Mathematics is a subject taught in every country. It is interesting to note that the word mathematics is plural in many western languages (among the ones that distinguish singular and plural names). This is the case, for instance, in French, German, Spanish, Catalan, etc. In Italian, the plural Matematiche was common at the beginning of the twentieth century, but now the singular Matematica is more popular. The plural name may hint to general facts. First, the fact that mathematics is conceived as a collection of different fields, distinguishing, for instance, pure mathematics and applied mathematics. Second, that mathematics is conceived in different ways in different cultures. Some examples of the second case are discussed in Bartolini Bussi and Sun (2018). The influence of values and cultures on the definition of curricula and curriculum reforms will be reconsidered in the cases reported in this chapter.

Although mathematics is usually conceived as a universal scientific subject, this is not true in general. Western mathematics is derived from the Greek approach to knowledge, and the structure of Indo-European languages. An astonishing example, mentioned in Bartolini Bussi and Sun (2018), is the fact that, in Maori language, whole numbers are verbs (actions) and not names or adjectives (Barton, 2008). The different ways of conceiving mathematics as a scientific subject have a strong influence on the way of teaching and learning mathematics. The national standards are not the same all over the world but point to different ways of considering the foundations of mathematics. We can illustrate this fact by referring to some exemplary cases.

Whole numbers are usually introduced as a multifaceted concept addressing cardinal, ordinal and measure aspects (Ma & Kessel, 2018; Bass, 2018). However, in Australia the Standards focus on a pattern-based approach (English & Mulligan, 2013). As defined in our studies, mathematical pattern involves any predictable regularity involving number, space, or measure. Examples include friezes, number sequences, measurement, and geometrical figures. By structure, we mean the way in which the various elements are organised and related. Thus, a frieze might be constructed by iterating a single ‘unit of repeat’; the structure of a number sequence may be expressed in an algebraic formula; and the structure of a geometrical figure is shown by its various properties What we call structural thinking is more than simply recognising elements or properties of a relationship; it involves having a deeper awareness of how those properties are used, explicated, or connected (p. 30).

This statement is mirrored in the Australian Curriculum: Mathematics of 2019, as from the Foundation stage, patterns and algebra are integral part of the curriculum. For instance, the Australian Curriculum reads for the foundation year:Footnote 1

patterns and algebra;

sort and classify familiar objects and explain the basis for these classifications;

copy, continue and create patterns with objects and drawings:

  • observing natural patterns in the world around us;

  • creating and describing patterns using materials, sounds, movements or drawings.

This focus is different from that of other countries, where patterns and algebra are introduced in higher grades. This means that in Australia, foundations of mathematics are reconceptualised around algebra and patterns rather than on other approaches that align with Piaget’s ideas on whole numbers. Algebra is considered a part of the foundation of mathematics rather than a development of arithmetic.

Algebra also features in the early mathematics curriculum in the Soviet Union (after Davydov) and China. This has been discussed by Mellone and colleagues (2019) during their elaboration on the construct of cultural transposition, as a process activated by researchers, educators, and teachers who deconstruct those educational practices adopted in other cultural contexts in order to reconsider the issues of educational intentionality, which is the background of any educational practice (p. 201).

A first example of educational activity focuses on the ‘problems with variation’, which is considered one of the most significant mathematics education tools in Chinese primary schools (Bartolini Bussi et al., 2014). The second example relates to the visionary mathematics curriculum for pupils attending the first grade, which is proposed by Davydov (1982). In both cases an algebraic approach through quantities is presented from the very beginning with extended references to pictorial equations. Figure 6.1 shows a representation from a Chinese textbook (45 + X = 75) in Bartolini Bussi & Sun (2018, p. 65) and from a standard Davydov representation of the relationship A + X = B.

Fig. 6.1
Two model diagrams of subtraction in Chinese and Russian curricula has a line marked as 75, a part of the line as 45, and the remaining with a question mark. B has three lines of different lengths, B, A, and X.

Two common examples of representing subtraction in Chinese and Russian curricula

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The two representations are very similar, and one may wonder whether Davydov’s curriculum and the Chinese curriculum have been developed independently of each other. The answer is no. In Shao, Fan, Huang, Ding and Li (2012), a careful historical reconstruction shows that a Russian educator, Ivan Andreevich Kairov, had a strong influence on the development of the Chinese curriculum in the second half of the twentieth century. Prior to the Revolution Russian schools were similar to schools in Germany. However, after revolution John Dewey’s ideas were introduced and for a short period had a strong influence on building new school systems. This approach was abandoned in the mid-thirties of the nineteenth century.

The importance of Algebra in the Chinese approach to whole numbers is so strong that some Chinese scholars (e. g. Liping Ma, personal communication) prefer to address the problems of variation theory, algebraic in fact, as a part of arithmetic. This approach to algebra is fundamentally different from the pattern-based approach, mentioned in the Australian curriculum.

The short examples provided above show that there are very complex relationships between the mathematics curriculum and different cultures. Figure 6.2 summarises some different variables involved (Bartolini Bussi & Martignone, 2013).

Fig. 6.2
A flow chart has implemented, attained, and intended mathematics curriculum, mathematics standards, implicit philosophy, social and political institutions, traditions, and economic organization.

A general scheme adapted from Xie and Carspecken (2008), as represented in Bartolini Bussi & Martignone (2013)

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In what follows, we briefly report on some examples that have been presented in the Conference, drawing on the papers submitted and published in the proceeding (Bartolini Bussi et al., 2018; Gooya & Gholamazad, 2018; Milinkovic, 2018). In all three cases, mathematics is considered very important in the three cultures, although the curricular choices may be different.

Italy: The Most Recent Global Curriculum Reform

The most recent global curriculum reform in mathematics in Italy took place at the beginning of the new century. In this section, a short summary of the process is given, as it was complex and intertwined with the political processes of the last twenty years (see Ciarrapico, 2002). In the year 2001, the UMI (Unione Matematica Italiana), in collaboration with the Italian Commission for Mathematical Instruction (CIIM), published a mathematics curriculum (Matematica, 2001) for primary and junior secondary school (grades 1–8) that had some influence on the curricula issued by the government in the following years. A few years later the UMI published a mathematics curriculum for grades 9–13 (Matematica, 2003, 2004). In 2001, the Ministry of Education Tullio De Mauro, an internationally acknowledged scholar of Italian language, introduced to Parliament a proposal of curriculum for pre-school, primary school and junior secondary school.

The document for mathematics was based on Matematica 2001. Three guidelines were taken into account in the elaboration: the essentiality, that is the identification of the fundamental epistemological aspects of mathematics (founding nuclei), with the intention of a quantitative reduction of the contents in favour of a better quality of learning, the progressiveness of the objectives along the entire primary and secondary school trajectory, since the mathematical goals are reached only in the long term, the continuity with the recent past, which takes into account successes and failures of past experiences.

The curriculum was organised by four thematic nuclei each with specific content (number, space and figures, relationships, data and forecasts), and three others, called process ones, which do not have their own content, because they are transversal to the first four: arguing and conjecturing, measuring, posing and solving problems. Mathematics, an essential component of the formation of the citizen, highlights two fundamental functions, the instrumental and the cultural: mathematics, therefore, is an essential tool for understanding reality and for everyday life. A formal mathematics, devoid of reference to reality, would in fact be a pure ‘play of signs’, but even a purely instrumental mathematics, without a global vision, would risk of being fragmentary and not very incisive. These two aspects, although with different nuances, have been recognised for many years as fundamental goals of mathematical teaching. De Mauro’s decree was issued but not implemented because of new elections with a change in the political majority. However, for mathematics, the influence of the debate around the decree was strong and strengthened by the subsequent publication of Matematica 2003 and Matematica 2004.

A paradigmatic example of the influence of the UMI curricula can be seen in the Indicazioni Nazionali National Guidelines (2012) that mentioned the construct of mathematical laboratory, elaborated in the intended curriculum Matematica 2003:

In mathematics, as in other scientific disciplines, the laboratory is a fundamental element, understood both as a physical place and as a moment in which the student is active, formulates his hypotheses and monitors the consequences, designs and experiments, discusses and argues his own choices, learn to collect data, negotiate and build meanings, leads to temporary conclusions and new openings the construction of personal and collective knowledge. In primary school it possible to use the game, which has a crucial role in communication, in the education to respect shared rules, in the development of strategies suitable for different contexts. (Indicazioni Nazionali, 2012, p. 60; translated by the authors)

The quotation above is taken from the Italian National Guidelines from pre-primary to grade 8. Laboratory activity is considered a general methodology not only for the scientific disciplines but for every subject matter as “it is the working method that best encourages research and planning, involves pupils in thinking, creating and evaluating shared and participated experiences with others, and can be activated both in the different spaces and occasions within the school and by enhancing the territory as resource for learning” (translated by the authors, Indicazioni Nazionali, 2012, p. 35).

A recent document (Indicazioni Nazionali e Nuovi Scenari, National Guidelines and New Scenarios) prepared by the Committee for the implementation of the National Guidelines has again focused on the importance of the laboratory and, in particular, of the Mathematical Laboratory:

the laboratory can also be a gym to learn how to make informed choices, to assess the consequences and therefore to assume responsibility, which are central aspects for the education to an active and responsible citizenship. (Indicazioni Nazionali e nuovi scenari, 2018, p. 12; translated by the authors)

In the Indicazioni Nazionali (2012), the term ‘laboratory’ is a reference to the teaching of scientific disciplines: the spirit of the laboratory activity is maintained, with reference to ICT, to the history of mathematics, to mathematical modelling and to students’ agency.

Bartolini Bussi et al. (2018) reported the features of the Mathematical Laboratory in the Italian intended curriculum, as stated in different documents, together with an example of implementation, that had some effects at a broader level.

A mathematics laboratory is not intended as opposed to a classroom, but rather as a methodology, based on various and structured activities, aimed to the construction of meanings of mathematical objects. A mathematics laboratory activity involves people (students and teachers), structures (classrooms, tools, organisation and management), ideas (projects, didactical planning and experiments). We can imagine the laboratory environment as a Renaissance workshop, in which the apprentices learned by doing, seeing, imitating, communicating with each other, in a word: practicing. In the laboratory activities, the construction of meanings is strictly bound, on one hand, to the use of tools, and on the other, to the interactions between people working together (without distinguishing between teacher and students). It is important to bear in mind that a tool is always the result of a cultural evolution, and that it has been made for specific aims, and insofar, that it embodies ideas. This has a great significance for the teaching practices, because the meaning cannot be only in the tool per se, nor can it be uniquely in the interaction of student and tool. It lies in the aims for which a tool is used, in the schemes of use of the tool itself. The construction of meaning, moreover, requires also to think individually of mathematical objects and activities. (Matematica, 2003, p. 26; translated by the authors)

The reference to the Renaissance workshop is clearly taken from the history of art, that is evident in many museums and exhibitions all over the country, hence is part of the umbrella cultural themes mentioned in the Fig. 6.2. A question arises: to what extent is the idea of a mathematical laboratory implemented across the country, at different school levels? A report of the effects of the Indicazioni Nazionali in a large sample of schools (grades 1–8) was prepared in December 2017 by the National Committee in charge of monitoring the experiments for all subjects. The conclusions are realistic and strongly support the need for investment in teacher development (Indicazioni nazionali e nuovi scenari, 2018).

This document is just the starting point of a necessary reflection on teacher development in Italian schools. Teacher development had not been compulsory but realised on a voluntary base in the Italian system of instruction for decades. Only recently, for the first time in all schools, a mandatory three-year programme (2016–2019) of teacher development has been issued. The issue of laboratory (including the mathematical laboratory) needs to be a major focus of teacher development to overcome the transmissive attitude and to foster students’ agency in the near future.

The lack of an institutional teacher education program for secondary schools had strong influences also on the teachers’ perception on the need for continuing education. The situation is very different from primary school where an institutional University program has been realised for more than twenty years.

Serbia: Changing the Perspective of Mathematics as a Subject

In the Serbian education system, the position of mathematics as a school subject was and continues to be extremely high. In grades Kindergarten to 10th, all students have mathematics classes with variations on the number of lessons per week in grades 9 and 10. In their final years (grades 11 and 12), math is present in all Gymnasiums, Technical Vocational Schools and Economical Vocational Schools. It has a special position in Mathematical High Schools. Finally, there are Mathematical Grammar Schools, enrolling about 1% of the student population. In the present reform, numerous high schools are forming special classes directed toward Informatics. Following worldwide trends (OECD, 2019), Informatics is taught from the first grade.

Reforms of curriculums in Serbia followed worldwide trends often with a few years delay. We present here two examples.

Example 1

In 1970, geometry teaching was predominantly based on Set Theory, starting with definitions of all geometrical objects and relations between them. Geometrical objects were defined as sets of points, while facts concerning them were defined as relations. Geometric objects (line, line segment, angle etc.) and facts concerning them (shapes, relations, measurement, etc.) were defined and analysed using language and apparatus of the theory of Sets.

In the 1985 state textbook (exclusively used at the time), one can read the following definition:

Polygon is a set of points in a plane which is a union of the set of points of a polygonal line without self-intersecting points and a set of points within that line. The polygonal line is called the boundary of the polygon. […] The domain of the polygon is a set of polygon’s points which do not belong to the boundary. Thus, the domain of a polygon is a difference between the set of points of the polygon and the set of its boundary. (Adnadjevic et al., 1985, pp. 30–31)

In the following reform – called mid reform – a new textbook proposed a somewhat less formalised introduction of the concept: “a polygon is a figure defined by a polygonal line, even if in the preceding lines in the textbook, the figure was introduced as a union of a closed line and a domain determined by it” (Micic & Jockovic, 2002, p. 35).

So far, the concept of the polygon was taught in Grade 5. As a result of the current reform, the approach to the concept of polygon is significantly relaxed without proposed formal definitions in the curriculum. Its introduction is postponed to Grade 7. It is expected that pupils explore and form the idea of the polygon through the practice of drawing a closed broken line without intersections and a plain domain determined by it.

Example 2

Serbia had the same historical pattern of development to the concept of function in the school curriculum. In 1985, first the concept of a binary relation was introduced in grade 5 as a subset of the Cartesian product. The function was defined as a special type of binary relation. In the next period, the general concept of a function was postponed until secondary school. In grade 7, the concept of dependency of two variables was introduced (direct and indirect proportionality) within realistic problem situations (e.g. change of temperature during the day). In the following grade, the linear function was introduced as a special type of functional dependency. In the current reform, in the curriculum for grades 5 to 8, the approach taken in the preceding curriculum was preserved, including a categorical demand that the general concept of function should not be mentioned. These are the facts that illustrate directions of changes in succeeding reforms of the math curriculum is Serbia.

We might summarise the situation as follows: referring to Fig. 6.2, we might say that, on the one hand, the curricular reforms depended on background ideas and, on the other hand, contributed to changing the perspective on mathematics as a subject in society.

Iran: Two Aspects of the Educational Systems Affected by Curriculum Reforms

Traditionally, mathematics has always had special place in the heart and mind of the Iranian people. It is assumed that the main essence of the mathematics that has been created and developed there was to solve practical problems that society asked for and gradually move towards more and more abstraction. Thus, mathematicians have served as one of the main pillars of Iranian culture and civilisation. This is evident in Iranian art, architecture, literature and poetry.

The formal education system in Iran was established in 1920 by adapting the French education system that was highly centralised. The centralisation comprised of all aspects of schooling including the mathematics curriculum. Further, since the 1960s, there has been one single national textbook for each school subject and with some tolerance, the textbooks were mainly considered as curriculum guides at the national level (Gooya, 1999). Despite the possible limitations that such centrality could pose for the system, it assured the accessibility of all students to educational resources.

Another characteristic of the education system in Iran is that, in general, schools are segregated from grade 1 to 12 – with some exceptions, including rural and nomad schools. Within this structure, mathematics plays different roles compared to a co-education system. The reason is that girls have never been compared with boys and thus in recent history, gender has not played a similar role in Iran as was the case in many other systems which kept girls away from mathematics.

The Driving Forces Behind Various Mathematics Curriculum Changes

Each education system has been, and will continue to be driven by various forces that are not necessarily rooted in education (Furinghetti et al., 2013). Iran is not an exception. Since the establishment of the formal education system in Iran, there has always been strong political and social forces behind almost all changes in education systems in general and mathematics curriculum changes in particular. As an example, the first mathematics curriculum of the first formal education system went through various contextual and content modifications, to accomplish the societal needs and political wills (Gooya, 2010). After World War II and the emergence of the “New Math era”, many education systems, regardless of their different cultural and social backgrounds and needs, adopted the New Math curriculum.

The approach of the New Math was to move towards internationalised mathematics curriculum or as Bishop (1990) and Clemens and Elerton (1996) have pointed out, an implicit form of modern colonisation. Another feature of the New Math was that its target population was not all students, but mainly designed for elites (Clements & Ellerton, 1996). Therefore, the majority of students showed resistance to that. Nevertheless, the history of mathematics curriculum provided much convincing evidence to show that internationalised school mathematics curriculum is more at the theoretical level than real world of schools which means ‘neutral” or ‘value and culture free’ mathematics curriculum cannot exist in the practical world of schooling (Chevallard, 2007; Bishop, 1997).

The social readiness and the new political establishment were two main driving forces for another major mathematics curriculum change. The expectation was to design a more meaningful curriculum by looking at the cultural, societal and national needs, and try to make a well-rounded integration between new findings in the mathematics education field at the global level, and having better understanding of the local opportunities to design a whole new curriculum.

After the revolution of 1979, the educational branches at secondary level and its mathematics curriculum was revised in 1992 by a study covering 10% of volunteer students, and the process went on until full implementation of the new curriculum in 1998. The theoretical perspective of the new curriculum was based on ideas taken from constructivism along with ‘integrated approach’ from different perspectives. The mathematics curriculum of the first-year senior secondary was designed and developed for all students. After the first year of secondary school, students had to choose their branch and strand, so the mathematics textbooksFootnote 2 of each strand were written based on that. The major change happened in the mathematics curriculum/textbooks in the Human Science strand. The focus of this curriculum was mathematics for those who had not much experience of enjoying and seeing the usefulness and applicability of mathematics in their own field. The main purpose was to provide students with opportunities to experience the beauty and usefulness of mathematics in a practical sense. The first year of Mathematics and Physics strand (grade 10) had two mathematics textbooks as Mathematics 2 and Geometry 2. The second year (grade 11) had two textbooks including pre-calculus, and algebra and probability. The integrated approach in the latter showed how deterministic and stochastic aspects of mathematics are related.

During mathematics curriculum change, one of the driving forces was the mathematics performance of the Iranian students in TIMSS- 2007Footnote 3 that was much lower than that expected. Along with this, a new tendency was shaped at policy-making level to look at the factors contributing to the school mathematics curriculum of the “successful” countries as well. Due to the rush for full implementation within a short period of time, it was decided to do parallel evaluations of grade 1 (Kabiri, 2011), grade 2 and grade 6 (Kabiri, 2012), grade 3 (Kabiri, 2013) and grade 7 (Gholamazad, 2013) at the intended and implemented curriculum levels. This decision was made to partially compensate for the lack of the trial implementation of the newly written textbooks. The findings of these evaluations identified major challenges associated with these textbooks (Gholamazad, 2015). However, the findings were only used for cosmetic revisions of the first drafts of these textbooks.

Yet, another major education change started in 2011 and consequently, school mathematics curriculum underwent a radical change, including approach, content organisation, and context. This sudden decision for change caused the Iranian education system to face a number of unexpected challenges; those that might explicitly or implicitly affect school mathematics curriculum in other situations as well. The forces behind these challenges were not necessarily educational in nature, yet have had enough power to distract the direction of change.

Considering Values and Goals

Values and culture have played a strong role in mathematics curriculum changes since the establishment of the formal education system in Iran. The very first mathematics curriculum included specific skills that the traditional workforce and the new bureaucracy requested. For instance, the traditional Iranian accounting system called ‘siagh’Footnote 4 arithmetic and teaching base 10 abacus was included in the mathematics (arithmetic) curriculum, as well as simple concepts of banking and book keeping (Gooya, 2010). Finally, the pedagogy was mainly teacher-centred and drill and practice. Overall, the major driving force of the first mathematics curriculum was political; believing that joining the international community and moving towards modernisation is not possible with a largely illiterate society.

Another salient feature is that despite the internationalised mathematics of New Math, the new reform in Iran was not limited to the imported approach of it and, instead, the content organisation and pedagogy adapted were influenced by the traditional and national style. Within the New Math curriculum, two parts of the Iranian mathematics curriculum, namely Euclidian geometry and trigonometry, remained in the same traditional manner, with separate national textbooks for each. This showed that despite the great hegemony that the New Math imposed on almost all mathematics curriculum around the globe, the influence of Iranian values and culture, was still significant.

To summarise, in Iran there was no single driving force behind the mathematics curriculum changes during the last hundred years; however, political determination is visible. Any extremism for curriculum change and ignorance towards local values and cultural context may lead to challenges. The Iranian experience shows that main driving forces for mathematics curriculum reform included political determination, international mathematics curriculum movements, international studies (TIMSS), new theories of learning, and new research findings in the fields of mathematics and mathematics education.

The lesson we learned in Iran from various mathematics curriculum changes is to avoid ‘radicalism’ or ‘extremism’ and choose a moderate approach to include local culture and tradition, as well as being connected with the global scene and international research findings.

Concluding Remarks

The three cases presented in this chapter show the important place that mathematics had in formal education of grades 1 to 12 in various education systems. These cases have also depicted some major driving forces behind mathematics curriculum changes that are not necessarily mathematical in nature and yet, should be carefully considered. We refer to the position of mathematics in the structure of educational systems; the traditions; the changes in the social, political and economic institutions; and the implicit or explicit models of mathematics teaching and learning.

The scheme introduced in Fig. 6.2 hints at many of the above variables and may be used to study a curriculum reform, analysing the deep roots of the teaching and learning mathematics in a given culture, but also used to study the effect of a curriculum reform on society. Hence the scheme, by reverting the arrows, may suggest how a curriculum reform may impact on the general background for instance influencing the implicit/explicit philosophies of mathematics and the learning theories.

Looking at all the three examples, it was clear that differences were to be considered. The important message for the international community is that the global perspective and local production are different from ‘internationalisation’ of mathematics curriculum which considers school mathematics as culture and value free. Any sort of extremism in mathematics curriculum design brings about a heavy and sometimes very costly load on education systems of every country. Theories have an important role to play in developing and designing mathematics curriculum. However, it is necessary to modify global theories to fit the local situations by considering the cultural, societal, and values at local levels.

This chapter has reviewed mathematics curriculum reforms in three education systems. In these cases, values and cultures, either explicitly or implicitly, have been crucial. The analysis of these reforms allows us to speculate that values and culture have and will continue to play a salient role in mathematics curricula globally.