This chapter looks at the knowledge and skills profile in mathematics that upper secondary students are expected to develop across the different mathematics programmes in the six focus systems and England (United Kingdom). It reviews official national curricula regulating the teaching of mathematics, and example assessments or examinations from mathematics courses across the different systems to provide insights on the competencies systems expect their students to develop and apply. It reviews each system’s mathematics curriculum, assessment and examinations it focuses on learning objectives, content knowledge, problem-solving skills and mathematical reasoning.
Mathematics for Life and Work
A Comparative Perspective on Mathematics to Inform Upper Secondary Reform in England
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6. The knowledge and skills profile across mathematics programmes internationally
Copy link to 6. The knowledge and skills profile across mathematics programmes internationallyAbstract
Key insights: Knowledge and skills profile across mathematics programmes internationally
Copy link to Key insights: Knowledge and skills profile across mathematics programmes internationallyVocational and lower-level mathematics programmes sometimes set limited expectations for the development of higher-order competencies
Vocational and lower-level mathematics programmes tend to emphasis fluency with routine procedures with less mathematical reasoning, yet there are country examples that demonstrate that mathematical reasoning and problem solving can be integrated across all programmes.
The breadth and depth of England’s A level mathematics is very high
The content of national mathematics programmes at 16-18 in general education in England – A levels – stand out as being particularly broad from a comparative perspective.
Mathematics GCSE in England sets a comparatively high level of demand
The foundation and higher-level mathematics tiers for GCSE in England (typically studied at 14-16) appear more similar in breadth and depth than levels or tiers in other systems.
The breadth and depth of GCSE maths in England (typically studied at 14-16) is comparable to some mathematics programmes studied until 18 or 19 in other systems.
Reviewing mathematics curricula across systems and programmes
Copy link to Reviewing mathematics curricula across systems and programmesIndividuals’ mathematics skills are influenced by the content and learning goals that they are expected to engage with and acquire during upper secondary education. For this report, mathematics curricula experts reviewed upper secondary mathematics programmes across the six focus systems, as well as England. Biographies of the two experts are available in Annex C. The aim of this review was to develop insights – although limited – around how different systems expose individuals to the competencies contributing to mathematical skills.
Sources of evidence
Official curricula
The main sources of evidence for the review were the official national curricula regulating the teaching of mathematics across each system. The official curricula were used to identify the learning objectives and content across the different mathematics programmes and courses. It should be noted that the curricula in upper secondary education can only provide a partial perspective on learning. Mathematical education starts at an early age and the results of the previous stages significantly influence learning outcomes at subsequent levels. For a list of all the mathematics curricula and programme documents reviewed please see Annex B. Readers should be aware that shortly after this review was undertaken, Singapore revised its A-Level mathematics syllabuses.1
Past or example examination questions
Assessments and examinations for upper secondary students provide a practical insight into the competencies that systems expect their students to develop and apply. The curricula review also analysed past or example assessments or examinations from mathematics courses across the different systems. The contrast of the curricula and practical assessment items also provided the opportunity to compare and contrast systems’ learning objectives with the actual questions that students are expected to solve.
Readers should be aware that during COVID-19 and sometimes in subsequent years, many systems adjusted the range and depth of content on which students were assessed to account for disruptions to learning and examination preparation related to COVID-19 (OECD, 2020[1]). Some of the examination papers reviewed might not provide a full reflection of all the content and depth of learning upon which systems typically assess their students. For a list of all the mathematics assessments and examinations reviewed please see Annex B.
Limitations
This curricula review provides a single perspective on mathematics across the different systems. There are many other factors which shape the teaching and learning of mathematics in education systems, and it would be misleading to form overall judgements based on this single source of evidence. In particular, the teaching tradition of the subject, teachers’ professional competencies, the content of textbooks and other supporting materials constitute equally important factors shaping mathematical education in any system.
Dimensions of the review
The chapter’s review of curricula and assessment items focused on four dimensions:
1. Learning objectives: the overarching ideas which should guide the pedagogy, providing an essential entry point to each system’s curricula.
Within the overall learning objectives, the review focused on features of mathematics curricula which are commonly present in most systems, and reflect broad consensus on the important concepts in modern mathematics curricula, including:
2. Content knowledge: the range of content knowledge covered.
3. Problem-solving skills: the skills required for students to connect the context of a problem with mathematics and solve it.
4. Mathematical reasoning: the use of proper reasoning and assumptions to arrive at results that can be trusted across a wide range of real-life contexts (OECD, 2018[2]).
The last three concepts: content knowledge, problem-solving skills and mathematical reasoning are discussed below.
Content knowledge
The content for teaching and learning defines the set of concepts, relations between them and procedures which form the basis for teaching students the method of mathematics. A review of content knowledge across a range of countries is challenging because traditional mathematics terms such as algebra, geometry and numbers have diverse meanings across different systems. To avoid misunderstandings, the OECD’s Programme for International Student Assessment (PISA) has defined four content areas:
quantity
change and relationships
space and shape
uncertainty and data
These four areas were used in PISA 2012 and 2022 when mathematics was the main domain. They were chosen to reflect mathematical phenomena that underlie many types of problems, the general structure of mathematics and the major strands of typical school curricula (OECD, 2018[2]). They are explained in further detail in Box 6.1.
Box 6.1. Categories of mathematical content knowledge areas used in the PISA 2022 Mathematics Framework
Copy link to Box 6.1. Categories of mathematical content knowledge areas used in the PISA 2022 Mathematics FrameworkQuantity
Quantity incorporates the quantification of attributes of objects, relationships, situations and entities in the world, understanding various representations of those quantifications and judging interpretations and arguments based on quantity. To engage with the quantification of the world involves understanding measurements, counts, magnitudes, units, indicators, relative size and numerical trends and patterns. Aspects of quantitative reasoning include number sense, multiple representations of numbers, elegance in computation, mental calculation, estimation and assessment of reasonableness of results.
Change and Relationships
The natural and designed worlds display a multitude of temporary and permanent relationships among objects and circumstances, where changes occur within systems of interrelated objects or in circumstances where the elements influence one another. In many cases these changes occur over time and, in other cases, changes in one object or quantity are related to changes in another. Some of these situations involve discrete change; others change continuously. Some relationships are of a permanent, or invariant, nature. Increasing literacy for change and relationships involves understanding fundamental types of change and recognising when they occur in order to use suitable mathematical models to describe and predict change. Mathematically this means modelling the change and the relationships with appropriate functions and equations, as well as creating, interpreting and translating among symbolic and graphical representations of relationships.
Space and Shape
Space and shape encompass a wide range of phenomena that are encountered everywhere in our visual and physical world: patterns, properties of objects, positions and orientations, representations of objects, decoding and encoding of visual information, navigation and dynamic interaction with real shapes as well as with representations, movement, displacement and the ability to anticipate actions in space. Geometry serves as an essential foundation for space and shape; but the category extends beyond traditional geometry in content, meaning and method by also drawing on elements of other mathematical areas such as spatial visualisation, measurement and algebra. For instance, shapes can change and a point can move along a locus, thus requiring function concepts. Measurement formulas are central in this area. The recognition, manipulation and interpretation of shapes in settings that call for tools – ranging from dynamic geometry software to Global Positioning Systems (GPS) and to machine learning software – are included in this content category.
Uncertainty and Data
In science, technology and everyday life, variation and its associated uncertainty is a given. It is a phenomenon at the heart of the theory of probability and statistics. The uncertainty and data content category includes recognising the place of variation in the real world, including having a sense of the quantification of that variation and acknowledging its uncertainty and error in related inferences. It also includes forming, interpreting and evaluating conclusions drawn in situations where uncertainty is present. The presentation and interpretation of data are key concepts in this category.
Source: OECD (2018[2]), PISA 2022 Mathematics Framework Draft.
Mathematical reasoning
The unpredictability and diversity of today’s world means that learners who are only equipped with ready-to-use procedures will find this insufficient when they are faced by unexpected and unfamiliar contexts and situations. The ability to adopt knowledge (e.g. operations, equations, formulae, etc.) to new situations is now a necessity to deal with everyday problems. Students’ ability to adapt existing tools to new problems can be supported by developing their mathematical reasoning.
Mathematical reasoning (both deductive and inductive) involves evaluating situations, selecting strategies, drawing logical conclusions, developing and describing solutions and recognising how those solutions can be applied (see Box 6.2 for a further explanation of mathematical reasoning) (OECD, 2018[2]). When confronted with a problem which does not look like anything practiced in their classroom, students with strong mathematical reasoning skills are more likely to attempt to solve the problem. In contrast, students with weak mathematical reasoning might not attempt to solve previously unseen problems, believing that it requires the reproduction a common memorised procedure.
Problem solving
Mathematical problem solving is key to successful application of mathematics in real life. Mathematical problem solving is based on students:
1. formulating situations mathematically
2. employing mathematical concepts, facts, procedures and reasoning
3. interpreting, applying and evaluating mathematical outcomes, eventually leading to appropriate corrections of the model
Models used in a school setting should avoid artificially constructed situations, such as calculating areas of a grass lawn in the shape of an equilateral triangle, which do not occur in real life. Instead, problems should always be authentic i.e. related to problems that students are likely to experience in real life, such as calculating changes to stopping speed and distance of vehicles in different weather conditions.
When teaching mathematics, ensuring that students develop the underpinning mathematical understanding is essential. Sometimes in countries, mathematics curricula introduce very advanced mathematics, for example differential and integral calculus or even differential equations. In these situations, students in upper secondary education focus on being able to apply these sophisticated tools, without necessarily achieving the underpinning understanding of how they work. Quite often, the same problems can be successfully analysed by using simpler mathematical concepts, such as the graph of a quadratic function or by referring to elementary inequalities. A good lesson of this approach is the well-known historical example of the Greek Aristarchus (310-230 BCE) who, using simple bare-eyed astronomical observations and similarity of triangles, was able to find good approximations to the sizes and distances from the Earth of the Moon and Sun.
Box 6.2. Mathematical reasoning in the PISA 2022 Mathematics Framework
Copy link to Box 6.2. Mathematical reasoning in the PISA 2022 Mathematics FrameworkThe ability to reason logically and to present arguments in honest and convincing ways is a skill that is becoming increasingly important in today’s world. Mathematics is a science about well-defined objects and notions which can be analysed and transformed in different ways using ‘mathematical reasoning’ to obtain conclusions about which we are certain. Through mathematics, students learn that through using appropriate reasoning they can reach results and conclusions which they can trust to be true. Further, those conclusions are logical and objective, and hence impartial, without any need for validation by an external authority. This kind of reasoning, which is useful far beyond mathematics, can be learned and practiced most effectively within mathematics.
Two aspects of mathematical reasoning are especially important in today’s world and in defining the PISA items. One is deduction from clear assumptions (deductive reasoning), which is a characteristic feature of mathematical process. The second important dimension is statistical and probabilistic (inductive reasoning). These days, at the logical level, there is frequent confusion in the minds of individuals between the possible and the probable, leading many to fall prey to conspiracy theories or fake news. From a technical perspective, today’s world is increasingly complex and its multiple dimensions are represented by terabytes of data. Making sense of these data is one of the biggest challenges that humanity will face in the future. Our young people should be familiarised with the nature of such data and making informed decisions in the context of variation and uncertainty. In PISA 2022, approximately 25% of the score points in mathematics focus on assessing mathematical reasoning.
Source: OECD (2018[2]), PISA 2022 Mathematics Framework Draft.
Depth and breadth
The review looked at curricula in terms of breadth and depth. Both are important for students’ ability to identify the mathematical implications of a given situation, find and apply the appropriate mathematics knowledge to solve a problem and check their answer.
Breadth of content – refers to the range of content covered i.e. the proportion of concepts and procedures students are expected to master. In the review, systems with extensive breadth expect students to master a very broad range of knowledge and skills, while those with initial breadth focus on a narrower range of concepts and procedures.
Depth – refers to how deeply students are expected to acquire knowledge and skills within specific areas. In the review, programmes with extensive depth expect students not only to be familiar with and apply competencies but also to develop understanding of their purpose, use and application. In contrast, programmes with initial depth require less depth of application and understanding. Depth can be inferred from the types of analysis that examination papers require related to specific concepts or procedures.
Programmes’ breadth and depth is shaped by their learning goals, student profile and future pathways. For students intending to study mathematics at higher levels, both significant breadth and depth will be important while for others, a solid grasp of operations required in daily life is most important.
The review of national curricula and examinations conducted for this report broadly categorised the breadth and depth of different systems’ mathematics programmes as extensive, moderate and initial. It should be noted that this categorisation is broad, approximate and partial, based on a review of available materials, such as the curriculum documents and recent national examinations. Examination papers change over years and the same content is not always assessed every year. To fully understand the experienced breadth and depth of curricula, knowledge of teaching and learning practices and prior learning are also important.
The knowledge and skills profile across different upper secondary mathematics curricula
Copy link to The knowledge and skills profile across different upper secondary mathematics curriculaTable 6.1 provides an overview of the curriculum and assessments for certification reviewed for this report. The text below highlights the main insights from this review.
There are significant similarities in the concepts and procedures upper secondary students are expected to acquire across countries
Overall, while there are significant variations in the descriptions of mathematical content across different systems, there is a high degree of commonality across concepts and procedures taught to upper secondary students in the six focus systems, as well as England. More broadly this reflects the relative consistency in the mathematical concepts across different systems’ curricula. There are also significant similarities in the learning objectives. In many cases, the importance of problem-solving skills and mathematical reasoning are reflected in the learning objectives.
Vocational and lower-level mathematics programmes sometimes set limited expectations for the development of higher-order competencies
Most systems offer more than one mathematics programme. Typically, this includes different mathematics programmes for general and vocational students, reflecting the broader organisation of upper secondary education in the country. In Austria, for example, there is one mathematics programme for general students and three mathematics programmes for vocational students. Countries also often provide mathematics options at different levels for example, in Ireland, students choose between higher, ordinary and foundation levels of mathematics.
Some countries expect all students enrolled in a programme, including across different levels, to demonstrate higher-order competencies such as mathematical reasoning and problem solving. In Ireland for example, all levels of the Leaving Certificate Established/Vocational from foundation through to higher expect students to acquire and demonstrate problem-solving skills. In Poland, curricula reforms have also sought to integrate greater mathematical reasoning for all students as discussed in Box 6.3. Yet in other systems, vocational and lower-level mathematics options give less space to higher-order competences such as mathematical reasoning and problem solving, instead placing greater focus on fluency in reproducing the trained algorithms.
Box 6.3. Integrating a greater focus on mathematical reasoning in Poland
Copy link to Box 6.3. Integrating a greater focus on mathematical reasoning in PolandCurricula reform
In 2008-2009, Poland undertook a series of curricula and structural reforms. This included defining new learning outcomes for each stage of education, indicating the main objectives of each subject and defining the requirements of central assessments and evaluations.
The reformed curriculum strengthened teacher autonomy and teachers’ responsibility for the learning process by defining learning outcomes broadly. The country’s national examinations were aligned with the new curriculum. For all subjects, including mathematics, this meant a shift from knowledge reproduction to a stronger focus on reasoning. One important change to examinations is that all exams are expected to include types of problems that students have never confronted in textbooks or in class. Being exposed to such problems encourages students to understand that mathematics is not about memorising set procedures to expected problems but identifying the mathematics in unseen and unexpected situations. Putting students in the position of being able to discover solutions for themselves, even for relatively simple problems, is empowering and motivating for students.
The changes were implemented gradually, first being announced publicly three years before their implementation. Teachers, students and parents were informed about the changes with the opportunity to see and learn about the new curriculum and examination prior to implementation. Preparations included teacher training and a public television campaign to inform students and their parents. Between 2009 and 2012 the mean Polish students’ mathematics score in PISA went from 495 to 518 score points. This improvement was mostly associated with improved responses to PISA items measuring more complex, analytical thinking.
Mathematical reasoning in the curriculum and examinations
In Poland, all students, across the extended and basic scope mathematics courses for general students, and the mathematics course for vocational students, are expected to demonstrate mathematical reasoning skills. Notably:
All general students, including those in the basic scope programme, are expected to prove and justify their solutions in the national exams. In terms of problem solving, they are expected to apply various mathematics tools to solve both practical and theoretical problems.
Students in the extended scope programme are also expected to design and justify multistep strategies, and to apply various mathematical tools to solve both practical and theoretical problems.
Students in the vocational programme are expected to solve practical problems focused on the specific occupational area of their vocational programme.
Source: Jakubowski, M. (2021[3]), Poland: Polish Education Reforms and Evidence from International Assessments, Springer, Cham, https://doi.org/10.1007/978-3-030-59031-4_7; Marciniak, Z. (2015[4]), Reviewing Polish education reform in the late 1990s—possible lessons to be learned.
Table 6.1. Overview of mathematics curricula and examinations in focus systems
Copy link to Table 6.1. Overview of mathematics curricula and examinations in focus systems|
|
Programme and course |
Balance of breadth and depth |
Content knowledge |
Mathematical Reasoning |
Problem solving |
|---|---|---|---|---|---|
|
Austria |
Academic secondary schools (AHS) |
Moderate breadth & depth |
|
Core competencies stress understanding of concepts and relations between them necessary for mathematical reasoning. |
Students expected to interpret results of their mathematical operations. Must understand functions as mathematical models and work with them appropriately. Application of mathematics to problem solving implied. Context of applications quite narrow. |
|
Higher vocational education and training (BHS) |
Moderate breadth; initial depth |
Similar to scope above; some clusters of applications include logarithms, variation of a random variable, etc. Strictly related to specific applications. |
Fluency with routine mathematical operations. Practical tasks require some invention with mathematical tools. |
Examination tasks focus on practical problems. Sometimes the mathematical tool is clear or explicitly indicated, limiting the assessment of problem-solving skills. |
|
|
Intermediate vocational education and training (BMS) |
Initial breadth & depth |
Basic computational tools to perform tasks in the learning objectives (left box). |
Reproduction of trained procedures. Mathematical reasoning is absent. |
Primarily production of computational procedures to solve typical classes of problems. |
|
|
Apprentice-ships/dual system |
Initial breadth & depth |
Focus on calculations needed for a particular profession. |
As above. |
Problems common to a specific profession. |
|
|
British Columbia |
Grade 11 - Foundations of Mathematics |
Moderate breadth & depth |
|
Mathematical reasoning explicit e.g., logic, conjecturing, inductive and deductive thinking, proofs, game/puzzle analysis, counter-examples. |
Includes mathematical problem-solving skills including methods of proving solutions are correct. |
|
Grade 11 - Workplace Mathematics |
Initial breadth & depth |
|
Focus on trained procedures. Mathematical reasoning not mentioned. |
Problem solving focused on a class of practical problems. |
|
|
Grade 12 - Calculus |
Extensive breadth; initial depth |
|
Content is similar to the calculus requirements in a formal engineering course. Mathematical reasoning in this area needs deep understanding of very complex mathematical ideas. |
Problem solving restricted to standard applications of calculus for optimisation problems. |
|
|
British Columbia |
Grade 12 - Apprentice ship Mathematics |
Moderate breadth; initial depth |
|
Focus on trained procedures. Mathematical reasoning not mentioned. |
Problem solving focused on a class of practical problems. |
|
Denmark 1 |
The Higher Technical Examination Programme (HTX) A-level |
Extensive breadth & depth |
|
Students are required to provide explanations and proofs. |
Students are expected to apply mathematics procedures to solve real life problems. |
|
The Higher General Examination Programme (STX) B-level |
Extensive breadth; moderate depth |
|
As above. |
As above. |
|
|
EUD |
Initial breadth & depth |
|
Mathematical reasoning covered in curriculum goals. |
Problem solving mostly related to financial literacy. |
|
|
England |
GCSE Foundation |
Moderate breadth; extensive depth |
|
Justifying results and proving is required, especially in geometry. Some exam problems require designing a simple strategy. Most exam items ask to show working, but these are mostly reproductions of standard procedures. Students are not asked to give arguments justifying statements. |
Problem solving is stressed in both practical and intra-mathematical contexts. The modelling cycle is set out in the learning outcomes. |
|
Higher GCSEs |
Moderate breadth; extensive depth |
As above and in addition:
|
Mathematical reasoning is explicitly listed among the objectives and also appears in the content description – using vectors in proofs. In exams there are questions where students have to provide arguments to justify their observations or make a judgement and explain their reasoning. |
As above. |
|
|
A levels |
Very extensive breadth; extensive depth |
|
Mathematical reasoning is explicitly present in the exams with students having to provide justifications and proofs. In the case of the area of the differential and integral calculus area most problems require just a reproduction of a standard procedure. |
Understanding and applying the problem-solving cycle is one of the main objectives. Some exam items are good illustrations of problem solving. |
|
|
Ireland 2 |
Leaving Certificate Established / Vocational – Foundation |
Initial breadth & depth |
|
Mostly reproduction of routine procedures. |
Supported problem sol-ving in practical situations. |
|
Ordinary |
Moderate breadth & depth |
As for the Foundation course plus:
|
Mathematical reasoning is present in geometry. |
Problem solving in more abstract contexts is also present. |
|
|
Higher |
Extensive breadth; moderate depth |
As for the Ordinary level plus:
|
Strong presence in geometry, also in other sections. |
As above. |
|
|
Leaving Certificate Applied |
Initial breadth & depth |
Focus on most routine applications of mathematical tools in a few real-life contexts. |
Covered in the learning objectives. |
Simple problem-solving skills. |
|
|
New Zealand |
NCEA Apply algebraic procedures in solving problems (Level 1) |
Moderate breadth & depth |
|
Represented by extended abstract thinking. |
Mathematical problem-solving skills in practical and purely mathematical contexts. |
|
NCEA Investigate a situation involving elements of chance (Level 1) |
Moderate breadth & depth |
|
Present in strategy of designing a probabilistic model and in justification of conclusions from the model. |
Demonstrated mostly by relating the real-world situation to mathematical tools. |
|
|
Apply calculus methods in solving problems (Level 2) |
Moderate breadth & depth |
|
Mostly reproduction of set of procedures. |
Mostly reproduction of set of procedures. |
|
|
Use statistical methods to make an inference (Level 2) |
Extensive breadth, moderate depth |
|
Present in strategy of designing a probabilistic model and in justification of conclusions from the model. |
Demonstrated mostly by relating the real-world situation to mathematical tools. |
|
|
Apply integration methods in solving problems (Level 3) |
Extensive breadth, moderate depth |
|
Use of advanced mathematics; exam items require routine applications of standard procedures. |
Problems solving mostly intra-mathematical; some items are standard applications. |
|
|
Use critical path analysis in solving problems (Level 3). |
Initial breadth & depth |
|
Not mentioned. |
Problem solving as designing a strategy; relatively narrow scope of application. |
|
|
Singapore 3 |
Express Mathematics O-Level. |
Moderate breadth and depth |
|
Required to provide explanations. |
Emphasis on mathematical problem-solving skills. |
|
A-Levels Mathematics H1 |
Moderate breadth; extensive depth |
|
Included as ability to analyse mathematical situations and construct logical arguments. |
The central focus of the curriculum is development of problem-solving competence. |
|
|
Mathematics H2 |
Extensive breadth; moderate breadth |
|
Develop thinking, reasoning, communication and modelling skills through a mathematical approach to problem solving. |
Develop thinking, reasoning, communication and modelling skills through a mathematical approach to problem solving. |
|
|
Mathematics H3 |
Extensive breadth & depth |
|
Very strong presence of formal proofs: proof by mathematical induction, proof by counterexample, proof by contradiction, proof of existence, the pigeonhole principle. |
Problem solving techniques in abstract contexts. |
Notes: 1A new curriculum in Mathematics (A, B, and C-levels) is in effect since August 2024; 2Depending on the examination year reviewed some of the curriculum content from the syllabus may not have been assessed, this does not mean that it has or will not feature on another assessment. Examination papers for the years 2020 through to 2023 should be viewed with caution as adjustments were made due to the impact of Covid. Some of the curriculum content from the Higher level .eg. co-ordinate geometry of the Circle does not feature but is part of the curriculum content and can and is assessed; 3New A-level mathematics syllabuses were implemented in 2024. The revised A-Level Maths syllabuses can be found accessed here: https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses; https://www.seab.gov.sg/home/examinations/gce-a-level/a-level-syllabuses-examined-for-school-candidates-2025. New syllabuses were not available at the time of this review and so information presented in the table and across the chapter focuses on the previous syllabuses. Content in the new 2024 H2 maths syllabus has been slightly reduced. The new H3 mathematics syllabus will be implemented in 2025.
Learning objectives
Appreciation of mathematics is frequently included in learning objectives
The most common learning objective across the national curricula is that students acquire sufficient knowledge of concepts and procedures within the content scope of the curriculum. Many systems also include an appreciation of the value of mathematics and developing confidence in mathematics in their learning objectives. In Ireland, for example, all students taking Leaving Certificate mathematics, regardless of the level, are expected to develop a productive disposition, which is the habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence, perseverance and one’s own efficacy (Department of Education, 2015[5]). Singapore’s learning expectations are perhaps the most developed in this direction. From when students begin N-Level mathematics at age 12 they are expected to build confidence and foster interest in mathematics. Students taking H2 A-Level mathematics are expected to experience and appreciate the nature and beauty of mathematics and its value in life and other disciplines (Table 6.1).
Students’ anticipated future pathways shape learning objectives
Some mathematics programmes are shaped by expected future student pathways. In some systems, this is explicitly stated in the learning objectives. For example, in Austria, learning expectations for VET programmes focus on acquiring skills to perform mathematical operations in occupationally-relevant areas. In Singapore, the different levels of the A-Level programme differentiate between students that are expected to progress to social sciences and business-related studies in tertiary education – for whom H1 is intended – and those who are expected to progress to science, technology, engineering and mathematics studies and who are recommended to enroll in H2 (Table 6.1).
Content areas
There is significant similarity in the content learning areas covered
The similarities in the learning content across the focus systems reflects broad similarity across countries internationally. However, the focus and intended future pathways of different programmes sometimes shapes how content is taught across different systems. For example, in Austria, the 5-year Higher Vocational Education and Training programme (BHS), focuses on different applications of mathematics for practical situations, reflecting the programme’s vocational focus (Table 6.1).
England’s maths A level covers a very wide range of concepts and procedures
Systems that expect their students to master an extensive range of concepts and procedures tend to be higher level papers for general upper secondary students. Programmes and courses in this group include the H2 and H3 A-Level programmes in Singapore, the A and B level programmes in Denmark, the Higher level in Ireland, and the A level in England. Of all the reviewed programmes, the content of maths A level in England has the greatest breadth. A level mathematics goes beyond the content typically specified in upper secondary mathematics programmes internationally by including statistical distributions, statistical hypothesis testing, integration and numerical methods. England’s A level also includes content that is normally covered in physics, such as quantities and units in mechanics, kinematics, forces and Newton laws, in other systems.
It is notable that the other examples of programmes with extensive breadth are all the higher tier or levels of different mathematics programmes. In contrast, A level mathematics in England does not provide students with the option of different levels. Overall, the A level sets a very high bar in terms of the range of procedures and concepts that students are expected to master – greater than the other systems - and provides students with limited options to study mathematics aside from this extensively broad programme. In this context, it might be expected that A level mathematics caters to a small, very able, minority of the 16-18 cohort.
Learning expectations in modular programmes are more dispersed
The modular nature of the curricula in British Columbia and New Zealand sometimes makes reviewing curricula breadth more challenging because expectations are spread widely across a range of different courses, which students might or might not take. Most of the courses in British Columbia cover the full range of the four content areas covered by the PISA 2022 mathematics framework. (Box 6.1). While the modular programmes naturally cover a narrower set of content, the breadth of some specific concepts is still high. For example, the Calculus programme for Grade 12 students in British Columbia (Canada) sets out a high degree of breadth in its learning objectives within the area of Change and relationships.
The degree of modularity in New Zealand’s curriculum is even more pronounced (and is one of the reasons that all the achievement standards are being reviewed as part of the National Certificate of Educational Achievement (NCEA) Change Programme, see Chapter 5) which means that most assessment specifications cover only one or two mathematics content areas. The dispersion of different content and learning aims across units raises questions about students’ ability to acquire a solid base in key mathematics skills. Some research suggests that modularisation reduces time for deeper, cumulative learning that occurs over a longer period (Hayward, 2007[6]).
Mathematical reasoning and problem solving
Countries interpret mathematical reasoning in different ways
While mathematical reasoning is broadly reflected in the reviewed curricula, there are diverse interpretations. In some systems, mathematical reasoning supports students to be ready to flexibly adopt the mathematics concepts they know to attack any problem they encounter. This interpretation is closer to how mathematics is assessed in PISA, where students are expected to be able to see and solve mathematical situations across a broad range of contexts. For example, in the HTX and STX mathematics courses in general education in Denmark and some statistical mathematics courses as part of the NCEA in New Zealand, students are required to develop their own strategies to solve problems and justify their solutions mathematically. In other systems, mathematical reasoning is understood more as fluency in applying sometimes very sophisticated mathematics tools to deal with fairly routine problems.
England’s maths GCSE expects extensive depth
England’s GCSE mathematics programmes (typically studied at 14-16) stands out as expecting students to master concepts and procedures at significant depth. In contrast, most of the systems requiring students to master mathematical content at extensive depth in this review were higher level programmes in the final years of upper secondary.
GCSEs in England are provided at two different levels – foundation and higher. In practice however, the review conducted for this chapter suggests that the depth expected across the two levels is comparatively similar. In other systems with different levels of mathematics, such as foundation, ordinary and higher mathematics in Ireland, there is much greater difference in the depth of procedures and concepts that students are expected to master across the levels. Foundation mathematics GCSE in England is intended to be the most accessible mathematics course in the country’s upper secondary education, yet the depth of mastery expected may make it a relatively demanding course, especially given its limited duration (2 years) and the young age of its typical students (14-16). The high level of challenge might be one reason why the greatest proportion of young people taking mathematics at 16-18 in England – in 28.8% in 2022 - are re-sitting GCSE mathematics or equivalent qualifications to achieve a passing grade (Department for Education, 2024[7]).
Vocational and lower-level programmes tend to emphasis fluency with routine procedures
In many of the programmes and courses of applied mathematics, either as vocational upper secondary programmes or as separate applied courses, there is an emphasis on developing fluency with routine procedures with limited mathematical reasoning. Focusing problem solving on the repeated practice of a constrained range of procedures seems to be more pronounced in vocational and professionally-oriented courses. The VET mathematics tests in Austria, for example, provide practical problems but often the mathematical tool is explicitly indicated or is obvious. This means that the assessment primarily focuses on assessing the reproduction of procedures practiced at school. Similarly, while Grade 12 Calculus in British Columbia provides a very high degree of breadth, the problem solving that students are expected to do is restricted to standard applications of calculus problems (Table 6.1).
Some courses integrate problem solving and mathematical reasoning from the start of upper secondary, and for all levels and mathematics options
Most curricula and examination papers provide some coverage of problem solving, often with expectations for problem solving distributed across different levels. In Ireland, students taking mathematics at foundation level are exposed to problem solving in practical situations with support to identify solutions, while at the ordinary level, expectations for problem solving are extended to abstract contexts. In New Zealand, NCEA level 1 (which typically corresponds to Year 11, the first year of upper secondary education) mathematics includes an assessment "Apply algebraic procedures in solving problems”, whereby problem-solving skills are required in both practically and purely mathematical contexts. In British Columbia, the Grade 10 numeracy assessment that students are required to take demonstrates problem solving in authentic real-life contexts, for example calculating car breaking distances in different weather conditions. Similarly, in New Zealand at NCEA level 1, mathematical reasoning is assessed in an example of an internal assessment for the course “Investigate a situation involving elements of chance”. The course requires students to develop their own investigation of change, analyse data and draw conclusions. Box 6.3 sets out reforms in Poland that included integrating mathematical reasoning across all programmes.
How do breadth and depth of upper secondary mathematics programmes in England compare internationally?
Copy link to How do breadth and depth of upper secondary mathematics programmes in England compare internationally?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 (Table 6.1). 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[8]). 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 (Vidal Rodeiro, 2022[9]), 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 (Vidal Rodeiro, 2022[9]). 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[10]). 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.
Both GCSEs and A levels have reasonably good coverage of mathematical reasoning and problem solving
One of the aims of this review was to look at how far countries expect students to acquire and demonstrate the practical application of mathematics in unfamiliar contexts and problems i.e., mathematical reasoning skills. Both maths GCSE and A level in England were considered to have good coverage of mathematical reasoning, with students required to design simple solution strategies from GCSE Foundation level. In GCSE Higher level, expectations for mathematical reasoning were higher with students having to provide arguments to justify their observations or form a judgement with reasoning. In A level mathematics, mathematical reasoning is clearly expected with students required to provide justifications and proofs. Other systems with strong coverage of mathematical reasoning include H3 mathematics in Singapore’s A-Level with a very strong presence of formal proofs, higher mathematics in Ireland with a particularly strong presence in geometry and in the HTX and STX programmes in Denmark.
Key insights and policy pointers: the knowledge and skills profile across mathematics curricula internationally
Copy link to Key insights and policy pointers: the knowledge and skills profile across mathematics curricula internationallyThis chapter has identified several common trends in the expectations that systems set for mathematical knowledge and skills. In line with these trends, several policy pointers can be identified around how systems, and England in particular, set expectations for mathematics knowledge and skills in upper secondary education.
1. Vocational and lower-level mathematics programmes sometimes set limited expectations for the development of higher-order competencies
Across countries, the tendency is to include only limited expectations or a limited notion of higher-order skills like mathematical reasoning and problem solving in upper secondary mathematics programmes pitched at a lower level or in vocational education. Yet, evidence from some systems show it is possible to include mathematical reasoning and problem solving from very early in upper secondary education and for all types of students, such as Ireland’s foundation mathematics. This can be achieved through simple yet unexpected contexts and problems that require students to draw on the mathematics tools they are familiar with (which might include simple addition, subtraction, multiplication and division) to develop a strategy and justify their solution through mathematical workings.
Policy pointers for England:
Set ambitious yet achievable expectations for all learners in 16-18 mathematics.
Policies to increase participation in mathematics across the 16-18 cohort in England will need to address how to provide mathematic options that respond to the learning levels and future ambitions across the full cohort. It will be important that these options provide all students with the support and expectation to develop key competencies that they will need in adult life – such as mathematical reasoning and problem solving.
Draw on the examples of systems which provide expectations for problem solving and mathematical competence for all students.
In developing mathematics for more students, England might look at the practical examples of the kinds of questions and assessments that students in lower level and vocational programmes are required to engage with in some systems to demonstrate problem solving and mathematical reasoning. For example, in Ireland’s foundation level mathematics, students engage with problem solving through scaffolded questioning and support to find the solution. Similarly, the statistical units at NCEA level 1 in New Zealand where students are required to develop probabilistic models, provide an example of ensuring that all students access mathematical reasoning and problem solving.
2. The breadth and depth of England’s A level mathematics is very high
The review of mathematics programmes presented in this chapter and other previous comparative reviews have highlighted the very high level of breadth and depth set by this A level (Ofqual, 2012[8]). At the same time, analysis presented in Chapters 7 and 8 of this report shows that 15-year-old students in England do not have more negative views of mathematics than their peers in other systems. In fact, often in terms of enjoyment and the value that students place on mathematics, 15-year-olds in England frequently have more positive views than in other OECD systems (OECD, 2023[11]).
While it is important to set high expectations for all, and to stretch the most able young mathematicians, England might consider how far the high demand of A level mathematics is at least partly fuelling national perceptions that mathematics is unnecessary and difficult. While Core Maths seems to cater well to the needs of students who do not pursue A level mathematics, the share of students achieving the qualification has never increased beyond 2% of the cohort since its introduction in 2015 (Department for Education, 2023[10]). In the absence of any popular differentiated mainstream options or levels, A level mathematics might be contributing to the societal acceptance that it is ok not be good at mathematics (because the level of demand set by A level mathematics is beyond the achievement of the average student).
Policy pointers for England:
Draw on international examples of how other high performing systems provide different mathematics options to cater to the needs and interests of the full student cohort up to 18.
Consider providing greater choice and diversity for mathematics at 16-18 to provide a range of different options at different levels of breadth and depth to meet different needs and interests across the student cohort.
Note that providing diversity in post-16 mathematics options does not necessarily require major structural changes and England could consider providing greater diversity within existing mathematics programmes. The latter is the approach that many of the focus countries reviewed in this report take, providing levels within wider mathematics programmes. For example, Denmark, Ireland and Singapore provide three different levels of mathematics within their main upper secondary mathematics programmes for general education.
3. Mathematics GCSE in England sets a comparatively high level of demand
The mathematics programme that nearly the full cohort of 14-16-year-olds in England take – GCSE maths – seems to set a relatively stretching benchmark. The breadth and depth of maths GCSE is comparable to other mathematics programmes reviewed for this report – such as the basic scope mathematics in Poland and the H1 mathematics in Singapore. Yet, students taking GCSEs are comparatively younger than their counterparts in Poland and Singapore and have just two years to cover the GCSE content. In addition, young people taking GCSE mathematics are also studying and taking examinations in around 9-10 other subjects.
While setting high and demanding expectations for students is important, it is also critical that expectations are achievable and reflect the mathematics that young people are likely to need in work and education post-schooling. There is likely space to consider the breadth and depth of maths GCSE to ensure that expectations are accessible for all learners. Almost a third (29.3%) of 16-18-year-old students studying mathematics are re-sitting their mathematics GCSE (UK Government, 2023[12]). These data lend weight to the view that the GCSEs set a particularly high level of demand that some students in practice need four years to reach. While GCSE mathematics is provided at two different levels – foundation and higher – from a comparative perspective, there seems to be limited differentiation across these two levels.
Setting a high level of demand in GCSEs might contribute to perceptions of mathematics as unnecessary and too difficult, which in turn reduces participation at 16-18.
Policy pointers for England:
Review the skills and knowledge that young people need in mathematics and the expectations that maths GCSE sets.
England might review young people’s needs for mathematics skills in post-16 education and the workplace and how far these needs are aligned with the expectations set by GCSEs. The high share of young people not passing GCSE mathematics – almost a third in 2023 – might suggest that the bar is unachievably high for some students.
Consider introducing a more accessible mathematics programme pitched at a lower level than GCSE foundation mathematics.
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 with maths post-16, especially among young people who are required to continue trying to pass GCSE mathematics after 16 through re-sits.
Consider creating longer mathematics courses that span 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 for mathematics is that students must cover content and develop deeper mathematical reasoning skills over a comparatively short course – two years in each case. The fact that a third of students spend an additional two years studying and trying to pass GCSE mathematics might create the case for providing similar content and expectations over the full duration of upper secondary i.e. four years, for some students.
Table 6.2 sets out the key insights from this chapter around the review of the knowledge and skills profile across upper secondary programme. It also identifies potential policy pointers for systems, and England specifically, based on these insights and comparative country practices.
Table 6.2. Overview of key insights and policy pointers for the knowledge and skills profile across mathematics programmes internationally
Copy link to Table 6.2. Overview of key insights and policy pointers for the knowledge and skills profile across mathematics programmes internationally|
Key Insights |
Policy Pointers for England |
Country Examples |
|---|---|---|
|
1. Vocational and lower-level mathematics programmes sometimes set limited expectations for the development of higher-order competencies |
When setting expectations, consider ways in which competencies like mathematical reasoning and problem solving can set accessible challenges for all students. |
Ireland – Foundation level mathematics provides supported problem solving Poland - Basic vocational schools provides expectations for mathematical reasoning in practical problem solving |
|
2. The breadth and depth of England’s A level mathematics is very high |
Assess how far A level maths meets national skill needs and review mathematics options at 16-18 to consider providing a range of different options at varying levels of breadth and depth to meet different needs and interests across the student cohort. |
All focus systems (with the exception of Austria) provide mathematics at different levels in upper secondary education |
|
3. Mathematics GCSEs in England set a comparatively high level of demand |
Consider introducing a more accessible mathematics programme pitched at a lower level or that spans the full duration of upper secondary education. |
Austria, British Columbia (Canada), Ireland, New Zealand and Singapore – all provide examples of programmes with more accessible breadth and depth than GCSE foundation mathematics |
References
[7] Department for Education (2024), Academic year 2022/23: Level 2 and 3 attainment age 16 to 25, https://explore-education-statistics.service.gov.uk/find-statistics/level-2-and-3-attainment-by-young-people-aged-19#contact-us (accessed on 15 October 2024).
[10] Department for Education (2023), Post-16 maths participation for pupils ending KS4 in 2018/19 - Ad hoc statistics.
[5] Department of Education (2015), Mathematics Syllabus: Foundational, Ordinary & Higher Level.
[6] Hayward, G. (2007), “Modular mayhem? A case study of the development of the A-level science curriculum in England”, Assessment in Education: Principles, Policy & Practice, Vol. 14/3, pp. 335-351.
[3] Jakubowski, M. (2021), Poland: Polish Education Reforms and Evidence from International Assessments, Springer, Cham, https://doi.org/10.1007/978-3-030-59031-4_7.
[4] Marciniak, Z. (2015), Reviewing polish education reform in the late 1990s—possible lessons to be learned.
[11] OECD (2023), PISA 2022 Online Education Database.
[1] OECD (2020), Lessons for Education from COVID-19, https://doi.org/10.1787/0a530888-en.
[2] OECD (2018), PISA 2022 Mathematics Framework Draft.
[8] Ofqual (2012), International Comparisons in Senior Secondary Assessment.
[12] UK Government (2023), “16 to 19 funding: maths and English condition of funding”, https://www.gov.uk/guidance/16-to-19-funding-maths-and-english-condition-of-funding.
[9] Vidal Rodeiro, C. (2022), The impact of GCSE maths reform on progression to A level, Cambridge University Press & Assessment, https://www.cambridgeassessment.org.uk/Images/687723-the-impact-of-gcse-maths-reform-on-progression-to-a-level.-.pdf (accessed on 15 October 2024).
Note
Copy link to Note← 1. New A-level mathematics syllabuses were implemented in 2024. The revised A-Level Maths syllabuses can be found accessed here: https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses;https://www.seab.gov.sg/home/examinations/gce-a-level/a-level-syllabuses-examined-for-school-candidates-2025 . New syllabuses were not available at the time of this review and so information presented in the table and across the chapter focuses on the previous syllabuses.