Changing the structure of AS/A levels in
Mathematics and Further Mathematics
An MEI discussion paper
1. Background
MEI is an independent curriculum body for mathematics and has developed one of the
current A level specifications for Mathematics and Further Mathematics, administered by
OCR. MEI has managed the national Further Mathematics Support Programme, and its
predecessors, since 2005.
Mathematics is unique, both through its importance for large numbers of students going on
to further study and employment in STEM and non-STEM fields, and in having a second
A level (Further Mathematics) which is dependent on the first A level (Mathematics).
MEI has written this discussion paper to contribute to national thinking about re-developing
A levels in Mathematics and Further Mathematics so that more students are better prepared
for the mathematical demands of further study and employment.
1.1
Forthcoming changes in the structure of A levels
New A levels in some subjects are to be developed for first teaching from September 2015.
At the time of writing this paper, it is intended the new A levels will have the following
attributes.
(a)
(b)
(c)
1.2
All the A level content should be assessed at the end of the course.
Assessment should include more open-ended and more synoptic questions.
AS levels continue to be half the content of an A level but the AS assessment
does not contribute to A level grades.
The current A levels in Mathematics and Further Mathematics
The current A levels in Mathematics and Further Mathematics each consist of six units. Four
of the A level Mathematics units are compulsory and, typically, so are two of the A level
Further Mathematics units. There is a choice for the remaining six units for the two A levels,
with some units in the applied mathematics strands able to count towards either
Mathematics or Further Mathematics (but not both).
The current structure has been in place since 2004, with the numbers of students taking
AS/A level Mathematics and AS/A level Further Mathematics increasing very significantly
year-on-year since then1. The current structure replaced the Curriculum 2000 A levels,
which had led to a catastrophic drop in numbers taking A level Mathematics2. This very
large drop illustrates the unfortunate consequences that can occur when a major reform,
1
Between 2004 and 2012 combined A level Mathematics and Further Mathematics numbers rose
from 58 508 to 98 937, a 69% increase and A level Further Mathematics numbers rose from 5 720 to
13 223, an increase of 131%. (JCQ)
2
Numbers prior to 2004 for A level Mathematics and Further Mathematics combined had been
reasonably stable for 10 years at around 70 000 until a disastrous drop in 2002 following the
Curriculum 2000 changes, when numbers fell by nearly 20% from 66 247 to 53940.
http://mathstore.ac.uk/headocs/7430_porkess_r_moremathcompgrads.pdf
1
though introduced with the best of intentions overall, is not sufficiently well thought through
for the effects it may have on particular subjects, especially mathematics.
In terms of increasing participation, the current A level structure for Mathematics and Further
Mathematics, supported by the Further Mathematics Support Programme, has been a major
success. This has been greatly appreciated by university STEM departments as significantly
more undergraduates have studied more mathematics in preparation for their degree
courses. However, there is still a long way to go before the supply of students with AS/A
level Mathematics and Further Mathematics qualifications reaches a level that would satisfy
the demand from universities and employers.3
A problem that has been identified with the current Mathematics A level is the high
proportion of students that achieve the top grades. About 45% of students achieve grades A
and A* in A level Mathematics4; this is a far higher proportion than for other mainstream
A level subjects. This lack of differentiation among the most able A level Mathematics
students makes it difficult for universities to select the best mathematicians. However, a
2012 DfE research report5 found that the impact of GCSE grade on progression to A level
study was highest for Mathematics, with high progression only from students with A and A*
grades at GCSE. The report also found that ‘Maths shows one of the largest differences
between GCSE A* and A grade pupils achieving an A* at A level’, indicating that it is already
one of the most demanding A levels; its grade profile reflects the cohort of students that
takes it, rather than its difficulty relative to other A levels. Making A level Mathematics more
demanding would enable greater differentiation at the top end, but if it becomes more difficult
to achieve the highest grades it is likely that fewer students will choose to study it.
Increasing the difficulty of A level Mathematics could also cause some schools and colleges
to encourage mathematically-able students to choose other subjects, where they are more
likely to succeed in achieving a top grade.
The current A levels were not designed with a view to all of the assessment taking place at
the end of the two-year course. For instance, a student taking A levels in Mathematics and
Further Mathematics currently takes twelve units, each with a 1.5 hour examination, spread
across two years. An examination load of this size, at the end of the second year of the
course, would be an extremely unattractive burden for students and would almost certainly
lead to a decrease in uptake of Further Mathematics. As well as reducing the level of
mathematics achieved by students progressing to university, this would reduce the ability of
universities to identify the most able students.
The number of students choosing to take Further Mathematics is especially sensitive
because, while taking it is in the interest of many students, Further Mathematics is not an
entry requirement for the vast majority of courses for which it is extremely beneficial. This is
because many university admissions tutors still lack the confidence to specify it in their offers
or recommend it in their prospectuses, fearing that it may deter students from applying. Any
changes that make Further Mathematics seem less attractive, and hence reduce the number
of students taking it, will make it less likely for universities to specify or recommend it. This
would reduce the incentive for students to choose it, resulting in a downward spiral of
demand and supply that would have a disastrous effect on uptake.
3
‘Mathematical Needs: Mathematics in the workplace and in higher education’ http://www.acmeuk.org/media/7624/acme_theme_a_final%20%282%29.pdf
4
JCQ figures
5
Subject progression from GCSE to AS Level and continuation to A Level, DfE 2012
2
1.3 The benefits of modularity
The MEI Structured Mathematics scheme was first introduced in 1990. The underlying
philosophy behind this first modular scheme was given in the introduction to the
specification.
‘Our decision to develop this structure, based on 45-hour Components, for the study of
Mathematics beyond GCSE stems from our conviction, as practising teachers, that it will
better meet the needs of our students. We believe its introduction will result in more people
taking the subject at both A and AS, and that the use of a greater variety of assessment
techniques will allow content to be taught and learnt more appropriately with due emphasis
given to the processes involved.’6
We remain convinced that a well-designed modular course has the following benefits and we
would like to see as many as possible of these benefits preserved under the new structure.




Encouraging uptake of AS and A level Mathematics and Further Mathematics due to
the flexibility of the course for students – they can stop after AS or continue further
after receiving feedback on progress so far.
Enabling appropriate assessment for each module.
Facilitating innovation through the development of new units, responding to changes
in the way mathematics is used in higher education and the workplace.
Allowing centres some flexibility in the organisation of teaching.
2. Optimising change in Mathematics A levels
Changing AS/A level Mathematics and Further Mathematics could and should present a real
opportunity to improve them, making them a better preparation for higher education and
employment by developing more valid assessments and embedding the use of digital
technology into teaching and learning. However, if changes are rushed, the unintended
consequences could be very serious, putting at risk the dramatic gains in uptake of the past
eight years.
The factors involved and a possible timetable for bringing about effective change are
discussed in this section.
2.1
Risk to uptake from rapid structural changes
The structure of Mathematics A levels would need to change significantly to meet
requirements (a), (b) and (c) for the new A levels, described in 1.1 above.
The structural changes needed should be considered very carefully, otherwise there is real
potential for disaster, as happened with Curriculum 2000, which resulted in a 20% fall in A
level Mathematics numbers2,7.
To allow new A levels in Mathematics and Further Mathematics to be available for teaching
from September 2015, they would need to be submitted to Ofqual for accreditation by May
2014. This would require many fundamental structural decisions to be agreed in the next
few weeks. The risks of attempting to make significant structural changes to Mathematics A
6
MEI Structured Mathematics specification
A world-class mathematics education for all our young people, Vorderman et al, 2011
http://www.conservatives.com/news/news_stories/2011/08/~/media/files/downloadable%20files/vorder
man%20maths%20report.ashx
7
3
levels on this timescale are simply too great. Changes should be delayed to avoid
jeopardising recent healthy gains, and to allow time for effective development.
2.2
Missed opportunities that would result from changes in 2015
There is a danger that rapid change in the short-term would mean missing opportunities to
make major improvements to A level Mathematics in the medium-term.
Major changes to Mathematics A levels will require new teaching and learning resources. If
changes are made in 2015, it is inevitable that the production of new resources will be
hurried, with no time for them to be informed by developments such as the Cambridge
University Mathematics Education Project (CMEP). Rushed development of textbooks and
other resources compromises their quality. The need for new teaching resources would also
necessitate significant investment, so that schools and colleges would not want to make
further investment in resources for several years.
The sequential nature of mathematics means that AS/A level Mathematics must build on
what happens at GCSE, but new Mathematics GCSEs will not be in place in time to inform
new A level Mathematics specifications for 2015.
The welcome intention to greatly increase participation in the study of mathematics post-16,
in line with Michael Gove’s ambition that “….we should set a new goal for the education
system so that within a decade the vast majority of pupils are studying maths right through to
the age of 18.”8 will require both increased participation in AS/A level Mathematics, and the
development of new level 3 mathematics pathways for those students who have achieved a
grade C or higher at GCSE but for whom AS/A level Mathematics is not the right option. To
ensure a coherent level 3 mathematics provision to meet the needs of all students, new A
levels for Mathematics and Further Mathematics should be developed in parallel with
qualifications for the new pathways.
2.3
The assessment of AS/A level Mathematics
The 2012 IPSOS Mori research for Ofqual indicated that there is little dissatisfaction from
higher education with the current content of Mathematics A level, but there are concerns
about its assessment.
“Another recurring message from those at higher education lecturing in Mathematics, and
Mathematics-based STEM subjects such as Physics and Engineering, was that the
Mathematics A level has generally the right content but students have not been given the
time to gain proficiency in using mathematical tools….. It was felt that the current
assessment system was partly to blame for this situation as students learned to apply a
technique in only the limited cases which had been demonstrated to them, and which they
knew tended to come up in assessment, rather than across a broad range of problems which
requires a more discerning approach and requires students to really understand the reasons
behind using a certain mathematical approach.”9
Ofqual’s review of standards over time in A level Mathematics has shown that Assessment
Objectives are not interpreted consistently between awarding bodies and that there is
insufficient emphasis on mathematical reasoning (Assessment Objective 2).
8
Michael Gove, Secretary of State for education, Royal Society, June 2011
Fit for Purpose? The view of the higher education sector, teachers and employers on the suitability
of A levels (April 2012)
9
4
“Many of the pure papers in both 2004 and 2007 contained too great a proportion of highly
structured questions, with very few unstructured questions requiring the construction of
extended arguments. It was therefore difficult to see how the minimum requirement of 30 per
cent of the overall marks could be allocated to AO2. This issue remained the same between
2004 and 2007 and so cannot be said to affect demand over time. Indeed, the same issue
was raised in the reports on the previous standards reviews in GCE mathematics.
Reviewers also judged that the coverage of AO2 was not consistent between awarding
bodies and that it affected the demand across awarding bodies in 2004 and 2007.”10
It is also the case that Assessment Objective 4, on comprehension, is widely ignored in the
current specifications. It includes ‘read critically and comprehend longer mathematical
arguments or examples of applications’. The MEI specification is the only current
specification to address this, even though mathematical comprehension skills are strongly
valued by higher education. MEI has shown this objective can be assessed effectively and it
should be addressed by all specifications.
The new Assessment Objectives being used for the current GCSE Mathematics have
resulted in improvements to its assessment. Revised Assessment Objectives for A level
Mathematics, which are consistently interpreted by examiners and enforced by Ofqual, could
result in improvement in A level assessment.
In order for new Assessment Objectives to result in improved teaching and more valid
assessment, they must be interpreted in the same way by all awarding bodies and changes
to the current styles of assessment must be communicated clearly and exemplified for the
benefit of teachers.
Serious discussion is also required about the place of coursework, especially in numerical
and statistical analysis involving the use of digital technology.
2.4
Updating A level Mathematics to reflect the impact of digital technology
The current AS/A level Mathematics and Further Mathematics courses do not reflect the
increasing use of digital technology to apply mathematical and statistical techniques in
higher education and industry. This means the current courses are scarcely fit for purpose
now, and certainly will not be in the medium or longer term. Curriculum development to
embed technology properly into A level Mathematics and Further Mathematics cannot be
done in time for 2015, but could be introduced subsequently.
2.5
Demand from universities
Universities must be honest about the mathematics qualifications they would like students to
have before embarking on different degree courses, and their offers must reflect this. The
experience of the Further Mathematics Support Programme has shown that demand from
universities is a powerful lever on students’ choices, and on the behaviour of schools and
colleges. If A level Mathematics and Further Mathematics courses become more
demanding at the highest grades, so that they differentiate better between the most able
students, demand from universities can help ensure that students do not decide against
choosing them.
10
Review of standards in GCE mathematics in 2004 and 2007 (March 2009)
5
2.6
Suggested timetable
To make the change to new AS/A levels in Mathematics and Further Mathematics effective,
while minimising the chance of disastrous unintended consequences, we suggest that, for
the reasons explained in section 2.1, changes should be delayed until 2016.

Limited changes could be introduced in 2016, to address items (a), (b) and (c) in
section 1.1; two possible models for how this might be achieved are set out in
appendix 1.

Changes to address section 2.2 above, and to integrate the use of technology into
the curriculum and further improve the validity of assessment, could then be
introduced on an on-going basis. This would enable AS/A level Mathematics and
Further Mathematics to evolve over time, with changes being properly trialled and
piloted. A good example of how this can work is the new ‘Further Pure Mathematics
with Technology’11 unit, which was developed for first teaching from September 2012
within the current OCR(MEI) A level Mathematics specification.
3. Conclusion
The existing A levels in Mathematics and Further Mathematics could and should be
improved, to increase the validity of assessment and to differentiate more effectively
between the most able students. There is no straightforward way to amend AS/A levels in
Mathematics, and especially in Further Mathematics, to meet the requirements for new
A levels, as set out in section 1. MEI believes that, because of the potential for very serious
unintended consequences, which could result in fewer students choosing to study
Mathematics and Further Mathematics, and to ensure best use is made of the opportunity to
improve the qualifications, as explained in sections 2.2, 2,3 and 2.4, changes should be
delayed until 2016. In the longer term, curriculum development in A level Mathematics and
Further Mathematics should become an on-going process, so that the curriculum can evolve
to reflect changes in the needs of universities and employers, and developments in
technology.
11
See http://www.mei.org.uk/?section=teachers&page=fpt for details
6
Appendix 1
Possible models for new A levels in Mathematics and Further Mathematics
In this section two possible ‘limited change’ models for revised A levels for teaching from
2016 are considered. Both of these models allow end of course assessment and also retain
the possibility of some applied components counting towards either Mathematics or Further
Mathematics (but not both), which is greatly valued by students and teachers. Even these
limited changes would require careful consideration and consultation to identify their possible
unintended consequences.
There is no way of knowing how many students will choose to take AS at the end of year 12
when it no longer contributes to A level grades. Allowing a compensatory AS to students
who do not achieve an A level grade, but who show sufficient grasp of half the content to
have obtained an AS if they had entered for one, could go some way to encouraging
students to start on, or continue with A level Mathematics and Further Mathematics.
1.
Combined pure papers for A level Mathematics and A level Further Mathematics
Possible scheme of assessment
A level Mathematics
A level Further Mathematics
 3 hour paper covering current C1-4
 2½ hour paper covering two current
content
Further Pure units or an agreed core
 Two 1½ hour applied papers (as now)
 Four optional units (as now) each with a
1½ hour paper
AS level Mathematics
AS level Further Mathematics
 2 hour paper covering current C1and 2
 The same structure as now with three 1½
content
hour papers including FP1
 One 1½ hour applied paper (as now)
Advantages
Disadvantages
 No change in content for A level
 Some awarding bodies currently allow a
Mathematics.
choice of second Further Pure unit –
some centres may have to change
 Most centres will experience little or no
content for Further Pure.
change in content for A level Further
 Further Mathematicians sit eight papers
Mathematics.
at the end of the course and so are likely
 Allows greater synoptic assessment of
to experience exam clashes.
pure content in A level Mathematics and
Further Mathematics.
 Total assessment for Further
Mathematics A level is 8½ hours; this is
 AS can be taught in year 12, as now, for
likely to be significantly more than other
both Mathematics and Further
A levels.
Mathematics. Students can stop after
one year but still get a qualification.
 The 3 hour paper for A level Mathematics
covers two thirds of the content and
 If there were a common core for Further
counts for two thirds of the marks, but
Mathematics, universities would know
has half the total examining time – will
that all students with A level Further
this have any impact on overall results?
Mathematics have studied the same
‘core’ material.
Reducing the length of the optional papers to one hour each would reduce total examining
time and improve balance between pure and applied options. However, this would make it
more difficult to incorporate open-ended questions in these assessments.
There could be an agreed common core for the 2½ hour Further Mathematics paper, or the
core for each of the awarding bodies could be based on their current units.
7
2.
Increase the compulsory content of A level Further Mathematics
Suggested scheme of assessment
A level Mathematics
A level Further Mathematics
 3 hour paper covering current C1-4
 3 hour paper covering three current
content
Further Pure units or an agreed core
(half the A level content)
 Two 1½ hour applied papers (as now)
 Three optional units each with a 1½ hour
paper
AS level Mathematics
AS level Further Mathematics
 2 hour paper covering current C1and 2
 The same structure as now with three 1½
content
hour papers including FP1
 One 1½ hour applied paper (as now)
Advantages
 No change in content for A level
Mathematics.
 If there were a common core for Further
Mathematics, universities would know
that all students with A level Further
Mathematics have studied the same
‘core’ material.
 Allows greater synoptic assessment of
pure content in A level Mathematics and
Further Mathematics.
 Can still teach AS Mathematics and AS
Further Mathematics in year 12 (as
model 1).
 Examination time for Further
Mathematics is reduced compared with
model 1.
Disadvantages
 Choice within A level Further
Mathematics would be reduced (many
students and teachers value this choice).
 There is less opportunity for interchange
of units between Mathematics and
Further Mathematics, which will make it
harder for centres to offer Further
Mathematics.
 The 3 hour Further Mathematics paper
covers half the content but has less than
half the examining time.
 The 3 hour paper for A level Mathematics
covers two thirds of the content and
counts for two thirds of the marks, but
has half the total examining time.
 Some of the optional units currently
taken by small numbers of students
would not survive (including higher
mechanics and statistics units).
 Centres will have to change what they
teach for A level Further Mathematics,
which may deter them from offering it.
Reducing the length of the optional papers to one hour each would reduce total examining
time and improve balance between pure and applied options. However, this would make it
more difficult to incorporate open-ended questions in these assessments.
There could either be an agreed common core for the 3 hour Further Mathematics paper, or
the core for each of the awarding bodies could be based on their current units.
8