This chapter provides a summary of the full report, highlighting policy insights and pointers for the provision of mathematics in upper secondary education in England (United Kingdom). The chapter summarises analysis from the OECD Programme for International Student Assessment (PISA) and the Survey of Adult Skills to look at mathematics skills of young people during and at the end of upper secondary education. It also explores perceptions of the subject among students, and influences from the labour-market and tertiary education on demand for mathematics skills. It summarises key findings about how mathematics is provided in upper secondary education across a selection of focus countries.
Mathematics for Life and Work
A Comparative Perspective on Mathematics to Inform Upper Secondary Reform in England
Abstract
Mathematics skills shape individuals’ outcomes throughout their lives – influencing the jobs that they can access, management of their household finances and even their health and overall well-being. Low numeracy skills are associated with an increase in an individual’s risk of being unemployed or inactive by around 7%, while high numeracy skills are associated with an average hourly income 13% greater than low numeracy skills. Mathematics shapes financial literacy which influences decisions that carry major consequences for individuals’ lives such as calculating mortgage repayments, tax returns and pension investments. In the United Kingdom, decisions linked to poor numeracy skills have been estimated to cost the average individual GBP 460 annually (Rose Martin et al., 2014[1]).
The influence of mathematics extends to the quality of individuals’ decision making and their health outcomes. Mathematical skills have been found to predict errors of judgement in decisions linked to analysing probability, with more numerate individuals being less swayed by how information is presented or misleading reasoning (Liberali et al., 2012[2]). People with strong numeracy skills are more likely to take strategic, rationally defined risks (Pachur and Galesic, 2013[3]; Jasper et al., 2013[4]). According to data from the OECD’s Survey of Adult Skills (PIAAC), adults with high numeracy are 65% less likely to report poor health than those with low numeracy (proficiency at Level 1 or below) (compared to 30% for literacy skills) (Jonas, 2018[5]).
Given the wide range of ways in which mathematical literacy shapes individuals’ outcomes, the cumulative impact of low maths skills across an economy is likely to be significant. A study from the United Kingdom estimated that in 2012, low levels of adult numeracy cost the country the equivalent of around GBP 20.2 billion or 1.3% of the country’s Gross Domestic Product (GDP) every year. On average across OECD countries, statistical estimates suggest that skills have a powerful effect on growth. Based on historical trends, improvements in mathematics and science performance of 15-year-olds in the Programme for International Student Assessment (PISA) are estimated to be associated with an increase in annual GDP per capita growth rates (OECD, 2010[6]).
In England (United Kingdom), around one in six young people choose to continue studying mathematics until 18. Data from 2012-17 PIAAC shows that the country’s young adults have low numeracy skills (OECD, 2019[7]). In a national review of mathematics achievement in the United Kingdom, the report’s author Sir Peter Williams, the then Chancellor of Leicester University, cited the cultural challenges around the subject in the country:
The United Kingdom remains one of the few advanced nations where it is socially acceptable – fashionable, even – to profess an inability to cope with mathematics. (Department for children, schools and families, 2008[8])
Concerns about the country’s engagement and performance in mathematics have been the concern of successive governments. In 2015, GCSE mathematics – a national qualification at 16 – was revised to increase the level of demand1 and a new mathematics programme for 16-18-year-olds, Core Maths, was introduced to boost achievement and participation at the end of schooling (Department for Education, 2013[9])2. Yet in 2023, less than 2% of young people achieved the new Core Maths qualification (Department of Education, 2024[10]). In October 2023, the then Government (the 2022 to 2024 Sunak Conservative Government) announced plans to make mathematics compulsory up to 18, as part of a policy to tackle the country’s “anti-maths mindset” (GOV.UK, 2023[11]).
As part of the policy response to transform how maths is viewed and experienced in England, and related achievements, the country requested this OECD report. The report explores how education systems internationally deliver mathematics in upper secondary education, and outcomes such as the share of students studying maths and performance. It explores several factors that shape how, when and which students study maths across upper secondary internationally, such as:
The requirements that different systems set for the study of mathematics in upper secondary education.
How systems cater to the needs of different students and their future ambitions in the provision of maths.
Similarities and variations in the level of demand, depth and breadth of mathematics across different systems.
How students and their parents across different societies view mathematics and how this might influence patterns of engagement with maths and performance.
The demand for mathematics in the labour market and tertiary education across systems.
Box 1.1 sets out the methodology and data used in this report. This chapter brings together the key insights and policy pointers for England from across the full report.
This report was produced by the OECD Above and Beyond: Transitions in Upper Secondary Education project at the request of England (United Kingdom) to look at how education systems internationally deliver mathematics education in upper secondary education.
What is upper secondary education?
Upper secondary education refers to ISCED 3, in the International Standard Classification of Education (ISCED). Some of the defining features of upper secondary include the increasing range of options and differentiation in content students can engage with, and the preparation it provides for individuals to either enter work or tertiary education (UNESCO Institute for Statistics, n.d.[12]).
Six focus systems
The report draws on data and information from across all OECD countries, yet focuses on the mathematics curricula, programmes and outcomes in six systems, as well as England (United Kingdom): Austria, British Columbia (Canada), Denmark, Ireland, New Zealand and Singapore. These systems were chosen because they all achieve comparatively high performance in mathematics yet are also diverse upper secondary systems and reflect a wide range of cultural contexts.
Data
The report draws on a rich pool of data sources, from the OECD and beyond, notably countries’ national data. The OECD surveys - the Programme for International Student Assessment (PISA) and the Survey of Adult Skills (PIAAC) - are used extensively to analyse performance and perceptions towards mathematics of 15-year-old students and young adults.
The starting point for this work was the popular perception that in England it is ok not to be good at mathematics. This perception has been captured in research, which suggests that comparative to other countries, parents in the United Kingdom attach less importance to their children’s achievements in mathematics. Statements like, “Don’t worry dear, I could never understand mathematics at school either”, are suggested to be more commonplace than elsewhere (Cockcroft, 1982, p. 62[13]). Research also suggests that the view that it is acceptable to perform poorly in mathematics is prevalent in English‑speaking countries more generally (Sam and Ernest, 2000[14]). Some studies have established a clear contrast in level of expectations between Chinese parents, and parents in Canada and the United States (Cai, 2003[15]; Cao, Bishop and Forgasz, 2007[16]).
Mathematics is not a relative weakness in England
Contrary to the perception that it is ok not to be good at mathematics in England, the country’s young people do not perform, on average, lower in mathematics compared to reading comparative to other countries. In Grade 4 (around 9.5 years), England's students have an average score of 556, the third highest among OECD countries. In Grade 8 (around 13.5 years), the performance of English students falls slightly (515) yet remains above the average of other OECD countries such as Finland, Norway, and Portugal (IEA, 2019[17]).
England’s data on individuals’ performance at 15 in PISA, and among 16-24-year-olds in PIAAC shows that individuals’ performance in reading and literacy and maths and numeracy were comparatively similar, relative to other countries, in 2022 and 2012 (Table 1.1 and Table 1.2).
OECD countries and Singapore
Above OECD average
Not significantly different from the OECD average
Below OECD average
Note: Maths and Reading rankings order OECD countries and jurisdictions under analysis, as well as Singapore, according to their PISA scores. The countries in bold are the comparator countries in this study. Scores across the two domains must not be directly compared, but only in relative terms, i.e., by looking at the ranking or the score difference with the overall average for the respective domain.
Source: OECD (2023[18]), PISA 2022 Online Education Database, https://www.oecd.org/en/data/datasets/pisa-2022-database.html (accessed on 23 January, 2024).
OECD countries and Singapore
Above OECD average
Not significantly different from the OECD average
Below OECD average
Note: Numeracy and Literacy rankings order OECD countries and jurisdictions under analysis, as well as Singapore, according to their PIAAC scores. The countries in bold are the comparator countries in this study. Scores across the two domains must not be directly compared, but only in relative terms, i.e., by looking at the ranking or the score difference with the overall average for the respective domain.
Source: OECD, (2012, 2015, 2018[19]) Survey of Adult Skills (PIAAC), https://www.oecd.org/skills/piaac/data/ (accessed on 20 September, 2023).
Across PISA and PIAAC data, maths and numeracy tend to be a strength in systems with strong upper secondary vocational systems, such as Austria, the Netherlands and Switzerland. This might suggest that the emphasis on technical skills and their widespread integration across vocational education helps to promote maths achievement in these systems.
Low numeracy scores among England’s young people in 2012 were likely to be driven by low participation in, and completion of, upper secondary education
In England, completing upper secondary education has one of the strongest associations with numeracy proficiency for young adults (16-24), increasing numeracy scores by 48 points on average (Figure 1.1). However, in 2012, the country’s young people had some of the lowest numeracy scores. This is explained by the comparatively high share of young adults who had not completed upper secondary education nor were enrolled in. In 2012, the share of 16-24-year-olds not in education and not having achieved at least upper secondary education in England (14.7%) was higher than the OECD average (10.7%) and was more than the double the share in Canada (5.9%), Ireland (5.6%), Poland (4.9%) and Singapore (1.1%) (OECD, 2012, 2015, 2018[19]).
Mean difference in numeracy proficiency between those attained at most upper secondary education and those who attained below upper secondary education in OECD countries and Singapore
Note: Chile, Greece, Israel, Lithuania, New Zealand, Singapore, Slovenia, and Türkiye have 2015 data (Round 2). Hungary, Mexico, and the United States have 2017 data (Round 3); For the cohort of 16-24 years-old, the values reflect the difference between those in education or with at least upper secondary and those not in education and lower upper secondary. For the cohort of 16-24 years-old, there is no data concerning Korea and Singapore; Missing OECD average. Data is only reported for Canada as whole rather than British Columbia but data for specific provinces is not available in the PIAAC data.
Source: OECD. (2012, 2015, 2018[19]), Survey of Adult Skills (PIAAC) (2012, 2015, 2018) https://www.oecd.org/skills/piaac/data/ (accessed on 26 June 2024).
At 15, students in the United Kingdom have more positive attitudes towards maths than their peers across the OECD on average
Data about how 15-year-olds and their parents in the United Kingdom view mathematics contradicts perceptions that mathematics is not liked or viewed as particularly important in the country.
Over two in five students in the United Kingdom say that mathematics is one of their favourite subjects
In 2022, 43% of 15-year-olds in the United Kingdom reported that mathematics was their favourite subject, above the OECD average (39%) (Figure 1.2) (OECD, 2023[18]).
OECD countries and Singapore
Note: Countries are ordered by the percentage of respondents who answered "agree" or "strongly agree," in ascending order.
Source: OECD (2023[18]), PISA 2022 Online Education Database, https://www.oecd.org/en/data/datasets/pisa-2022-database.html (accessed on 26 June 2024).
15-year-olds in the United Kingdom feel well prepared to tackle maths problems and challenges
In 2022 in the United Kingdom, 15-years-olds on average expressed positive attitudes towards mathematics with higher self-efficacy and lower anxiety than across the OECD on average (Figure 1.3) (OECD, 2023[18]).
OECD countries and Singapore
Note: Mathematics self-efficacy: Students' belief in their ability to handle mathematical tasks effectively and overcome difficulties; Mathematics Anxiety: Encompasses students' thoughts and feelings about themselves in relation to mathematics, including feelings of helplessness and stress when dealing with mathematical tasks.
Source: OECD (2023[18]), PISA 2022 Online Education Database, https://www.oecd.org/en/data/datasets/pisa-2022-database.html (accessed on 26 June 2024).
Many 15-year-olds in the United Kingdom perceive maths to be important for their future
Across OECD countries, the United Kingdom stands out with the third highest level of student instrumental motivation for studying mathematics (OECD, 2013[20]). Instrumental motivation to learn mathematics is related to students’ drive to learn the subject based on their perception that it is useful for them and their future studies and careers (see (Eccles, 2002[21]; Miller and Brickman, 2004[22])). Similarly, many parents in the United Kingdom also believe mathematics to be important. In 2012, 15-year-olds in the United Kingdom were among those with the highest share of parents who believed that studying mathematics is important (between 94% and 95% of parents) (OECD, 2013[23]).
Few countries demonstrate an overwhelming “culture of maths”
As well as suggesting that students and their parents in the United Kingdom have comparatively more positive views towards mathematics, data also seems to suggest that no country has an overwhelming predisposition or culture of liking maths (perhaps with the exception of Singapore). Even in countries with very high performance and high participation, students report low enjoyment and/or negative self-beliefs towards mathematics such as Austria, Japan and Korea (Table 1.3).
This seems to suggest that, as England looks to improve participation and performance in maths at 16-18, there is no intrinsic cultural reason why this should be more difficult or challenging that in other systems. Although of course historical patterns of participation in maths influence perceptions.
↑ Improved since PISA 2012
↓ Worsened since PISA 2012
Above OECD average
Below OECD average
Note: Green indicates the country performs better than the OECD average; Pink indicates the country performs worse than the OECD average. The arrows indicate the evolution since PISA 2012, where comparable data exists.
Source: OECD (2013[20]), PISA 2012 Online Education Database, https://www.oecd.org/en/data/datasets/pisa-2012-database.html (accessed on 20 September, 2023); OECD (2023[18]), PISA 2022 Online Education Database, https://www.oecd.org/en/data/datasets/pisa-2022-database.html (accessed on 23 January, 2024).
High enjoyment and positive views towards maths might suggest latent demand at 16-18 in England
Students in the United Kingdom report comparatively high enjoyment, self-efficacy and low anxiety related to mathematics (Table 1.3). Across OECD countries, students in United Kingdom report greater maths enjoyment than in Ireland and the Netherlands, higher self-efficacy than in Japan, Korea and Estonia and lower anxiety than in Canada, Japan, or Singapore (OECD, 2013[20]; OECD, 2023[18]). Despite the positive views that young people have towards mathematics at 15, participation in mathematics post-16 in England is much lower than in many countries where students have more negative views towards the subject at 15. In 2021, only 16.5% of upper secondary students were doing a post-16 or Level 3 mathematics programme in England (Department for Education, 2023[24]).
One of the factors affecting participation in mathematics at 16-18 in England is likely to be the limited range of options and levels from an internationally comparative perspective (see Policy Insight 3). The main option for 16-year-olds to continue studying the subject is mathematics A level. Yet, the international mathematics review undertaken for this report found that the breadth and depth covered by A level maths in England is very extensive (see Chapter 6). Young people entering A level mathematics typically have very high prior achievement in mathematics. In 2015, England introduced Core Maths to meet the needs of a wider range of young people who wish to continue studying maths until 18 but participation remains low. In 2023, just 1.9% of 19-year-olds achieving a post-16 or Level 3 qualification achieved Core Maths and it represented just 12% of post-16 or Level 3 maths qualifications (see Policy Insight 3).
In contrast, many of focus systems discussed in this report, notably British Columbia (Canada), Denmark, Ireland, New Zealand, and Singapore provide a far wider range of different mathematics levels and options to serve varied interests, needs and future aspirations among students. Importantly, the diverse range of options to continue engaging with mathematics until the end of upper secondary creates a perception and expectation that the subject is for everyone. In these systems, mathematics is not compulsory for the duration of upper secondary education for all students, yet participation rates are high, with around at least half of young people continuing to study maths until the end of upper secondary, and sometimes almost all students choosing to do so.
Communicate broadly, that students in England, especially up to 16, and consequently the education system, perform well in mathematics from a comparative perspective.
This might help to address societal views that mathematics skills in England are low and that this is acceptable.
Explore ways to ensure that students post-16 can continue to build on their positive beliefs and engagement in mathematics to boost participation at 16-18 (see Policy Pointers for England 3).
Given the comparatively low share of 16-18year-olds studying mathematics in England, this report sought to explore the policies and practices that other systems employ to promote high participation in the subject. As well as looking at policies around the subject, the report also looked at the drivers to study mathematics, notably tertiary selection.
In many countries, all or nearly all students, study mathematics until the end of upper secondary education
The high participation in mathematics across the focus countries for the duration of upper secondary education contrasts with the situation in England (Table 1.4). In England, in 2020/21 the share of students studying for a post-16 qualification in mathematics was just 16.5% (Department for Education, 2023[24]). In 2018/19, a third (29.8%) of students studying mathematics at 16-18 were those who did not at least a Grade 4 in GCSE mathematics and are required to continue studying the subject to re-sit their GCSE mathematics (Department for Education, 2023[25]).
Note: 1 England data are for 2018/19; 2 In England, since 2014/15, most pupils are required to continue to study mathematics at post-16 as part of their study programme if they have not achieved a Grade 4 or above in GCSE mathematics (or an equivalent qualification). 3 The share of young people studying Level 3 programmes in England’s Qualification Framework. Level 3 includes programmes which correspond to the end of upper secondary education. This number does not include young people re-taking GCSE (GSCEs are a Level 2 programme). 4 In British Columba, the share of the cohort studying mathematics at the end of upper secondary education is not available. However, there is data on the number of enrolments on different mathematics options in Year 12 (see Chapter 5). 5 Data for New Zealand shows the share of students assessed in 14 or more credits in mathematics at Level 3 NCEA in 2021.
Source: Statistics Austria: (2024[26]), https://www.statistik.at/en/statistics/population-and-society/education (accessed on 07 February 2024); Denmark Statistics (2024[27]), https://www.dst.dk/da/Statistik/emner/uddannelse-og-forskning/fuldtidsuddannelser/ungdomsuddannelser (accessed on 07 February 2024); Ireland State Examination Statistics (2023[28]), https://www.examinations.ie/statistics/?l=en&mc=st&sc=r19 (accessed on 07 February 2024); NZQA (2023[29]); Department for Education (2023[25])
High participation in mathematics can be achieved in the absence of compulsory requirements
Given the unequivocal importance of mathematics for individuals and the economies in which they live, all the focus countries, as well as England, set some formal requirements for young people to undertake some form of mathematics in upper secondary education. Yet among the focus countries, only Austria requires the full student cohort i.e. both general and vocational students, to study mathematics for the full duration of upper secondary education. In most other cases, students may be able to stop mathematics before the end of upper secondary education, or the requirement to study maths throughout upper secondary education only applies to specific programmes in either general or vocational education (Table 1.5).
Source: Federal Ministry of Education, Science and Research of Austria (2024[30]), https://www.bmbwf.gv.at/Themen/schule/schulsystem/sa/ahs.html (accessed on 21 January 2024); Federal Ministry of Education, Science and Research of Austria (2024[31]), https://www.bmbwf.gv.at/Themen/schule/schulsystem/sa/bmhs.html (accessed on 21 January 2024); British Columbia Government (2023[32]), https://curriculum.gov.bc.ca/provincial/grade-10-numeracy-assessment (accessed on 21 January 2024); Denmark Ministry of Children and Education (2024[33]), https://eng.uvm.dk/upper-secondary-education/national-upper-secondary-education-programmes (accessed on 21 January 2024); Denmark Ministry of Children and Education (2024[34]), https://eng.uvm.dk/upper-secondary-education/vocational-education-and-training-in-denmark (accessed on 21 January 2024); Department for Education (2024[35]), https://educationhub.blog.gov.uk/2023/08/24/gcse-results-day-what-to-do-if-you-didnt-get-the-grades-you-were-expecting (accessed on 26 September 2024); National Council for Curriculum and Assessment (2024[36]), https://curriculumonline.ie/senior-cycle (accessed on 21 January 2024); NZQA (2023[37]), https://www.nzqa.govt.nz/ncea/subjects/literacy-and-numeracy/level-1-requirements (accessed on 21 January 2024); Singapore Ministry of Education (2024[38]), https://www.moe.gov.sg/secondary/schools-offering-full-sbb (accessed on 21 January 2024); Singapore Ministry of Education (2024[39]), https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses (accessed on 21 January 2024).
Mathematics is not required for students in general upper secondary education in Ireland and Singapore but in practice almost all students take it
Some systems achieve very high levels of participation despite mathematics not being compulsory. In Ireland, while the country’s upper secondary students taking the Leaving Certificate are technically not required to study mathematics, in practice, the vast majority – just over 98% in 2019, 2020 and 2021 – do. In Singapore, students enrolled in the A-Levels are also not technically required to take maths, although in practice, almost all do (Table 1.4). In both systems, tertiary education is an important influence on take-up of mathematics. In Ireland, most tertiary institutions require mathematics as part of the Leaving Certificate for entry (Institute Of Guidance Counsellors, 2017[40]). Similarly in Singapore, mathematics A-Level is required for a broad range of courses from business and accountancy to computer science and engineering, sciences and of course mathematics in tertiary education.
Maths is typically required for tertiary entry, except in England
In most countries across the OECD, selection into tertiary education requires, or is informed by, upper secondary certification (OECD, 2019[41]). In countries where mathematics is not compulsory until the end of upper secondary – such as Ireland, and Singapore, almost all tertiary programmes include some form of mathematics in their entry requirements. This contributes to nearly universal take-up of mathematics in both countries, and even widespread perception in Ireland that mathematics is compulsory. In contrast in England, achievement in mathematics is only required for tertiary programmes with high mathematical content such as mathematics, physics or engineering. As well as influencing the drivers for upper secondary maths, the narrow targeting of maths for tertiary entry to small group of subjects might contribute to, and perpetuate, the perception that maths is the preserve of a minority of talented mathematicians, rather than an important skill for everyone in society.
The competitive nature of tertiary selection in England and the narrow structure of upper secondary certification creates a disincentive for students to take mathematics
In England, students typically take just three or four A-levels at 16-18 meaning that each subject mark carries greater weight than individual subjects in other systems where students typically take 6-9 subjects in upper secondary education (Stronati, 2023[42]). For students, achieving a lower grade in one subject could result in missing a place on their desired tertiary course. In this context, students tend to choose the subjects where they are likely to achieve the highest marks. While A level students who achieve the highest grades in GCSE maths (Grades 9-7) frequently also achieve high grades in A level maths, for those with less than Grade 7/A, comparatively low A level results are more frequent. In 2017, holders of Grade 6/B in GCSE mathematics most frequently achieved Grade D (28.8%) in A level mathematics, closely followed by Grade E (28.5%) (Rodeiro and Williamson, 2022[43]). Rationally, A level maths becomes a difficult choice for all but the highest performing students in mathematics who also wish to pursue mathematics-related subjects in tertiary education.
School-level practices and the design of upper secondary programmes can also influence mathematics participation
Other factors like school-level practices can influence mathematics take-up. In Ireland, schools typically require that students study mathematics for their Leaving Certificate. Also in New Zealand, many schools encourage students to take mathematics up to Level 2 in the National Certification of Education Achievement (NCEA), typically in Year 12 (17). Another influence is the design of upper secondary education. In Singapore, students are required to take at least one subject from a contrasting discipline for their A-Levels. For students taking humanities and arts subjects, mathematics is a common choice for the contrasting discipline.
High participation in mathematics is not sufficient for strong performance
Data on the numeracy proficiency of 16-24year-olds with general and vocational qualifications provides the closest internationally comparative information on the numeracy skills of young adults after completing different kinds of upper secondary programmes (Figure 1.4). Graduates of vocational and general programmes in Austria, Denmark and Singapore demonstrate high numeracy skills. In contrast, upper secondary graduates from general programmes in Canada, England and Ireland have comparatively low numeracy skills. It is notable that vocational graduates in Austria and Denmark demonstrate higher numeracy proficiency than graduates from general upper secondary education in England and Ireland, given the typically lower skills that vocational students have in PISA and PIAAC. The data also shows that even in systems where there is high participation in upper secondary mathematics, such as British Columbia and Ireland, this is not a guarantee of high performance.
OECD countries and Singapore
Note: Chile, Greece, Israel, Lithuania, and Singapore have 2016 data (Round 2). Mexico, and the United States have 2019 data (Round 3); Missing OECD average; Data on individuals who have completed upper secondary education focuses on individuals for whom this is the highest level of education they have attained, meaning it does not include students with tertiary qualifications; New Zealand was not included as the data includes post-secondary diplomas, i.e. qualification that are post-secondary but are offered at ISCED level 3.
Source: OECD (2012, 2015, 2018[19]), Survey of Adult Skills (PIAAC) (2012, 2015, 2018), https://www.oecd.org/skills/piaac/data/ (accessed on 26 June 2024).
Students in vocational programmes in Austria and Denmark have significant exposure to science, technology, engineering, and mathematics
One of the factors driving the comparatively high numeracy of young adults from vocational upper secondary programmes in Austria and Denmark might be greater exposure to Science, technology, engineering, and mathematics (STEM) and maths skills. In England, the share of vocational upper secondary qualifications devoted to STEM is lower than continental European countries with historically strong vocational systems. In the United Kingdom, STEM subjects are the main area of study for less than a third (28.5%) of vocational upper secondary students compared with Austria (41.2%) and Denmark (43.7%), where STEM is main field of study for over two in five vocational students (OECD, 2022[44]).
Austria and Denmark also provide maths as both integrated, applied content and as a standalone discipline in vocational programmes:
Integrated, applied mathematics. In the Schools of intermediate vocational education and training (BMS) and the Part-time vocational schools (Apprenticeships/dual system) in Austria, mathematics is taught as an applied subject in relation to the occupational area of a student’s programme. This is also the case with the vocational education and training (EUD) programme in Denmark, where mathematics is included in around half of the available courses (around 100) in relation to the programme’s occupational focus. Singapore’s Polytechnics and Institute of Technical Education also provides a similar example where mathematics may be included in the course depending on its occupational focus.
Mathematics as a standalone disciple. Mathematics is also taught in the five-year Colleges of Higher Vocational Education and Training (BHS) in Austria, where all students are required to study mathematics throughout the programme, as a separate subject. Similarly in Denmark, all students in the vocational education examination qualifying for access to higher education (EUX) programme study mathematics as a separate subject, following the same curricula as general students and to similar standards. These programmes in Austria and Denmark provide direct access to tertiary education at ISCED 5 or 6.
Invest in levers that influence young people’s perceptions of mathematics’ importance and the drivers for taking it at 16-18.
Important levers include working with tertiary education to explore policies for setting expectations for mathematics for entry, the advice and guidance that students receive around continuing mathematics post-16 and helping to communicate its role for life outcomes, such as financial and health decisions.
Note that high participation in mathematics does not automatically translate to strong achievement.
Strong vocational systems seem to support strong mathematics achievement for students, with the more applied learning, technical learning perhaps helping to protect students from the risks of low achievement. England might work with countries such as Austria and Denmark to explore how mathematics in these systems is taught and organised.
Upper secondary mathematics in England sets high expectations in terms of breadth and depth from an international perspective
With the important caveats of the review and categoristion of upper secondary mathematics programmes undertaken for this report in mind (i.e. being approximate and partial in nature), the mathematics experts undertaking the review formed the opinion that:
GCSE mathematics – both foundation and higher levels provide moderate breadth and extensive depth.
A level mathematics – provides very extensive breadth and extensive depth.
A level maths content stands out as being particularly extensive in breadth
The international comparative review for this report found that the breadth of England’s A level mathematics qualification is greater than any other programme analysed in this review. Notably, some concepts included in the A level curriculum are not typically covered at the upper secondary level in other systems (e.g. statistical distributions, statistical hypothesis testing, integration and numerical methods) and others are typically covered in physics instead (e.g. quantities and units in mechanics, kinematics, forces and Newton’s laws).
The conclusion that A level mathematics covers considerable and unparalleled breadth is consistent with previous comparative work. Previous work identified mathematics A level as being the broadest and deepest programme compared with a set of international mathematics programmes (second only to Further Mathematics A level in England, not reviewed in this report) (Ofqual, 2012[45]). The same review noted the particularity of A level mathematics in including mechanics, which in other systems is covered by physics.
To some extent, the greater depth of A level mathematics reflects the narrower range of subjects covered by individual students at 16-18 in England. Since students typically only take three A level subjects, compared to six to nine subjects internationally, they have more time to cover more content within each subject. Yet, depth and breadth contribute to how demanding the subject is overall. Subject results demonstrate the degree of difficulty. In 2017, the most frequent grade in A level maths for students who had achieved Grade 7/A in GCSE mathematics (26.3%) was a Grade D (Williamson and Rodeiro, 2024[46]), suggesting that it is a challenging subject for students, despite the time they have for it.
Comparatively, foundation and higher-mathematics GCSE appear to set a fairly similar level of demand
Since the review of systems’ mathematics programmes for this report was based on a broad categorisation of mathematics courses, with limited space for more nuanced judgements, it is difficult to identify the exact, and relative differentiation of depth and breadth across foundation and higher GCSEs. However, both courses were categorised by the report’s review as being of moderate breadth and extensive depth. This is notable given both programmes are intended to be pitched at different levels to meet different needs. For most other systems, where there are different levels of mathematics catering to different abilities, this report’s review formed the view that there was a corresponding difference in breadth and/or depth. For example, in Ireland, foundation level mathematics was judged to be of initial breadth and depth, whereas ordinary level mathematics was judged to be of moderate breadth and depth and higher mathematics of extensive breadth and moderate depth. This suggests that the differentiation across different GCSE maths levels is not as significant as it is in other systems.
GCSEs set a comparatively high level of demand
The breadth and depth of both foundation and higher-level maths GCSE appears to be relatively stretching in a comparative context. This might partly reflect the revisions to GCSE mathematics in 2015 to provide a more rigorous basis in mathematics and to better support transitions into A level mathematics (Williamson and Rodeiro, 2024[46]). However, the perception that GCSE mathematics sets a comparatively high level of demand does not just reflect the depth and breadth of content and expectations but also that it is studied by relatively young students (14-16), for a comparatively short period of time (two years) and is typically studied alongside eight or nine other GCSEs. In comparison, some of the mathematics programmes reviewed for this chapter were found to have similar levels of breadth and depth as maths GCSE, while catering to an older age group and often over a longer period of time, notably the basic scope mathematics porgramme in Poland (for 15-19) and H1 mathematics in Singapore (16-18).
Other systems have mathematics programmes pitched at lower levels of breadth and/or depth than foundation GCSE. Notably, mathematics in Academic Secondary Schools (AHS) and Colleges of Higher Vocational Education and Training (BHS) in Austria; some of the Grade 11 and Grade 12 courses (with the exception of Calculus) in British Columbia (Canada); EUD in Denmark, foundation and ordinary level mathematics in Ireland’s Leaving Certificate; and O Levels and N Levels in Singapore (typically studied from 13-16). Many of these systems also achieve strong mathematics skills at 15 and by the end of upper secondary education (see Chapter 3). England might consider if the current foundation tier of GCSE maths meets the needs of all learners, especially those who have struggled with maths at earlier stages of education.
Looking across national and international data also seems to suggest that GCSE mathematics sets a high-level demand. In 2019, a third of young people taking GCSE mathematics in England (29.8%) did not achieve the benchmark required to pass (Department for Education, 2023[47]). Yet, PISA data suggests that, around the same age that they take GCSEs at 15, young people in England have comparatively strong mathematics skills. In 2022, on average, 15-year-old students in England performed above the OECD average. Furthermore, less than a quarter (23.3%) did not have basic mathematical skills, lower than the share who do not pass GCSE mathematics (see Chapter 3). As well as the demotivating signal that not passing mathematics sends to young people, students who do not pass mathematics are required to continue studying for and trying to pass their GCSE mathematics over 16-18.
Mathematics education at 16-18 in England provides comparatively few levels and options
In England, there are only two mainstream options for mathematics post-16: A level maths - which is very demanding; and Core Maths – which has very low uptake (only 1.9% of 19-year-olds achieved Core Maths in 2023 (Department of Education, 2024[35]). In contrast, the focus systems, British Columbia, Denmark, Ireland, New Zealand and Singapore provide mathematics content at least at three different levels, with comparative participation levels at each:
Denmark: upper secondary general students choose between A, B and C-level mathematics for general upper secondary students.
Ireland: students taking Leaving Certificate Established and Vocational choose between foundation, ordinary and higher-level mathematics.
Singapore: provides different levels for mathematics at 14-16 – Normal (Technical), Normal (Academic) and Express. Three levels of mathematics are also provided during A-Levels and at 17-19 - H1, H2 and H3.
Different levels of mathematics typically vary in terms of depth and breadth of content, skills and knowledge and prepare students for different future pathways in tertiary education and employment.
Contrasting with upper secondary maths in other systems, A level maths is taken almost exclusively by high achieving students and results are frequently very high
The entrance profile of young people to A level mathematics stands out in contrast to other A level subjects. Figure 1.5. Progression rates from GCSE to A Level by subject, by GCSE gradeshows the entrance profile of young people into mathematics and other large A level subjects – sciences, English, history and geography - based on their GSCE grades. In 2017, very few students achieving below Grade 7/A progressed to A level mathematics. In contrast, students achieving the highest grades in GCSE mathematics progress in far greater numbers.
Source: Williamson and Rodeiro (2024[46]), Progression from GCSE to A Level, 2020–2022.
In contrast to other subjects in England, A level maths has a quite unique results profile, with over two fifths (41.9%) of students achieving the highest grades – A/A* in 2022/23. This contrasts with other A level subjects. Even in science subjects such as Chemistry and Physics (which also have entrants with some of the highest prior achievement at GCSE), only slightly over a third of students achieve A*/A (Department of Education, 2024[48]). To some extent, the high results in maths may reflect the nature of the subject and how marks are awarded in contrast with more essay-based subjects where it may be less frequent to obtain full marks for an individual item. However, the entry profile of A level maths likely also influences this performance distribution as well as the subject’s breadth and depth.
In contrast, in systems where there is greater variation in the level of breadth and depth set by upper secondary mathematics programmes, this creates the possibility that results are more evenly spread across different grades and more in line with the distribution in other subjects. In Ireland for example, as Figure 1.6 shows approximately half of students taking mathematics higher, ordinary and foundation achieve Grades 1-3, similar to other subjects such as English and biology at higher and ordinary levels. In England, the fact that A level maths seems to cater to a small group of very capable mathematicians might limit options for other young people who are proficient in the subject to continue developing their skills post-16. Since students’ choices of study in tertiary education are closely connected to their A level subjects, this has the potential of reducing the overall pool of young people with mathematics, and STEM skills more generally, for England’s workforce.
Source: Ireland State Examination Statistics (2023[28]), Annual Exam Statistics - State Examination Commission (examinations.ie) (accessed on 16 September 2024).
In 2015, England introduced Core Maths to address the comparatively limited maths options post-16
Core Maths was developed to address a gap in mathematical skills for students not pursuing A level Maths but still needing maths for future studies or employment. While Core Maths is still a relatively new qualification, some early research shows that students, schools, employers and tertiary institutions are generally positive about the content and teaching (Homer et al., 2020[49]). However, the overall numbers studying Core Maths remains small. In 2022/23, 1.9% of 19-year-olds achieved a Core Maths qualification (Department of Education, 2024[35]). The low number of students taking Core Maths means that it has so far had a relatively limited impact on overall continuation of mathematics post-16. Over the past two decades, the share of young people achieving a Level 33 or post-16 mathematics qualification by 19 has almost doubled – from 7.25% in 2004/05 to 14.51% in 2022/23. Yet, only a quarter of that increase (25.58%) has been from Core Maths, with the increase being largely driven by an increase in A level mathematics (Department of Education, 2024[10]). Moreover, while the numbers of students taking Core Maths have increased over time since its introduction, increases seem to have plateaued in recent years. In the four years since 2020, entries have increased by around 1000 overall (MEI, 2024[50]).
While the content of Core Maths seems to effectively meet the needs of many learners, ultimately it struggles to find its place in a system where post-16 education and tertiary entry are centred on A level results. An important policy question for England is if students achieving below Grade 7/A in GCSE maths – and the economy more broadly - are well-served by the current maths provision post-16.
Consider if the current options post-16 provide sufficient diversity to cater to students’ needs.
With the introduction of Core Maths, England has two main maths options at post-16 / Level 3. While this provides some options, the focus systems in this report all provide more, typically three options, and sometimes more.
England might consider if wider choice (beyond A level and Core Maths) might be useful. While this may appear to be a significant structural change, it is notable that varied maths provision in the focus systems is typically not linked to having multiple, wholly separate programmes but providing greater diversity and choices within single programmes. For example, the Leaving Certificate in Ireland and maths A-Level in Singapore provides maths at three levels. Providing greater diversity within the main mathematics programme also signals that mathematics to 18 caters to the breadth of the cohort’s needs and aspirations. The current entrance profile and results for A level maths in England suggests it caters to a relatively small elite of high performing mathematicians.
Review Core Maths and opportunities for increasing participation.
The absence of different options for studying mathematics across the 16-18 phase in England stands out internationally. While early feedback and analysis of Core Maths suggest that it is effectively meeting the previous gap in maths options post-16, England might review why take-up of core maths is currently low. In particular, consideration might focus on how Core Maths could better align with the broader structure of post-16 study and the incentives for students to take the subject, especially the role of employers and tertiary education.
Consider how schools, teachers and national communication on education can convey the ways in which different future pathways after school require mathematics.
The potentially large numbers of students who currently do not take any post-16 mathematics, might suggest that these students are not cognizant of the ways in which they are likely to need mathematics in their lives post-schooling. Teachers, schools and national campaigns or communications on post-16 choices, careers and pathways could focus on communicating the relevance and value of mathematics across a range of contexts.
Consider introducing a more accessible mathematics programme at 14-16 pitched at a lower level than GCSE foundation mathematics.
The high share of young people not attaining a Grade 4/C in GCSE mathematics – almost a third in 2023 – and the contrast between these data and England’s PISA data - might suggest that the bar is unnecessarily high for some students. Several of the focus systems provide a mathematics programme or course at a lower level of demand than England’s foundation tier mathematics GCSE. These include systems with strong performance and participation in mathematics such as Austria, Denmark and Singapore. In England, providing mathematics that is more accessible could help to improve motivation and engagement, especially among young people who are required to continue trying to attain a grade 4 in GCSE mathematics after 16 through re-sits.
Consider creating longer mathematics courses that spans the full upper secondary period (14‑18).
The structure of upper secondary education in England is fairly unique internationally, being divided into two distinct phases - 14-16 and 16-18. Each phase is marked by high stakes national qualifications and exams, and can take place in different educational institutions. One of the consequences of this structure in mathematics is that students must cover content and develop deep mathematical reasoning skills over a comparatively short course – two years in each case. The fact that almost a third of students typically spend an additional two years studying trying to attain a Grade 4/C in mathematics GCSE creates the case for exploring providing similar content and expectations over the full duration of upper secondary i.e. four years, for some students.
This report has highlighted an apparent paradox around mathematics in England. The country’s students perform well in mathematics at 15, and express positive attitudes towards the subject. Yet, the share of young people choosing to take the subject at 16-18 is very low by comparative standards. One likely explanation for the sharp contrast between students’ perceptions of maths in England at 15 and their take-up post-16 of the subject is the nature of the offer available. The analysis presented in this report suggests that mathematics programmes across both 14-16 (GCSEs) and 16-18 (A levels) cover considerable breadth and depth. In contrast to the focus systems reviewed for this report, the breadth and depth of mathematics provision across upper secondary in England, combined with the relative absence of options catering to different needs across the student cohort, creates relatively narrow provision.
Having a numerate society and workforce, in particular with a stream of young people entering the labour-market with different types of mathematics skills, is important for a national economy. The comparative absence of options for many students who are proficient in maths to continue to study the subject post-16 in ways that are comparably valued to the option for the highest performers is an important policy question for England.
To promote greater and more equitable participation in maths post-16, it will be important for England to consider how the options that are available for students can be more responsive to the diversity of their strengths, interests and future aspirations. The currently narrow provision of maths post-16, alongside the narrow range of tertiary subjects for which it is required, likely contributes to (and might even actively create) the societal perception that maths is hard and unnecessary for most people.
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← 1. The content was revised in 2015, with the first examinations reflecting the revised content taken in 2017.
← 2. Since 2014 all post-16 students who have not achieved grade 4 in GCSE must continue studying maths, either studying towards GCSE grade 4-9 or towards an equivalent Functional Skills Qualification.
← 3. Level 3 in England’s Qualification Framework which typically includes qualifications completed post-16 such as A levels, T levels. and other post-16 vocational qualifications (GOV.UK, 2024[51]).