International Reform Movements in the Twentieth Century

At the beginning of the twentieth century, reforms of mathematics education took place in various European countries, concerning principally secondary and higher education. They aimed to develop a better transition between these levels and to adapt mathematics education to the increasing technological and scientific needs of the period. Mathematicians played leading roles in these reforms: for example, Felix Klein in Germany led the so-called “Merano Syllabus”, Henry Poincaré and Emil Borel among others in France contributed to the 1902 curricular reform (Weigand et al., 2017).

International exchanges promoted these reform movements and offered a basis to international co-operation for the development of mathematics education. Soon after its creation in 1908, the International Commission on Mathematics Instruction (ICMI) was engaged in facilitating this co-operation under the direction of Felix Klein. International comparative reports were created on the development of the teaching of various topics, contributing to the dissemination of reform ideas in Europe. World War 1 broke this collaboration, which was restarted only after World War II, in the 1950s.

Significant curriculum changes took place in many countries at this period. In the Western block, the so-called ‘Sputnik shock’ (the launch of the first Sputnik by the USSR in 1957) is often defined as a starting point: the technical competition of the cold war would give political motivation to invest in mathematics education in Western countries, and especially in the US. However, historical studies underline its multiple motivations. Mathematicians were actively discussing the necessity of adapting mathematics education to the development of modern mathematics since the beginning of the 1950s. Socio-economic changes due to industrial development in both the Eastern and Western block also created a need for specialists educated in mathematics. At the same time, the massification and democratisation of lower secondary education required to define a new function for primary education (Gispert & Schubring, 2011; Kilpatrick, 2012).

Various international organisations structured the debates on the reform of mathematics education. The CIEAEM, created by Caleb Gattegno in France in 1952, focused on the psychology of mathematics education, based on Piaget’s work and Bourbaki’s conception of modern mathematics. These two entries were connected by the notion of ‘structure’: mental structures on the one hand and mathematical structures on the other. ICMI, reorganised in 1952, focused more on the role of mathematics in post-war societies, on the increasing importance of applied mathematics and the consequences of the massification of educational systems. The OECD emphasised the economic needs of a revised mathematics education. It opened an office in Paris in 1958 to favour the development of mathematics and science teaching and organised several international meetings at the beginning of the 1960s. UNESCO contributed to the international discourse by holding conferences and publishing recommendations for mathematics curricula and teaching methods.

Several key elements can be identified in these discussions, which impacted reforms all around the world. One is the idea of aligning the school mathematics curriculum to the contemporary development of research mathematics, in its content as well as in its organisation. Emphasis was put on mathematical structures and sets as the basis of the construction of mathematics. Finally, the impact of Piaget’s results on the psychological development of children; and, especially concerning the lower grades, the intention was to implement methods of active pedagogy, mathematical games and manipulatives. However, the curricular reforms implemented in different countries are various and reflect local specificities. As Stanic and Kilpatrick (1992) suggest, New Math is “a label not so much for a cohesive set of reform proposals and activities as for an era during which a variety of reforms were undertaken” (p. 413).

In a report published by ICMI (Freudenthal, 1978) in Educational Studies in Mathematics, entitled “Change in mathematics education in the late 1950’s”, reforms of countries from five continents are discussed: Australia, Bangladesh, France, Great Britain, Hungary, India, Iran, the Netherlands, Nigeria, Poland, Sierra Leone, Sri Lanka, Sudan, Thailand, the United States and the West Indies. Some of these developmental projects were based on completely original material, others were mainly based on the implementation of other countries’ materials. In other cases, an international effort was made to support the development of original materials for some countries or regions, as in the case of the UNESCO Arab State Project and the Entebbe project in Africa. In most of the cases, these projects are characterised by Western dominance:

Even when attempts were made to produce original materials specifically for the countries concerned, for example, the Entebbe Project and the UNESCO Arab States Project, the writing teams were dominated mathematically and professionally, if not numerically, by Western authors who lacked any prior understanding of the educational systems of the countries concerned and, more importantly, of the social ethos that was manifested in the schools. (Kilpatrick, 2012, p. 567)

The Cold War political background played a role in certain international co-operation, for example, in the US projects implemented in Latin-America. The USSR had its own reform in the period, known as the ‘Kolmogorov-reform’, parallel with the Western reforms in many senses. However, some Eastern European countries were also engaged in the collaborations structured by the CIEAEM or the UNESCO during the 1960s and 1970s (Freudenthal, 1978; Karp & Schubring, 2014). Some countries integrated lessons from the international exchange with specific local developments; in some later reforms, especially from the 1970s, conclusions from the vivid social and professional debates and the experienced failures of the first wave ‘New Math’ reforms were also considered.

The ‘Mathématiques Modernes’ Reform in France

The complexity of the motivations and driving forces behind the New Math movement can be well observed in the case of the French reform. This reform appeared in the context of a unification and democratisation process of lower secondary school. In 1959, the age limit of obligatory schooling was prolonged to age sixteen; then, between 1959 and 1975, the lower secondary school was progressively transformed into a unified education for all until grade 9 (15–16-year-old students). In addition to this transformation of the educational system, discussions were held on the role of mathematics in society: the discourse on the technical, industrial and scientific needs met with the influence of structuralism, suggesting that mathematics can offer a universal language and a model of thinking for all (d’Enfert & Kahn, 2010). In this context, modern formal mathematics is considered not only as the best tool to form the students’ mental structures but also as a tool of democratisation, assuring that each student receives the same education, independently of their social background.

The ambition of modernising the content of mathematics education is particularly emphasised, and French mathematicians played a leading role in the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM) activities and the discussions about the reform of mathematics education. Among those involved were Jean Dieudonné, the leader of the Bourbaki group; Gustave Choquet, the first president of the CIEAEM who introduced the ‘new mathematics’ in his lectures at the University of Paris in the 1950s; and André Lichnerowicz, the future leader of the French reform commission (Gispert, 2010).

The French mathematics teachers’ association, the APMEP,Footnote 1 actively promoted the reform with the establishment of a dedicated commission and with the publication of recommendations during the 1960s. Whilst the 1950s heard from different voices in the debate and the axiomatic school became dominant in the 1960s, the APMEP emphasised the teaching of the same mathematics from the kindergarten until the university, based on the notion of set, on the modern algebraic language of mathematics, on the structures and the axiomatic method (Barbazo & Pombourcq, 2010).

From a pedagogical point of view, the emphasis was made on the promotion of active pedagogical methods (d’Enfert, 2010). These ambitions were influenced by Piaget’s psychological work and the aspirations to extend the renovation of mathematical content to the primary school level. The coherence of the project was confirmed by the conviction that the development of the children’s mental structures, described by Piaget, correlates with the mathematical structures described by Bourbaki.

Thus, a diversity of actors anticipated the reform, with various and sometimes conflicting ambitions. Many of these actors were represented by the Ministerial Commission, created in 1966 to prepare the reform, under the direction of André Lichnerowitz. It includes mathematicians and secondary school mathematics teachers and also physics and technology teachers, representatives of the industry and primary school teachers. Several experimentations accompanied the project since 1968. The new curriculum was introduced progressively, starting from 1969 for grade 6 and 10 (respectively the first year of the lower secondary and the high school) and from 1970 on for primary education. A general agreement accompanied the introduction of the first two years in lower and upper secondary education, but those of grades 8 and 9 provoked serious tensions and led to a crisis of the reform.

The coherence of the curriculum was assured by a hierarchic structure of the different mathematical domains, based on set theory and showing the influence of contemporary mathematicians’ work, especially the structure of Bourbaki’s Éléments de mathématique. However, while in the lower grades, these principles appeared in combination with activities like ‘practical exercises’ and ‘observations of physical objects’, a critical break can be observed between the curricula of grades 7 and 8.

The most representative example and also the theme which caused the most controversies was the curriculum of geometry. In the lower grades, the geometry curriculum contained observations and activities related to the physical reality, but the theme had minor importance, and was not recognised as ‘veritable mathematics’. The curriculum underlined that the study of ‘veritable geometry’ started from the eighth grade, as an example of axiomatic thinking. Classical synthetic geometry was completely eliminated and the main aim was not to study geometrical figures but to construct an algebraic tool to describe first the affine, then the Euclidian plane and space. Principal notions were projections, vectors, frames, transformations, etc.

According to the instructions, axioms and notions had to be introduced via physical observations, but once they were admitted, they had to be clearly distinguished from the physical word and every further theorem had to be deduced by formal demonstrations. The textbooks however barely gave help to this introduction, they contain principally an axiomatic–deductive treatise of geometry, where figures are only illustrations of the theorems described in a formal algebraic language (d’Enfert & Gispert, 2011).

The curriculum was soon criticised inside the Commission as well as by the general public. It was accused to serve only the interest of the elite, that is the future mathematicians, and not a wider audience, not even future engineers or students of experimental sciences. At the same time, the majority of mathematics teachers came from the earlier popular school system and were familiar with a practical approach to mathematics. For them, the mathematical content of the new curriculum, as well as the radical epistemological change, posed serious problems. Furthermore, because of the growing number of students in the lower secondary school, many teachers at this level were former primary school teachers without specific training in mathematics.

The commission anticipated the necessity of in-service teacher education. The network of IREMs (Institut de Recherche sur l’Enseignement des Mathématiques) was founded in 1969 with the intention to contribute to the continuous development of mathematics education, to organise in-service teacher training and to prepare resources for teachers. A variety of media, including television, was also deployed to promote the reform and prepare the teachers. However, these efforts were insufficient compared to the needs (Barbazo & Pombourcq, 2010).

The ‘mathématiques modernes’ reform process came to an end in 1972, when Lichnerovicz resigned, and the Commission finished its work. It was often considered as a failure. However, it exerted a long-term impact on French mathematics education in several senses. A new curriculum was introduced in 1977, eliminating the most controversial aspects of the recent reform, but the main structure of the curriculum and many elements of it remained. More radical changes were introduced in the 1980s whereby many characteristics of the New Math period disappeared. Problem solving and applications of mathematics became progressively more to the forefront (Gispert, 2014).

Several projects and institutions, created in the 1960s in order to support the reform process, continued to work, and became the main contributors to the discussion and debate around the problems of the reform as well as the first centres for didactics research. The experimentations of the National Pedagogical Institute went on and led to the publication of an innovative resource for primary schools, the ERMEL series. The IREM network continues to work today, and these institutes offer a forum for teachers and researchers from several domains to work in thematic groups, develop new material and teacher training sessions. Many of the first generations of French researchers in mathematics education started their work in the frame of this network. The Theory of Didactical Situations, created by Guy Brousseau in the late 70s (Brousseau, 1997), which is considered one of the early ‘big’ theories of mathematics education research, can be understood in many aspects as a reaction to the discourse and debates around the ‘mathématiques modernes’ (Dorier, 2018).

This French reform exerted significant influence abroad. As we saw earlier, several actors of the French reform played a leading role in international organisations and meetings. The documents of the French reform were disseminated in and beyond Europe; for example, the new curriculum and the related textbooks were adopted by former French colonies of Africa (Khôi, 1986). However, this influence was reduced in time and soon ‘counter-reforms’ were implemented, trying to come back to a more traditionalist view of school mathematics. But many elements remained, like the replacement of the old ‘arithmetic’ by the strand of ‘numbers and operations’ and the disappearance of quantities in favour of sets of numbers.

A Reform Movement from the Eastern Bloc: Varga’s ‘Complex Mathematics Education’ Reform

In the 1950s, Hungary was part of the Eastern bloc, under the political influence of the Soviet Union. This alliance also determined the development of the educational system. In the 1960s, however, a certain liberalisation and political opening towards the Western bloc increased the possibilities for educational developments. A reform movement in mathematics education was stimulated by a series of workshops given by Zoltán DienesFootnote 2 in 1960, and by a UNESCO conference on mathematics education organised in 1962 in Hungary (Halmos & Varga, 1978). The leader of the Hungarian reform movement, Tamás Varga, engaged in the international ‘New Math’ discourse following this conference. For example, he was invited to co-edit the report of the UNESCO conference with the Belgian Willy Servais (Servais & Varga, 1971), and was invited to various countries (the Soviet Union, Germany, France, Italy, the USA, Canada, etc.). He also regularly participated in international conferences and published in international mathematics education journals during the 1960s and 1970s.

Varga started an experimental project in 1963 in two classes of grade 1. In the following years, the experiment was progressively expanded to other schools and the lower secondary school level, reaching more than a hundred classes in the country. The project was conducted by a group within the National Pedagogical Institute and involved close collaboration with another group, namely the Hungarian Mathematical Research Institute, on the preparation of the newly created (high school level) special mathematics classes curriculum. In the early 1970s, a ministerial commission evaluated different experimental projects concerning mathematics education. They chose Varga’s project as the basis of the planned new curriculum. An optional version of the reform curriculum was introduced in 1974 before the reform became obligatory in 1978, in the framework of a general reform of Hungarian curricula.

In the Hungarian case, the frames of the educational system in which this reform arrived were established since 1946. Compulsory education was provided by the eight-grade, single-structure ‘basic schools’, comprising elementary (grades 1–4) and lower secondary (grades 5–8) education. During the 1950s and 1960s, the regulation of the educational system was extremely centralised, with detailed curricular instructions, while the communist ideology was imposed. From the late 1960s, the most significant change concerning the educational system was the launch of a slow liberalisation process (Báthory, 2001). The influence of the ideology was pushed into the background, pedagogical and psychological considerations were taken into account and differentiation, as well as teachers’ autonomy and liberty, began to be emphasised. This in turn played a crucial role in the preparation of the 1978 reform, and Varga’s project can be considered as pioneering in this sense.

The impact of the New Math movement can be observed on the Hungarian reform in many aspects. For example, the introduction of a coherent subject termed ‘mathematics’ instead of ‘arithmetic and measurement’; new mathematical domains introduced in early ages like sets or logic; the reference to Piaget’s psychology and Dienes’s mathematical games; the role of manipulative tools, etc. However, Varga was also critical of some aspects of other countries’ New Math reforms, especially with, what he considered, the excessive emphasis on mathematical formalism. His project is based on an epistemology of mathematics which is significantly different from the ‘Bourbakian’ epistemology, and rather influenced by Hungarian mathematicians’ ‘heuristic’ view on mathematics (Gosztonyi, 2016).

This tradition existed originally in the teaching of young mathematical talentsFootnote 3 and went back at least to the beginning of the twentieth century. Varga himself was in intensive personal contact with some representative mathematicians of this tradition (L. Kalmár, R. Péter, A. Rényi, J. Surányi, among others) since the 1940s; and they all supported, more or less actively, Varga’s later reform movement, which extended this approach for all students. These mathematicians, together with well-known thinkers like George Pólya or Imre Lakatos, represented a quite coherent, ‘heuristic’ epistemology of mathematics, closely related to questions of mathematics education.

This epistemological approach emphasised that mathematics is a human activity, developed through a dynamic of problems and attempted solutions, based on intuition and experience. Mathematical activity was seen as dialogical, and teaching mathematics as a joint activity of the students and the teacher, where the teacher acts as an aid in students’ rediscovery of mathematics. These mathematicians rejected excessive formalism, seeing formal language also as a result of a development. They described mathematics as a creative activity close to playing and to the arts.

The pedagogical and psychological background of the reform seems to be more complex. Together with Piaget’s influence, several Hungarian thinkers, representing a socio-constructivist approach, impacted on Varga’s conception, stressing the importance of visual intuition, among other things (Gosztonyi et al., 2018).

As with other New Math reforms, Varga sought to integrate new topics in mathematics education, and to present mathematics as a coherent science, organising the curriculum in accordance with modern mathematics. It involved basing notions on sets and relations, or the strengthened role of algebra. However, for him, it also meant introducing logic, combinatorics, probability or algorithmic thinking in primary and lower secondary school. He was internationally recognised for his work on the teaching of logic, combinatorics and probability – the specific domains studied by the Hungarian mathematicians supporting his movement.

The internal coherence of the curriculum was ensured by the parallel, spiral presentation of 5 big domains, all being present throughout the entire curriculum, with frequent and various internal connections amongst them. These were: (1) sets and logic; (2) arithmetic and algebra; (3) relations, functions and series; (4) geometry and measure; (5) combinatorics, probability and statistics. Another significant characteristic of Varga’s curriculum was its flexible structure with ‘suggested’ and ‘compulsory’ topics distinguished from ‘requirements’. This organisation essentially gave liberty to teachers, allowed differentiation amongst students, provided a rich and varied experimental basis to the progressive generalisation and abstraction of mathematical notions, and supported a learning process based on mathematical discovery, while elements of mathematical knowledge can emerge as tools during problem-solving situations.

Teachers’ adaptations to the new curriculum and to the related pedagogical expectations were supported by a series of textbooks and teachers’ guides prepared by those responsible for developing the curriculum. At the time it was the only available textbook series in Hungary. For the primary school, similarly to other countries in the New Math period, worksheets were available, intended for use only as partial resources alongside various activities. Official teacher’s guides served as primary resources for teachers. For middle school, there were textbooks provided, with (much less detailed) teacher’s manuals.

According to the handbooks, teachers had an important responsibility in the construction of long-term teaching processes, which were based principally on ordered series of problems. Mathematical concepts were constructed on a broad experimental basis, by discovering links and analogies among apparently different problems and by generalising progressively the knowledge related to concrete problem contexts. The importance of the use of various manipulative tools and representations was underlined (some of these tools being widespread at the time including the Dienes blocks or the Cuisenaire rods).

Various forms of classroom organisation were promoted in the guides (including individual and group work), but collective classroom dialogue was particularly emphasised. The guides offered advice regarding teacher questions and interventions that would enable teachers to react efficiently to students’ contributions. It was envisaged that this would help the advancement of the collective research project while leaving an important responsibility to students in the problem-solving process and the construction of mathematical knowledge.

As with many other reforms of the period, Varga’s reform provoked vivid public debates and was followed by an important correction during the 1980s. His former colleagues interpreted this as a failure, and they considered the obligatory introduction of the reform for all as the main reason of its rejection. According to them, the approach should have been disseminated progressively in the frame of a bottom-up process, as had happened during the (generally successful) experimentations – but this kind of slow diffusion was not politically supported. Although teacher education media were offered, these efforts were far from enough to prepare teachers for this radical reform and to settle the resistance. While a narrow circle of teachers (mostly colleagues of Varga and their disciples) continued to follow the approach in their teaching practices in the following decades and until today, the majority of Hungarian teachers did not adopt it or integrated only partial elements of the approach in their practice.

Despite that, Varga’s work remained influential in Hungarian mathematics education. An important continuity can be observed in the current curricula’s conception: the main structure and the content of the curriculum remained quite stable until today. Some of the textbook authors from his team were active until the 2010s, and their textbooks demonstrate continuity with the original versions of the 1970s – although other manuals are also available now. Most of the teacher trainers consider his ‘guided discovery’ conception still relevant and find inspiration in it, especially for primary level in-service teacher training (Gosztonyi et al., 2018).

Varga’s work also exerted a particular international impact, although not comparable to the leading Western European and American projects. Some of his works were translated in many countries in the Eastern bloc, and several of his publications also appeared in France, Canada, and the USA, especially in the domains of teaching combinatorics and probability in lower grades (Glaymann & Varga, 1973; Varga 1967, 1982; Varga & Dumont, 1973). His worksheets were translated into Italian and used mainly for teacher education,Footnote 4 and more recently, a Finnish primary school teacher association was created, inspired by his work.Footnote 5

At the same time, and in contrast to the French case, Varga’s reform did not lead to a dynamic research culture in mathematics education. After he died in 1987, only a few Hungarian researchers remained active. His work became, however, the catalyser of a newly emerging research movement in Hungary in recent years.

Realistic Mathematics Education in the Netherlands

Realistic Mathematics Education (RME) has become the main approach to mathematics education in the Netherlands and has also left its mark on mathematics education in other countries (Van den Heuvel-Panhuizen, 2020a, b). In this section, we give a short sketch of the development of the RME reform movement in the Netherlands and we take the perspective of the reform that happened in primary school mathematics education.

When from the early 1960s on, New Math gained world-wide influence, the Netherlands chose another direction to change the rather mechanistic approach to mathematics education that was common at that time. This approach was characterised by teaching fixed calculation procedures in a step-by-step manner with the teacher demonstrating for each step how to proceed, with real-world problems only used for the application of previously learned calculation procedures, and little or no attention for developing insight in the underlying mathematics of these procedures.

The reform that was an answer to this approach was initiated by the inception of the Wiskobas project in 1968 and was further enhanced by the establishment of the IOWO (Institute for Development of Mathematics Education) of which Freudenthal was the first director. The IOWO produced a broad variety of materials making this change to a new mathematics education possible, including rich tasks, themes, lessons, teaching sequences, and complete programs for various topics within arithmetic, measurement and geometry. The Special Issue of Educational Studies in Mathematics titled “Five Years IOWO” (Freudenthal, Janssen & Sweers, 1976) reflects the outburst of ideas in the initial period of the reform movement.

In addition to these design activities carried out in the early days of RME, the underlying theory was also given much attention. Freudenthal (1973) published his ground-breaking book Mathematics as an educational task, and Treffers (1978) brought out his first work on the goals and approaches to mathematics education according to Wiskobas. Other important research work that started at the end of the 1970s involved carrying out textbook analysis. Existing textbooks were commented on and critically examined from the perspective of the intended reform, which had a guiding function for the innovation.

In 1981, the Wiskobas work came to an end as a result of a decision of the government. The work on RME was continued by OW&OC (Mathematics Education Research and Educational Computer Center) and the newly established SLO (the Netherlands Institute for Curriculum Development). Characteristic for the 1980s were the many research activities and the various national and international publications that resulted from them. For primary education, important work was done for example by Adri Treffers (progressive schematisation), Leen Streefland (context and models for fractions), and Jan van den Brink (mathematical language and representations for early number). Furthermore, a new boost to theory development was given by Freudenthal’s (1983) Didactical phenomenology of mathematical structures and Treffers’ (1987) Three dimensions.

An important impetus for implementing RME in curriculum documents, textbooks and school practice was the establishment of the Netherlands Association for the Development of Mathematics Education (NVORWO) in 1982. One of the first actions of NVORWO was to prepare a national plan for primary mathematics education. In 1984, a draft version of this plan was submitted for consultation to almost three hundred experts in the field of primary school mathematics. It was proposed to give algorithmic digit-based calculation a less central position in favour of mental calculation, estimation and number sense, to aim more at applicability, and not to start with teaching students the most shortened forms of standard algorithms immediately, but begin with a notation using whole numbers.

This plan received much acclaim. In 1987, the findings resulted in the first blueprint for a national programme for mathematics education in primary school, the so-called ‘Proeve publications’ (e.g. Treffers, De Moor & Feijs, 1989). Later, the goals as described in the Proeve publications were officially given approval by the government by adopting as the national core goals for primary education (MoE, 1993). A further implementation in curriculum documents was possible through the development of the TAL teaching-learning trajectories for primary mathematics education commissioned by the Ministry of Education (e.g. Van den Heuvel-Panhuizen, 2001). Also, the Proeve and TAL publications with their descriptions of goals, examples of tasks and teaching methods served as beacons for textbook authors. This resulted in a noticeable change in the nature of textbooks. The market share of RME-oriented textbooks increased from 15% in 1987 to 75% in 1997 and reached 100% around 2004.

However, the educational climate changed remarkably around 2007 (see Van den Heuvel-Panhuizen, 2010). This change was prompted by the results from the 2004 PPON survey (the National Assessment of Educational Achievement) by CITO, the Netherlands Institute for Educational Measurement. The results showed that student performance in number sense, mental calculation and estimation had substantially improved since 1987, but that achievements for written algorithmic calculation had decreased. Although this was to a certain degree in line with the performance profile opted for twenty years earlier, these findings evoked much protest against the RME reform.

The complaints particularly came from a few mathematicians, who were in favour of returning to the mechanistic approach to mathematics education of forty years ago. A fierce debate arose in newspapers. After a commission established by the Royal Netherlands Academy of Arts and Sciences (KNAW) (2009) concluded there was no convincing empirical evidence for the claims on the effectiveness of traditional methods versus RME, the peace returned. However, the debate was not without consequence.

The market share of RME-oriented textbooks lost a few percentage points of its 100% position, and new editions of the RME-oriented textbooks that were still on the market included features of the mechanistic approach of the past by putting more emphasis on written calculation. Nevertheless, to date they do not focus on blindly training of procedures but aim at understanding by starting with a phase of transparent whole-number-based written calculation. By and large, the RME characteristics are still upheld in most current textbooks.

Another recent movement toward the past is the return to the original ideals of RME to give much attention to mathematical reasoning and problem solving (Wiskobas team, 1980). One example of this revival is the Beyond Flatland project, set up in 2015, to investigate how the Dutch primary school mathematics curriculum can be made ‘more mathematical’ by including activities on mathematical reasoning in the context of early algebra, dynamic data modelling, and probability.

The idea of already starting in primary school with mathematical problem solving through modelling and reasoning is also reflected in the advice given by the project team of teachers recently commissioned by the Ministry of Education to rethink and revise the current mathematics curriculum to have students better equipped for their future personal and professional life. The resulting plans are supported by NVORWO (2017) and also clearly show elements of the spirit of Wiskobas.

Mathematics Curriculum Reform in Japan

Reforms of mathematics education in European countries at the beginning of the twentieth century affected mathematics education in Japan. Mathematics had, until that time, been taught separately, broken down into domains of mathematics. In elementary school, the curricular focus was on arithmetic such as calculation and conversion of units of measurement, followed by the study of algebra, geometry (especially Euclidean) and analysis in secondary school. Japan was in the midst of modernisation in the early twentieth century and had just established an educational system and curriculum modelled on those of Western countries.

These reforms, such as the ‘Meran project’, were introduced in the 1910s and 1920s, but it took time for these to be reflected in practice, and they truly came into effect after the 1930s. These reforms sought to update traditional mathematics education to suit the development and interests of children. The curriculum was structured without dividing the academic domains of mathematics (Monbu-syo, 1931). Arithmetic and geometry were studied at the same time during elementary school and were not presented via axiomatic treatment by proof (Monbu-syo, 1935). The national textbooks in this age were introduced in the tenth International Congress of Mathematicians (ICM) convened in Oslo in 1936 (Kunieda, 1936). In the new teaching method, the so-called Life Arithmetic, which relates mathematics to daily life and experience, was taught at schools affiliated with the national normal schools and advanced private schools.

In the 1950s, in the years after World War II, US progressive education was introduced by the Occupation Command and incorporated into mathematics education as part of post-war educational reform. Thus, the relationship with children’s daily life and experience was emphasised for its teaching. Teaching content was reduced or taught in later grades. However, they were revised during Japan’s independence.

In the 1960s, the influence of New Math was also seen in Japanese mathematics education. New material such as set theory, algebraic structures and linear transformations was incorporated during secondary school lessons. This resulted in new difficulties, which were not limited to relating the new material to the old grade distribution but included structural changes and improvement to the mathematics curriculum as a whole. More specifically, the idea of set theory reconfigured conventional learning on the range of numbers and the meaning of operations from a different perspective, which required changes in elementary school mathematics. The New Math curriculum was incorporated into the revision of the national curriculum in the late 1960s (Monbu-syo, 1968).

In the 1970s, the mathematics curriculum was reorganised in reaction to New Math. Set theory and algebraic structure were removed from elementary and secondary school mathematics, and the teaching content was carefully selected to present foundational knowledge and skills. Elementary school mathematics content was divided into four categories: (a) numbers and calculations; (b) quantities and measurements; (c) geometrical figures; (d) mathematical relations (Monbu-syo, 1977). The first three categories corresponded to the subjects handled in mathematics that is, number, quantities, and shapes. Mathematical relations summarise the methods and ideas dealt with in mathematics and include functional concepts, statistical methods, and mathematical expressions related to other contents such as number, quantities, and shapes. This framework based on separately describing the subjects and methods was a characteristic way to systematise the content in mathematics curricula, and it was close to the categorisation of common standards of the National Council of Teachers of Mathematics (NCTM, 1989).

After 1990, the focus of the Japanese mathematics curriculum shifted from the content to learning processes and purpose. Objectives now included the new term ‘mathematical activity’ and the principle that students should learn math through ‘mathematical activities’ (Monbu-syo, 1998). It also includes new category types that fall under the heading ‘mathematics activity’: (a) activities to discover and develop the properties of numbers and figures based on learned mathematics; (b) activities using mathematics in everyday life and society; (c) activities that use mathematical expressions to clarify the basics, to explain them reasonably and to communicate them to others (Monbu-kagaku-syo, 2008). Table 4.1 summarises the changes in the post-war mathematics curriculum in Japan.

Table 4.1 Changes in the post-war mathematics curriculum in Japan
Full size table

The national curriculum was revised and announced in 2016 for elementary and junior high school, and in 2017 for high school. This revision will go into effect in 2020 in elementary schools, in 2021 in junior high schools, and in high school staring in the first year from 2022. This revision emphasises qualifications and literacies as the purpose of education. It is based on the idea of key competencies and the OECD-PISA framework. Kyouiku-Katei Kikaku Tokubetsu Bukai (KKKTB – Curriculum Study Group of National Council for Education), an advisory board for Ministry of Education, Culture, Sports, Science and Technology (MEXT), compiled three pillars of qualification and literacies for the foundation of the new curriculum in 2014 (KKKTB, 2014). The three pillars suggested to MEXT are: (a) what you know and what you can do (individual knowledge and skills)? (b) how do you use what you know and what you can do (application of thought, judgment, expression)? (c) how do you live a better life and relate to society and the world (promoting individuality, diversity, co-operativeness, attitudes toward learning and humanity)?

These qualifications and literacies became watchwords and were guided by both the desire to examine what children can do as a result of school education and the reason for teaching it, rather than focusing on what to teach. The mathematics curriculum has been organised using these three pillars as a framework. The objectives for this subject area have thus been rewritten from these three perspectives, and the teaching content is presented separately for the categories (a) individual knowledge and skills and (b) ability to think, judge, and express. This curriculum framework will be used in the further development of mathematics education in Japan.

Conclusions

Reforms of mathematics education are national phenomena, and they have been presented here as such. Nevertheless, as the examples of the movement of the beginning of the twentieth century and the reforms that originated after the Royaumont conference show, they are also closely related to international exchanges. International exchanges were both a resource and a consequence of national reform movements, and international organisations, especially ICMI, played a crucial role in this process since its foundation.

The cases included in this chapter have also been chosen because of their importance beyond their strict national sphere. The first two cases – France and Hungary – correspond to reforms that took place over a short period and were followed by counter-reforms, even if they deeply marked the mathematics curricula established after them, in their content, organisation and resources. The remaining two cases – The Netherlands and Japan – embrace a more extended period and show a continuity in the reform processes, with some back-and-forth movements but presenting a clear sustainability.

In all cases, well-known and highly regarded mathematicians were involved in the first steps of the process and, what seems more relevant but maybe remains less visible in the descriptions provided, is the intensive mathematical ‘engineering work’ (to use Freudenthal’s expression) undertaken to launch the reforms. This work was a collective enterprise leading to the construction of new mathematical contents, curriculum organisations and teaching resources. The efforts put by researchers and teachers to carry out this enterprise and the means initially provided to do so appear as a common trait of the four cases considered. They seem to have also generated a fertile milieu for the emergence of mathematics education research.