This chapter looks at the different types of mathematics programmes systems provide in upper secondary education. It shows that many systems provide programmes linked to the orientation of their upper secondary education - general or vocational. Countries also tend to provide different options for students in terms of the level of breadth and depth at which they study mathematics, frequently these options are linked to different future pathways. The chapter compares the options provided for upper secondary students in England studying maths and their peers in the focus countries. Finally, the chapter explores patterns between options and choices for studying mathematics and performance and participation in the subject.
Mathematics for Life and Work
A Comparative Perspective on Mathematics to Inform Upper Secondary Reform in England
Abstract
Nearly all systems provide mathematics at different levels of knowledge and skills
Different levels of mathematics mean that students can take mathematics options that respond to their strengths, needs and future ambitions. Different levels of mathematics often provide pathways to different tertiary courses, notably courses with more or less mathematical content.
For example, Ireland and Singapore provide three different levels of mathematics for the duration of upper secondary. In British Columbia, New Zealand and Singapore, different mathematics levels correspond to different durations, enabling some students to continue mathematics until the end final year of upper secondary and others to stop earlier.
Most systems explicitly design different mathematics levels and options to cater to students’ future pathways, and communicate this to students
For example, in Singapore, mathematics at H1 level is designed for students who will study business or social sciences in tertiary education, H2 for students intending to study tertiary courses like sciences, and H3 for students who are passionate about mathematics and are keen to specialise in the subject.
Mathematics in vocational education tends to be integrated as applied content but some systems also provide mathematics as a separate subject
In vocational systems, mathematics tends to be integrated as applied content in line with a student’s occupational focus.
However, Austria and Denmark also have upper secondary vocational options where students study mathematics as a separate, more school-based programme which provides direct access to tertiary education.
An overview of requirements for mathematics in upper secondary education and the different options and choices across systems is provided in Annex A.
Among the focus systems, and across OECD countries more generally, it is common to provide students with choices and options in the mathematics content that they study in upper secondary education (Stronati, 2023[1]). Typically, systems provide students with a choice in the level at which they study mathematics content and sometimes choice over the type of content and courses they take i.e. different mathematics options. These choices are rarely entirely distinct, with levels and content naturally interacting. More advanced levels typically require students to engage with a wider range of more advanced content, although some systems do offer different options of mathematics e.g. applied or theoretical mathematics and even beyond this, choices between calculus, statistics, etc. within the same level, such as British Columbia (Canada) and New Zealand.
Some degree of choice can be particularly important at the upper secondary level. Providing students with more choice, options and differentiated learning in contrast to lower levels of schooling is one of the defining features of upper secondary education (OECD/Eurostat/UNESCO Institute for Statistics, 2015[2]). Student choice over the level and mathematics content can help ensure that all students are able to achieve success and align mathematics content with their interests and future ambitions.
Providing different levels of mathematics is linked to the depth, breadth and demand of the content
Across the focus systems, British Columbia, Denmark, Ireland, New Zealand and Singapore provide mathematics content at different levels (Box 5.1. Examples of levels in upper secondary mathematics and how countries provide them). More broadly across OECD countries, providing courses at different levels is a common practice for both mathematics and national language. In Sweden, for example, students can select modules in core subjects at different levels to meet their requirements, such as Swedish 3 or Mathematics 2. In Finland, students can choose from Basic and Advanced Mathematics. Schools in Japan might offer up to six different options for mathematics, while in Korea they offer a choice between three options in mathematics (Stronati, 2023[1]).
Of the focus systems, the only country (aside from England (United Kingdom)) that provides limited student choice in terms of mathematics levels is Austria. This reflects the design of Austria’s upper secondary education as a highly structured system. In Austria, the area of specialisation is determined by the programme students take, of which there are many aligned to different student needs and future career paths, rather than choice and specialisation being determined by students themselves within the programme. Other systems in this category include Germany, the Netherlands and Switzerland (Stronati, 2023[1]).
Different levels respond to diverse levels of mathematics competence
Providing mathematics at different levels is important for two reasons. First, following widespread recognition that all young people should complete upper secondary education, participation in this level of education has increased across OECD countries, creating the need for upper secondary education to cater to a wide range of student needs and strengths. In all the focus systems at age 15, around the age at which young people transition into upper secondary education, there are wide variations in student performance in the Programme of International Student Assessment (PISA) (see Chapter 3) (OECD, 2019[3]). Upper secondary mathematics provision needs to meet the complex needs of the full range of students’ proficiency levels. Systems must provide students with the appropriate amount of support when they are struggling and the right amount of challenge to help them achieve their potential. Achieving these goals is particularly important in upper secondary education as it is the last opportunity that education systems have to influence the knowledge and skills development of the full student cohort before the end of formal schooling.
In countries where vocational education is a well-developed and recognised option with multiple programmes, such as Austria, the vocational offer can help respond to different levels of learning. In systems where vocational education is less developed, providing diversity within general education programmes is important. Ireland, British Columbia (Canada) and New Zealand do not have upper secondary vocational programmes for the typical age cohort at this level, according to the International Standard Classification of Education (ISCED) classification (OECD, 2020[4]). In this context, providing choices around the levels at which students engage with mathematics in general education programmes is essential. Denmark, Ireland, New Zealand and Singapore all provide students with three different levels at which they can learn mathematics (as set out in Box 5.1).
Different levels enable students to develop the depth and breadth of mathematics that they need for their future ambitions
One of the reasons that upper secondary education provides greater choice and diversity is because it is the moment when students are expected to start defining their interests and developing deeper and specific skills in areas that correspond with their future career ambitions. While all OECD education systems have some compulsory mathematics content – pointing to a belief that some mathematics in upper secondary education is essential – how deep those skills need to be and for which cohorts of students varies. Students intending to study science, technology, engineering and mathematics (STEM) or humanities in post-secondary education are likely to have different needs in terms of the depth and breadth of their mathematics knowledge and skills.
Providing students with choice over the level of mathematics that they pursue in upper secondary education enables them to develop mathematics competencies in line with their broader career ambitions. In Singapore for example, the curriculum for general upper secondary students taking A-Levels specifies that Higher 1 (H1) – the first level of mathematics – is intended for students studying business or social sciences in tertiary education whereas Higher 2 (H2) – the next level of mathematics – is intended for students who will study mathematics or sciences in tertiary education. Similarly in Ireland, higher-level mathematics is intended for students intending to pursue extensive mathematics in tertiary education.
Levels might interact with the duration of mathematics
Levels in British Columbia, Denmark and to some extent New Zealand, are slightly different because they correspond to grade levels or the number of years that a student will study mathematics for:
In British Columbia, students are required to do four credits of mathematics in either Grades 11 or 12 (or they can combine both) (British Columbia Government, 2023[5]). While mathematics in Grade 12 would naturally be expected to be more advanced than in Grade 11, students within the same year group do not have a choice over the depth and breadth at which they study mathematics in the same way as the levels in other systems provide.
In New Zealand, students can study mathematics at Levels 1, 2 and 3 of the National Certification of Educational Achievement (NCEA). Students tend to progress linearly through Levels 1-3 in the three years of upper secondary, but it is possible for learners in the same year group to study at different levels to best meet their needs. For example, a learner could work towards Level 2 over the last two years of upper secondary.
In Denmark, students can pursue mathematics for one year (Level C), two years (Level B) or three years (Level A). The three-year, Level A course covers the full duration of upper secondary education. General students can only take the one year, Level C option, if they study three foreign languages or more. In practice a very small share of upper secondary general students, 1.2% in 2022, take the mathematics Level C option which lasts for just one year.
These systems tend to provide less explicit options linked to different levels of mathematics, using the duration or grade level to reflect the complexity and depth of content. In systems where the full student cohort, either in theory or in practice, studies mathematics until the end of upper secondary education – such as Ireland or Singapore, providing varied levels seems particularly important.
Differentiated mathematics options are not the same as ability streaming at lower levels of schooling
Ability grouping refers to practices for selecting and sorting students with similar perceived abilities or performance into more or less fixed teaching groups, classes or schools. Ability streaming aims to reduce variations of ability between students for more tailored teaching (Ansalone, 2010[6]; Hornby and Witte, 2014[7]). However, in recent years ability streaming has been questioned because evidence suggests that it can be negative for lower and average achieving students while providing little to any benefit for high achieving students. Reasons why ability grouping practices might not support the learning outcomes of students in middle and lower ability groups include providing a less demanding curriculum and holding lower expectations (Chmielewski, Dumont and Trautwein, 2013[8]). Being placed in a lower tier class can also negatively influence students’ motivation (Hanushek and Woessmann, 2006[9]; Rubie-Davies et al., 2010[10]). At the same time, no clear results have been found to suggest that ability grouping consistently benefits high achievers either (Preckel et al., 2019[11]; Smale-Jacobse et al., 2019[12]; Francis, Taylor and Tereshchenko, 2020[13]). The potentially negative effects of ability streaming have led some countries where it was previously a common practice to take steps to reduce its use. New Zealand recently announced a policy position to end the practice of ability streaming (CORE Education, 2024[14]).
This report emphasises the value of providing mathematics at different levels in upper secondary education. It suggests that differentiated mathematics options in upper secondary education are substantively different from ability streaming at lower levels of schooling, and for these reasons are not necessarily at odds with the policy direction in a number of countries to end ability streaming. At the upper secondary level, differentiated mathematics classes:
Enable students to use and exercise choice in upper secondary linked to future pathways
While all students will need some mathematics in life after school, only those intending to pursue post-secondary education and careers with high levels of mathematical content will need very deep or broad mathematics. Providing students with mathematics options that clearly link to future pathways, as is the case in Ireland or Singapore, where higher mathematics levels are clearly indicated as being for students who wish to continue mathematics in tertiary education, provides helpful signaling for students.
Carry less risks for limiting student pathways compared with lower levels
Much of the research about the risks of ability grouping focuses on lower levels of education. In primary and lower secondary, ability grouping can carry risks because it may inaccurately identify students at lower performance levels, constraining their learning, expectations and even future pathways. In contrast, by the beginning of upper secondary education, most students have already developed key fundamentals in mathematics and indeed many will stop studying mathematics as a formal subject at the end of this stage of education. Provided all mathematics options do not limit access to subsequent levels of education, differentiated options at upper secondary likely carry less risks for students compared with lower levels.
Create space to provide a mathematics option focused on building key maths for students entering upper secondary with low skills
Research has highlighted the numerous risks that individuals with low mathematics skills experience on the labour market, and for decision making linked to their financial and even health outcomes (Jonas, 2018[15]) (see Chapter 2). Upper secondary is the last structured learning context where governments can shape how the full cohort spends their time. At this stage, one of education systems’ central priorities must be to be ensure that all young people complete upper secondary education with a solid basis in mathematics, both for the individuals concerned and the societies in which they live. Differentiated options create the space to provide dedicated mathematics options, perhaps with a different pedagogy, for students entering upper secondary with fragile mathematics skills.
Denmark – A, B and C-level mathematics for general upper secondary students
In Denmark, the three levels of mathematics are associated with different durations: Level C - one year; Level B - two years; Level A - three years. The main upper secondary general programmes (Higher General, Technical and Commercial Examinations (STX, HTX, HHX)) require at least Level B mathematics. Some students who take three foreign languages can take just one year of mathematics (Level C). The longer length corresponds to greater expectations in terms of the depth and breadth of content covered, and all the programmes have a comparatively high level of demand.
Level C: aims to develop basic mathematical competencies and acquire fundamental arithmetic and algebraic skills, as well an understanding of basic statistical concepts and data analysis.
Level B: aims to extend skills to topics such as calculus, trigonometry, and more complex algebra. Students solve problems, as well as analyse and interpret data using statistical methods. Students should also develop critical thinking and problem-solving skills.
Level A: aims to engage with higher-level mathematical concepts, including advanced calculus, complex numbers, and different equations. Students acquire a deeper understanding of abstract mathematical theories and proofs (Ministry of Children and Education, 2023[16]).
Ireland – Ordinary, Foundation and Higher level in the Leaving Certificate Mathematics – Established and Vocational
In Ireland, the three different levels of maths correspond to differences in depth and breadth. In all levels, students study mathematics for the full duration of upper secondary education (two years):
Foundational level: is intended to equip learners with the knowledge and skills required in everyday life. It is also intended to lay the groundwork for learners who may proceed to further studies in areas in which specialist mathematics is not required (Department of Education, 2015[17]).
Ordinary Level: focuses on fundamental mathematical concepts, including algebra, geometry, trigonometry and statistics. It aims to equip students with essential mathematical skills applicable to everyday life and various vocational pathways (Department of Education, 2015[17]). It may not cover more advanced topics required for higher-level studies in STEM fields.
Higher Level: covers more complex topics, such as calculus, differential equations and advanced algebra. Designed for students aspiring to pursue tertiary education in STEM-related fields or other disciplines that demand a deeper understanding of mathematics and more mathematically intensive career paths (National Council for Curriculum and Assessment, 2015[18]).
Singapore – H1, H2 and H3 mathematics in A-Levels
Singapore provides different levels for mathematics at 14-16 – Normal (Technical), Normal (Academic) and Express and at 17-19 as part of the A-Levels. Singapore’s three different levels for mathematics in A-Levels cover increasing degrees of depth and breadth and aim to respond to students’ different aspirations in tertiary education in terms of their future field of study. Students study mathematics for the duration of their post-secondary education (two or three years) regardless of which level they choose:
H1: provides students with a foundation in mathematics and statistics that supports study of business or social sciences at the university. Offers an opportunity to learn important mathematical concepts and skills in algebra and calculus, as well as basic statistical methods.
H2: prepares students for a range of tertiary education courses where a good foundation in mathematics is required, such as sciences. Develops mathematical thinking and reasoning skills that are essential for further learning of mathematics.
H3: designed for students with passion for, and ability in, mathematics. It aims to engage students who are keen to specialise in mathematics by providing more challenging problems to solve, including proving mathematical results. Designed for students who want to pursue mathematics in tertiary education (Ministry of Education, 2023[19]).
Source: Ministry of Children and Education (2023[16]), Upper Secondary Education Programmes; National Council for Curriculum and Assessment (2015[18]) Leaving Certificate Mathematics Syllabus: Foundational, Ordinary & Higher Level; Ministry of Education (2023[19]), A-Level curriculum and subject syllabuses, https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses
Providing a range of different mathematics content through options
In addition to providing choice over the level at which students take mathematics, some systems also provide choice over the actual content that they study. This is most notably the case in British Columbia and New Zealand where students can choose across multiple different courses. Ireland has also recently announced that students taking the Leaving Certificate Applied will have the choice between taking the existing Mathematical Applications course that they are required to take at present or the same mathematics subject that is taken for the Established and Vocational Leaving Certificate (Ireland Department of Education, 2022[20]). As noted earlier, the choice between levels and options tend to interact, with more advanced levels covering a greater range of content, at greater depth.
Mathematics in vocational education is covered as a separate and integrated subject in some systems
Among countries with separate vocational and general upper secondary programmes, some systems demonstrate two different approaches to the teaching and coverage of mathematics in vocational education:
Integrated, applied mathematics. Many systems provide a vocational option where mathematics content is shaped by a student’s specialisation and the occupational focus of their programme. For example, 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 / Erhvervsuddannelse (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 provide a similar example where mathematics may be included in the course depending on its occupational focus.
Mathematics as a standalone disciple. In contrast, in some vocational programmes, mathematics is taught as a separate discipline in a similar way as in general programmes. This is the case 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 EUX programme study mathematics as a separate subject, following the same curricula as general students and to similar standards (Table 4.2) (see also Chapter 4). Mathematics is compulsory at least at Level C (the one-year programme) for all 45 vocational programmes, while Level B (two years) is the requirement for most programmes. Students in the EUX might also take mathematics at Level A as an optional subject in some vocational programmes. These programmes in Austria and Denmark provide direct access to tertiary education at ISCED 5 or 6.
In Türkiye’s upper secondary vocational education, both approaches are combined. Mathematics is offered as a standalone discipline in all vocational fields, in the Grades 9 and 10 as mandatory, and in Grades 11 and 12 as optional elective. A form of integrated applied mathematics is also included in a “Vocational Mathematics” course in the fields of marketing and retail, and accounting and finance (and is also offered as an optional elective in office management and executive assistance) (Türkiye Ministry of National Education, 2024[21]).
Countries with strong upper secondary vocational education, especially where this is historically based on strong linkages with employers as in Austria, the Netherlands and Switzerland, often involve employers directly in determining the mathematics content and how it is taught for different vocational programmes. In Switzerland, where vocational upper secondary education is popular with 62% of students enrolled in 2021, the country’s professional associations play a central role in the design of mathematics curricula (Box 5.2).
In Switzerland’s upper secondary vocational programmes, the mathematical competencies of each programme depend on its specific occupational area. Some learners might spend a lot of time in school learning theoretical mathematics for professions where it is required, while others will spend more time learning applied mathematics in a work setting.
Multiple stakeholders are involved in the construction of the vocational education curricula in Switzerland, with the confederation and canton governments coordinating with professional associations. The employer and professional organisations have a central role in defining what is to be taught and how the teaching is organised in each vocational programme. The close relationship and coordination with industry and employers enables vocational programmes to respond rapidly to the needs of the labour market. This close and collaborative relationship between education and employers facilitates the integration of graduates from vocational upper secondary education into work and is considered to be one factor influencing Switzerland’s low rates of youth unemployment (Center on International Education Benchmarking, 2015[22]).
Source: Center on International Education Benchmarking (2015[22]), Gold Standard: The Swiss Vocational Education and Training System.
Options tend to give students the choice between more academic or applied mathematics
In comprehensive systems without distinct vocational and general programmes, like British Columbia and New Zealand, students still have a choice between more academic mathematics (which might be seen as a preparation in mathematics for tertiary education) and applied mathematics (which might provide students with mathematics knowledge and skills for the workplace). Students in these systems also have the choice to study statistical and modelling content (which might prepare students for some STEM subjects as well as social sciences in tertiary education). In Year 12 for example, students in British Columbia can choose from apprenticeship mathematics, calculus, computer science, foundations of mathematics, geometry, pre-calculus and statistics, among others (British Columbia Government, 2023[5]).
Extensive modularisation can limit coherence
The modular, credit-based systems of British Columbia and New Zealand provide students with significant choice across a range of options. The degree of potential variation across how students construct their programme of study is particularly significant in New Zealand where mathematics courses for NCEA Level 2 (typically taken by learners aged 16/17) may be comprised of 13 different Achievement standards, units which set the standard for assessment and count for credit (NZQA, 2023[23]). Subjects are assessed through a mix of internal and external assessments and teachers have significant flexibility in how the internal assessments are designed and which combination of internal and external assessments they will offer to their students. Reforms are currently being implemented to introduce a standardised structure of two internal and two external assessments for all courses, with this structure being implemented for NCEA Level 1 from 2024 (Box 5.3). Having fewer – but larger – standards available from which teachers and students can choose aims to create greater coherence for students and greater consistency across the student cohort.
Assessment in modular programmes, whereby learners can revise some content and then be directly assessed, can suit some learners better in contrast to final examinations at the end of one, two or even three years of study (Baird et al., 2019[24]). However, some teachers and subject experts feel that modularisation reduces the possibility for deeper, cumulative learning that occurs across a longer period. This concern is echoed by teachers across many key subjects, including mathematics (Baird et al., 2019[24]).
In New Zealand, the significant number of standards that students can take as part of the NCEA has led to concerns about subject coherence. Students might take a wide variety of different units that, together, do not constitute a coherent package of learning although. A Review of Achievement standards is being conducted as part of the NCEA Change Programme to introduce new standards that will eventually replace all the existing achievement standards.
For NCEA Level 1 mathematics, since the start of 2024, only four standards – which have been designed to be delivered collectively as a package – are available:
1.1. Explore data using statistical enquiry process
1.2. Use mathematical methods to explore problems that relate to life in Aotearoa New Zealand or the Pacific
1.3. Interpret and apply mathematical and statistical information in context
1.4. Demonstrate mathematical reasoning
By developing packages of fewer but larger standards, content is not removed but rather the focus is on ensuring the learning that is considered to be essential, which cannot be left to chance, is covered by all students. Rather than the topic- or concept-based approach of the previous standards, with each standard focusing on specific skills, the new standards take a competence-based approach. New Level 1 standards were implemented from the start of 2024 and new standards for Levels 2 and 3 are currently being developed along a similar competency-based approach.
Source: Ministry of Education (2023[25]), https://ncea.education.govt.nz/review-achievement-standards (accessed on 9 April 2024).
Note: In Singapore, with the implementation of full subject-based banding in 2024, secondary students in Singapore can learn each subject, including mathematics, at the level that best caters to their overall strengths, interests and learning needs. There will no longer be separate Express, Normal (Academic), and Normal (Technical) course of study, and the different levels of mathematics will be termed G1, G2 and G3 Mathematics, with G3 Mathematics being the most demanding. Please see https://www.moe.gov.sg/microsites/psle-fsbb/full-subject-based-banding/about-full-sbb.html.
Source: Federal Ministry of Education, Science and Research of Austria (2024[26]), https://www.bmbwf.gv.at/Themen/schule/schulsystem/sa/ahs.html (accessed on 22 January 2024); Federal Ministry of Education, Science and Research of Austria (2024[27]), https://www.bmbwf.gv.at/Themen/schule/schulsystem/sa/bmhs.html (accessed on 22 January 2024); Federal Ministry of Education, Science and Research of Austria (2024[28]), https://www.bmbwf.gv.at/Themen/schule/schulsystem/sa/bs.html (accessed on 22 January 2024); British Columbia Government (2023[5]), https://curriculum.gov.bc.ca/curriculum/mathematics (accessed on22 January 2024); Denmark Ministry of Children and Education (2024[29]), https://eng.uvm.dk/upper-secondary-education/national-upper-secondary-education-programmes (accessed on 22 January 2024); Denmark Ministry of Children and Education (2024[30]), https://eng.uvm.dk/upper-secondary-education/vocational-education-and-training-in-denmark (accessed on 22 January 2024); Department for Education (2018[31]), https://www.gov.uk/government/publications/get-the-facts-gcse-and-a-level-reform/get-the-facts-gcse-reform (accessed on 22 January 2024); National Council for Curriculum and Assessment (2024[32]), https://www.curriculumonline.ie/senior-cycle/senior-cycle-subjects/mathematics/ (accessed on 22 January 2024); NZQA (2024[33]), https://www2.nzqa.govt.nz/ncea/subjects/select-subject/mathematics-and-statistics/ (accessed on 22 January 2024); Singapore Ministry of Education (2024[34]), https://www.moe.gov.sg/secondary/courses (accessed on 22 January 2024); Singapore Ministry of Education (2024[35]), https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses (accessed on 22 January 2024).
This section looks at national data on participation in the mathematics options and levels offered by the focus systems. It aims to understand if, and how, the supply of these different levels and options influence participation in mathematics during upper secondary education and seeks to shed light on possible connections with performance.
In systems where mathematics levels interact with duration, around half of the cohort continues with mathematics until the final year of upper secondary
In British Columbia, Denmark and New Zealand, students choose the level and duration of their mathematics programme. Around half of the cohort in these systems decides to continue mathematics until the final year of upper secondary education (Table 5.2):
In British Columbia, 32 919 individuals took mathematics in Grade 12 in 2021/22.
In Denmark, 44.5% of the cohort in 2022 took Level A mathematics which covers the full duration of upper secondary education.
In New Zealand, over half of the cohort – 58% in 2022 achieved NCEA Level 3 mathematics, which students typically take in Year 13, their final year of upper secondary education.
Note: 1 Data for Denmark concern the three-years programmes (stx, hhx and htx); 2 Data for New Zealand, NCEA Level 3 refers to students assessed in 14 or more credits in mathematics in 2021; Data are unavailable for Singapore.
Source: NZQA (2023[36]), Department for Education (2024[37]), https://www.gov.uk/government/statistics/provisional-entries-for-gcse-as-and-a-level-summer-2022-exam-series/provisional-entries-for-gcse-as-and-a-level-summer-2022-exam-series#gcse-entries (accessed on 25 September 2024); Ireland State Examination Statistics (2023[38]), https://www.examinations.ie/statistics/?l=en&mc=st&sc=r19 (accessed on 25 September 2024).
New Zealand achieves high participation in mathematics and statistics until the end of upper secondary education
The NCEA numeracy requirement specifies that students must achieve 10 credits in numeracy at Level 1 or higher. Given the increasing depth and complexity at Levels 2 and 3, it might be expected that participation in subsequent levels would drop significantly. While the shares of students taking Levels 2 and 3 mathematics are lower than for Level 1, the overall fall is quite limited, with 58% of NCEA Level 3 students taking mathematics at Level 3.
One of the factors driving steady enrolments at Level 2 might be that some schools do not offer Level 1, with students having to enrol directly in Level 2. Schools might consider that the time and investment to achieve Level 1 is unnecessary since Level 2 is typically regarded as providing key mathematics and statistics competencies for the labour market. Discussions with national stakeholders underlined the general perception of the importance of mathematics and so schools might encourage students to take mathematics at least until Level 2, based on recognition of its importance for students’ future.
In British Columbia (Canada) participation in mathematics is high, even after requirements are fulfilled
Students in British Columbia must take 4 credits in mathematics in Grade 10, and 4 further credits across either Grades 11 or 12. As students take Grade 11 mathematics before Grade 12 mathematics, it is likely that students would complete the mandatory requirement in Grade 11. In line with this, Figure 5.1 shows that student enrolments in mathematics options are largely stable across Grades 10 and 11 and decline by around 35% in Grade 12. It is important to note that these data show the number of enrolments on each course and individual students may take multiple courses. Despite the fact that data likely includes some individual students taking multiple courses, the largely constant numbers in mathematics enrolments across Grades 10 and 11 suggest that most students complete their mathematics requirement in Grade 11, with the enrolments in Grade 12 being largely from students deciding to continue mathematics once it is no longer compulsory.
Note: “Fdn Math & Pre-calc”: Foundation of Mathematics and Pre-Calculus 10. Data supplied by Ministry of Education Child Care of British Columbia (Canada).
In Ireland the share of students taking higher-level mathematics is growing
In Ireland, over the past decade there has been a growth in students taking higher-level mathematics. Between 2013 and 2023, the share of students taking higher-level mathematics increased from 24.7% to 35.4% (Ireland State Examination Statistics, 2023[38]). The increase in higher-level maths might reflect the influence of policy changes to increase take-up and promote mathematics skills more generally in Ireland. Since 2012, students receive extra points (25 bonus points) if they take higher mathematics and achieve a mark of H6 or higher (out of 8 grades) in the calculation of their points for tertiary entry (Central Applications Office, 2023[39]). Another part of Ireland’s maths strategy has been to support the training and supply of mathematics teachers (Box 5.4).
Ireland’s Professional Diploma in Mathematics for Teaching was introduced in 2019 and supports post-primary teachers who would like to teach mathematics but do not currently have a qualification in the field of mathematics.
The two-year part-time, online programme is scheduled for out-of-school hours and is provided in partnership with multiple tertiary education institutions who deliver the course. The programme is free to eligible post-primary teachers and is fully funded by the Department of Education and Skills as part of the national strategy to support the implementation of the Teacher Supply Action Plan and improve standards in maths education. The national funding means that teachers choosing to re-train as mathematics teachers are not required to bear the costs of their training themselves.
Source: University of Limerick (2023[40]), Mathematics for Teaching (Level 8) - Professional Diploma, https://www.ul.ie/gps/course/professional-diploma-mathematics-teaching-level-8 (accessed 20 October 2023).
England stands out with little diversity for students in terms of levels or options in mathematics
Across the focus systems, England stands out in terms of the comparative absence of different levels for mathematics or choices over the content (at 16-18). This lack of diversity may limit the range of students who continue to study the subject when it becomes optional.
A level mathematics is taken almost exclusively by high achieving students
At 16-18, students entering post-16 education in England have the option to take a range of different qualifications. A levels are the main general upper secondary programme and inform tertiary selection. Typically, students must achieve at least 5 GCSEs (national certifications taken around 16) at Grades 4-9 to study A levels. This means that students studying A levels typically perform relatively well in GCSEs at 16. Yet, the entrance profile of young people to A level mathematics stands out in contrast to other A level subjects. Historically, A level mathematics has been taken by a small group of high achieving students. Across a sample of six different A level subjects, students completing mathematics and physics had the highest GCSE grades (Matthews and Pepper, 2007[41]).
Recent data – following recent reforms to A level mathematics – suggests that the subject continues to attract the highest achieving students. Figure 5.2. Progression rates from GCSE to A Level by subject, by GCSE grade shows the entrance profile of young people for 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 maths progress in far greater numbers.
Source: Williamson and Rodeiro (2024[42]), Progression from GCSE to A Level, 2020–2022.
In the past, teachers and students reported that one of the challenges for students to progress from GCSE to A level maths was that GCSEs did not provide adequate preparation (Rigby, 2017[43]; Rushton and Wilson, 2014[44]). Reforms to GCSE (and A level) mathematics in 2015 aimed to provide students with a stronger foundation in maths and better support the transition to A level. While data about the progression from GCSE to A level mathematics are still early1, initial results suggest that while reforms have supported the transition for the most able students, those who achieve Grade 6/B and below are not progressing at significantly greater rates than in the past. In 2017, progression from GCSE to A level maths increased by 5.5 percentage points for students achieving a Grade 7/ A or above at GCSE, compared with pre-reform, while for students achieving a Grade 4/C or above the increase was just 2.4 percentage points over the same period (Rodeiro and Williamson, 2022[45]).
While lower progression for students with lower grades might reflect a variety of factors, achievement data suggests that students who achieve Grade 6/B and below in GCSE maths frequently achieve relatively modest marks at A level. 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%)2. For students who achieved a Grade 5/C in GCSE mathematics, the most frequent grade in A level maths was a U (i.e. fail) (33.2%) (Rodeiro and Williamson, 2022[45]). Previous research surveying students who achieved a Grade B in GCSE maths but chose not to continue the subject at A level found that these students perceived A level maths to be “notoriously difficult” and were worried that they would have to devote too much time to studying for it, given the subject’s difficulty (QCA, 2006[46]). While these views were collected prior to revisions of maths GCSE and A level, these students’ perceptions seem to still be reflected in more recent achievement data.
A level maths has a unique results distribution compared with other subjects
In contrast to other subjects in England, A level maths has a quite unique results profile, with over two fifths (41.90%) of students achieving the highest grades – A/A* in 2022/23. Figure 5.3 shows that 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. 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, depth and the difficulty of assessment in the subject.
Source: Department for Education (2024[47]), https://explore-education-statistics.service.gov.uk/data-tables/fast-track/fcea8ac0-477d-40db-dfd4-08dbe03b87c8 (accessed on 16 September 2024).
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 5.4 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 elite 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, into England’s workforce.
Source: Ireland State Examination Statistics (2023[38]), Annual Exam Statistics - State Examination Commission (examinations.ie) (accessed on 16 September 2024).
In contrast to other systems, there are few maths options post-16 for the needs of the full cohort
The comparative lack of diversity in mathematics in England has also meant that A level mathematics must simultaneously meet the needs of very able mathematicians who are likely to continue studying the subject in tertiary education and who perhaps will need to use advanced mathematics in their careers, and those who need mathematics because it underpins other areas of interest, such as science (Matthews and Pepper, 2007[41])). Other systems expressly use their different levels of mathematics to cater to these different populations. In some systems, this also includes courses explicitly providing general mathematics skills for students who intend to study subjects with limited mathematical content in tertiary education, such as humanities or social sciences. Many of the focus systems explicitly describe their highest-level mathematics option as being for students who intend to continue mathematics in tertiary study. This is the case for H2 mathematics in Singapore, Higher-level mathematics in Ireland and some of the Grade 12 mathematics options in British Columbia such as Pre-Calculus and Calculus.
In 2015, England introduced Core Maths to address the comparatively limited maths options post-16
Policy goals that aim to achieve higher levels of participation in maths must consider the opportunities for young people across the cohort to continue studying mathematics since economies require a range of different types of skills at varying levels of specialisation. In England, Core maths focuses on practical applications of mathematics in daily life and professional contexts. It was developed in 2015 to address a gap in mathematical options for students not pursuing A level maths but still needing some 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[48]). Core Maths also seems to cater well to the group of students it was intended to serve – those achieving at least Grade 4/C in GCSE mathematics but not the highest performing students who are more likely to go onto study A level maths. In 2021/22, most (78%) students taking Core Maths had achieved Grades 5/B-C to 7/A in GCSE mathematics, with a few achieving Grade 4/C (8.6%) (Gill, 2024[49]).
However, the overall numbers studying Core Maths remains small. In 2022/23, 1.9% of 19-year-olds achieved a Core Maths qualification. The low number of students taking Core Maths means that it has so far had relatively limited impact on overall continuation of maths post-16. Over the past two decades, the share of young people achieving a Level 33 or post-16 mathematics qualification by 19 has doubled – from 7.25% in 2004/05 to 14.51% in 2022/23 (Department of Education, 2024[50]). Yet, as Figure 5.5. Share of young people obtaining Level 3 qualifications in shows, a very small share of that increase has been from Core Maths, with the increase being largely driven by an increase in A level mathematics.
Note: The data for the International Baccalaureate (IB) is not included, which explains the gap between the sum of the qualifications and the Total Level 3.
Source: Department for Education (2024[50]), https://explore-education-statistics.service.gov.uk/find-statistics/level-2-and-3-attainment-by-young-people-aged-19 (accessed on 11 October 2024).
While numbers of students taking the Core Maths have increased over time – with around 2 000 additional entries each year between 2016 and 2020. However, increases seem to have plateaued in recent years. In the four years since 2020, entries have increased by around 1000 overall (MEI, 2024[51]). Some of the factors related to low uptake are considered to be:
For many schools and colleges, the number of non-A level mathematics students is too small to justify offering an additional course.
The structure of Core Maths is out of sync with the typical structure and funding of post-16 qualifications which supports three or four linear A levels. This makes implementation difficult for schools and colleges.
Awareness and status of Core Maths (especially in contrast to A level maths) is relatively lower and weaker.
Tertiary education institutions do not recognise Core Maths in the same way as they recognise other qualifications and other non-A level qualifications receive more points for tertiary entry.
There is a shortage of mathematics teachers and challenges with teacher recruitment and retention more generally.
Changes in support and funding for Core Maths, with initial support provided temporarily (Lee, Deko and Hussain, 2021[52]; Department for Education, 2023[53]; Homer et al., 2020[48]).
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 levels. An important policy question for England is if students achieving below Grade 7/A – and the economy more broadly - are well-served by the current maths provision post-16. The focus systems reviewed in this report provide comparably more options and choices for all students to continue maths until the end of upper secondary.
The section below sets out policy pointers for countries, and England specifically, for providing levels and options in upper secondary mathematics.
1. Nearly all systems provide mathematics at different levels of knowledge and skills
Providing different levels at which to study maths is a common practice across OECD countries. Countries tend to provide at least two levels, and many provide three. In Singapore, for example, mathematics is provided at H1, H2 and at H3 levels and in Ireland mathematics is provided at foundation, ordinary and higher levels. The only focus countries that do not provide levels are Austria and England (at 16-18). In Austria, however, student diversity is catered for within the system’s design, with students already engaging in mathematics at different levels by virtue of the different general and vocational programmes.
Systems tend to have two slightly different approaches for providing different levels of maths. One, in those countries such as Ireland and Singapore where the full cohort are required to, or in practice do, study mathematics for the duration of upper secondary education, students are provided with a range of differentiated options. This approach is likely important for meeting diverse learning levels and future aspirations across the cohort. Second, in systems such as British Columbia, Denmark and New Zealand, where mathematics is not required for the duration of upper secondary, different options correspond to progressively longer duration, and correspondingly greater breadth and depth.
Policy pointers for England:
Consider if the current options post-16 provide sufficient diversity to cater for 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. These options vary in depth, breadth, content and sometimes also length. The diversity of this offer reflects the different types of mathematics skills young people need, depending on their future aspirations and pathways.
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 within the single programme. 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. England might consider introducing different levels of A level maths for students intending to study social sciences in tertiary education focused on statistics. A more foundational option could provide greater continuity with GCSEs for students who do not achieve Grade C/4 at GCSE, replacing the requirement for re-taking GCSE maths (see Chapter 6 for a discussion of GCSE maths).
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.
2. Most systems explicitly design different mathematics levels and options to cater to students’ future pathways, and communicate this to students
In upper secondary education, mathematics needs to cater to at least three distinct sets of different needs and student profiles. First, there are those who intend to continue the subject in post-secondary education (and most likely, for whom the subject is a strength). Second, are others who will need the subject to underpin learning in other areas, such as sciences or medicine. Third, are students who do not intend to study or work in mathematics-heavy areas but will need mathematics to support their general achievement and as a key life skill.
Most systems have explicitly designed different levels or options in mathematics to cater to these different groups of students. For example, H2 in Singapore, higher mathematics in Ireland and Pre-Calculus and Calculus in British Columbia are expressly designed for students who intend to pursue extensive mathematics in tertiary education.
Policy options for England:
Review mathematics options to ensure that they meet the needs of at least three different student profiles across 16-18.
The current mathematics A level seems to cater most directly for students wishing to pursue mathematics in tertiary education (see Chapter 6). The country might consider how far Core Maths is able to effectively meet the needs of other student groups. One option could be to adjust Core Maths’ design and content to be comparable to a full A level which might be better recognised by tertiary institutions. Within a full maths A level, different levels/options might include:
An intermediate option or level to cater for students intending to pursue social sciences, humanities and even some science post-school who need opportunities to continue developing their mathematical skills to support data and statistical analysis in their domains, as well as for life in general.
A foundational level course for those who do not achieve at least Grade C/4 at GCSE to continue building their core mathematics knowledge and skills (replacing GCSE re-sits).
England might work with tertiary institutions to ensure that course content for new levels reflects well the needs of tertiary education to promote their recognition (Chapter 8 discusses the demand for maths set by tertiary institutions).
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 but will clearly require these skills in their daily lives 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.
3. Mathematics in vocational education tends to be integrated as applied content but some systems also provide mathematics as a separate subject
Systems with strong vocational education tend to have comparatively strong outcomes in mathematics, particularly following completion of upper secondary education, reflected in the PIAAC data (see Chapter 3). In most vocational systems, mathematics is integrated as applied content in line with a student’s occupational focus. In Austria, a system with highly developed technical education, all students must study mathematics, even if the occupational field they are pursuing might involve more routine uses of mathematics such as catering or forestry. In other systems, such as Denmark and Singapore, mathematics may not always be included in the vocational programme, depending on its occupational focus (see Chapter 4).
In Austria and Denmark, upper secondary vocational students achieve particularly strong outcomes in mathematics. Notably, vocational upper secondary graduates in these systems achieve higher numeracy outcomes than students from general upper secondary education in England in PIAAC (see Chapter 3). One way in which these high numeracy skills are supported is likely the strong integration of mathematics and STEM-related content across Vocational Education and Training (VET) programmes. Another reason might be the requirement for some vocational students to study mathematics as a separate, more school-based programme in the five-year Colleges of higher vocational education and training (BHS) in Austria and the Vocational education examination qualifying for access to higher education (EUX) in Denmark. Both of these programmes that provide a direct pathway to tertiary education (see Chapter 4).
Policy pointers for England:
Ensure that there is strong integration of mathematics across all vocational upper secondary programmes, as applied content.
Explore the option of creating separate, more school-based mathematics requirements for some vocational students to provide robust preparation in mathematics and a potential pathway to technical and mathematics-related content in tertiary education.
Table 5.3 provides an overview of key insights, policy pointers and practical country examples for England’s consideration.
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← 1. The 2015 revised GCSE examinations were first sat in 2017 and the 2015 revised A level examinations were first sat in 2016.
← 2. Grades are awards from A-E, with tertiary entrance typically requiring Grades A-C.
← 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[54]).