Mathematics education does not take place in a vacuum. It is greatly influenced by and must reflect or even anticipate changes in the educational and social system. (Howson, 1978, p. 183)

A critical part of mathematics education in any system of schooling is the curriculum, which includes the prescribed mathematics content in the form of syllabuses and or standards and pedagogical approaches. Over the span of the last two centuries although the content of the school curriculum has expanded to meet the changing needs of society, there has been stability in the structure even as waves of reform have swept across the surface (Kilpatrick, 1996). The fact that more than fifty countries participating in the Trends in Mathematics and Science Studies from 1995 till the present concur on the core mathematical knowledge that is tested at the fourth and eighth grade levels suggest that most countries appear to have the same basic mathematical content knowledge taught by certain grade levels (Linquist et al., 2017). This is not unexpected as the canonical school mathematics curriculum, a result of curriculum developers in one educational system simply copying what is being done in another, was developed in Western Europe in the aftermath of the Industrial Revolution and adopted practically in every country during the twentieth century (Howson & Wilson, 1986). The adoption was either voluntary or via colonisation.

Despite its seemingly structural stability, school mathematics curriculum has made significant shifts in emphasis periodically. These shifts have been consequences of national or international initiatives resulting from the evolution of mathematics content knowledge, developments in learning theories and needs of societies (Howson et al., 1981). Chapters 32 and 33 illustrate two such cases that provide food for thought about underlying reasons that result in reforms in the teaching and learning of mathematics in a nation or in a wider context. The purpose of this chapter is to augment Chaps. 32 and 33 and provide some background of reforms in mathematics education as to glean knowledge about the why and what of these reforms.

School Mathematics Reforms

The Era of “New Math”

The “New Math” reform, sparked by the successful launch of an unexpected first satellite into orbit around the earth by the Soviet Union in 1957, began in the US. With mathematicians taking a deeper interest in what to teach in the schools a revamp of the content was carried out and new topics like modular arithmetic, set theory, abstract algebra, etc. were introduced (Hayden, 1981). Groups of teachers guided by mathematicians wrote the “New Math” textbooks, and by 1960, New Math curriculum materials were available for use in schools in the US. In a rush to put the books in the classrooms, many other aspects of implementing a curricular change were lacking.

As such in the US, the reform was short lived and by the early 1970s New Math was dead (Klein, 2003). This was so as social issues overtook curricular ones. It is apparent from the book Why Johnny can’t add: The failure of the new math (Kline, 1973) that, in haste to implement New Math, teachers had to enact it without knowledge and understanding whilst the wider society was ignorant of the need for change. Schoenfeld (2004), in his reflection on the New Math, aptly notes that:

Specifically, it provides a cautionary tale for reform. One of the morals of the experience with the New Math is that for a curriculum to succeed it needs to be made accessible to various constituencies and stakeholders. If teachers feel uncomfortable with a curriculum they have not been prepared to implement, they will either shy away from it or bastardise it. If parents feel disenfranchised because they do not feel competent to help their children, and they do not recognize what is in the curriculum as being of significant value (and what value is someone trained in standard arithmetic to see studying ‘clock arithmetic’ or set theory?) they will ultimately demand change? (p. 257)

Elsewhere the reform made its way over time. In some places it did not take root (for example Hungary) while in others it did influence the intended school mathematics curriculum (for example in Singapore). In Hungary, “news of the sputnik shock and the subsequent boom of the New Math with its fresh wind and dust storms did not reach us [them] before some years later” (Halmos & Varga, 1978, p. 225). When it did, the local (Hungarian) reform in school mathematics was already well underway with mathematicians, teachers and the wider public working together towards “efficient mass education in mathematics” (p. 227), that included “working on miscellaneous problems” (p. 227) and “mathematical literary – reading formulae, graphs, statistical tables, understanding what is behind them” (p. 227). The approach adopted by the local reform and the content focus was affirmed when news began circulating about the shortcomings of New Math – typified by the claim “why Johnny can’t add” (p. 228).

In Singapore, up to the late 1950s, several mathematics syllabuses were in use as schools were vernacular in nature with the Chinese, Indian, Malay and English schools (where the medium of instruction was Chinese, Indian, Malay and English respectively) adopting their curricula from China, India, the Malay Archipelago and Britain respectively. The first local set of syllabuses for mathematics for use in all schools, both primary and secondary schools, was drafted in 1957 and published in 1959 (Lee, 2008). A revision of this set of syllabuses took place in the early 1970s in response to the New Math reform that was traversing the world.

While the primary mathematics curriculum was added an outcomes-based approach, the secondary school mathematics curriculum included ‘modern mathematics’ topics such as modular arithmetic, set theory, transformations, data representations and analysis. However, by the end of the decade it was replaced as globally the curriculum was no longer in tandem with the University of Cambridge Examinations Syllabuses (Lee, 2008) that was adopted by the Ministry of Education (MoE) for mathematics. It may be said that this was perhaps due to the colonisation of Singapore by the British from 1819 till 1963 that impacted many adoptions by Singapore’s Education System, a significant one being the Cambridge Examinations. As mentioned in the introduction, school mathematics curriculum is canonical and therefore world trends que curriculum adoptions in many countries, including Singapore.

Realistic Mathematics Education (RME)

As the “New Math” reform was making its way into countries around the world, in the Netherlands the emergence of an initiative, Realistic Mathematics Education (RME), led by mathematics didacticians in 1968 ensured that the Dutch mathematics education was not influenced by the formal approach of the ‘New Math’ movement. Freudenthal (1983), a mathematician with deep interest in mathematics education, introduced the method of didactical phenomenology that made RME a domain-specific instruction theory for mathematics.

Based on the teaching principles of RME, a number of local instruction theories and paradigmatic teaching sequences focusing on specific mathematical topics have been developed over time (see van den Brink, 1989; Streefland, 1991; de Lange, 1987). Design research guided the local instruction theories (Gravemeijer, 1994). With the availability of technological tools for mathematics instruction, the development of local instruction theories included technology (for details, see Drijvers, 2003; Bakker, 2004; Doorman, 2005). Almost five decades on, RME is still work in progress (van den Heuvel-Panhuizen, 2020). It may be considered as a research and development venture that directly inputs into mathematics classroom instruction. The RME approach has completely influenced the intended school mathematics curriculum in the Netherlands through the textbooks for both primary and secondary schools (van den Heuvel-Panhuizen & Drijvers, 2014).

During the 13th International Congress on Mathematical Education (ICME) held in 2016 during the Thematic Afternoon on “European Didactic Traditions”, RME was illuminated, through presentations by the traditional owners of RME and others who have adopted or adapted aspects of RME in their countries, states, or classrooms. An outcome of the afternoon is a 366 pages publication that has documented the adoptions and adaptations. This is a testimony to the impact of RME in many countries beyond the Dutch mathematics classrooms (van den Heuvel-Panhuizen, 2020). The publication notes that making acquaintance with RME was in most cases the result of a personal encounter at a gathering of mathematicians or mathematics educators somewhere in the world. This suggests that the spread of RME has been an educational endeavor and not a politicised one. International collaborations have also led to RME-based textbooks in the US, “Mathematics in Context” (Wisconsin Centre for Educational Research and Freudenthal Institute (2010) and a mathematics education reform, known as Pendidikan Matematika Realistik Indonesia (PMRI), based on RME in Indonesia for more than two decades, that began in 1994 (Zulkardi et al., 2020).

The ‘Model’ Method: A Pedagogical Reform in Primary School Mathematics

In Singapore, school mathematics curriculum reforms have been driven by both global trends and national needs. A reform that arose out of a national need to improve the learning of mathematics was a pedagogical reform in primary school mathematics that has had outreach internationally. A study carried out in 1975, revealed that 25% of students, after six years of primary school, failed to meet the minimum numeracy level by the standards of the MoE (MoE, 2009a). This and similar findings for other school subjects prompted the Prime Minister in August 1978 to call for a review of the education system that resulted in formulation of a New Education System (NES) (MoE, 1979). The NES was implemented in 1981. The goal of the NES was to provide improved education for every child in the system.

As part of the NES, the establishment of the Curriculum Development Institute of Singapore (CDIS) in June 1980 was an important milestone. Its main function was development of curriculum and teaching materials. It was directly involved in the implementation of syllabuses and systematic collection of feedback at each stage of implementation so that subsequent revisions and refinements would be strategic (Ang & Yeoh, 1990). Among the various project teams, at the CDIS was the Primary Mathematics Project (PMP) team. The task for this team was to produce instructional materials for the teaching and learning of primary mathematics with effective teaching approaches and professional development of teachers (MoE, 2009a). In 1981, the team administered diagnostic tests of basic skills of mathematics to a sample of 17000 Primary 1 to Primary 4 students. The findings were dismal with more than half of Primary 3 and 4 students doing poorly on items that tested division, 87% of Primary 2 to 4 students could solve word problems when key words like ‘altogether’ and ‘left’ were provided but only 46% of them could solve word problems without key words (MoE, 1981).

The PMP team comprising experienced educators from schools and the MoE, together with expertise of international consultants, produced the new primary mathematics curriculum in 1981. The curriculum adopted a concrete–pictorial–abstract approach for the teaching and learning of mathematics. This approach provided students with the necessary learning experiences and meaningful contexts, using concrete manipulatives and pictorial representations to construct abstract mathematical knowledge (CDIS, 1987). In the new primary mathematics curriculum, the ‘model’ method, a heuristic to solve word problems, was included. Theories that underline the method and the method, are discussed and described respectively, elsewhere (see Kaur, 2019a).

The concrete–pictorial–abstract approach pervaded the design of the materials and pedagogy of teaching primary mathematics henceforth. The ‘model’ method, helps students visualise the abstract mathematical relationships and the varying problem structures through pictorial representations (Kho, 1987). The method, initially meant for upper primary school grades (4–6) in 1983, is now an essential feature of the Singapore primary mathematics curriculum and introduced to students in primary 1 (MoE, 2009a).

The stellar performance of Singapore students in International Benchmark Studies (Kaur et al., 2019), Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA), has drawn a lot of attention to the method that is often referred to ‘Singapore Math’. Several countries, including Brunei, Thailand, South Africa, and various states in North America have attempted to adopt the method by customising Singapore textbooks for their use and initiate reform in their mathematics classrooms (see Sim, 2014; Teng, 2014; Tham, 2014). However, successful adoption would need teachers to have a good grasp of the pedagogy and sound mathematical knowledge. This is crucial, as when the PMP team set forth to develop the materials there was equal emphasis placed on the professional development of teachers alongside them.

Era of Problem Solving

In the US, the aftermath of the New Math era, was a ‘Back to Basics’ turn, i.e. the pre-New Math curricula was re-adopted. The outcome of this turn, as noted by Schoenfeld (2004) was:

By 1980, the results of a decade of such instruction were in. Not surprising, students showed little ability at problem solving—after all, curricular had not emphasized aspects of mathematics beyond mastery of core mathematical procedures. But performance on the “basics” had not improved either. (p. 258)

In response to the poor performance, the National Council of Teachers of Mathematics (NCTM) in the US published An agenda for action in 1980. This agenda called for problem solving to be the focus of school mathematics (NCTM, 1980). The agenda was timely as more evidence appeared about the dismal performance of US students in the Second International Mathematics Study (SIMS) in the mid-1980s:

There is one consistent message. Students from the United States, regardless of grade level, generally lag behind many of their counterparts from other developed countries in both mathematics and science achievement. That, perhaps, is the only consistent message. (Medrich & Griffith, 1992, p. 29)

However, as to what ‘problem solving’ entailed was again a contentious issue. It is apparent that the agenda for action, followed by the curriculum and evaluation standards (NCTM, 1989) and principles and standards for school mathematics (NCTM, 2000) were sources of continued debate on mathematics education in the US (Schoenfeld, 2004). The present common core state standards in mathematics appear to be where the focus is at in the US at present (CCSSM, 2010).

As noted in the preceding sections, in Hungary and the Netherlands, ‘problem solving’ was already part of their school mathematics curricula much earlier than the intent of the US. Nevertheless, the agenda was timely as globally there was an emerging interest in problem solving as it essentially emphasised the acquisition of mathematical knowledge to solve non-routine problems or doing realistic mathematics through applications, modelling and mathematisation (de Lange, 1996). This emphasis also stemmed from the need to prepare students to be competent citizens for democratic life as opposed to qualifying for the future work force (Keitel, 1993).

In Singapore, in the early 1980s, there was also the ‘Back to basics’ turn as basic numeracy skills of students across the school system continued to decline following adoption of Modern Maths which was done in haste to keep abreast of global trends. However, the ‘Back to basics’ turn is best known as the ‘Mathematics for every child’ reform in Singapore (Kaur, 2019b). This reform was in sync with the New Education System (MoE, 1979) that was implemented in 1981. The 1980 NCTM’s agenda for action drew attention of educators in Singapore and a decade later in 1990, after careful deliberations by educators at the Institute of Education, MoE and classroom practitioners the framework for school mathematics curriculum was detailed with mathematical problem solving as its primary goal. This framework, shown in Fig. 34.1, has been steadfast for the last three decades (MoE, 2018).

Fig. 34.1
An illustration of the mathematics curriculum framework. A pentagon with the text mathematical problem-solving in the center and attitudes, skills, concepts, processes, and metacognition in the sides.

Singapore school mathematics curriculum framework. (MoE, 2018, p. 15)

Full size image

The framework makes apparent that for students to be mathematical problem solvers they must acquire conceptual knowledge, mathematical skills, mathematical processes, have good attitudes for learning and be metacognitive. This framework is robust as it also aligns with the student outcomes of twenty-first-century competences, shown in Table 34.1, which are: a confident person, a self-directed learner, an active contributor and a concerned citizen (Wong, 2016).

Table 34.1 Student outcomes (twenty-first-century competences) and components of the Singapore school mathematics curriculum framework
Full size table

‘Looking Forward’ and ‘Looking Back

In this section, we examine the key issues related to reform in school mathematics curriculum illuminated in Chaps. 32 and 33.

The OECD appears to have envisioned and mapped the future direction for school mathematics curriculum 2030. As noted in Chap. 32 by Taguma et al., the future is unpredictable. However, by being cognisant of the current trends “we can learn – and help our children learn – to adapt to, thrive in and even shape whatever the future holds” (Taguma et al., Chap. 32 Abstract). The OECD Learning Compass 2030, formulated after extensive research carried out by the OECD, has seven elements. The elements are: 1. student agency/co-agency; 2. core foundations; 3. transformative competences; 4. knowledge; 5. skills; 6. attitudes and values; 7. Anticipation–Action–Reflection cycle. Details of these elements are in Chap. 32.

Implications of the Learning Compass 2030 for mathematics curricula stem from future workplace demands in an environment that is highly automated. In a nutshell, we note that content and practices in mathematics lessons must:

  • go beyond core foundations in numeracy and disciplinary knowledge (e.g. number systems, geometry and operations) and place a greater focus on contemporary topics such as statistics, data analysis and computational thinking;

  • engage students with epistemic knowledge of maths as part and parcel of the work students do in their mathematics lessons;

  • engage students in interdisciplinary tasks so that they apply mathematical knowledge in authentic settings;

  • facilitate the development of skills needed for life-long and self-regulated learning.

This is so that our students are future-ready for the decade ahead of us! In incorporating the above in the school mathematics curriculum, a unique challenge confronting the custodians of the curriculum would be about what to keep or replace in the present curriculum in view of the time allotted for mathematics instruction.

As every leaner must reach his or her potential in mathematics there is a need to cater to individual learning needs. For this, curriculum experts and learning scientists need to work hand-in-hand and prepare curriculum guides that illuminate learning trajectories in mathematics. Lastly, for the vision 2030 to be carried out, teachers must be developed and society be kept abreast of the rationale for change in school mathematics content and practices.

Some elements of the OECD Learning Compass 2030 are already present in the Singapore school mathematics curriculum as of the “Values-based, student-centric phase (2012-present)” of mathematics education of Singapore (Kaur, 2019b). The twenty-first-century competences framework 2010 (MoE, 2009b) was put forth by the Curriculum 2015 committee set up in 2008 to study twenty-first-century skills and mind-sets needed to prepare future generations in Singapore for a globalised world framed this phase.

The framework led to a review of all mathematics syllabuses for Singapore schools in 2010. The revised syllabuses of 2012 (MoE, 2012) implemented in 2013, made explicit that learning mathematics is a twenty-first century necessity and it is a key fundamental in every education system that aims to prepare its citizens for a productive life in the twenty-first century. It also noted that for Singapore as a nation the development of a highly skilled and well-educated manpower was critical to support an innovation- and technology-driven economy. The goal of the national mathematics curriculum was to ensure that all students achieve a level of mastery of mathematics that will serve them well in their lives, and for those who have the interest and ability, to pursue mathematics at the highest possible level.

The syllabuses placed heightened emphasis on the role of learning experiences for mathematics learning. They stated that:

Learning mathematics is more than just learning concepts and skills. Equally important are the cognitive and metacognitive process skills. These processes are learned through carefully constructed learning experiences. For example, to encourage students to be inquisitive, the learning experiences must include opportunities where students discover mathematical results on their own. To support the development of collaborative and communication skills, students must be given opportunities to work together on a problem and present their ideas using appropriate mathematical language and methods. To develop habits of self-directed learning, students must be given opportunities to set learning goals and work towards them purposefully. A classroom rich with these opportunities, will provide the platform for students to develop 21st century competencies. (MoE, 2012, p. 22)

In 2011, nation-wide professional development of mathematics teachers was carried out to prepare them for the implementation of the 2012 revised mathematics curriculum. The implementation of these syllabuses began in 2013. Following a six-year cycle of review and revision of mathematics curriculum, in 2018 the next revision of the syllabuses has taken place. The 2018 revised school mathematics curriculum has included emphasis on “epistemic knowledge of maths as part and parcel of the work students do in their mathematics lessons”. This again affirms the impact of global initiatives on the school mathematics curriculum in Singapore.

McCallum, in Chap. 33 presents the product of the ‘math wars’, the Common Core State Standards in Mathematics in the US. The math wars began with the publication of NCTM’s Curriculum and Evaluation Standards for School Mathematics in 1989 (NCTM, 1989). Through the lens of mathematicians and mathematician educators, what appeared to be a resolved product, the Common Core State Standards in Mathematics (CCSSM, 2010), have raised challenges for the robust enactment of it.Views of mathematics amongst the curriculum writers and enactors have led to two significant perspectives – one on what mathematics is, and another on how it is learned.

Aptly, McCallum has surfaced a duality that merits attention of mathematician educators and researchers. The sense-making stance, a process of people making sense of mathematics, and the making-sense stance where the mathematics content is structured in ways that it makes sense. This duality calls for both the stances to be examined simultaneously as in the works of Ball and Bass (2003) and Iszák and Beckmann (2019) elaborated in the chapter.

On one hand, a pitfall of standards, as in the CCSSM, is that the subject may be reduced to a list of items, what Schoenfeld (1992) refers to as ‘bite-sized’ pieces, which, when enacted without coherence and fidelity may lead a learner to view “mathematics as facts and procedures”. On the other hand, McCallum also cautions that sense-making and making-sense could be futile if learning materials are inappropriately organised as in the two examples Hiebert et al. (1996) and Illustrative Maths (2017) he cited in the chapter. It looks like there is much work ahead for all, mathematicians, mathematician educators, researchers, curriculum designers, and teachers to curtail desired outcomes of the standards. Just like the RME in the Netherlands, this could be work in progress for the next few decades!

In Singapore, the sense-making stance has dominated instruction for a long while. However, in the revised school mathematics curriculum (MoE, 2018) for implementation in 2020, the making-sense stance has been initiated through big ideas, not meant to be authoritative or comprehensive, namely notations, diagrams, proportionality, models, equivalence, measures, invariance and functions for secondary schools, and notations, diagrams, proportionality, models, equivalence and measures for primary schools.

The curriculum notes that there are two orientations to mathematics learning that are relevant to the design of the syllabuses. They are: (i) learning mathematics as a tool that places emphasis on using mathematics to solve problems; and (ii) learning mathematics as a discipline that places emphasis on understanding the nature of mathematics illuminating the practices of mathematicians. As every review of the curriculum in Singapore is guided both by internal needs and by global trends, one may speculate that the ‘math wars’ in the US has initiated the making-sense stance in the curriculum in Singapore. For the initiative to take root, development of teachers, curriculum materials and research on the what and how of BISM (Big Ideas in School Mathematics) have started and in due course there will be lessons to share with the mathematics education fraternity.

Concluding Remarks

Curricula reforms are inevitable as systems must continuously strive for improvements. The stimulus for reform can be external or internal to a system. The examples in this chapter together with Chaps. 32 and 33 provide us insights of a few reforms in school mathematics. Chapter 33 illuminates tensions in intentions of the curriculum at the micro-level that may lead to dire consequences of the outcomes of intentions. Such micro-level deliberations are important as not all such deliberations are in the purview of many curriculum policy initiators or others involved in the process of enacting a reform. What is significant is the research aspect warranted for inputs into the efficacy of a reform.

Reforms like the RME and the ‘model’ method are each examples of reforms that grew out of the needs of an educational system. Their positive impact on the learning of mathematics merited adoptions in other countries. Global trends and societal needs have also fuelled reforms and “the era of problem solving” is one that though may be said to have a formal launch in the US, was a much needed one world-wide. The theme helped many educational systems to chart directions of their school mathematics programs. These directions have not been unique. In Singapore “problem solving” is still the primary core while in the USA, other developments have over shadowed it. In a similar vein the OECD Learning Compass 2030 has ignited global reform for stakeholders to adopt or adapt in responsible ways.