An Evolution of Mathematics Curriculum

The increasing availability of international data on student performance has provided the opportunity for researchers to examine the influence of mathematics curriculum across countries on students’ achievement (Schmidt et al., 2001[1]). Further analysis of mathematics curricula is needed to better understand global educational trends and outcomes.

This chapter describes the key findings from the international mathematics curriculum studies carried out by the OECD Future of Education and Skills 2030 project. It focuses on the following aspects of curriculum development for which systematic comparative data are available: i) how mathematics curricula are evolving across countries, particularly on content coverage; and ii) how countries are integrating some of the so-called 21^st-century competencies in their mathematics curricula. The findings invite a reflection on some gaps identified between current curricula and aspirations for the future, which require further investigation.

Comparison of curricula worldwide, and the subsequent association of curriculum with student achievement, has only become possible in the last 30 years with the increased participation of nations and jurisdictions in international assessments such as the Trends in International Mathematics and Statistics Study (TIMSS) and the Programme for International Student Assessment (PISA). As part of TIMSS in 1995, Schmidt et al. (2001[1]) studied the structure of the mathematics curricula across the participating countries. TIMSS was the earliest opportunity for the comparison of student achievement, through large-scale testing of students across over 50 countries. The study demonstrated a clear relationship between the structure of mathematics and science curricula and achievement of students, as measured by TIMSS assessments, and showed significant differences among the national curricula analysed by structure and content.

Twenty-five years later, participating countries in the OECD Future of Education and Skills 2030 (E2030) project agreed to look deeper into how mathematics curricula have changed and how they still need to evolve to prepare students to meet the demands of the 21^st Century, such as societal changes including technological advances. This section outlines the key findings from the E2030 Mathematics Curriculum Document Analysis (MCDA) study (Schmidt et al., 2022[2]) with respect to curriculum content.1

Recognising the evolving transformations in society, a future-oriented mathematics curricula needs to continue to include an education about formal mathematics (formal ideas, concepts, algorithms and procedures that shape the discipline), but also include opportunities for children to develop the type of subject-specific reasoning that equips them to develop their understanding and application of mathematics (mathematics, statistics, geometric and algorithmic reasoning) in real life. This is reflected in the MCDA Framework, which was used for cross-country comparison of curricula. The framework includes the content in Table 2.1, organised around the key focus areas covered in mathematics curriculum in the first eight years of education: quantity, space and shape, change and relationships, statistics, probability and data.

Figure 2.1 provides a visual comparison of content coverage across the 19 participating countries/jurisdictions between 1995 (content included in the curricula of top-performing countries in TIMSS-95) and 2020 (MCDA study).

The curriculum content coverage comparison shows which topics formally appear in the curriculum standards of the MCDA participating countries/jurisdictions, as well as the grade year in which they are included. Overall, the study reveals:

• a relative stability in mathematics curriculum coverage patterns over this 25-year period;

• a noticeable increase in topics related to statistics;

• the predominance of mathematical literacy in curriculum standards (discussed later in this chapter), particularly quantitative reasoning, real-world applications, and 21^st-century competencies (albeit at varying degrees of emphasis);

• the rare inclusion of algorithmic reasoning (also part of mathematical literacy) and non-linear statistical models.

Other important findings emerged – related to the accompanying analysis of textbooks – which will be discussed later in this chapter. In the sequence, we discuss the rise of mathematical literacy and of statistics. These changes are significant and understandable given many changes to the societal, political and academic landscape that influence curricula development. The impetus for change in modern curricula comes from many sources, as seen in Chapter 1, including globalisation, access to and use of digital and other technologies, emerging opportunities and demands of workplaces and wider citizenship, concerns over social inequity and learnings from research in mathematics and statistics education. Participatory citizenship also demands greater ability to apply academic knowledge of mathematics and statistics to situations, from managing spending and borrowing, to interpreting advertising data, to designing houses for functionality, to planning and booking travel, to comparing quotations for services, to evaluating data about educational and health facilities.

The concept of numeracy, which first appeared in the British Crowther Report in 1959 (Kus, 2018[3]) has grown significantly in importance over time. This increase has been driven by the recognition that applying mathematics is not only a key predictor of individual success, but also an essential driver of national economic performance. As numeracy became more politically and socially significant, it began to influence mathematics curricula worldwide. This evolution in the role of numeracy has led to its deeper integration into curricula, reflecting societal demands for a more future-ready workforce.

On an individual level, it is well established that knowledge of mathematics and statistics is positively associated with personal life outcomes. High achievement in numeracy at school exit predicts employability, income and socio-economic status, as well as health and well-being outcomes in adulthood (Bruine de Bruin and Slovic, 2021[4]; Bregant, 2016[5]). Numeracy and higher income are also positively associated with life satisfaction, in general (Bjälkebring and Peters, 2021[6]). At a societal level, numeracy contributes significantly to national economic growth and stability. Higher levels of numeracy contribute to more efficient labour markets, as workers are better equipped to handle tasks requiring calculation, measurement as well as data analysis, technological proficiency, and critical thinking. In turn, this enhances productivity across various industries, from finance to healthcare and manufacturing. Furthermore, a numerate workforce is essential for addressing complex societal challenges, such as climate change, public health and economic inequality, all of which require data-driven decision-making (OECD, 2019[7]).

The growing importance of numeracy has reshaped educational approaches, with a focus on equipping students with skills applicable to real-world contexts. The OECD Learning Compass for Mathematics 2030 (OECD, 2023[8]) defines numeracy as the ability to interpret, assess and communicate mathematical information and ideas in a variety of contexts. In line with this, numerate students are expected to apply their mathematical understanding not just in school, but in everyday life. Often referred to as "mathematical literacy" in the PISA assessments, numeracy requires students to engage with mathematics and statistics in realistic situations, many of which may extend beyond their immediate experiences. These contexts provide opportunities for students to broaden their perspectives, but they also increase the cognitive demand by requiring them to draw on prior knowledge and apply multiple strategies for problem solving. In this way, numeracy education, with its emphasis on contextual problem solving, represents a greater challenge than traditional mathematics instruction, which has, in the past, often developed concepts and skills in isolation from application.

In the past, mathematics education often focused heavily on rote memorisation and the mastery of standard procedures, such as long division or algebraic manipulation. However, the shift towards numeracy in recent years reflects a broader aim: to equip students with the ability to apply their mathematical understanding to dynamic and often unfamiliar situations. This evolution in pedagogy has brought about the inclusion of more complex, open-ended tasks and problem-solving activities in curricula, encouraging students to think critically and flexibly.

However, the integration of numeracy into curricula presents its own set of challenges. Educators now need to balance the development of traditional mathematical skills, such as manual calculation or proof-based reasoning, with newer competencies that emphasise real-world application and digital fluency. In today’s classrooms, content that once focused on repetitive manual calculations is being reconsidered. With the increasing accessibility and importance of digital tools, students are learning how to use technology – such as spreadsheets, graphing software, statistical modelling programs and generative AI – to analyse data, simulate real-world scenarios and make informed decisions. This shift has sparked debates about the relevance of traditional topics like written algorithmic or advanced algebra, which some argue can be outsourced to technology. Instead, curricula are gradually prioritising tasks that allow students to engage with mathematical reasoning and problem solving through the lens of digital tools, enabling them to work more efficiently and creatively in data-driven environments.

While the discipline of mathematics has a long history dating back many thousands of years with contributions from many civilisations and cultures, the discipline of statistics is relatively new. Although data collection, such as census-taking, dates back to the Roman Empire, the development of ways to makes sense of such data began in the 18^th Century and emerged strongly as a modern approach in the late 19^th Century. Statistics emerged as a discipline in the 1960s and 1970s, with increasing formalisation of statistics education in curriculum in the 1980s, reflecting the importance of encouraging students to engage more deeply with practical applications of statistics, rather than merely learning abstract concepts. This focus on the applicability of statistics to various fields of knowledge through statistical reasoning and real-world data analysis underscores the growing importance of statistics in modern curricula.

This has provided strong support for inclusion in the curriculum of two aspects of statistics: statistical literacy and probability (Shaughnessy, 2019[9]). Literacy involves reasoning about the data-based reports of others, by applying critical thinking about the methods used (sampling, measures, treatment of confounding variables, etc.) and the efficacy and significance of the findings. Variability is at the heart of statistical enquiry, and all results must be treated with a degree of uncertainty. Probability began as a field of mathematics. The models applied in statistics, such as types of distributions, are founded on probability. However, in the real world, the probability of most events, such as a given weather condition occurring, cannot be established theoretically. The statistical enquiry cycle is enacted with large samples by experiment to predict probabilities. The relationship between theoretical and experimental probability in school curricula is not uniformly agreed, even among statistics educators. Student misconceptions about probability abound in the research literature on this topic, leading some commentators to advocate that probability should be treated informally in primary school, using a more experimental than theoretical approach.

Burrill and Pfannkuch (2024[10]) presented four areas of development for statistics curriculum going forward, as a summary of 50 papers by experts in the field. “Data Science” represents the interface between statistical data analysis and programming to create explanatory models. “Social Statistics” positions statistical investigation and literacy as fundamental to student exploration of issues that are important to society, and advocates for students to become agents for informed societal change. “Contexts for learning” explores the potential for technological data display tools to allow new ways for students to explore real contexts that are of relevance to them. “Visibilising Statistical Concepts” is about the development of new ways of supporting students to develop concepts, such as using technology to illustrate variability in samples.

Despite strong arguments for statistical education at all levels, there are considerable differences in adoption across countries and jurisdictions (Burrill and Biehler, 2011[11]) In the competition for curriculum space, important statistical ideas are often lost, particularly at elementary/primary levels. Statistics can be frequently relegated to the category of non-essential, in contrast to mathematical topics, usually number-related, that are designated as “basic”, despite the importance of statistical literacy to citizenship and workplaces.

Education systems participating in the MCDA study were asked to identify the number of content topics from the framework included in their curriculum standards, as well as the grade levels in which they are expected to be taught. The results underline the various curriculum choices countries make about coverage of these topics, the order in which they are introduced (by grade) and how long they remain in the curriculum. Distinct design choices emerge from a cross-country comparison regarding the focus of curriculum (breadth, depth, balance) and its organisation.

Figure 2.2 illustrates the number of topics from the TIMSS 1995 benchmarking list across all 19 countries/jurisdictions intended to be covered at each grade according to their 2020 curriculum standards. A notable finding is the remarkable expansion in the range of topics introduced in the curriculum, particularly from Grade 4, across participating countries/jurisdictions: as students progress through grades, the variability in the number of topics covered tends to increase.

The range remains wide in the following grades, further expanding in Grades 7 and 8. Grade 8 shows the greatest number of topics intended to be covered across curricula. This seems to reflect the cumulative organisation of the mathematics topics in curricula (with many early topics remaining present in late grades but at a more advanced levels), as seen in Figure 2.1. The wide range of topics planned for Grades 7 and 8 may also signal the potential pressure to prepare students for high school more broadly, and an intention to gear the curriculum towards higher-stakes examinations later. A cross-country analysis provides further insight into these trends (Table 2.2).

Table 2.2 provides an insightful comparison of the range of topics covered across different countries/jurisdictions, categorised into three groups: those below, within, and above the middle inter-quartile range. A few key trends and patterns emerge from these data.

When looking across grade levels, most countries, including Australia, Estonia, Korea, Lithuania and the United States, consistently fall within the middle inter-quartile range across grades. This reflects a balanced approach to curriculum coverage, where the number of topics remains relatively consistent and aligned with the global trends observed in this set of countries/jurisdictions.

Japan, Argentina and Hungary often fall below the middle inter-quartile range, especially in the earlier grades (Grades 1-5). This suggests a more focused or streamlined approach in their mathematics curriculum, where fewer topics are covered, favouring in-depth coverage rather than breadth of coverage, possibly focusing on foundational concepts.

Countries like Latvia, Norway, and Sweden appear frequently above the middle inter-quartile across multiple grades. These countries tend to cover a much greater number of topics compared to the majority of other participants, potentially reflecting a broader curriculum in earlier education. This wide range of topics seems to have contributed to some of the shifts observed in the analysis of curriculum standards in 2020 compared to 1995. A number of topics that in the past were introduced much later in the curriculum started being included in earlier grades, such 2-D geometry, measurement, whole numbers and percentages.

Figure 2.3 allows for an appreciation of the evolution of mathematics curriculum in Japan and Sweden as well as their curriculum design choices, which diverge from the global trends observed.

Japan and Sweden provide a clear contrast in the number of topics covered in their curriculum standards: Japan introduces few topics in early grades and maintains that same approach with a very focused curriculum in Grades 7 and 8. Sweden, on the other hand, clearly shows an extensive range of topics to be covered throughout the curriculum. This illustrates the variations in national priorities when it comes to mathematics education, particularly around the introduction of advanced topics in middle school, including different emphasis placed on depth versus breadth in topic coverage.

The extent to which topics included in curriculum documents are mandatory varies among education systems. Just as countries differ in the coverage of content topics in their curriculum, they may also differ on what is regarded as core/essential learning vs. optional, and on whether such a distinction is needed. These decisions depend largely on their goals, but also on their national and local traditions.

For example, some countries may include a wide range of topics in their curriculum while indicating that not all topics are mandatory. Such non-mandatory topics may be categorised as optional or recommended. They may be added with the intention of giving schools and teachers the flexibility to decide whether to include them in their teaching. This allows for a more adaptive curriculum that can cater to regional needs, school priorities, or even student interests, but it may also create long curriculum documents. It is important that these decisions take into account the local traditions and the likely reactions of teachers to avoid the risk – real or perceived – of content overload (OECD, 2020[12]).

To this end, it makes sense that some education systems explicitly identify their “core learning” or “essential knowledge” in their curriculum documents – e.g. “common core” in Brazil, Costa Rica and the United States; “core components” in the Netherlands; “essential learning” in Portugal – as they signal to teachers what every child should know by the end of a given learning cycle (OECD, n.d.[13]).

In mathematics curricula, the inclusion of optional content can be formalised in different ways:

• Advanced mathematics options: In countries like Singapore and Australia, while there is a core set of topics that every student must cover, there are also optional advanced topics, typically at later grades in primary education and in secondary education. For example, students pursuing higher-level mathematics might encounter additional topics such as calculus, complex numbers or discrete mathematics, which are not required for all students but are offered as elective or specialised topics (Ministry of Education Singapore, 2023[14]; OECD, n.d.[13]).

• Elective pathways: Countries like Finland and the United States offer differentiated pathways in mathematics. Students can choose between standard, advanced or honours-level math courses, each with varying content depth. The more advanced pathways include topics that are optional for students who are not on the specialised or advanced math track (OECD, n.d.[13]; National Agency of Education., 2020[15]).

• Enrichment programmes: In some countries, such as Australia, the United Kingdom, and the United States, there are enrichment opportunities that allow high-achieving students to explore additional mathematical topics that go beyond the standard curriculum. These topics, while not mandatory, are designed to deepen mathematical understanding and may be part of extracurricular programmes or offered to students who excel in mathematics (Massachusetts Institute of Technology (MIT), n.d.[16]; Australian Maths Trust, n.d.[17]; Piggott, 011[18]; Millennium Mathematics Project, University of Cambridge, 2023[19]).

This balance between mandatory and optional topics allows countries to maintain rigorous national standards while providing schools with the flexibility to adapt the curriculum to meet local and student-specific needs.

While the previous section takes stock of mathematics topics included in curriculum standards, this section describes the results of the E2030 Curriculum Content Mapping (CCM) exercise, which provides a supplementary picture of countries’/jurisdictions’ mathematics curricula. In particular, it shows how countries embed various 21^st-century competencies into their lower secondary mathematics written curricula. This question has been partially examined in Chapter 1, which presented the findings for how various competencies are being incorporated across learning areas, including mathematics. The findings in this section are specific to the mathematics curricula of the participating countries/jurisdictions2.When mapping their mathematics curriculum, participating education systems in the CCM exercise used a content framework that includes topics typically present in mathematics lower secondary curricula (Table 2.3, first column). Curriculum experts then rated to what extent each of the various 21^st-century competencies indicated (Figure 2.5, columns 1-28) are intended to be targeted in the teaching of these mathematics topics. They used a 4-point colour-coding scale ranging from “not present” (lightest colour) to “main target” in the curriculum (darkest colour). The results (most frequent rating across 14 countries/jurisdictions) are shown in the “heat map” below (Table 2.3 and Table 2.4) with darker blue cells indicating competencies that are more explicitly targeted across curriculum content items.

As expected, numeracy and problem solving stood out as the most strongly targeted competencies across content topics in mathematics curricula. This was followed by a strong rating of critical thinking and literacy as two other key competencies intended to be developed in the teaching of most mathematics topics, although to a lesser degree. The heavy emphasis on literacy may seem counter-intuitive, as this may be thought to be a more natural competency in language learning, but the results highlight the unequivocal value of literacy and language proficiency for conceptual understanding, for representation of mathematics problems, for interpretation of data, and for problem solving in any discipline/learning area (Caponera, Sestito and Russo, 2016[20]; Jiban and Deno, 2007[21]; Beal, Adams and Cohen, 2009[22]).

These competencies, together with data literacy and ICT/digital literacy are heavily represented, particularly among topics related to data and probability, confirming a growing emphasis in curriculum on statistics (as observed in early grades from the MCDA study) and a growing interest in helping students develop competencies that are essential for learning, working and living in digital environments. This is reinforced separately, too, by the emphasis placed on computational thinking/programming/coding in the teaching of “concepts related to programming, data science, computational thinking”.

The predominance of these foundational cognitive competencies remains at the heart of current mathematics curricula. Furthermore, the results recognise opportunities for teachers to include in their teaching:

• socio-emotional skills, particularly collaboration, self-control/self-regulation, persistence;

• meta-cognitive skills, such as learning to learn;

• self-initiated actions/dispositions, which are captured by the findings related to agency, responsibility and action. To some extent, the Anticipation-Action-Reflection cycle (OECD, 2020[23]) included in the study mirrors this layering of priorities with “reflection” as a primary target for the teaching of problem solving (numbers) and data-related topics, followed by “action”. This may indicate the areas in which mathematics curricula lean themselves more easily to concrete, real-world applications.

A surprising finding is the lower emphasis placed on creativity in mathematics, in spite of its recognised importance as a competency for the future (World Economic Forum, 2016[24]; Azzam, 2009[25]), given its role in advancing knowledge in any field, including in mathematics and technology. Other areas, such as financial literacy and global citizenship, that are less frequently emphasised, also point to opportunities for further integration of practical, real-world skills into mathematics teaching.

The MCDA study provides an opportunity for exploring how the intended curriculum in mathematics (curriculum standards) compares to the taught curriculum (as represented by textbooks) in participating countries/jurisdictions with respect to key mathematics-related competencies, namely: quantitative reasoning, higher-order thinking skills, and selected 21^st-century competencies. This section describes the results of such comparisons.

Figure 2.4 and Figure 2.5 depict the emphasis placed on various types of quantitative reasoning across countries/jurisdictions, as expressed in their mathematics curriculum standards (Figure 2.4) and the corresponding emphasis in representative textbooks (Figure 2.5) used in these participating education systems. While the scales of these graphs are not strictly commensurable, they each represent the explicit or implicit emphasis placed on the given mathematics-related competencies, and as such, they shed light on the alignment – or misalignment – between what education systems intend to teach and what ends up being taught in classrooms.

In areas such as mathematics (e.g. basic numerical reasoning) and geometric reasoning, the charts show a closer alignment between the curriculum standards and textbooks. This suggests that in these areas, what policymakers intend to be taught is more faithfully being translated into textbooks.

Traditional mathematics reasoning (such as in algebra, arithmetic and number theory) receive the strongest emphasis both in curriculum and in the textbooks. This type of quantitative reasoning seems to be well-integrated into both the intended and taught curriculum, possibly reflecting the chief priority across countries to offer students a strong foundation in these essential skills.

The moderate emphasis placed on geometric reasoning in curriculum is closely mirrored in textbooks, recognising geometric concepts as essential but not overly dominant in these first eight grades of mathematics education. The emphasis on statistical reasoning is modest but consistent both in the curriculum standards and in textbooks, reflecting an increasing recognition of the importance of data literacy and statistics in contemporary education.

The cross-country analysis of curriculum standards also shows a significant emphasis placed on algorithmic reasoning in the intended curriculum, perhaps linked to computational and algorithmic thinking in modern mathematics education. This is contrasted by a much weaker emphasis on algorithmic reasoning in textbooks in general, showing that these skills may not be adequately reinforced through textbook exercises.

The Mathematics Curriculum Document Analysis Study (MCDA) report emphasises the importance of developing 21^st-century competencies through mathematics education, particularly by engaging students in quantitative reasoning and in higher-order thinking, including both higher-order-real-world applications (HoRw), and higher-order math-world applications (HoMw) (Schmidt et al., 2022[2])

HoRw tasks are set in real-life contexts that require students to sift through complex data, identify what is relevant, and formulate problems mathematically. These problems often have multiple solutions, reflecting the complexity of real-world decision-making.

In contrast, HoMw tasks are situated within mathematics itself, where students must use deductive reasoning and connect various mathematical concepts – such as recognising theorems and constructing proofs – to find a solution. Both types of tasks focus on fostering reasoning skills that go beyond simple computation, encouraging deeper mathematical thinking and problem solving.

The study found that while countries/jurisdictions emphasise higher-order applications in their curriculum standards (albeit at varying degrees), a profound gap of those intentions exists while analysing their mathematics textbooks for Grade 8, used as a proxy for “taught curriculum”. The textbooks, which are recognised as being widely used in these countries/jurisdictions, overwhelmingly fail in providing students sufficient opportunities to develop higher-order thinking skills, particularly those related to real-world applications. The textbooks analysed were found to be dominated by computational exercises and word problems of little value to help students develop the types of higher-order thinking/reasoning expressed in curriculum documents.

The contrasting figures highlight a significant discrepancy between curriculum standards and textbooks, particularly in the competencies of information use (linked to digital literacy), persistence/resilience in the face of difficulties, and systems thinking (the ability to think holistically beyond isolated parts). These are all important skills that allow students to develop their higher-order thinking. Their weak presence in textbooks reinforces the gap in higher-order thinking both in mathematics and in real-world applications observed in the study and discussed earlier.

A misalignment is also observed – albeit to a lesser degree – in critical thinking and reflection, which receive greater emphasis in curriculum standards compared to what is found in textbooks. Consistency between the intended curriculum and textbooks was only found for communication skills, not surprising given the finding that textbooks are dominated by word problems (Schmidt et al., 2022[2])

The gaps observed suggest that while education systems recognise the importance of these skills, students may not be receiving adequate opportunities to practice and develop them in the classroom. To bridge this gap, textbooks and instructional materials need to be updated to better reflect the competencies emphasised in the curriculum, and teachers need to receive adequate preparation in line with the intended curriculum.

Figure 2.6 and Figure 2.7 allow for a comparison of the emphasis placed on a range of 21^st-century competencies in mathematics curriculum standards and in textbooks.

The contrasting figures highlight a significant discrepancy between curriculum standards and textbooks, particularly in the competencies of information use (linked to digital literacy), persistence/resilience in the face of difficulties, and systems thinking (the ability to think holistically beyond isolated parts). These are all important skills that allow students to develop their higher-order thinking. Their weak presence in textbooks reinforces the gap in higher-order thinking both in mathematics and in real-world applications observed in the study and discussed earlier.

A misalignment is also observed – albeit to a lesser degree – in critical thinking and reflection, which receive greater emphasis in curriculum standards compared to what is found in textbooks. Consistency between the intended curriculum and textbooks was only found for communication skills, not surprising given the finding that textbooks are dominated by word problems (Schmidt et al., 2022[2])

The gaps observed suggest that while education systems recognise the importance of these skills, students may not be receiving adequate opportunities to practice and develop them in the classroom. To bridge this gap, textbooks and instructional materials need to be updated to better reflect the competencies emphasised in the curriculum, and teachers need to receive adequate preparation in line with the intended curriculum.

Education systems use a variety of means for assessing students’ learning, which then are fed back into their own system for evaluating how closely students are demonstrating the types of learning intended and prioritised in their mathematics curriculum. The most immediate level of assessment takes place in classrooms, where teachers get to evaluate on an individual basis to what extent students’ performance reflects their teaching. They can use formative assessment methods like quizzes, observations and in-class assignments as well as summative assessment methods such as examinations and grades (Black and Wiliam, 2018[26]).

Making inferences from students’ learning in a way that allows for comparisons across groups of students, classrooms, schools and regions often requires a certain level of standardisation. This is captured by national standardised testing, often administered at selected time points (grade levels), allowing for some diagnostic understanding and monitoring of how closely students’ performance meets given learning goals/standards, including some manoeuvring from a policy perspective for targeted interventions at critical stages of the learning cycle, by age or grade level (Shepard, Penuel and Pellegrino, 2018[27]; Shepard, 2019[28]).

Content topics in mathematics, given its hierarchical and formal nature, seem to allow for standardisation more easily than other learning areas, such as humanities. Standardised testing may be used at varying degrees in high-stakes assessments, such as high school exit and university entrance examinations. These are placed at the end of compulsory education, at which point the opportunities for reversing trajectories of low achievement in mathematics are very limited (Brookhart, 2020[29]); their suitability for a global assessment of one’s learning has also been questioned.

Challenges related to assessment – and assessment frameworks – remain. While there is a strong tradition of standardised testing of basic knowledge and skills, standardised assessment of higher-order thinking and other competencies emphasised in maths curricula presents significant methodological challenges (OECD, 2013[30]). Higher-order thinking by nature requires students to arrive at their own solution to problems. Wide variation in responses makes it difficult to assess such skills in a standardised way; similarly for the assessment of other competencies, such as critical thinking, creativity and collaboration (Darling-Hammond, 2020[31]). Nevertheless, countries are trying to address these limitations by adopting diverse approaches to assessment. In Sweden, for example, the assessment of mathematical reasoning includes not only written but also oral testing Box 2.1). This is an innovative way to address the difficulties of aligning forms of assessment to broader curriculum goals, and it requires a thoughtful approach – as well as well-prepared educators – to ensure that open assessments remain consistent and fair.

International assessments of students’ learning, such as PISA and TIMSS, provide additional opportunities for education systems to monitor how closely students’ learning reflects their own national/local priorities, as stated in their curricula documents.

The PISA 2022 mathematical framework addresses some of the concerns of assessing not only what can be easily measured (content knowledge), but also highly valued competencies for mathematics, such as applying mathematical knowledge to unfamiliar situations and to solve real-world problems or demonstrating proficiency in mathematical reasoning.

To bring an international perspective to the reflection on how intended curriculum compared to achieved curriculum, Table 2.5 displays the pattern of curriculum content coverage of four PISA top-performing countries/jurisdictions: Hong Kong (China), Japan, Korea, and Chinese Taipei (China)3. While strong inferences cannot be drawn from such exploratory comparison, it allows for an appreciation of curriculum features that these countries seem to have in common. For example:

• All four education systems’ curriculum standards align with global trends on content areas covering the full range of topics in the MCDA benchmarking framework.

• Japan and Korea concentrate on fewer topics in early grades compared to Hong Kong (China) and Chinese Taipei (China).

• They each show a structured, staggered expansion in the range of topics introduced from early to late grades with some noticeable steps: an expansion occurs around Grades 3 or 4; another in Grade 5; with a fuller range of topics being covered by Grades 7 and 8. Japan’s coverage patterns show the most gradual choices compared to the other systems.

Table 2.5 provides a supplementary picture of this by overlaying countries’ curriculum coverage (given by the range of topics) with the latest PISA performance in mathematics. This table draws on Table 2.2, only this time with the PISA performance levels identified.

In the table, the four top-performing education systems mentioned above are now seen in the context of comparison to other countries/jurisdictions. All of them maintain a somewhat selective to moderate range of topics covered across grades, with some convergence towards fewer topics in later grades. This seems to reflect these systems’ preference to offer a narrower but focused curriculum, allowing students sufficient time to build solid foundations in learning in early grades before introducing new or more advanced content. This is in line with earlier cross-country findings that identified focus, rigour and coherence as important characteristics of mathematics curriculum in high-achieving systems (Schmidt et al., 2001[1])

A few countries also performed above the OECD average in mathematics while including a broader range of content topics in their curriculum: Latvia and Sweden (in early grades), together with Estonia, the Netherlands, and New Zealand (from Grade 4). Some countries with below average performance in PISA, namely Argentina, Greece and Kazakhstan also display similar patterns of content coverage as the top performers, underlying the complexities of arriving at direct inferences about the gaps between curriculum standards and student learning. Strong factors explaining discrepancies between mathematics curriculum and student outcomes include the mismatch between:

• written curriculum and textbooks (Schmidt et al., 2022[2]; Valverde and Schmidt, 1997[32]; Schmidt et al., 2013[33]);

• curriculum and teacher preparation (Darling-Hammond et al., 2017[34]; Shulman, 1986[35]);

• curriculum and assessment (Brookhart, 2020[29]; Pellegrino, 2018[36]).

Other gaps remain to be explored that go beyond the scope of this chapter, as they pertain to various other dimensions of curricula. For example, as the relevance of mathematics extends far beyond the classroom and is becoming increasingly important for societies not only for progress in STEM-related fields, but also for individuals in many aspects of daily life (as discussed in Chapter 1), to what extent mathematics curriculum can integrate broader expectations held by multiple social actors (expected curriculum) remains to be explored. Or to what extent intended curriculum compares to students’ lived experiences of learning in classrooms (experienced curriculum)? And how to incorporate new research evidence from learning sciences as well as new approaches to teaching and learning in the digital era as classroom environments start to incorporate artificial intelligence? These questions will continue to offer opportunities for further exploration and for continuous evolution of mathematics curricula with important implications for self-directed and personalised learning, teacher professional development, and new forms of assessment of learning, among others (OECD, 2020[37]).

In retrospect, this chapter reviewed the evolution of mathematics curriculum in various countries/jurisdictions, highlighting the growing emphasis being placed on mathematical reasoning and statistics. It explored how countries structure and distribute content across grade levels, showing variations – even among high performing systems – in their curriculum design, with some choosing a focused curriculum with fewer topics (suggesting deep learning as a priority) while others offer a broad curriculum from early years (perhaps putting emphasis on respecting teachers’ culture of professional autonomy and choice of topics).

The chapter also revealed which key 21^st-century competencies are already being explicitly integrated into lower secondary education in various systems. An emphasis on core cognitive and meta-cognitive foundations was observed with numeracy and problem solving, followed by critical thinking and literacy being strongly represented in the curricula analysed. Data and ICT/digital literacy are also now embedded in mathematics curricula, expressing the importance placed on helping students develop skills required for them to navigate technology-rich environments in school, work and life.

The exploration of gaps in mathematics curricula across education systems revealed concerning discrepancies between curriculum standards and textbooks, which are widely used in mathematics education. This is particularly concerning when it comes to the fostering of higher-order thinking and real-world applications, as textbooks overwhelmingly fail to offer adequate opportunities for students to develop these important competencies. Other important 21^st-century competencies well-articulated in curriculum documents are often very poorly represented in textbooks, particularly information use, persistence/resilience and systems thinking, all of which are essential for students to develop proficiency in mathematics reasoning and to comfortably navigate the digital world.

Finally, when looking at how intended curriculum translated into achieved curriculum, the analysis uncovered the complexities of aligning assessment to broader curriculum goals, including not only content mastery but also future-oriented competencies. It also shed some light on important intervening factors that link curriculum to student learning, as these are more likely to reflect implemented curriculum (textbooks, teaching practices, opportunities to learn) rather than intended curriculum. The following chapter will address a series of challenges in curriculum design and implementation, offering examples of how countries are trying to address these challenges in mathematics curriculum.

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