting data, rather than encouraging substantive involvement with “real life” problems. There is also concern over the comparatively large amount of time spent on GCSE coursework in relation to the amount of timetabled time for the subject itself. We are aware that the QCA has amended the coursework marking criteria in response to perceived teething troubles, but the Inquiry still feels that there is sufficient concern to merit a review of current requirements. This needs to be considered alongside Recommendation 4.4. |
| Recommendation 4.3
The Inquiry recommends that there should be an immediate review by the QCA and its regulatory partners of the quantity of coursework in GCSE mathematics and, in particular, the data handling component, with a view to reducing the amount of time spent on this specific element of the course. (See, also, Recommendation 4.4) |
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| 4.16 | More generally, there has been considerable disagreement among respondents regarding the appropriate treatment of the Handling Data strand of the PoS for Key Stage 4 (Higher). Basic Probability is clearly seen as part of the mathematics core, but some have argued that Handling Data should be absorbed into the using and applying mathematics strands in number and algebra and in shape, space and measures. Others have argued that the roles of Statistics and Data Handling are so fundamentally important, both in other disciplines and in the workplace, that, in the long term, these topics need to be found their own timetable niche – perhaps embedded in the teaching of other disciplines – rather than taking up a substantial part of the mathematics timetable that used to be available for practice and reinforcement of fluency in core mathematics techniques. In addition, the function of GCSE Statistics is thought by many respondents to be unclear. The majority view is that it is not sensible for pupils who achieve a good GCSE Mathematics pass in year 10, or earlier, either to discontinue the study of mathematics altogether in year 11 or to study GCSE Statistics as an additional GCSE replacing formal study of mathematics in this year. Conversely, it is suspected that some pupils are entered for GCSE Statistics because it is seen as a softer option than GCSE Mathematics itself in terms of grade attainment, rather than for sound educational reasons. |
| 4.17 | The Inquiry strongly believes that knowledge of Statistics and Data Handling is fundamentally important for all students and would wish to see these topics continue to be given due emphasis and timetable allocation. However, we believe it would be timely – in the context of a radical re-think of future 14- 19 mathematics pathways within the general structure that may emerge following the 14-19 Working Group review – to reconsider the current positioning of Handling Data within the GCSE mathematics timetable, where it occupies some 25 per cent of the timetable allocation. Many respondents believe the current mathematics curriculum at Key Stage 4 to be overloaded. We have no doubt that much of the concern expressed to us about the perceived decline of fluency with core mathematical operations reflects the pressure on the mathematics timetable that has resulted from the inclusion of this significant element of Handling Data. |
| 4.18 | We have also received a number of responses arguing that the teaching and learning of Statistics and Data Handling would be greatly enhanced if they were more closely integrated with the other disciplines that rely heavily on these topics, such as biology and geography. We support this view and believe it to be timely to begin to review this issue in the context of the general philosophy of the approach to 14-19 learning programmes emerging from the Tomlinson review. This prompts our next recommendation, which should also be considered in the context of our longer-term recommendations about future pathways set out later in this chapter. |
| Recommendation 4.4
The Inquiry recommends that there should be an immediate review by the QCA and its regulatory partners of the future role and positioning of Statistics and Data Handling within the overall 14–19 curriculum. This should be informed by: (i) a recognition of the need to restore more time to the mathematics curriculum for the reinforcement of core skills, such as fluency in algebra and reasoning about geometrical properties and (ii) a recognition of the key importance of Statistics and Data Handling as a topic in its own right and the desirability of its integration with other subject areas (see, also, Recommendation 4.11). |
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| 4.19 | In terms of usable skills, although GCSE grade C is the minimum societal expectation, evidence to the Inquiry suggests that employers are often less than happy about the mathematical abilities of recruits with GCSE, even when the grade obtained is at least a C. The perception of the level of mastery signified by a grade C has been further damaged by the claim in an article in the Daily Express in the summer of 2003 that some students were achieving the grade on the basis of 15 per cent raw marks. More generally, evidence to the Inquiry and the findings of the report Mathematical Skills in the Workplace suggest that GCSE Mathematics itself now seems to many employers to be an inadequate preparation for the growing mathematical needs of the workplace. The perception is that students are learning most of their mathematics in a vacuum, with little attention given to any sort of mathematical modelling, or to a range of problems set in real world contexts and using real data. In addition, the report Mathematical Skills in the Workplace makes clear that there is serious concern that students have little exposure to how ICT can be used to enhance each of these aspects of mathematics, even though employers today increasingly want a combination of mathematical skills harnessed to ICT skills. In terms of the appeal of the subject to students, evidence from focus groups run by the QCA for the Inquiry reveals that for many students, GCSE Mathematics seems irrelevant and boring and does not encourage them to consider further study of mathematics. At the same time, many respondents have impressed on us the dangers of also losing the attention and interest of some of the most able because of the perceived lack of depth and challenge in the standard curriculum. |
| 4.20 | The Inquiry is acutely aware of the dangers of diluting the essence of the discipline of mathematics by inappropriate attempts to make everything immediately “relevant” and by the use of clearly unrealistic versions of “real” problems. That said, we believe that the time has come for a radical re-look at longer-term options for 14-16 mathematics provision that do provide sufficient appropriate pathways for those who need motivating more through perceived practical relevance. We shall later make recommendations directed at beginning this process. In the meantime, we believe that there is an immediate action to be taken in relation to the needs of the most mathematically able. |
| 4.21 The Inquiry believes that it is vitally important to provide appropriate challenge for the mathematically more able and motivated. We also accept the view of the overwhelming majority of respondents to the Inquiry that current provision is failing in this respect. Some respondents to the Inquiry have suggested that the more able students should be catered for by accelerating their exposure to material covered at higher qualification levels. The overwhelming majority of respondents disagree. The prevailing view is that what is required is deeper challenge and exposure to more open-ended problem solving with material from the student’s current qualification stage. The Inquiry supports this latter view. We have an open mind about whether such provision should be statutory and whether it should lead to a formal qualification. | |
| Recommendation 4.5
The Inquiry recommends that the QCA and its regulatory partners should be funded to develop an extension curriculum and assessment framework for more able pupils at Key Stages 3 and 4. This extension curriculum should be firmly rooted in the material of the current Programmes of Study, but pupils should be presented with greater challenges. These should involve harder problem solving in non-standard situations, a greater understanding of mathematical inter-connectedness, a greater facility in mathematical reasoning (including proof) and an ability to engage in multi-step reasoning and more open-ended problem solving (see, also, Recommendation 4.11). |
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FSMQs and AS Use of Mathematics: concerns over key skills and Application of Number (AoN) |
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| 4.22 | The Inquiry also believes that the action is vital to provide appropriate challenge and motivation for those who need and want to continue the study of mathematics post-16, but are primarily motivated by seeing the relevance of mathematics in the context of a range of real-world applications. In this connection, many respondents have indicated to the Inquiry that there is insufficient awareness and use of the FSMQs and AS Use of Mathematics qualifications. In particular, respondents have indicated that there is scope for more level 2 FSMQs, to cover a wider spectrum of mathematics. In particular, it is argued that a level 2 Use of Mathematics should be developed along lines similar to the existing AS Use of Mathematics. The Inquiry has not had the time or resources to consider this in depth. However, we do believe it would be timely to conduct a review of all these issues and we suggest a way forward in Recommendation 4.7. |
| 4.23 | Despite the rapidly increasing numbers making use of FSMQs, take up remains
comparatively small. Despite some very positive reports, the Inquiry does
not feel that there is sufficient experience of their use for it to be able
to judge clearly the merits or otherwise of the current portfolio of FSMQs.
However, the Inquiry has become aware of a number of seemingly unnecessary
current obstacles to delivery and further take-up. These include:
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| 4.24 | The Inquiry accepts that prima facie FSMQs have much to offer, particularly in the context of a re-design of 14-19 mathematics pathways. It would therefore clearly be highly desirable to have greater experience of their use as part of the process of working towards a richer portfolio of 14-19 pathways. However, we accept that this is unlikely to happen without at the very least a concerted campaign to raise the profile and acceptance of these qualifications. More generally, we are concerned that provision of Application of Number has not been developed within a coherent framework together with FSMQs and AS Use of Mathematics. We are concerned about this potentially lack of coherence and believe that it would now be timely to review this whole portfolio of provision as a prerequisite to the re-design of more practically oriented pathways within a new 14-19 structure. A specific way forward is detailed in Recommendations 4.6 and 4.7. |
| 4.25 | The Inquiry has received a significant number of responses raising serious concerns about the implementation of the key skills agenda and particularly the AoN component. While there are doubtless instances where successful implementation is taking place, the messages we have received are overwhelmingly negative. The Inquiry is aware of the danger of being over-influenced by strongly expressed views and is conscious of the fact that it has had neither the resources nor the expertise to conduct independent studies or surveys in relation to many of the issues raised. However, the messages have been consistent enough for us to be convinced that this whole area requires at the very least a thorough and radical review. |
| 4.26 | One key issue around the delivery of AoN has been whether delivery should be separate or integrated with the students’ other courses, particularly those of a vocational nature. The Inquiry shares the view of many respondents that for many students at this stage of their education, particularly those who have made firm vocational choices, integration of the mathematics with the vocational subject would be highly desirable. In practice, however, evidence to the Inquiry makes clear that many teachers on non-mathematical courses have found it very difficult to provide satisfactory delivery of AoN. Many teachers of vocational subjects who are not mathematics specialists are not confident in their understanding of how mathematics can be used to enhance their own areas of work. They typically have even less confidence in teaching mathematics to their students, who also work from a very low level of mathematical understanding. This seems to be especially true of students on Modern Apprenticeships. Many of these students may need to address problems with their basic numeracy skills before moving on to AoN. The Skills White Paper in England announced that the services provided by the Key Skills Support Programme will continue to be available to practitioners in schools, colleges and work-based training from 2004-05. The Inquiry acknowledges the efforts that are being made here, but continues to be concerned that the issue of the vocational teachers’ actual skills and confidence levels in mathematics are not being fully addressed. |
| 4.27 | We understand from respondents to the Inquiry that, in practice, in most FE Colleges the delivery of AoN is currently the responsibility of specialist mathematics staff, many of whom would regard themselves better employed teaching other aspects of mathematics where their specialist skills are more crucial. Where this is the case, this has clearly resulted in tensions for local managers in reconciling teacher preferences and learners’ needs and effectively deploying specialist teaching resources. Such tensions have often been difficult to resolve, although in many cases local solutions have been found. In some cases, mathematics specialists have shared the teaching load with vocational or other subject specialists. In others, specialist teachers have provided a resource to support and advise other teachers. Overall, however, the Inquiry is clear that there is a continuing serious short-term problem with teaching delivery of AoN. We cannot see an immediate solution. However, longer-term we believe that effective support for integrated delivery and for enhancing the mathematical and mathematics teaching skills of specialists in vocational subjects can and should be provided through the national infrastructure for the support of teaching of mathematics. (See, Chapters 5 and 6.) |
| 4.28 | Separate from the issue of teaching delivery, many respondents to the Inquiry are concerned that the mathematical content of AoN is too narrow; in particular, there is concern about what is seen as the superficial approach to the component relating to collecting and interpreting data. The narrowness of the content doubtless reflects the original conception of limiting the mathematics to core numeracy in order not to burden students with unnecessary content. However, the concern has been raised that this may have resulted in the too rigid exclusion of closely related and relevant mathematics that in many cases would help individual students with their vocational specialisms and other studies. This, in turn, is seen as an obstacle to students fully appreciating the relevance of application of number element of key skills to their interests and course of study. The Inquiry notes this widespread concern, but has not had the resource or expertise to make a definitive judgement. |
| 4.29 | There has been some concern that some of the requirements of portfolios have made them difficult to complete. Also, there are concerns that the form in which evidence is required may often be too structured and inflexible. This current inflexibility, together with problems of integrated teaching delivery referred to above, is felt to lead in many cases to poor integration of key skills and to encourage stand-alone key skills activities. We note, also, that concern about the external tests for AoN has been voiced by representatives of those involved in delivering the the work-based route. In particular, it has been argued that the tests are too academic. The Inquiry notes that the Skills White Paper measures represent a response to these and other concerns about the key skills external tests. The measures offer support for key skills teaching and learning, more accessible assessment and more equitable funding. The Inquiry notes that the QCA and its regulatory partners have taken these views into account in their recent review. As a result of the review, the key skills assessment arrangements will remain unchanged in England, with a continuing use of both test and portfolio evidence. In Wales, assessment from September 2004 will be based on a portfolio only model. In Northern Ireland, an operational pilot of a portfolio model with a task-based external element will be implemented from September 2004. A further important factor in the appeal and value of key skills qualifications has been, and will continue to be, the attitude of universities. The current position is that some 33 per cent of the total of 45,974 courses on offer in HE for entry in 2003 accepted the key skills tariff points. |
| 4.30 | Another concern communicated to the Inquiry is that the AoN qualifications lead to a serious distortion of the way in which qualifications are deemed to be equivalent to each other. AoN can be taken at levels 1-4 in schools, Sixth Form Colleges and in Colleges of Further Education. Level 3 AoN is only a small subset of the mathematics provision at the level 2 end of GCSE Mathematics. Similarly, level 2 AoN is only a very small subset of the entirety of mathematics at the level 1 end of GCSE Mathematics. However, the impression has been given that level 2 AoN can be thought of as equivalent to a GCSE ‘pass’ in Mathematics and that a level 3 AoN can be thought of as mathematical attainment beyond GCSE. Respondents to the Inquiry are clear that GCSE Mathematics and level 2 AoN are not fully equivalent in mathematical content and should not be thought of as equivalent on this basis. From the perspective of having an unambiguous understanding of mathematics qualifications, we therefore accept that there is a problem in both GCSE and level 2 AoN being defined as level 2 qualifications. In the same way, level 2 AoN is not nearly as mathematically demanding as a level 2 FSMQ. The fact that AoN demands are not appropriate at their stated level of the NQF is seen by a number of respondents as potentially bringing the framework into disrepute. They note that the level 3 AoN qualification contains no mathematics above the equivalent of grade B GCSE, and only one item at that level; they also note that the mathematics of the AoN level 3 qualification corresponds to the bottom end of level 2. This leads to considerable confusion amongst users, who, not unnaturally, assume that all mathematical qualifications at level 3 include mathematical material at the same level. Respondents also views with concern the Universities and Colleges Admissions Services (UCAS) tariff of 20 points for level 3 AoN. This is the same tariff as for grade A performance on a level 3 FSMQ, which does represent genuine mathematics achievement at this level. Decisions on the current allocation of qualifications to levels within the NQF are the statutory responsibility of the regulatory authorities. This prompts the following recommendation. |
| Recommendation 4.6
The Inquiry recommends that QCA and its regulatory partners undertake a comparative review and make appropriate re-designations as necessary, to ensure that claimed equivalences of levels of mathematics qualifications are well founded. |
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| 4.31 | In our increasingly technological and information-rich society, mathematical skills are becoming more and more important. Rather than decreasing the need for mathematics, as evidenced in the Mathematical Skills in the Workplace report, the rise of information technology has increased the range of mathematics needed to perform competently in the workplace. The majority of respondents are clear that AoN does not deliver the full range of mathematical skills and knowledge that this report shows to be essential in the work-place across many important sectors of the modern economy. The Inquiry accepts this, but, in fairness to the developers, also recognises that AoN was not designed to achieve these ends. However, the fact remains that evidence to the Inquiry from focus groups organised by QCA on behalf of the Inquiry makes clear that AoN is disliked by many students and by many provider institutions and that there is a widespread perception – which the Inquiry reports rather than endorses – that being in possession of an AoN qualification rarely results in candidates having transferable mathematics skills of any worth. Some respondents to the Inquiry have been much more positive about the extent to which FSMQs have the potential to impart worthwhile, transferable mathematical skills. In view of the limited take up thus far of FSMQs, we can again only report, rather than endorse, this perception. |
| 4.32 | The Inquiry has also been told that present funding regimes for colleges create greater incentive to provide AoN at the expense of FSMQs. If this really reflects national LSC funding rules rather than local management decisions, this would seem to the Inquiry to be a somewhat perverse incentive. Piecemeal development has led to patchy provision at levels 1 and 2 and we are persuaded that it is unhelpful to consider numeracy and AoN to be distinct from mathematics itself. There is a need for a more coherent and comprehensive approach. Currently, in FE colleges both the provision of AoN and the widespread use of GCSE resits stand in the way of such an approach. Gaps and overlaps in mathematics provision and qualifications at levels 1-3 were reviewed by the QCA in 2002 and the findings made available to the Inquiry. The Inquiry believes that it would now be timely to ask the QCA and its regulatory partners to extend this work into a general review of problems with the delivery, content and assessment of AoN and the availability to students of FSMQs and AS Use of Mathematics, with a view to feeding into work on the design of future 14-19 mathematics pathways. This would also provide an opportunity to explore and promote greater use of ICT in the delivery of future developments of these courses. |
| Recommendation 4.7
The Inquiry recommends that the QCA and its regulatory partners undertake an immediate review of current problems of delivery, content, assessment and availability of courses at levels 1– 3 provided by FSMQs, AS Use of Mathematics, AoN and Adult Numeracy. The aim of the review should be to identify scope for improvements in and potential rationalisation of this provision, including opportunities for more systematic integration of ICT in teaching and learning, as part of the longer-term design of a new 14–19 pathway structure for mathematics (see, also, Recommendation 4.11). |
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Concerns relating to GCE Mathematics |
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| 4.33 | Although GCE has historically been regarded in some quarters as a gold standard, there have been a number of serious concerns for some time. The Dearing Reforms tried to give more rigour to A-level mathematics and tried to demand prerequisite achievement at the upper end of GCSE Mathematics. However, respondents to the Inquiry have overwhelmingly reported that some of the Dearing recommendations, especially those of a more generic nature reflected in the Curriculum 2000 AS plus A2 model, have had very negative consequences for mathematics. The Inquiry is convinced that the serious problems for Mathematics in 2001 and the subsequent two years arose because the curriculum model imposed for all subjects worked to the detriment of mathematics. The numbers of students studying A-level Mathematics decreased within one year by 20 per cent as a direct result of the implementation of Curriculum 2000 and has stayed at this level the year after. |
| 4.34 | In the view of the Inquiry, the seriousness of this cannot be underestimated. The numbers continuing with GCE mathematics post-16 provide the supply chains for mathematicians, statisticians, scientists and engineers in higher education, research and employment. This supply chain is key to the strategy for tackling the problems identified in SET for Success, as well as providing an increased supply of future qualified mathematics teachers. It is vital that ways be found to restore the numbers not only to the levels of two years ago, but to increase them significantly. Far too few achieve level 3 qualifications in mathematics in England and Wales. |
| 4.35 | Respondents have also wished to challenge the current arrangement whereby GCE mathematics attracts the same UCAS tariff as any other GCE at either AS or A-level. This is seen as unhelpful on two counts. First, there is clear evidence that mathematics does not present a level playing field in terms of attaining grades and a clear perception that mathematics is hard. It is argued that an incentive is needed to counteract this. Secondly, mathematics is unique in providing the key underpinning of so many other disciplines. It is argued that this needs to be formally recognised in order to encourage greater involvement with mathematics post-16. In particular, it is noted that the AEA in Mathematics currently attracts no UCAS points at all thus providing no incentive to enter for the qualification other than for love of the subject itself. We understand that UCAS are currently reviewing this issue. |
| Recommendation 4.8
The Inquiry recommends that the effects of the introduction of the revised specifications for GCE be closely monitored by the QCA and its regulatory partners as a matter of high priority and that funding be made available to support this. If there is no significant restoration of the numbers entering AS and A2 mathematics within the next two or three years, the Inquiry believes the implications for the supply of post–16 qualified mathematics students in England, Wales and Northern Ireland to be so serious that consideration should be given by the DfES and the relevant devolved authorities to offering incentives for students to follow these courses. One possible form of incentive could take the form of financial incentives to HEIs to include AS or A-level mathematics as a prerequisite for certain degree courses. Another possibility might be to offer financial incentives directly to students following such course in HEIs, possibly through fee waivers or targeted bursaries. |
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| 4.36 In addition to these considerable concerns about the organisation of the curriculum and the serious effects of the Curriculum 2000 changes, there are also serious concerns about the frequency of assessment of material in GCE AS and A-level Mathematics. This is felt by many respondents to hinder the development of the learning and understanding of mathematics at this level. It is the consensus view that far too much time is devoted to examinations and preparing for examinations – “teaching to the test” – and that this is at the expense of the understanding of the subject itself. Many identify the problem as the splitting of the subject matter of A-level mathematics into six separately examined modules. This is seen as having the effect of splintering the unity and connectedness of the mathematics to be learned at this level. It is felt that this fragmented presentation makes it virtually impossible to set genuinely thought-provoking examination questions that assess the full range of mathematical skills. It is also felt that the style of short examination papers results in a race against the clock that adversely affects weaker candidates. We are aware that the criteria for GCE mathematics have just been reviewed and changed, and we appreciate that there is a natural desire for some stability in the system. However, we have received such strong representations on this issue that we nevertheless make the following recommendation. | |
| Recommendation 4.9
The Inquiry recommends that the QCA and its regulatory partners conduct an immediate review of the frequency and style of current GCE assessment, with a view to reducing the time spent on external examinations and preparation for examinations. |
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| 4.37 | In terms of student choices and the general perception of the subject, AS and A-level Mathematics are the mainstream qualifications available at this level, but do not attract enough students to study some level 3 mathematics in post-compulsory education. Many respondents have commented that the distribution of grades for A-level mathematics presented in Chapter 3 suggests that the more able students entered for A-level mathematics are insufficiently challenged and the least able are frequently overstretched. In the majority of subjects, the distribution of A-level grades is roughly bell-shaped with relatively few candidates at the extreme grades A or E. However, historically in A-level mathematics, grade A is the modal grade and the distribution of grades is virtually a straight line down to the lower grades. In terms of students’ and teachers’ perception of the subject, many respondents believe that, for other than the mathematically clearly very able students, there is a tendency for schools to see choosing mathematics A-level as higher risk in terms of outcome than many other disciplines. To add to this perception, it is clear that many weak students do not complete the course in GCE Mathematics and many of those who do complete are not classified on their examination performance. At the other end of the scale, A-level Mathematics is felt not to discriminate sufficiently amongst those awarded the highest grades in the subject. University mathematics departments have made clear to the Inquiry that they are often unsure of the real value of a grade A pass at A-level. |
| 4.38 | Following the revision of the GCE criteria for Mathematics in response to the Curriculum 2000 debacle, many respondents are in no doubt that A-level Mathematics has been made easier for the very best candidates. In terms of the potentially most able mathematics students, the Inquiry believes that far too few able candidates are entered for AS or A-level Further Mathematics because their schools or colleges do not have sufficient resources to provide these courses. The same appears to be the case for the AEA in Mathematics, although the original intention of AEAs was that they would not require additional teaching. There are many students who would benefit from studying Further Mathematics or the AEA in Mathematics, but who are currently denied the opportunity. Candidates who have studied Further Mathematics or the AEA in Mathematics are likely to be much more confident with the inner workings of the subject. University departments in all subjects identified as vulnerable in the Roberts SET for Success report would benefit greatly if more candidates were qualified at this level. Further Mathematics and the Advanced Extension Award in Mathematics (redesigned if necessary) are the courses that could and should provide the extra stimulation for the top fifteen per cent or so of the A-level mathematics cohort of students and the Inquiry is deeply concerned that the current system is not able to make adequate provision for this important cohort. |
| Recommendation 4.10
The Inquiry recommends that there should be an immediate review by the DfES, LSC and the relevant devolved authorities of measures that could be taken to support and encourage current GCE course provision for the most able mathematics students. In particular, we believe there is a need to ensure that there are no funding disincentives in schools and colleges for providing access to Further Mathematics and the Advanced Extension Award in Mathematics We also believe that consideration should be given employing the same incentives as suggested in Recommendation 4.8. |
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| 4.39 | The higher education sector and the learned and professional societies have made clear to the Inquiry their serious concerns about the interface and transition between A-level mathematics and university courses heavily dependent on mathematics, such as degree courses in mathematics and statistics, or in physics, electronics, engineering and economics. In the shortterm, the Inquiry believes that Higher Education has little option but to accommodate to the students emerging from the current GCE process. Many are, of course, already doing this through, for example, the provision of first year enhancement courses. Longer term, we would hope that there would be significant changes resulting from Recommendations 4.5 and 4.10 and the future re-design of 14-19 pathways. More generally, we would hope that there would be significant positive consequences of the greater interaction of HE with schools and colleges proposed in Chapters 5 and 6. |
Concerns with Adult Numeracy |
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| 4.40 | There is some concern that employers are not yet fully recognising the
new Adult Numeracy qualifications. It has also been impressed on the Inquiry
that adults want to learn mathematics for a variety of reasons, often not
concerned with gaining qualifications. Respondents to the Inquiry have expressed
some concern that, at present, test questions tend to reflect traditional
“school mathematics”, in the sense of testing mathematical procedures
posed as contextualised problems with multiple choice answers. It is felt
that these tests do not necessarily fit well with the idea of individual
adult learner plans and properly exploit adult learners’ contexts. It
is also felt that the present tests at levels 1 and 2 disadvantage ESOL learners
and those with dyslexia or dyscalculia, or low levels of literacy. Many
respondents feel that:
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| 4.41 | Respondents to the Inquiry are clear that the adult numeracy strategy
is a challenging and demanding one for teachers and learners alike. Progress
could easily be undermined by:
In Chapter 6, we suggest that the national infrastructure for the support of the teaching of mathematics include specific support for teachers of adult numeracy. |
Possible Future Pathway Models for Mathematics 14-19 |
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| 4.42 | In conjunction with the Advisory Committee on Mathematics Education (ACME), the Inquiry ran a series of workshops attended by a wide range of stakeholders concerned with possible future mathematics 14-19 pathways. These workshops considered the ways in which mathematics is embedded in educational pathways in other countries and tried to stimulate initial constructive thinking about an appropriate future structure for 14-19 mathematics pathways in England. |
| 4.43 | As a result of these and other extensive consultations, the Inquiry believes
that the following principles should guide the construction of a future pathways
approach to mathematics provision 14-19 in the UK:
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| 4.44 | In addition to requiring adherence to these principles, respondents to
the Inquiry are clear that in developing pathways it will be essential to
be clear about the positioning in the pathways of the following key mathematical
developments:
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| 4.45 | Subsequent discussion will use ‘pathway’ to describe progression in mathematics, with the understanding that the mathematics is merely a component, along with other specialist and optional components, of the larger curriculum pathways envisaged by the 14–19 Working Group. Each pathway should be clear in what it offers as a core of mathematics and how it is applied, and it should also be an adequate preparation for the next stage of progression. Much work will be needed to develop the Tomlinson proposals into a coherent curriculum and assessment regime. The Inquiry has had neither the time nor the resource to attempt to begin to do this for the mathematics component of such a curriculum and assessment regime. |
| 4.46 | The approach we have adopted is therefore the following. We outline, on the basis of suggestions made during the consultation process, schematic versions of some of the different models and approaches put to us for consideration. These indicate, in broad-brush terms, possible future pathways that are guided by the principles summarised above and designed to remedy the perceived defects of the current structure detailed in this chapter. Each of these models and approaches has its supporters among one or other significant grouping of the mathematics community. |
| 4.47 | It would be inappropriate for the Inquiry to express a clear preference for one model or approach rather than another, although we are inclined to believe that Figure 4.2 below will provide something close to the desired pathway structure for mathematics. We believe that intensive curriculum development, trialling, feedback and modification will be essential to ensure that the new structure is workable and better than the system it is designed to replace. The construction of pathways depends on both curriculum and assessment considerations and future political imperatives. |
| 4.48 | A system based primarily on equity might seek to opt for a single pathway, at least to age 16 and possibly to age 19. In such a model, all students study precisely the same mathematics curriculum, but progress at different rates. Students are then credited for the mastery of the stage they have reached by the chosen age at which the pathway ends. Sweden has adopted essentially this approach. In the Swedish model (see Figure 4.1, which we present schematically, without discussion of the programme content) the mathematics curriculum can be thought of as blocks A,B,C,D,E fitting end to end and forming a continuum up to the standard required for entry to study mathematics at university (D, or D+E). Students learn at different rates, and are certified as successfully completing one or more of the fixed number of partitioned subsets A,B,C and D that make up the continuum. Only relatively few students master the whole curriculum. The majority leave secondary education having achieved a number of ‘stepping stone’ credits along the mathematical pathway. We note that support for this approach runs counter to the support the Inquiry has received for the extension rather than acceleration approach discussed earlier (see paragraph 4.21 and Recommendation 4.5). |
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| 4.49 | In contrast to the above schematic, Figure 4.2 presents a more detailed possible model of 14–19 pathways. This starts from the assumption that the present Key Stage 3 Programme of Study should form the common basis for all students, prior to the age of 14. It then maps out a number of possible routes through 14–19, five potential pathways from age 14, increasing to seven from age 16. Each pathway varies in content, difficulty and abstraction and is designed to enable students to follow the one best suited to their needs. The model emphasises relative speeds of progression and the nature of the levels of the mathematics components on different pathways. The model allows for movement between the pathways. |
| 4.50 | In Key Stages 4 and post–16, all courses shown in the figure have the title “Mathematics” followed by a code. The names associated with the codes are descriptive only. The model emphasises relative speeds of progression and the nature of the levels of the mathematics components along the different pathways. Mathematics in levels 1 and 2 of this qualifications framework would be drawn from the Key Stage 4 PoS, but with not all students expected to make equal progress. The way the intended curriculum is delivered and assessed might differ from pathway to pathway, with more emphasis on applications in some parts and more emphasis on abstract reasoning in others. In this model, students learn to tackle problems appropriate to their current level of mathematical understanding and motivation. The pathways are designed so that individual students would be able to maintain interest in the subject and to make steady and continuous progress as they move to age 18 or 19. Each student should be on a pathway that is accessible and provides meaningful challenges to the student at each stage. |
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| 4.51 | At level 3, students would elect to do mathematics as a minor or as a major subject, and perhaps some additional mathematics beyond that. Mathematical techniques, applications and mathematical reasoning would be developed through a continuum which allows some variation in the applications encountered and the way mathematics is used to model real problems. The aim would be for an increasing number of students to progress to both levels 2 and 3 by age 19. A small percentage of students might only progress to level 1. |
| 4.52 | The degree of mathematical content, difficulty and abstraction increases as one moves down the figure and along each pathway from left to right. All courses from level 1 upwards would develop calculation in a variety of contexts, and, as appropriate, would introduce aspects of algebra, geometry and application of mathematics in a varying mix for different student groups. Entry level would focus mainly on numbers and measures and simple applications. The Extension courses would be for those who absorb mathematics easily and seek a greater understanding of the subject. Students on these courses would study mathematics at greater depth and at greater levels of abstraction, but based on the same curriculum content at a given level. Extension courses would concentrate more on reasoning, proof, chains of logical reasoning, multi-step problem solving and a range of harder and sometimes open-ended problems. An example of an extension curriculum at level 2 is set out in Making Better Use of Mathematical Talent, published by the Mathematical Association, 2003. |
| 4.53 | Mathematics E at level 3 would be the nearest equivalent to the current GCE Further Mathematics and the AEA in Mathematics, although it would be a new hybrid with its own distinctive features. There was a very strong positive response to the Inquiry in favour of providing mathematics courses at this level. Mathematical Literacy at levels 2 and 3 would be the nearest equivalent to the higher tier end of GCSE Mathematics (KS4 Higher) and A-level Mathematics, respectively. These would concentrate on the study of a wide range of mathematical ideas, techniques and application, but not developing rigour or harder problem solving to the same extent as on the extension pathway. At level 2, both Mathematics ML and Mathematics E would be worth a double award in the qualifications framework (in line with Recommendation 4.2). Quantitative Literacy level 2 would be the nearest equivalent to working at around the current C grade level of GCSE Mathematics (KS4 Foundation), but with a greatly different emphasis. The course would also encompass level 1 as a fall back position. |
| 4.54 | Application of mathematics (involving number and algebra, measures and geometry) to analyse substantial real world contexts would be stressed, and appropriate ICT would be used to analyse realistic data and fit models. Students would also learn about multiplicative and proportional reasoning. They would also learn to communicate mathematical ideas to others. QL level 3 would develop this approach further, building on more mathematical content that goes beyond that currently in the Key Stage 4 Programme of Study. All QL courses would develop the philosophy and pedagogy pioneered by Free Standing Mathematics Qualifications and AS Use of Mathematics. It would be important here to make full use of the power of ICT to analyse real data using appropriate mathematical models. This is the sort of course designated as ‘Techno-Mathematics’ by the authors of the report Mathematical Skills in the Workplace. |
| 4.55 | Numerical Literacy courses would only go as far as level 1. They would aim to provide familiarity with the most basic ideas in number, measures, algebra and geometry and how these are used in elementary application and in making geometrical models and patterns. They would play the role of a stepping-stone to mathematical understanding that might begin to unlock doors in training or employment, and in further and higher education. At level 3, a mathematics course is proposed for all students progressing to level 3 from level 2. This would follow the pattern of the French Baccalaureate, in which there is mathematics provision on all the designated academic routes (the sciences, the social sciences and the humanities), and also on the vocational and technological routes. The possible course in Public Understanding of Mathematics (possibly to include Science and Technology) could provide a form of continuing exposure to mathematics for those with academic aspirations that do not include technical use of mathematics, but for whom society would wish – given that many will have influential and opinion forming roles in their future careers – to understand the role of mathematical ideas in human culture, the development of science and technology and as an instrument for social and economic change. |
| 4.56 | The level 3 Statistical Methods course would be akin to current AS Statistics and would be an appropriate pre-requisite for those intending to progress to courses in HE which are heavily statistical in nature. Many respondents to the Inquiry have indicated that such a course would fill an existing serious gap in the qualifications framework. The symbol (T) in figure 4.2 denotes that some transition material would have to be mastered to make the indicated transition from one pathway to another. Other transitions might be possible. Students might wish to make a transition after starting on a particular pathway, but then would have to realise that there could be a cost to making such a transition and that extra effort might be required to make the transition successfully. |
| 4.57 | We do not believe it would be desirable to indicate rigidly pre-determined
destinations for each of the pathways. However, in very broad-brush terms,
with considerable cross-over, we would see the following kinds of destinations
as corresponding to the pathways as we move down the figure:
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| 4.58 | There are a number of possible variants of the model. One of is shown in highly simplified schematic form in Figure 4.3, based around two fundamental courses Mathematics and Use of Mathematics. The key idea is that there is scope to develop both a level 1 and a level 2 ‘Use of Mathematics’ course to complement the level 3 AS ‘Use of Mathematics’ that exists at present. Starting from these, Figure 4.3 then shows the following common pathways: Mathematics L2 to Mathematics L3 (minor, major and beyond) or to Use of Mathematics L3; (Use of Mathematics L1 to) Use of Mathematics L2 to Use of Mathematics L3. |
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| 4.59 | A fourth possibility is to develop two or three distinct pathways from a notionally accepted common curriculum up to age 14. A version of this fourth model is shown below in Figure 4.4. This proposes three distinct programmes from age 14. These are referred to here descriptively as Entry-Vocational (EV), Vocational-Technical (VT) and Technical-Academic (TA). The shared letters indicate the desirability of allowing for subsequent movement; they do not necessarily indicate identical content. All the courses would progress from a common core of mathematics at Key Stage 3, which would act as an effective foundation for all students. However, those students who do not complete the whole of Key Stage 3 by age 14 would not be obliged to continue repeating the same material until it is mastered. |
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| 4.60 | None of the models or approaches presented here has any current validity or preferred status for the Inquiry. They are simply intended to show how actual mathematical pathways could be constructed in line with the principles enunciated earlier and with the aim of overcoming the perceived deficiencies of the current structure. We indicated earlier that a great deal of work will be needed to develop such ideas into a coherent curriculum and assessment regime that will provide appropriate mathematics pathways within the general structure that emerges from the work of the 14–19 Working Group. We understand that the final 14–19 Working Group proposals will be available in Autumn 2004. The Inquiry therefore does not believe it would make sense to try to select a preferred set of mathematics pathways and to work out every detail of the curriculum and assessment for such pathways ahead of understanding the Government’s response to the 14–19 Working Group proposals. |
| 4.61 | However, whatever emerges as a new 14–19 structure, the Inquiry is clear that we shall need to develop some or all of the elements and components of the models discussed above and to begin to address the major deficiencies identified in the current framework. We believe, therefore, that it is vital to begin work immediately on detailed further curriculum and assessment development based around these pathways models. The aim should be to carry out a cycle of trialling, feedback and modification of two or three variants of these models in time to inform a future decision on the preferred way forward for mathematics in the context of the overall 14–19 structure. We would suggest that this work should be completed by the end of 2007. |
| 4.62 | We also firmly believe that in this process of development it will be vitally important to involve as wide a range of the mathematics community as possible. We have been struck in the course of this Inquiry by the energy and commitment of the mathematics community in responding to issues raised. In particular, the outline models we have presented have emerged from significant groupings of the community. All this informs the following major recommendation. |
| Recommendation 4.11
The Inquiry recommends that funding be provided to the QCA and its regulatory partners to commission, through an open bidding process, up to three curriculum and assessment development studies of variants of these pathway models and approaches, including trialling, feedback and modification and an assessment of the workload implications. These studies should take on board developments arising from Recommendations 4.4, 4.5 and 4.7. The aim of this exercise will be to inform the selection of a preferred pathway model to form part of the reformed 14–19 structure in England and possible parallel developments in Wales and Northern Ireland. Given the importance of ensuring the widest possible involvement and commitment of the mathematics community to the outcome, the Inquiry recommends that the regulatory authorities work in partnership with ACME and mathematics community representatives from Wales and Northern Ireland, and that the DfES and relevant devolved authorities provide appropriate funding to support this. |
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