The first forum took place in 2000. Then, two important researchers from the Freudenthal Institute: Doctor Marja van den Heuvel-Panhuizen and Doctorandus Martin Kindt, were invited to give a series of lectures at Forum on the Theory and Practice of Realistic Mathematics Education. This first forum was held in two places: in Beijing and in Changchun. Different from the lectures given by Freudenthal, who focused more on the theoretical parts of RME, Van den Heuvel-Panhuizen's and Kindt's presentations mainly emphasised the application of RME in mathematics curriculum and classroom teaching, for example, they addressed how to teach and learn arithmetic in primary education and geometry in secondary education. Another difference with the lectures given by Freudenthal was the audience. In 1988 a small number of scholars and researchers from universities and research institutes attended the lectures, while in the forums held in 2000, hundreds of educators and in-service mathematics teachers were able to participate. In addition, instead of mainly focusing on lecturing such as Freudenthal did, Van den Heuvel-Panhuizen and Kindt adopted another way of communicating with their audience. They saved sufficient time and provided many opportunities for the audience to ask questions freely. The people in the audience asked what they wanted to know most and the speakers answered the questions by illustrating what happens in Dutch classrooms. Both the content in the presentations and the ways in which it was presented highly inspired the audience. The mathematics teachers, particularly, showed great interests in the the-

1On March 7, 2018, Xiaoyan Zhao successfully defended her PhD thesis at Utrecht University.

ory and application of RME. This forum also reflected that the curriculum reform in mathematics education in China began to step forwards to another stage: from involving researchers and scholars and focusing on the theoretical part of RME to paying more attention to mathematics teachers' understanding and the application of RME. Although, generally speaking, it is widely accepted that there is a gap between research results and teachers' practice and that it can be very difficult for researchers and teachers to be on the same wavelength, in this forum, both groups agreed with the ideas of RME and showed common interests and attention.

From then on, the academic exchanges in RME between China and the Netherlands were not only limited to the researchers' level, but extended their impact to teachers by evoking them to reflect on mathematics education and change their classroom practice. More and more mathematics teachers in primary and secondary education began to read the book *Mathematics as an Educational Task*. During the same period, the book *Revisiting Mathematics Education: China Lectures* was translated into Chinese and published. The two books, together with the first Forum on the Theory and Practice of Realistic Mathematics Education, helped people in China get a better understanding of RME and pushed the cooperation and exchange in mathematics education between China and the Netherlands to a peak in that year.

The second Forum on the Theory and Practice of Realistic Mathematics Education was held in 2013 in Beijing. Professor Marja van den Heuvel-Panhuizen and Doctor Michiel Doorman were invited to give lectures in Beijing, Chengdu and Jinan. Van den Heuvel-Panhuizen started working at the Freudenthal Institute in 1987 and made a great contribution to the development of RME in the post-Hans Freudenthal time. In the second forum, she gave three speeches in which she addressed the principles of RME, recent research in mathematics education from an RME perspective, and research on RME textbooks used in the Netherlands. The Chinese audience, after more than 10 years, got a second chance to gain more insight in RME and receive an update of the research done at the Freudenthal Institute. While Van den Heuvel-Panhuizen was explaining the principles of RME, the Chinese audience noticed that 'free productions', which was one of the key features of RME about 20 years ago, was not mentioned this time. Instead, the 'mathematisation under the guidance of the teacher' was introduced. In fact, the idea of 'free productions' was hard to be understood or put into practice in the Chinese educational context. Whereas 'mathematisation under the guidance of the teacher' was closer to the situation in China, because it not only affirms students' primary role of learning mathematics—mathematisation is considered to be the starting point of mathematics learning and teaching—but also emphasises the importance of teacher' guidance during the process of mathematisation. As a result, this idea was quickly accepted and supported by the Chinese audience.

While looking back at the process of evolvement of RME—one of the most important theory systems in mathematics education—it is clear that RME is not a finished or closed system, but is continuously open to new developments and innovations according to the ever-changing society and accumulated experiences of people. Just as a famous Chinese saying goes, it keeps pace with the times. Only when a theory can move along with the latest changes in the environment that it is generated from and serves to, it can have lasting vitality and the power to extend without limit in both theoretical and applicable aspects. Van den Heuvel-Panhuizen's interpretation of such characteristic of RME deepened our understanding of RME, which made us feel more respectful to this theory.

Doorman had been doing research in RME and its application in secondary mathematics education for nearly 20 years. In this second forum, he introduced the curriculum design and teaching practice of calculus under the guidance of RME, which aroused great interest in the audience. It is almost 30 years since calculus was included in the mathematics curriculum in secondary education in China. But there was not really a clear teaching trajectory for calculus. The main problem was that the distinction between the educational goals of learning calculus in secondary school and in tertiary education was not clear. It seemed that what students have learned in secondary school was repeated in tertiary education. For example, the teaching trajectory of calculus in secondary school starts with introducing the slope of the tangent of a graph, the average velocity and the instantaneous velocity. The same path is also taken for teaching calculus in tertiary education. Doorman pointed out the educational value of learning calculus in secondary school from the perspective of mathematisation. This means helping students to understand and analyse the relationship between finity and infinity by using various and rich contexts. It also means facilitating them to quantify the relationship rather than simply remembering the definition of the derivative, calculating it and using it to solve problems without context. In his lecture Doorman first showed in a trace graph how a hurricane approached the mainland. Then, the next step of teaching was asking students to explain how the trace graphs changed by studying the points of tangency and the slopes of the tangent. In this way, together with other contexts, the definition of limit is introduced step by step. Along these lines, it was clearly illustrated how to use RME to guide the teaching in secondary education.

Organising forums on the theory and practice of RME in China means building a platform for more researchers and mathematics teachers in China to meet and discuss with the successors of Freudenthal and gain insight in RME as a theory system in mathematics education, its application in the curriculum and in teaching, and the relation to how teachers and students perform in practice. Such forums are held every few years in China with the purpose of updating the knowledge of RME. This continuous attention to and regular updating of knowledge regarding RME reveal that there is a special interest in RME in China, especially in the context of the mathematics curriculum reform. These forums in which Chinese participants are informed about RME in the Netherlands, are not common for the cooperation in the field of mathematics education between China and other countries. Therefore, these forums reveal that there is a special interest in RME in China; in particular this is the case in the context of the mathematics curriculum reform in China.

#### **10.2 The Influence of RME in the Chinese Context**

It has been 30 years since the academic communication in mathematics education between China and the Netherlands started in 1985. In the years since then, RME research done by Freudenthal and other researchers at the Freudenthal Institute have provided rich and precious experience, which is considered an important resource by Chinese scholars and educators for developing mathematics education in China. Particularly, RME exerted an effect on the latest curriculum reform in mathematics education in China in terms of policy making, new textbook design and change in classroom teaching.

#### *10.2.1 The Influence of RME on Curricular Policy Making*

In China, the curricular policy is embodied in the publication of curriculum standards by the Ministry of Education of the People's Republic of China (MOE). There is no doubt that these standards, being the only applicable document formulated according to the intention of the educational policy, is of great importance. Scholars and researchers are largely responsible for establishing them. After being proved by an evaluation team organised by governmental authorities, the curriculum standards released by MOE had a statutory status.

In the publication *Mathematics Curriculum Standards for Full*-*time Obligatory Education*2 (MOE, 2001), one can find detailed information about (1) fundamental ideas about mathematics and mathematics education, (2) the objectives of mathematics education, (3) mathematical content, and (4) suggestions for instruction, assessment, and the design of mathematics textbooks and other materials. From all these descriptions in the Curriculum Standards, it is evident that its design was obviously influenced by RME, because many keywords and expressions which echo the basic characteristics of RME had never appeared in similar official documents before 2001. Several examples are provided in the following.

Mathematics is the process that people understand and describe the external world qualitatively and quantitatively, progressively conceptualise and generalise rules or theory, and put them into practice. (MOE, 2001, p. 1)

Here, mathematics is defined as a process, as a process of human's activities. It is consistent to how mathematics and mathematisation are interpreted in RME. In Curriculum Standards it is also stated:

Students are the owner of their mathematics learning [and]

mathematics teaching has to be built on students' cognition development and the experience they have already got. (MOE, 2001, p. 2)

2Hereafter referred to as 'Curriculum Standards'.

Such expressions put students in the central place in all teaching activities and emphasise students' existing knowledge and experience as the starting point of teaching activities. These ideas are different from the traditional view of mathematics as a discipline and the traditional teacher-centred educational style, but more related to the connotation of 'realistic' in RME.

The content learned in mathematics is realistic, […], should be potential to let students be actively involved in observation, doing experiment, conjecture, proving, reasoning, communication and other mathematics activities. (MOE, 2001, p. 2)

This sentence outlines the relationship between mathematics learned in school and the realistic external world around students. In this way, the sentence answers to the fundamental question that where the mathematics learned in school is from and what it serves for. This point is highly relevant to one of the basic characteristics of RME that mathematics is a human activity.

Another significant feature of RME, which does not belong to the classical teaching philosophy in China with its emphasis on teachers and teaching, but which is reflected in Curriculum Standards, is that

practical activity, initiative and independent exploration, cooperation and communication are all important approaches of learning mathematics […] (MOE, 2001, p. 2)

Although, Bloom's taxonomy was the basic requirement of mathematics teaching in China for a long time, in the new Curriculum Standards, in addition to 'knowing', 'understanding', 'remembering', and 'applying', three new requirements are added, namely 'undergoing', 'experiencing', and 'exploring'. Clearly, these three words are all associated with mathematisation. Therefore, we can conclude that many elements in the Curriculum Standards are interrelated with RME.

#### *10.2.2 The Influence of RME on Textbook Design*

Figure 10.1 shows a page from the Chinese textbook *The Primary Mathematics for the New Century* (Liu, 2014). The pictures at this page refer to the theme that is addressed here. Such a picture is called a 'context picture'. Based on the information provided in this picture, especially information related to numbers and shapes, a series of questions are posed one by one, and the last question of the series is intended to be answered by an explicit conclusion about the concept that is at issue.

In contrast to what is shown in Fig. 10.1, where one context is used, generally two or three contexts around one mathematics theme and their corresponding questions series are used. In this way, the core of what has to be learned is approached from different perspectives. This example makes clear that Chinese textbook designers acknowledge the importance of context and mathematisation. Although there is still much room for improving the choice and design of rich context problems, and providing opportunity to students to explore problem situations by themselves, the

**Fig. 10.1** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014); the page is meant for Grade 3 and illustrates the structure used in such a textbook: starting with a context problem followed by a series of questions

recently designed textbook series pay much attention to students' mental and cognitive development, which is totally different from the traditional textbooks that start from definition and examples.

Generally speaking, starting with a context problem, followed by a series of questions to lead students to what they are supposed to learn, is the basic style in most textbooks series used in primary and secondary mathematics education after the curriculum reform. Until the 21st century, this way of structuring textbooks did not exist in the history of mathematics textbook design in China. To a great extent, this change was inspired by the RME idea of using context problems and mathematisation. In addition, mathematics textbooks designed within the RME approach were other important resources for the textbook design in China. For example, the textbook *Mathematics in Context* (Wisconsin Center for Education Research & Freudenthal Institute, 1997–1998) served as a model for Chinese reformed textbooks.

The style of a context followed by series of question series is not only adopted in textbooks designed for primary education, but also in textbooks for secondary education, see Fig. 10.2.

As well as in the style of structuring textbooks, the influence of RME on textbook design can be found in the aspect of content. Like in RME, in Chinese textbooks attention is paid to connecting mathematics to reality. For example, besides knowing and understanding the number system, students should also develop knowledge about daily life numbers, see Fig. 10.3.

Another clear example of adapting the content can be found in the domain of geometry. The content of geometry in traditional primary school textbooks in China involves mainly measurement, including the definition of area and volume with the main focus on calculation. So, for a long time, the concept of space which is so important for students was not included in the mathematics textbooks. However, after the curriculum reform, there was a big change in this approach. In the newly designed textbooks the concept of space was added, which is illustrated by pages from primary school textbooks shown in Figs. 10.4, 10.5, 10.6 and 10.7. From these textbook pages, one can see the impact of RME. In Fig. 10.8 this influence can also be recognised in a textbook used in junior high schools.

In China, the focus in mathematics teaching and learning has been always on problem-solving. However, what were considered as problems here were mainly bare number problems and simple word problems. Moreover, such problems were grouped into certain types of problems. Therefore, a textbook starts with a sample problem which represents a type of problems. By reflecting and generalising the way of solving this sample problem, students are expected to know how to solve that type of problems. Then, exercises aimed at enhancing students' ability to solve particular types of problems are provided in textbooks. Since exercises in textbook are always considered to be not enough, more exercises and learning material follow. The result is that Chinese students generally become the most hard-working students. The fixed procedure of grouping problems into types and solving these types of problems is effective in getting a good result in examination. However, in this approach to mathematics education not enough attention is given to prepare students for daily life and their future professional career. The question of what students can

#### **Exploration of the Pythagorean theorem**

In Fig 1-1, the upper part of a telegraph pole needs to be connected to the ground by a steel rope.

If the upper part of the pole is 8 m away from the ground and the touching point of the steel rope on the ground is 6 m away from the pole, how many metres the steel rope needs to be.

In a rectangular triangle, if two arbitrary sides are determined, the third side is also determined. Between the three sides, there is a certain relationship. In fact, people in ancient time have already found there is a particular relationship between the square of each side. Let us go to explore together!

(1) Draw a rectangular triangle on the grid paper and measure its three sides. See what the relationship between the square of the three sides is, and discuss what you find with your classmates.

(2) In Fig 1-2, what are the square of the three sides? Is it fit to the rule you just discussed? How do you calculate? Discuss with your classmates. How about the rectangular triangles in Fig 1-3?

(3) If the legs of a rectangular triangle are 1.6 and 2.4 unit respectively, do you think the rule still apply? Give your reason.

**Fig. 10.2** Page from the textbook *Mathematics* (Ma, 2013) designed for junior high school; the page is meant for Grade 8 and illustrates the structure of starting with a context which is followed by a series of questions

**Fig. 10.3** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014) meant for primary school Grade 4; in this page, the students have to learn bigger numbers

**Fig. 10.4** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014); the page is meant for Grade 1; the students have to observe objects from different perspectives

**Fig. 10.5** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014); the page is meant for Grade 5 and addresses the content of cubic figures and their nets

## **2 Orientation and location**

### **East, South, West and North**

Go to the playground, and discern east, south, west and north. Make a note of what you see in which direction in Fig 1 in appendix 1.

**Fig. 10.6** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014) meant for Grade 2; the page addresses the content of orientation and location

**Fig. 10.7** Page from the textbook *The Primary Mathematics for the New Century* (Liu, 2014) meant for Grade 4; the page addresses the content of simple spatial reasoning

When we observe a object from different perspectives, normally what we see are different.

In mathematics lesson in primary school, we have learned to discern different images of a object from the front, the left and the top. For example, in Fig 1-18 the cubic figure is composed of some cubes. Its images from the front, the left and the top are shown in Fig 1-19 respectively.

**Fig. 10.8** Page from the textbook *Mathematics* (Ma, 2013) meant for junior high school Grade 7; in this page students have to observe objects from different perspectives

get from mathematics education except preparation for an examination has bothered us for a long time, even when facing the great performance of Shanghai students in PISA (Programme for International Student Assessment). At the beginning of the curriculum reform, we realised that it was needed to change both fundamental ideas of mathematics education at the level of educational policy making, and textbook design with respect to the structure of the textbooks, their content, and their connected materials. However, at a moment that the focus of teaching and learning was on solving types of problems for having high scores in tests, it was not easy to find the right direction to change mathematics education. At that moment, RME from the Netherlands provided a good example for us.

#### *10.2.3 The Influence of RME on Classroom Teaching*

Based on the principle of considering students as the owner of their mathematics learning, Chinese mathematics teachers are gradually shifting from a role as authority in class to a role as organiser, facilitator, and co-operator in students' mathematics learning. The proportion of instruction is reducing; while the proportion of students' exploration, communication, and cooperation is increasing. There is no doubt that it is a huge challenge for Chinese teachers, especially considering the deep-rooted Confucian culture in China that highly values the dignity and authority of the teacher.

From Hans Freudenthal, to Jan de Lange, to Marja van den Heuvel-Panhuizen, to Martin Kindt, and to Michiel Doorman, all the researchers at the Freudenthal Institute had put the emphasis in their lectures on students' active involvement in the learning process rather than on teachers' teaching in the sense of demonstrating what the students have to do. Especially, they underlined that teachers can reduce their unnecessary interventions by providing appropriate context problems which offer students opportunity for mathematisation. Many examples mentioned in these lectures have become classical cases used in China for mathematics teachers' professional development. By analysing and reflecting these cases, many Chinese mathematics teachers get a better understanding of RME, and try to change their former teaching practice of direct transmission. The two books written by Freudenthal are very popular among mathematics teachers in primary and secondary schools in China. Moreover, the book chapter written by Xiaotian Sun (2003) published in the book *The Development of Mathematics Curriculum from an International Perspective*, in which he vividly illustrated how RME by Dutch teachers is implemented in real classrooms, also became a resource for Chinese teachers to think over and learn from during the process of putting ideas of curriculum reform into practice.

Nowadays, students have become active learners in their mathematics class rather than learning passively. Instead of immersing themselves into exercise individually, students are given more change to voice their thought and to discuss with and learn from each other. Communication and exchange between teachers and students, between students and their peers are now mainstream in mathematics classes in China. It is delighting that Chinese students' ability of expressing themselves has been largely improved. The ability of articulating one's understanding is an important external indicator of the ability of thinking. Students' improvement in this aspect has been considered as one of the main achievements of the curriculum reform, which echoes the previously formulated expectation of the curriculum reform. Of course, this achievement is the result of multiple endeavours. One of them obviously is the contribution made by researchers from the Freudenthal Institute to bring RME from the Netherlands to China.

For a long time, China has put herself in the position of a student in the international class of mathematics education. Although Chinese students perform very well in PISA, there is still a long way to go for us as a student to learn from other countries. The improvement in curriculum reform in China is definitely related to the advanced international experience of mathematics reform worldwide. Scholars and researchers in China have done many studies on curriculum standards, textbooks and teaching practice in tens of countries, which provide important nutrition to improve curriculum reform in China. In this paper, we focus on reflecting the profound influence of RME, Hans Freudenthal and other researchers at the Freudenthal Institute in the Netherlands on the development of mathematics education in China. In our view, RME from the Netherlands has provided great power for the growth of the mathematics curriculum in China. At the same time, the application of RME in China also provides evidence of RME's power beyond the boundary of its cultural context.

**Acknowledgements** The authors of this chapter are very grateful to Xiaoyan Zhao for helping to translate their manuscript into English.

#### **References**

De Lange, J. (1987). *Mathematics, insight and meaning*. Utrecht, the Netherlands: OW&OC.

**Open Access**This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated.

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## **Chapter 11 The Enrichment of Belgian Secondary School Mathematics with Elements of the Dutch Model of Realistic Mathematics Education Since the 1980s**

#### **Dirk De Bock, Johan Deprez and Dirk Janssens**

**Abstract** In search for alternatives for the failed New Math movement of the 1960s and 1970s, Belgian mathematics educators looked with great interest to the Dutch model of Realistic Mathematics Education (RME), developed by Hans Freudenthal (1905–1990) and his team at the University of Utrecht. In this chapter, we primarily focus on how, from the mid 1980s until the mid 1990s, valuable elements of that model were integrated in Belgian secondary school mathematics. At that time, the influence of Dutch mathematics education on Belgian curricula was quite substantial, but some form of collaboration between the communities of mathematics teachers in both countries already existed since the early 1950s. However, from the 1950s until the 1970s, school mathematics in both countries evolved largely independent of each other. In Belgium, the structural New Math approach, with Georges Papy (1920–2011) as the main figurehead, became dominant in school mathematics, while the modernisation of school mathematics in the Netherlands was strongly inspired by Freudenthal's RME model emphasising the role of applications and modelling.

**Keywords** Georges Papy · Hans Freudenthal · Modelling and applications · New math · Realistic mathematics education · Uitwiskeling

D. De Bock (B)

D. Janssens e-mail<EMAIL_ADDRESS>
© The Author(s) 2020 M. van den Heuvel-Panhuizen (ed.), *International Reflections on the Netherlands Didactics of Mathematics*, ICME-13 Monographs, https://doi.org/10.1007/978-3-030-20223-1\_11

189

Faculty of Economics and Business, KU Leuven, Leuven, Belgium e-mail<EMAIL_ADDRESS>
J. Deprez · D. Janssens Department of Mathematics, KU Leuven, Leuven, Belgium e-mail<EMAIL_ADDRESS>
#### **11.1 Papy and Freudenthal: Opposite Views on Mathematics Education in Neighbouring Countries**

With the creation of the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM) in the early 1950s, international, or at least Western European, collaboration in mathematics education was launched. According to Caleb Gattegno (1911–1988), the founder of the CIEAEM, national organisations of mathematics teachers had to take a leading role in international exchange. Gattegno stimulated the creation of such organisations and set a good example by creating the Association for Teaching Aids in Mathematics, now the Association of Teachers of Mathematics (ATM) in the United Kingdom in 1952. Following Gattegno's call, the Société Belge de Professeurs de Mathématiques (SBPM)/Belgische Vereniging van Wiskundeleraren (BVW) was founded in 1953 andWilly Servais (1913–1979), one of the main CIEAEM personalities of the time, became its first president (Miewis, 2003; Vanpaemel, De Bock, & Verschaffel, 2012). The SBPM/BVW brought together a few hundred mathematics teachers from both linguistic communities (Dutch and French) and from all school networks (state schools and Catholic schools). It started its own (also bilingual) professional journal *Mathematica & Paedagogia* (*M&P*) and in his first editorial, Servais held a strong plea for international openness and exchange. He wrote: "Mathematics as a truly universal language has, by its nature, an international vocation; we will open the columns of our journal to colleagues in other countries" (Servais, 1953, p. 4).1 Servais' plea was received favourably by the international mathematics and mathematics education communities of the time. Several famous authors, most of them members of the CIEAEM, submitted contributions to *M&P*. The Belgian journal rapidly became a forum for national and international exchange in mathematics education (De Bock & Vanpaemel, 2015b).

This spirit of internationalisation was also present in the articles of the SBPM/BVW, in which the establishment of relationships with foreign associations of mathematics teachers and other international organisations sharing similar goals, was included as an important objective. In *M&P* 1, Servais presented the international network of the SBPM/BVW, including professional organisations of mathematics teachers in France, United Kingdom, Switzerland, Italy and Germany. Later, as communicated in *M&P* 5, this network was expanded through collaboration with the National Council of Teachers of Mathematics in the United States and with LIWENAGEL and WIMECOS, two associations of mathematics teachers in the Netherlands at the time (see, e.g., Maassen, 2000). The collaboration with the Dutch was realised in three ways: the associations exchanged and reviewed each other's journals and entrance exams (for future students of civil and military engineering), board meetings were mutually attended and mathematics educators of both countries gave lectures at conferences of the fellow associations. We do not claim there was a real interaction or mutual influence between Belgian and Dutch mathematics education at that time, but at the professional organisations' level, both communities

1In this chapter, all translations into English are by the authors.

of mathematics teachers were regularly informed about what happened in the other country.

During the 1950s, several leading mathematics educators from the Netherlands (Freudenthal, Van Hiele, Vredenduin, …) contributed to *M&P.* Of particular interest was a contribution by Luke N. H. Bunt (1905–1984) who wrote about an interesting development in the Netherlands. Bunt was invited at the SBPM/BVW conference of 1959 to report about an introductory course on probability and statistics he had developed with a team of six mathematics teachers and with which he had experimented in the alpha streams of Dutch secondary schools (Bunt, 1959). Statistics was a blind spot in Belgian mathematics education at that time and Bunt's main target group, future students in economics, psychology and other social sciences, was mostly neglected. Bunt presented a pragmatic approach, which was also in contrast with the more systematic and rigorous approaches that were generally applied in Belgium. He deliberately started with provisional definitions, definitions that are incomplete from a scientific point of view. For example, he first defined the probability of an event as the ratio between the number of favourable and the total number of outcomes in the case of equally likely events. Based on that definition, he proved the main calculation rules for probabilities. Later on in his course, when the need arose to cover more situations, Bunt presented a new definition, based on the limit of relative frequencies and without further explanation, he stated that "for probabilities based on this new definition, the previously proven calculation rules remain valid" (Bunt, 1959, p. 38). As a consequence of his pragmatism, Bunt was able to arrive at the basic ideas of hypotheses testing in a limited number of lessons.

By the end of the 1950s and in the 1960s, Belgian school mathematics was gripped by New Math or 'modern mathematics', a structural approach to mathematics teaching that was officially launched at the Royaumont Seminar (1959) and then spread worldwide (De Bock & Vanpaemel, 2015a). One of the main objectives was narrowing the gap between school mathematics and mathematics as a scientific discipline. New Math not only led to new mathematical content—sets, relations, logic, mathematical structures (groups, rings, …), linear algebra and topology—but also to a modernisation of teaching aids, for example, the use of Venn diagrams, arrow-graphs and colour conventions. Proper notations and symbols, the use of the right jargon and theory development received a lot of attention. Barriers between mathematical subdomains (algebra, geometry, trigonometry,…) were largely eliminated and geometry education was redirected towards transformation and vector geometry. The main architect and uncontested leader of Belgian New Math was Georges Papy, professor of mathematics at the Université libre de Bruxelles, who was, at that time, also influential at the international level, for example, as president of the CIEAEM in the 1960s (Bernet & Jaquet, 1998). After some years of experimentation (Fig. 11.1), coordinated by the Centre Belge de Pédagogie de la Mathématique/Belgisch Centrum voor Methodiek van de Wiskunde, from 1968 on, New Math became compulsory in the first year of all secondary schools in Belgium (and from then on gradually in the subsequent years). It was one of the most drastic educational reforms that Belgium had ever seen. A few years later, New Math was also introduced at the primary level. For

**Fig. 11.1** Papy and one of his students experimenting with New Math, 1960

about 20 years New Math was the dominant paradigm for the teaching and learning of mathematics in Belgium.

Also in the Netherlands, some mathematics educators fell for the charms of New Math, among them the logician and Royaumont participant Piet Vredenduin (1909–1996), who positively reported about Papy's experiments in *Euclides*, the journal of the Dutch associations of mathematics teachers (Vredenduin, 1967). But from the outset, New Math was also strongly criticised. Freudenthal summarised his critique in two words: "anti-didactic inversion" (Freudenthal, 1973, p. 103), expressing that an end product of mathematical activity, the most recently composed structure of mathematics, is taken as a starting point for mathematics teaching (Fig. 11.2). Although Freudenthal could not prevent that also in the Netherlands, a New Math inspired curriculum for the secondary level was introduced in 1968, the implementation was less radical and New Math only lasted for a few years. New Math was never introduced in Dutch primary schools. In 1971, the IOWO (Institute for the Development of Mathematics Education) was founded with Freudenthal as its first director. With a staff of 37 people, Freudenthal put into practice his ideas about mathematics education which resulted in Realistic Mathematics Education (RME), the Dutch answer to New Math and to traditional, mechanistic approaches, both for the primary and secondary level (La Bastide-van Gemert, 2015). The features of RME include the use of rich contexts and realistic situations, that is, problem situations which students can imagine, in order to develop mathematics, the use of students' own productions as well as researched activities encouraging students to move from informal to formal representations, less emphasis on algorithms and more on sense-making and the use of guided reinvention.

**Fig. 11.2** Papy and Freudenthal (caricatures by Léon Jesmanowicz, 1971)

#### **11.2 Critique on New Math in Belgium and Search for Alternatives**

In 1974, the SBPM/BVW was restructured on a linguistic basis into the SBPMef (Société Belge des Professeurs de Mathématiques d'expression française) and the VVWL (Vlaamse Vereniging Wiskundeleraars), with respective journals *Mathématique et Pédagogie* and *Wiskunde en Onderwijs*. The SBPMef did not continue some form of cooperation with the community of Dutch mathematics teachers, but instead started some networking with the French and later also with French speaking mathematics teachers in Switzerland. Hence, at least at the level of professional organisations, cooperation and even exchange of information between the French speaking Belgian community and the Dutch stopped. For the purpose of this chapter, we will further focus on the evolutions in Flanders.

Although at that time and until the late 1980s, the VVWL had become a fortress of New Math proponents (including, e.g., De Bruyn, De Munter, Holvoet, Laforce, Verhulst, Vermandel, Warrinnier), cooperation with the Dutch continued formally and Vredenduin became the main contact (Holvoet, 1996). Except for a number of articles by Vredenduin himself, mostly about logical issues, *Wiskunde en Onderwijs* at that time rarely published articles by Dutch authors or contributions informing its readership about ongoing evolutions in the Netherlands. The work of Freudenthal and his team was ignored by the official association of mathematics teachers in Flanders.

The stable position of New Math in Belgium during the late 1960s and 1970s and the absence of critique in official fora, does not imply that the whole mathematics education community in Belgium was unconditionally in favour of Papy's approach. Instead, Papy's method and its implementation at school had divided this community into two camps, the so-called 'Papy'ists' and 'anti-Papy'ists' (Colot, 1969), but that latter group was not in power at that time and remained relatively silent in public fora. This silence was broken on March 11, 1980, when Albert Pirard and Paul Godfrind, professors at respectively the Université de Liège and the École Royale Militaire, published an opinion article in *La Libre Belgique*, a main newspaper of the French speaking Belgian community. The title of the article, "The Disasters of Modern Mathematics", set the tone. Among other things, the authors stated: "We now see in the entrance exam [for future students of engineering, *added by the authors*] candidates who have almost never practiced geometry" (Pirard & Godfrind, 1980, p. 15). They heckled the tough and abstract approach to geometry in the third year of secondary school in which, for example, the length of a segment was defined as the class of all congruent segments and an angle was briefly described as "a rotation that has lost its centre" (ibid.). They further argued that modern mathematics is absolutely useless and that a total aversion prevailed among students and science teachers. They advised "urgently to leave the abstract language and aberrations of modern mathematics and to return to a realistic, concrete and basic teaching of mathematics" (ibid.). However, as far as we know, the impact of Pirard and Godfrind's article in Flanders was limited.

In Flanders, the public debate opened only a few years later, in 1982, when the pedagogue and teacher educator Raf Feys published a virulent pamphlet "Moderne Wiskunde: Een Vlag op een Modderschuit" (Feys, 1982), in which he firmly criticised the fundamental principles of New Math and the way it was introduced at the primary level. In his close contacts with schools, Feys did not see the appearance of the promised fascinating world, but "artificial results in a fake reality" (ibid., p. 3) and also little enthusiasm in children, but "more disgust, disorientation and desperation" (ibid., p. 3). Feys not only criticised New Math, he also suggested how mathematics education at the primary level should evolve and the model he had in mind was the Dutch RME. Instead of taking the structure of mathematics as a starting point, mathematics education should start from and gradually develop, the intuitive, informal and real-world knowledge and skills of the children. Feys' pamphlet focused on primary education and had most impact at that level, but thanks to media attention, it also echoed at the secondary level.

A follow-up event and important step towards a broad and open societal debate in Flanders was initiated by the Foundation Lodewijk de Raet, which on April 30, 1983 organised a colloquium in Brussels titled What Kind of Mathematics for 5–15 Year Olds? (Stichting-Lodewijk de Raet, 1983). Nearly 150 people participated, including representatives of mathematics education from primary to university level, teacher educators, members of the Inspectorate of Education and of mathematics curriculum committees. In a lecture by Freudenthal, participants were confronted with developments abroad (including the decision of some German states to prohibit by law set theory at school because "it made children mentally ill" (Stichting-Lodewijk de Raet, 1983, p. 4) and they learned about the Dutch alternative, illustrated by plenty of IOWO materials. On the opposite side, Roger Holvoet (1938–1998), professor of mathematics at the University of Leuven and fervent Papy-ist, minimised elementary school students' difficulties with typical New Math elements and confirmed his confidence in the current update and innovation of mathematics education for the primary level. Another speaker at the colloquium, Jan Vermeylen, mathematics teacher at a Flemish secondary school and board member of the VVWL, showed himself more critical of the ongoing modernisation: "Little has been realised from the beautiful dream, finding an easy way to learn mathematics" (Stichting-Lodewijk de Raet, 1983, p. 12). He argued that, in most study streams, we should teach mathematics being as useful as possible, based on students' experiences and interests. He further stated that, for this purpose, set theory is not strictly prohibited, but in most cases, unnecessary and even harmful. Clearly, the colloquium revealed diverging points of view. But more importantly, it definitely made it clear that an adjustment of New Math had to take place and that "the learning materials of the IOWO and the new Dutch textbooks inspired by the RME approach, could no longer be neglected" (Stichting-Lodewijk de Raet, 1983, p. 2).

At the secondary level in Flanders, another development took place. In 1983, Dirk Janssens, recently appointed as professor in mathematics education at the University of Leuven, launched the idea to start a new journal for mathematics teachers. A group of his former students and newly started mathematics teachers (Deprez, Eggermont, Gyssels, Kesselaers, Remels, Roelens and Roels) responded positively and started the journal entitled *Uitwiskeling* (Fig. 11.3), with the first issue being published in 1984. *Uitwiskeling* is an untranslatable neologism connecting 'wiskunde' (the Dutch word for mathematics) with 'uitwisselen' (the Dutch word for 'to exchange'). The name of the journal refers to the idea of creating a forum where mathematics teachers can exchange ideas and discuss questions related to the practice of mathematics education. So, the action of *Uitwiskeling* was basically a constructive one: not directly criticising New Math, but searching for and sharing teaching resources that motivate pupils and stimulate their active participation in the learning process. *Uitwiskeling* had some fixed columns, the most important ones being "Cobweb", intended for questions and answers, hints, ideas, suggestions, reports of lessons and other short contributions from the readership; "Under the Magnifying Glass", a larger article in which members of the editorial board scrutinised and elaborated a part of the curriculum or an aspect of mathematics education; and "Guide to the Library", in which articles and books (in most cases from abroad) that were considered useful for classroom practice were identified, summarised and discussed. Soon, *Uitwiskeling* reached a large audience of Flemish mathematics teachers and became a channel through which these teachers learned about the new developments in mathematics education in other countries. Special attention was paid to developments in the Netherlands, Germany, and the French speaking part of Belgium, in which at that time Nicolas Rouche (1925–2008), professor of mathematics education at the Université catholique de Louvain and his Groupe d'Enseignement Mathématique (GEM), founded in 1978, became very influential. The journal *Uitwiskeling* still exists (www.uitwiskeling.be)—currently, that is, 2019, the 35th volume is running—and the angle of incidence remained unchanged: the practice of mathematics education and the confrontation of that practice with new ideas from didactics of mathematics.

**Fig. 11.3** Cover *Uitwiskeling*, November 1987 (design: Abdon Van Bogaert)

### **11.3 How During the Middle 1980s and 1990s New Developments in Neighbouring Countries Reached the Community of Flemish Mathematics Teachers**

#### *11.3.1 Rounding off the Rough Edges of New Math*

During the middle 1980s, the call for change became ever louder in Flanders and an official response was therefore inevitable. A first modification of the curricula for the secondary level started in 1983 and lasted until 1988. It was a modest reform that mainly rounded off the rough edges of New Math (Roels, 1995). We briefly describe some new accents in the programmes for the catholic schools (which is the largest network of schools in Flanders). Changes in the other networks ran more or less in parallel. For the first two years, a more intuitive approach to arithmetic was proposed: the different types of numbers and number operations were no longer defined in a set-theoretic environment and many proofs were eliminated. So, it became permitted again to introduce negative numbers with reference to temperatures below zero or to profit and loss and, for the rational numbers, teachers were allowed again to refer to the fractions that pupils had learned at the primary level. The time saved on theoretical issues was spent on practicing the operations and on solving equations and word problems. The plane geometry of the first two years, which was affine, remained unchanged, but the curriculum committee chose a radically different approach for metric plane geometry, which is part of the programme from the third year on. The length of a segment and the measure of an angle were accepted as primitive concepts (such as, e.g., points or straight lines) in combination with a few intuitively

**Fig. 11.4** Papy's (1967) version of the Pythagorean theorem ('Hoofdstelling' = Main theorem; 'Als' = If; 'dan' = then)

accessible axioms, and thus the need to define them fell away. Partly because of that, the extensive study of isometries (reflections, translations, rotations, …) was shortened so that more time and attention could be spent on the classical plane figures (triangles, quadrilaterals, circles, …) and their properties. The Pythagorean theorem, reduced to a special case of the formula for the norm of a sum of vectors during the New Math period (Papy, 1967) (Fig. 11.4), regained its central position of the past. Finally, in analytic geometry, the equation of a straight line could again be introduced without relying on vector spaces. The introduction of this new geometry programme for the third year was accompanied by a large-scale action of in-service teacher education (Janssens & Roels, 1985).

At the upper secondary level, the curriculum change was more radical. In solid geometry, in Flanders a topic traditionally reserved for study streams with a strong mathematics package (6 to 8 h per week), the synthetic perspective was revalued. The curriculum stated that solid geometry should be seen as an extension of plane geometry and no longer as a part or application of linear algebra. The synthetic approach, meant for developing students' spatial skills, had to precede the analytic treatment. Special attention had to go to the sketching and understanding of planar representations of spatial situations. It was further recommended to also include problems about area and volume of solids. But a new curriculum cannot succeed without new and appropriate teaching materials. An attempt to provide such materials was undertaken by *Uitwiskeling* (Deprez, Roelens, & Roels, 1987b) and developed in more detail in an in-service teacher education course (Deprez, Eggermont, Janssens, & Roelens, 1987a). One of the sources of inspiration was the work of the Dutch HEWET team (led by De Lange and Kindt) who had developed teaching materials, in line with the RME philosophy, on various mathematical topics for pre-university education (for an overview, see, e.g., De Lange, 1987). This work was realised under the umbrella of OW&OC (Mathematics Education Research and Educational Computer Centre), the successor of the IOWO since 1981. *Uitwiskeling* was also influenced by publications of the GEM (Groupe d'Enseignement Mathématique) and of the French IREMs (Instituts de Recherche sur l'Enseignement des Mathématiques). A pragmatic-eclectic approach to solid geometry was presented, starting intuitively with a phase of exploration and investigation. In that phase, it was suggested, students had to work in groups on problems that are challenging, but easy to understand and imagine, such as, for example, "What types of plane figures can occur when one intersects a cube and a plane?" This phase should then result in a series of statements about possible mutual positions of straight lines and planes, of which students have to check the correctness. Next, the correct statements could be accepted as starting points—axioms—for the further development of solid geometry. Later on, based on the results of that synthetic phase and on what students have learned in the fourth year in their lessons on plane geometry (e.g., about the scalar product), space coordinates and vectors could be introduced and analytic descriptions of straight lines and planes could be deduced. Finally, a number of richer problems about solids should be investigated, using both synthetic and analytic tools.

Another important change at the upper secondary level related to the approach of analysis (calculus) in study streams with 2–4 hours of mathematics per week. This new approach was not a mere weakening of the corresponding curriculum part for study streams with 6–8 hours of mathematics per week, but tried to meet the specific needs of students who had, for some reason, chosen a limited package of mathematics in their final years of secondary school. The idea was to opt for a less formal approach by skipping the topological foundation and by introducing the concepts of continuity and limit in an intuitive-graphical way. The time saved had to be spent to derivatives and integrals and to applications of these basic concepts of calculus. Because the meaning(s) of these latter concepts was central, calculation techniques were limited to polynomial and rational functions. Another novelty was a change in the order of integral calculus for these study streams: to allow a more insightful and motivating approach, the definite integral was introduced first, as the (oriented) area under the graph of a function, before the concept of primitive function (or indefinite integral). Initially, for this important curriculum change, little or no didactical support was provided for the teachers involved, but quite soon, *Uitwiskeling* spent an issue on this new approach (Deprez, Gyssels, & Roels, 1985) and later, this was further developed in an in-service educational course for teachers (De Bock et al., 1986). This course presented a quite radical interpretation of the new curriculum: the authors immediately started with derivatives, continuity was omitted and limits were, to some extent, integrated into the section on derivatives. That section on derivatives was largely inspired by two Dutch HEWET cahiers (Kindt & De Lange, 1984, 1985) in which the derivative was distilled from different real-world contexts in which (rate of) change had to be measured. The idea of 'conceptual mathematisation', that is, mathematisation as a way to introduce mathematical concepts (De Lange, 1987), was quite innovative in Flanders at that time.

#### *11.3.2 A Second Wave of Changes*

The curriculum changes of the middle 1980s were positively welcomed by most of the Flemish mathematics teachers. The conviction grew that priority in school mathematics should be shifted from mathematics as a static, rigorous deductive system to a meaningful and useful activity related to the broader world and society. Of course, parts of the curriculum that had remained unchanged in the first curricular modification were now assessed more critically by the teachers. Therefore, a second adjustment became inevitable. The introduction of a new structure for Flemish secondary education in 1989 provided the opportunity for this second wave of curriculum changes. In terms of content, changes were in line with the foregoing: typical New Math elements, such as sets and relations, were further reduced, mathematical structures were no longer explicitly addressed and the ambition to set up a global deductive system was abandoned. The treatment of geometry in the early years of secondary school became metric from the start and the notion of area and corresponding calculations were no longer neglected. Another important innovation was the extension of combinatorics and probability to statistics, now also including descriptive statistics (data analysis) and the testing of hypotheses as a preview of the application of statistics in practice (Carbonez & Veraverbeke, 1994; Kesselaers & Roelens, 1992). Finally, the modernisation of analysis that had started with the first curricular modification in study streams with a limited package of mathematics was now added to the study streams with a strong mathematical component. The idea was to start with sequences as a basis for a mathematically rigorous, but at the same time more intuitive and dynamic approach to the concepts of continuity and limit (De Bock et al., 1992). In these study streams the time spent on continuity and limits was also diminished: derivatives and integrals were considered the core concepts of secondary school analysis and had to receive maximum attention.

This second wave of programme changes was not limited to the content of secondary school mathematics, but also brought a number of didactical innovations. First, and perhaps most importantly, there was the role given to modelling and applications. In contrast with the previous period in which mathematics was basically taught as an autonomous discipline, the applicability of mathematics in other domains was now strongly emphasised. These domains did not only include the traditional areas of application (such as physics), but also biology, economics and other social sciences. Once again, inspiration was found in the Dutch RME materials, in particular in the HEWET cahiers. In contrast to the classical role of applications in mathematics education (i.e., applying a pre-designed mathematical method in another domain), also the idea of conceptual mathematisation (as explained above) enjoyed increasing attention in Flanders: mathematical ideas that are developed from diverse contexts, get a richer meaning and may subsequently be applied more easily in new domains (De Lange, 1987). Emphasising the applied side of mathematics also fitted with the belief that it is motivating for students to realise that mathematics is closely related to their own living environment. Second, more attention was given to (guided) self-discovery and active learning processes in the teaching and learning of mathematics. Contemporary research in educational psychology had shown that effective learning is based on constructive processes, mediated and guided by adequate and supportive intervention strategies (De Corte, 1996). Applied to mathematics education, it means that students should not only be confronted with 'end products' of mathematical activity (i.e., 'finished' mathematical texts), but should also have the opportunity to go through the process of mathematical discovery and to authentically (re-)build pieces of mathematics themselves. Third, the growing importance of graphing calculators and computers was recognised, not only as powerful calculation tools, but also as means for authentic mathematical exploration, discovery and simulation (Cleve, De Bock, & Roelens, 1993). These new technological tools made a more graphical approach to mathematics—and even an approach based on multiple representations—achievable in mathematics classrooms at the secondary level.

The above-mentioned curricular innovations were again supported by the largescale action of in-service teacher education Mathematics Taught By Applications (De Bock, Janssens, Roelens, & Roels, 1994; Janssens, 1993; Roels et al., 1990). The idea was to integrate the modelling perspective into the study of elementary functions, matrices, derivatives and integrals. In line with the objective to activate pupils during the mathematics classes, an active involvement of the participants was promoted and therefore, the sessions were perceived as real working sessions. During these sessions, groups of upper secondary mathematics teachers were confronted with real, and hence rather complex, modelling problems. In order to find appropriate solutions, they had to go through the whole modelling cycle (see, e.g., Verschaffel, Greer, & De Corte, 2000). An interesting case concerned the journey of the drilling rig called 'Yatzy' on the river Scheldt near Antwerp. The passage of the Yatzy under a high-voltage cable, taking into account the tides of the river, created considerable tension—both on board the platform, and among the teachers who participated! This authentic and large-scale modelling exercise, in which several mathematical models were integrated, also caught the attention of Dutch colleagues: the case was published in *Nieuwe Wiskrant* (De Bock & Roelens, 1990) (Fig. 11.5), the mathematics education journal published by OW&OC, which in 1991 was renamed as Freudenthal Institute after its founder. By that time, Flemish mathematics educators had become very familiar with the design principles of Dutch realistic mathematics education, and therefore exchange was no longer one-way traffic!

#### *11.3.3 Consolidation*

History continued and soon a new development took place. In 1989, the Flemish Government became responsible for educational matters. To promote and control the quality of education, it was decided to develop attainment targets for mainstream education at the primary and secondary level. The targets were designed as minimal objectives, which the government considered necessary and attainable for school children at these levels. These objectives referred either to knowledge, to skills or to attitudes. The attainment targets were approved by the Flemish Parliament and

**Fig. 11.5** Cover *Nieuwe Wiskrant* with the *Yatzy*, 1990

gradually implemented as parts of the curricula of the different educational networks (of Catholic and publicly run schools) from 1997 on; and they are still valid now. In this third wave of curriculum changes, the innovations of the previous phases, such as, e.g., the role of modelling and applications, the importance of a constructivist vision on learning and a meaningful implementation of ICT tools, were consolidated and continued. Remarkably, modern mathematics was no longer an issue and even elementary set theory was definitively removed from the new curricula: none of the attainment targets still referred to these icons of New Math! More importantly, a number of vertical learning trajectories were identified in the attainment targets. We briefly exemplify three such trajectories throughout students' secondary school careers.

A first vertical learning trajectory refers to statistics (Deprez, Roels, & Roelens, 1992). Nowadays, statistics starts in the first two years of secondary school with the analysis, representation and interpretation of real data and with some basic elements of probability related to fractions. Hence, a bridge with what students had learned at the primary level is made. In the middle two years of secondary education, the attainment targets state that students have to be able to select by themselves representations that are most appropriate in a given situation and should learn about different measures for the central tendency and the spread of a set of data. In addition, students learn to use probability trees to solve more complex probability problems. In the final two years, all students study the normal distribution, and students in a study stream with a strong mathematical component are introduced to confidence intervals or the testing of hypotheses, with a strong emphasis on conceptual understanding over technical fluency (see above). This development is in line with those in many countries, especially in the Netherlands, which played a pioneering role in statistics education since the 1950s and which has served as a model for Flanders (Bunt, 1959; Garst, 1990; Zwaneveld, 2000).

A second vertical learning trajectory refers to functions (Eggermont & Roels, 1997). While in the New Math period, this topic immediately started with an abstract and technically advanced definition ('a special type of subset of the Cartesian product of two sets') in the first year in secondary school, now students first encounter tables, graphs and formulas as representations of various types of meaningful relationships (e.g., proportional and inverse proportional relationships). Gradually, more functional skills are developed (e.g., transforming the graph of a function), the level of abstraction is raised and several classes of functional relationships are studied, leading to the notion of a 'real function of a real variable'. A more general and abstract definition of the concept of function, treated as an independent mathematical object, only occurs in the final years of secondary education. That way, students of different ages learn about and work with different aspects of functions that are adapted to their situation. This approach was closely related to the Dutch 'TGF analysis' (tables, graphs, formulas), promoted from the late 1980s on by the HAWEX team (led by Kindt, Roodhardt, Van der Kooij and Van Reeuwijk), but also to contemporary developments in the United States (National Council of Teachers of Mathematics, 1989).

A third vertical learning trajectory refers to solid geometry (Deprez & Roels, 2000; Op de Beeck, Deprez, & Roels, 1997; Thaels, Eggermont, & Janssens, 2001). The emphasis came to lie on gaining insight into spatial objects and their planar representations. During the New Math period, this component of geometrical thinking was completely absent in the first four years of secondary education because it did not fit with an axiomatic approach at these levels. Now in primary school, pupils learn about solids by seeing and doing, most often in realistic contexts. They gain insight in such objects on the basis of three-dimensional models or on the basis of planar representations. This learning trajectory is continued in the first years of secondary school. At that level, the (re)construction of situations in space starting from a planar representation is developed further and some attention is already paid to argumentation related to properties of solids. Area and volume of elementary solids (cubes, cuboids and cylinders) are also part of the curriculum. In the middle years of secondary school, students learn to build more precise arguments about straight lines and planes in space, but this argumentation is always embedded in concrete problem situations, for example, about planar sections of solids. In the final two years of secondary school, for those students following a study stream with a strong mathematical component, a more structured—analytical—approach to solid geometry is developed. The development has a sound mathematical basis using a modern axiom system based on points and vectors. But the system works only at the background and can eventually be discussed at the end of the course. For some students, this gives the opportunity to sketch the way to 4-dimensional geometry (Deprez et al., 1987a, b). In the new view on solid geometry for the first four years of secondary school, the influence from scholars of the Freudenthal Institute (see, e.g., the HAWEX and HEWET materials) and from Nicolas Rouche and his team is also very prominent.

#### **11.4 Some Topics that Underwent a True Metamorphosis**

In this section, we describe in some detail a number of topics that, compared to New Math, have been drastically changed and in which the influence of the RME approach is particularly clear. The first topic relates to exponential and logarithmic functions. The treatment of exponential and logarithmic functions during the 1960s and 1970s in Belgium was a textbook example of Freudenthal's (1973) notion of 'anti-didactical inversion'. In the sixth year of secondary education, first, the natural logarithmic function was defined as an integral function of 1/*x*. Next, logarithmic functions with other bases were introduced as multiples of the natural logarithmic function and finally, the exponential functions were defined as the inverses of these logarithmic functions with arbitrary base. At the end, an aha experience was evoked: for rational exponents, the 'new' exponents coincide with the ones students had previously met in the fourth year. Needless to argue that this approach, although mathematically logical, had many educational disadvantages. The RME alternative is to start with exponential functions as models for exponential (or cumulative) growth, a context that gives a concrete meaning to the exponent (time) and proved to be very useful to understand various problems situations related to this class of functions (De Lange & Kindt, 1984a, 1986). The functions *x* → 2*x* and *x* → 0.5*x* (a model for negative growth) serve as prototypes of, respectively, increasing and decreasing exponential functions, and logarithmic functions are introduced as their inverses (and thus no longer vice versa). The transition from rational to real exponents is handled intuitively. The context of growth also proves to be very helpful for reasoning about logarithms and their properties (e.g., the fundamental theorem of logarithms was clarified as "time needed for doubling + time needed for tripling = time needed for multiplication by six"). Furthermore, it is 'proved' that the slope function of an exponential function is proportional to itself and the natural exponential function (and the number *e*) is introduced as the exponential function for which the proportionality factor equals 1 (hence, as the function that is equal to its derivative). A lot of additional applications are given (logarithmic scales, drawing log-log graphs, the Carbon 14 dating method, …).

A second topic that underwent a thorough metamorphosis was trigonometry. In fact, classical trigonometry was not so much influenced by New Math—those reformers generally showed little interest for this type of 'applied mathematics'—and was still taught in a rather mechanistic (pre-New Math) way, focusing on trigonometric formulas and the (technical) solution of trigonometric equations and inequalities. The new curricula separate the geometric part about angles and the solution of triangles, taught in the third and fourth year, from the functional part—in which the arguments of sine and cosine are real numbers—that is part of the fifth and sixth year curriculum. In the functional part, trigonometric functions are seen as models for periodic phenomena, such as, for example, tides (De Lange & Kindt, 1984c, 1985). The 'sine model' and its main characteristics (amplitude, period, and horizontal and vertical translation) receive ample attention. These model characteristics are systematically explained and graphs of generalised sine functions are drawn using a grid frame. This system not only allows to approach some more complex modelling problems, but also provide a framework for the graphical solution of trigonometric equations and equalities.

Matrices are a third example. In the New Math period, this topic was part of a linear algebra course. After some practicing of computational techniques, matrices were studied in a very abstract way, with much emphasis on aspects related to properties of operations leading to the identification of underlying algebraic structures. Also, here, the HEWET materials provided an alternative for a more concrete point of view, connecting matrices to graphs and different types of contexts, such as distance and connectivity, population dynamics, consumer behaviour … (De Lange & Kindt, 1984b). In this vision, much attention is paid to the contextual interpretation of a matrix, its square and its product with another matrix. The widespread availability of ICT tools makes it also possible to perform calculations with big matrices that arose from real contexts. That way, matrices become a powerful tool for modelling various application problems in which blocks of numbers are involved. These problems originate from different disciplines (biology, economics, other social sciences, …) and proved to be a better introduction to abstract algebra than an immediate start with abstract structures.

We conclude this section with the fourth example: the new approach to differential and integral calculus. In this domain, the versatility in meanings of the concept of derivative and (definite) integral, that one cannot possibly understand based only on a definition, was given a central place. Meaning depends on the context in which these concepts occur, or, as Freudenthal (1973, p. 513) wrote: "What the differential quotient and the integral of a function mean depends on what the function means, and this can be many different things." Basically, the derivative is meant for measuring change (Kindt & De Lange, 1984, 1985). To arrive at a solid understanding of the concept of derivative, conceptual mathematisation was promoted. On the basis of different contexts in which change occurs and has slightly different meanings (speed of an animal, population growth, marginal cost, …), the notions of average and instantaneous change are explored and interpreted graphically as, respectively an average slope (slope of a secant) and the slope in a point (slope of the tangent). The transition from average to instantaneous change can be clarified on the basis of an intuitive concept of limit. This pre-formal phase is meant to equip the derivative with a rich and flexible meaning, and only at a later moment, a formal-analytical quantification is presented (as, respectively, a difference quotient and a differential quotient or derivative). In line with Poincaré's (1908, p. 83) words that "mathematics is the art of giving the same name to different things", the derivative appears as a common name for rate of change. The didactical track continues with the differentiation rules, first deduced for polynomial functions, but gradually expanded when other classes of functions are involved, and ending with various new applications, e.g., problems related to optimisation, in which, thanks to the rich pre-formal phase, the derivative is now recognised and applied more easily.

Unfortunately, directly usable HEWET or other RME materials for the teaching of integral calculus were not available, but it proved possible to design a course in the same spirit, that is, giving a central place to the versatility of meaning of the concept of (definite) integral (De Bock, 1990; De Bock et al., 1994; Roelens, Roels, & Deprez, 1990). Inspiration was, among other sources, found in the work of German mathematics educators (see, e.g., Kirsch, 1976). Although integrals are defined geometrically—directly as the (oriented) area under the graph of a function or by means of lower and/or upper sums—their meaning in different contexts and, hence, the type of problems that can be solved using integrals, is emphasised. Three types of such problems are identified. A first type is labelled 'reconstruction problems': starting from the rate of change of a variable, the variable itself can be reconstructed. So, for example, the area under a graph of flow rate of a river, velocity or marginal costs enables to reconstruct, respectively, water volume passing, distance travelled or total costs. A second type relates to summation. Integrals are not only approximated by sums (the idea of 'numerical integration'), but also vice versa: sums of a large number of terms can be approximated or idealised by integrals. Sometimes, it is also helpful to perceive a magnitude (e.g., a volume, a surface area or the length of an arc) as a sum to discover how that magnitude can be calculated as the integral of some function. Third, integral problems are also related to averaging continuously changing magnitudes. Definite integrals with variable upper limit lead to the concepts of integral function and anti-derivative or indefinite integral. Finally, the link between the concepts of (anti-)derivative and (definite) integral is established (leading to the so-called 'fundamental theorem of calculus').

#### **11.5 Conclusion**

The Dutch RME model enriched Flemish mathematics education in secondary schools during the 1980s and 1990s and its positive influence is generally recognised. However, this does not mean that mathematics education in Flanders nowadays is a copy of that Dutch RME model (as embodied in, e.g., the HEWET and HAWEX materials). Elements of the more traditional approach, focusing on calculation drill and algebraic techniques, as well as more structural elements, focusing on a logical organisation of content and on proof and argumentation, are still essential parts of Flemish mathematics curricula and of classroom practice, although their importance has decreased. In our view, this has resulted in a more or less balanced approach to mathematics education in Flanders. This specificity of Flemish mathematics education, which was the result of multiple influences, is probably one of the reasons why an orthodox version of the RME model could never be implemented. For example, a project in the late 1980s in which some of the HEWET cahiers were 'translated' to the Flemish context, proved to be unsuccessful, likely because Flemish teachers missed a clear structure, the provided modelling problems were sometimes too open and computational skills received insufficient attention. It may also be one of the reasons why the Math Wars, that originated in the 1990s in the United States, have had very limited impact in Flanders (while they have been an important educational-political issue in the Netherlands).

#### **References**

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### **Chapter 12 Echoes and Influences of Realistic Mathematics Education in Portugal**

**João Pedro da Ponte and Joana Brocardo**

**Abstract** This chapter traces the connections between Realistic Mathematics Education (RME) and Portuguese developments in mathematics education in terms of research studies and curriculum development. The basis for this work is a literature review of papers and other documents, with special attention to the period 2005–2015, and research studies organised by mathematical topic. Although there is no research group in Portugal that is perfectly aligned with RME principles and curriculum materials, noticeable influences may be seen in the frequent references made in some research groups to key RME ideas, notably the importance of students working from tasks in meaningful contexts, the role of representations and models to support students' thinking, and the levels of students' mathematical activity. This is most noticeable in conceptual frameworks for developmental research studies in the area of number and in the use of realistic contexts in task design, and it is also apparent in the official 2007 Portuguese curriculum document.

**Keywords** Tasks · Representations · Numbers · Algebra · Geometry

#### **12.1 Introduction**

Realistic Mathematics Education (RME) has a clear influence in Portugal, both in research and in curriculum development. Portuguese mathematics educators began to know about RME ideas from reading Freudenthal (1973) and from their participation in two international meetings, that of PME (International Group for the Psychology of Mathematics Education) and of CIEAEM (International Commission for the Study and Improvement of Mathematics Teaching), which both took place in

J. P. da Ponte (B)

Instituto de Educação, Universidade de Lisboa, Lisboa, Portugal e-mail<EMAIL_ADDRESS>
J. Brocardo

Escola Superior de Educação, Instituto Politécnico de Setúbal, Setúbal, Portugal e-mail<EMAIL_ADDRESS>
the Netherlands in 1985. Since then, contacts have been frequent, in other international meetings, in study visits made by Portuguese researchers to the Freudenthal Institute (FI) in Utrecht, and also through the visits made by researchers of the FI to Portugal to attend mathematics education conferences,1 to participate in activities of research and evaluation projects, and to deliver seminars to doctoral students.2

Research in mathematics education in Portugal began its development from the 1980s, with several doctoral degrees obtained abroad (in the United States and the United Kingdom), but only in the 2000s did it become more intensive, with doctoral degrees offered at several Portuguese universities (mainly in Lisbon, Aveiro, and Braga). It was around these doctoral programmes that the most important research groups developed, with the association of several schools for higher education (such as those in Lisbon, Setúbal, and Viana do Castelo).

In this chapter, we give an account of the main influences of RME in Portugal, with special attention for the last 10 years (2005–2015). The chapter is constructed from a revision of doctoral theses, edited books, articles published in mathematics education national and international scientific journals (*Quadrante*, *BOLEMA*, *Relime, Uni*-*Pluri/Versidad*), articles published in the teacher journal *Educação e Matemática* of APM (Associação de Professores de Matemática3) and communications in proceedings of the national mathematics education research meetings SIEM (Seminário de Investigação em Educação Matemática) and EIEM (Encontro de Investigação em Educação Matemática). The chapter contains two main sections, one concerning research studies (subdivided in mathematical topics) and another concerning curriculum development. In each section, the studies and documents are highlighted in which the influence of RME ideas appears to be stronger, seeking to identify the main contributions to theory and practice of mathematics education as well as the aspects in which this influence may be traced. The chapter concludes with a summary of RME influence in our country.

#### **12.2 Influences on Research Studies**

#### *12.2.1 Whole Numbers and Operations*

The notion of number sense inspired several researchers of the Developing Number Sense (DNS) project to develop an alternative approach to teaching whole numbers and operations that includes certain central RME ideas. This project is a central reference in the mathematical domain of number in Portugal. Here the fundamental

1Koeno Gravemeijer has been at EIEM in 1997 (Castelo de Vide), Marja van den Heuvel-Panhuizen at CIEAEM in 1997 (Setúbal), Rijkje Dekker, Koeno Gravemeijer and Jean-Marie Kraemer at 'Mathematics Education: Paths and Crossroads in Memory of Paulo Abrantes' in 2005 (Lisbon), and Henk van der Kooij at EIEM in 2006 (Monte Gordo).

2Especially Koeno Gravemeijer and Jean-Marie Kraemer.

3Association of Mathematics Teachers.

influences can be identified related to the role of counting, place value and standard algorithms, the role of the context of tasks and the use of models in mathematics learning.

Also in this perspective, in a book published by the Portuguese Ministry of Education to support the activity of infant education teachers, Castro and Rodrigues (2008) indicated that children may use counting to compute. From a sequence of tasks, they exemplified how the activity of counting may evolve from one to one counting to the re-invention of informal mental strategies based on counting, one of the RME big ideas (Beishuizen & Anghileri, 1998). Still in the same vein, Rodrigues (2010), in a study addressing the development of number sense in children from three infant schools, concluded that it is from the numerical sequence and from counting competences that children develop other numeral competences. The children that she studied used counting as an informal strategy that, based in diversified experiences in meaningful numerical contexts, was progressively structured, discovering counting in patterns and jumps.

#### **<IP_ADDRESS> Place Value and Standard Algorithms**

The tradition of focussing numerical learning in the early use of place value aiming at the quick construction of standard algorithms was strongly questioned by several RME authors. Brocardo, Serrazina and Kraemer (2003), following this trend and basing their argumentation on authors such as Gravemeijer (1991) and Fosnot and Dolk (2001a) highlighted the need to link structurally the development of computation methods and techniques to the construction of numbers, their organisation and structure. In order to achieve this, all these authors argued that it is necessary to delay the learning of algorithms the early introduction of which they viewed as hindering an adequate development of number sense. Brocardo and Serrazina (2008), reflecting on the work carried out by the DNS project, strongly influenced by Gravemeijer (1991, 1994, 2005), Treffers (1987, 1991) and Buys (2008) on the teaching and learning of numbers and operations, suggested that the algorithms must not be the central focus of the curriculum, and that students must learn them in a long journey based in developing number sense.

Two teaching experiments carried out as doctoral theses underscore this perspective. In the first, carried out in a Grade 2 class, Ferreira (2012) was strongly inspired by the notion of landscape of learning of Fosnot and Dolk (2001a, b) in which students were invited to construct their ideas and strategies based on the analysis and manipulation of numbers as a whole. As proposed by RME, this teaching experiment did not include explicit references to place value and emphasised a holistic approach to number, with the development of 'horizontal' written calculation strategies. The author concluded that the four students studied managed to solve the proposed addition and subtraction problems without using the standard algorithm. Instead, they began to reason arithmetically, using familiar relations between numbers and between addition and subtraction, as indicated by Blöte, Van der Burg, and Klein (2001) and Gravemeijer (2005).

In another teaching experiment, in a Grade 3 class, Mendes (2012) got support on the theoretical ideas of Fosnot and Dolk (2001b) related to multiplication. The big ideas that Mendes assumed in her work included unitizing (numbers are used to count not only objects but also groups), the distributive property (realising that 7 × 5 can be solved by adding 5 × 5 and 2 × 5, or any combination of groups that add up to seven groups), the associative property (2 × {5 × 9} = {2 × 5} × 9) and the commutative property (52 × 2 = 2 × 52).

#### **<IP_ADDRESS> The Role of the Context of Tasks and the Use of Models and Manipulative Materials**

The influence of RME ideas regarding the context of tasks inspired the production of classroom materials in the DNS project (Brocardo et al., 2005; Brocardo & Serrazina, 2008) and of research studies (such as Delgado, 2013; Mendes, 2012; Rodrigues, 2010). In these studies, the context of tasks was carefully planned as a starting point and source for modelling. Attention was paid to the clarity of the written text, to the accompanying images that should not be simple illustrations, and to how the task might arouse students' curiosity. Explaining the way the project team thought about the context of tasks, Brocardo and Delgado (2009) underlined the idea of Freudenthal (1968) that mathematics must be learned as a process of mathematising reality and, if possible, in the process of mathematising mathematics itself. They also indicated the important characteristics of the context of tasks so that it would become a situation that lends itself to mathematising, as Fosnot and Dolk (2001b) suggested—to allow the use of models, to make sense for students, to create surprise and arouse questions.

The tasks with contexts that appeal to models are one of the RME influences that it is possible to identify in many research studies carried out in Portugal (such as Delgado, 2013; Ferreira, 2012; Mendes, 2012). An example is the 'Drinks machine' task (Brocardo et al., 2005), inspired by a task developed by Kraemer and Paardekooper (1998) in which cans with drinks of different flavours are put in a machine with a maximum capacity of twenty cans for each variety. There are horizontal limits that organise the stacks of cans in groups of five, allowing for connections with the string of beads and the empty number line model structured from five in five (Fig. 12.1).

The exploration of contexts such as fruit boxes or tile tessellations are paradigmatic examples of contexts associated with the rectangular model widely used to structure multiplication. Another example was provided by Rocha and Menino (2009) who presented a task chain inspired by Fosnot and Dolk (2001b) with an underlying progression aimed at using the rectangular model in successive phases of abstraction. To achieve that, the context used began by allowing a counting of the objects that are all visible (fruits shown on a box), then allowing to count all the objects of a half of a rectangular array (two equal curtains in which one has all patterns visible and the other does not), and, finally, situations in which counting was discouraged by covering part of the rectangular patterns with objects or people leading students to think in terms of lines and columns and begin to use the rectangular model, thus leaving addition and using multiplication.

2005)

Mendes (2012) used this type of contexts in her research focussed on learning of multiplication, and in the design of the sequence of tasks, she sought that the numbers involved, carefully framed from task to task, would appeal to multiplicative numerical relationships, as suggested by Treffers and Buys (2008) (Fig. 12.2).

The results of this research indicated that the chosen context for the multiplication tasks contributed to consolidating the use of multiplicative procedures. She also concluded that students who established connections among the contexts and the numbers of sequential tasks constructed procedures based on that relationship, and there were multiplicative procedures used by students (such as the use of relationships involving doubling) that were induced by the numbers used in the tasks. In her study, focussed on the practices of two teachers, Delgado (2013) concluded that in an initial phase of the study the teachers mainly valued contexts of tasks that were close to

**Fig. 12.2** Example of one multiplicative context used by Mendes (2012)

students' daily life situations, to motivate them and lead them to engage in solving the tasks. As the study progressed, the teachers also began to recognise the importance of contexts in developing meaning for the numbers and operations associated with them, as underlined by Fosnot and Dolk (2001b) when referring to the characteristics of the contexts of tasks that foster the development of number sense.

In Portugal, there is not a strong tradition of using manipulative materials, despite the fact that all official curricula since 1975 refer to the importance of manipulating objects and using structured materials such as MAB or the abacus. RME influenced a reflection on the use of such materials, in contrast with the use of models, in particular the empty number line. Brocardo et al. (2005) made a detailed analysis of the potential of the empty number line, and followed RME perspectives in reflecting on the potential and limitations of MAB. They argued that usual materials for learning computation with whole numbers are often difficult for many students, provide few opportunities for the use of informal strategies, and do not promote the evolution of mental computation strategies. Brocardo and Serrazina (2008) referred to the comparison that Beishuizen (2001) made between the use of the one hundred square and the empty number line, highlighting the importance that the model must lead the students to think in the strategy that they use and the computations that they do. Therefore, they criticised the use of the one hundred square because it is possible to calculate using mechanical procedures such as "add 10 is to come down one line and read the number" or "take away 5 is to move 5 numbers backwards and see where we stop" that allows to arrive at a result doing no thinking at all. Several tasks produced in the DNS project discuss the advantages of the empty number line, indicating that its use enables children to have an image of jumping forwards and backwards keeping the multiples of ten as landmarks and thinking in numbers and their relations, calculating with the head. Use of the empty number line began to spread since 2005 and was recommended for the first time in an official curriculum of the Ministry of Education (Ministério da Educação, ME, 2007). In later research studies focussed on teaching and learning numbers and operations (such as Delgado, 2013; Ferreira, 2012) and in some textbooks, the use of the empty number line model became usual.

#### *12.2.2 Mental Calculation*

The concept of mental arithmetic referred to by Buys (2008) and widely used in the Netherlands gained certain acceptance in Portugal. Brocardo and Serrazina (2008) indicated that for the DNS project mental calculation implied dealing with number values (not with digits), using elementary calculation properties and number relationships, and allowing for the possible use of suitable intermediate written notes. This notion aligns well with that of Buys and is adopted by many other Portuguese researchers. Several booklets edited by the Portuguese Ministry of Education to support the implementation of the 2007 Portuguese mathematics curriculum adopted this concept of mental calculation and exemplified it with appropriate tasks for how it can be developed.

Several studies were undertaken in the last decade focussing on the development of mental calculation. These studies emphasised the development of mental calculation strategies supported by a careful articulation of sequences of tasks (for example, Delgado, 2013; Ferreira, 2012; Mendes, 2012; Pinto, 2011). Following RME perspectives, these authors considered the written methods as specific developments of the mental strategies that children learn, which should be organised in a continuous progression. To ensure that this development happens, it is important that students learn to use mental calculation strategies in a flexible way. These strategies may be organised in three big groups: (i) stringing strategies, in which the operations are movements along the counting row, (ii) splitting strategies, in which operations are performed by splitting and processing the numbers based on the ten's structure; and (iii) varying strategies, based on arithmetic properties.

The teaching experiment conducted by Mendes (2012) included several mini lessons with mental mathematics strings (as suggested by Fosnot & Dolk, 2001a, b). This researcher concluded that the articulation between the numbers used in the problems and the numbers used in the mental mathematics strings was important since students were able to adapt from each other the procedures that they used. She also concluded that the numbers used in the mental mathematics strings and the way they were constructed enabled students to use numerical relationships based on multiplication properties. The studies of Pinto (2011), Ferreira (2012), Mendes (2012) and Delgado (2013) provided evidence that an approach to mental calculation that articulates the knowledge of number facts and strategic methods has big potential to develop students' numerical competences. In fact, the students studied evolved in a significant way, using adequate strategies, and in many cases flexibly adapted to contexts and numbers.

Brocardo (2011) proposed the setting up of goals for mental calculation to attain at the end of each school cycle based on the distinction of three categories of mental calculation proposed by Buys (2008). Referring to ideas of the TAL team,4 she indicated that this distinction demystifies the idea that such calculation cannot include written notes, and clarified that mental calculation is not just automatic calculation. She also presented a proposal for the development of mental calculation that included several RME ideas. Underlining that mental calculation work must be systematic and intentional, she exemplified that mental mathematics strings (Fosnot & Dolk, 2001a, b) may contribute to constructing basic numerical knowledge important for the development of mental calculation strategies from Grades 6 to 12. Brocardo (2011) proposed exploring open tasks, in which students discover interesting numerical patterns that may lead to efficient calculation techniques that students get for their use. She also suggested that systematic work in constructing a web of relationships, as recommended by Kraemer and Van Benthem (2011). Such a web is based on the idea of beginning from a known fact such as 4 × 5 is 20 and constructing all multiplicative relations connected to it. In her proposal, Brocardo (2011) emphasised that the teacher has a key role in selecting tasks that arouse students' curiosity and leading them to develop mental calculation, in distinguishing situations in which it is appropriate to use the calculator from those in which that does not make any sense and in assuring that students use mental calculation always when appropriate.

#### *12.2.3 Rational Numbers*

Teaching and learning rational numbers has attracted significant attention from Portuguese researchers in mathematics education, with two particularly noticeable influences—from the Rational Number Project (Behr, Harel, Post, & Lesh, 1992) and from RME. We analyse here the influence of RME.

#### **<IP_ADDRESS> Mapping Students' Difficulties**

In a literature review on teaching and learning rational numbers that has been an important reference for research on this topic in Portugal, Monteiro and Pinto (2006) discussed students' difficulties in working with fractions and their strategies in solving problems. The presense of RME ideas is particularly noticeable in the point

4The TAL team was responsible for developing a teaching-learning trajectory for calculation with whole numbers in primary school (Van den Heuvel-Panhuizen, 2008). Kees Buys was one of the members of this TAL team.

on students' strategies, as the authors underlined the idea that mathematics must be reinvented by students in a progressive process of generalisation and formalisation (De Lange, 1996; Gravemeijer, 1991, 1997; Streefland, 1986, 1991; Treffers, 1991). Monteiro and Pinto (2006) indicated that, in the perspective of RME, students' learning is based on informal strategies for solving tasks, from which they develop concepts and connections among concepts, in a mathematising process. They also referred to horizontal mathematising and to modelling of real situations through the use of symbols as well as to vertical mathematising as a path internal to mathematics (Gravemeijer, 1997). Monteiro and Pinto (2006) presented the ideas of Keijzer (2003) regarding mathematising processes (modelling, symbolisation, generalisation, formalisation and abstraction). They pointed out that, to bridge the gap between concrete and abstract, students need tools such as visual models, schemas, and diagrams that work for them as supports for thinking (Streefland, 1993). They also indicated that symbols may become objects of thinking, constituting images for more abstract levels of understanding (Streefland, 1991).

Monteiro and Pinto (2006) recalled the ideas of Streefland (1986) about the algebraic structure of rational numbers being the support for the most common view on the teaching of fractions. In their perspective, that explains why operations appear in the Portuguese curricula in a given order and why the algorithms to compare, add and multiply fractions have so much weight. They considered that it is the role of the teacher to provide students with opportunities to reinvent mathematics, instead of seeking to make accessible to them 'ready-made' mathematics.

#### **<IP_ADDRESS> Use of Representations and Models**

Based on a teaching experiment, Ponte and Quaresma (2011) studied the development of Grade 5 students' understanding of the notion of rational number, ordering and comparing rational numbers, and equivalence of fractions. In this study, a fundamental idea was the simultaneous use of different representations, as well as of different meanings, kinds of magnitudes, and kinds of task. The authors referred to the 'iceberg model' of representations of Boswinkel (see Webb, Boswinkel, & Dekker, 2008) that suggests that students need a large amount of experience with different informal and preformal representations as a basis to construct a meaning for formal mathematical representations. In this way, the students used pictorial representations as a support for their work with the more formal representations of decimal numerals, fractions, percentages, and mixed numerals. Based on ideas from Streefland (1991), who underlined that the work on fractions must be done based on their names, such as 'a half', 'a third', 'a quarter', and so on, the authors considered that verbal representation, which is sometimes neglected in research, fulfils a fundamental role in the work with rational numbers, notably in oral communication. In addition, echoing ideas from Gravemeijer (2005), through all the teaching unit, as a starting point for constructing concepts, they valued the students' intuitive and informal strategies as well as their prior knowledge. In their results, they pointed out that students tend to begin by using simultaneously verbal and pictorial representations that enable the interpretation of the information in the statement of the task and support the reasoning to get to the solution.

The study of Ventura (2014), carried out with Grade 5 students, aimed to understand their evolution in learning the concept of rational number, based on a teaching experiment that was based essentially on the use of the numerical bar and the numerical line, and at the same time to ascertain the potential of this approach. In this study, the notion of model (based on Van den Heuvel-Panhuizen, 2003), fulfilled a key role as a representation of a problematic situation that reflects essential aspects of the mathematical concepts and that, therefore, constitutes a tool for solving problems. Ventura concluded that the students evolved in their learning of the concept of rational number, and indicated that, as a problem-solving strategy, many of them began to use the numerical bar as a 'model of', and later used it as a 'model to' reason (Gravemeijer, 2005; Streefland, 2003).

Guerreiro and Serrazina (2014) also studied students' strategies in solving problems involving rational numbers, but in this case in Grade 3. They based themselves on the perspectives of Fosnot and Dolk (2002) that indicate that students must understand important ideas and progressively refine their strategies in order to make them more efficient. Guerreiro and Serrazina assumed as central the notion of mathematical model, regarded as a tool for problem solving, and referred to examples such as "ratio tables, double numerical lines, clocks, grids and percent bars" (Fosnot & Dolk, 2002, p. 83). In the perspective of Guerreiro and Serrazina, these models may support students in generalising, going beyond what is specific in each situation. Building on ideas of Fosnot and Dolk (2002) and Gravemeijer (2005), Guerreiro and Serrazina considered that models emerge from situations experienced by students that evolve towards mathematical models of numerical relationships, becoming mathematical tools.

In other studies involving teaching and learning rational numbers in several grade levels, Ponte and Quaresma (2011, 2014b), Quaresma and Ponte (2012) and Guerreiro and Serrazina (2015), also based themselves on the model of different levels of mathematical activity of Gravemeijer (2005) and on the iceberg model of Boswinkel (Webb et al., 2008) to indicate the need for working from contexts meaningful for students and to assume that, in a first phase, an emphasis must be placed on informal representations that students already know, in order to introduce then, gradually, more formal new representations and working processes with rational numbers.

#### **<IP_ADDRESS> Learning Multiplication and Division**

The study of Pinto (2011) analysed the development of the multiplication and division sense in Grade 6 students in working with rational numbers through a teaching unit. This unit involves the exploration of multiplication and division of rational numbers in meaningful contexts, based on RME principles. The unit values solving problems with contexts meaningful for students, their written productions, the development of models of the situations, the mathematical connections, the interactions in the classroom, and formative and regulatory evaluation. Underlying this unit is a hypothetical learning trajectory for multiplication and division of rational numbers that emphasises the development of multiplicative reasoning and operation sense in an integrated way. Building on RME researchers (such as Fosnot & Dolk, 2002; Freudenthal, 1973, 1983, 1991; Treffers, 1987, 1991; Treffers & Goffree, 1985), Pinto assumed that the study of rational numbers in school must begin based on fair sharing contexts related to students' reality and be oriented towards a constructive mathematising process. She also valued ratio problems (quoting Streefland, 1991, 1993) and suggested ratio tables as models for comparing fractions, especially in the case of fractions that are difficult to compare without applying rules. She noted, however, that students need to work with the other meanings of fractions (ratio, partwhole, measure and operator). In her analysis, she also assumed an important role for the notion of model as a learning support in moving from concrete to abstract knowledge (Van den Heuvel-Panhuizen, 1996, 1998; Van den Heuvel-Panhuizen & Wijers, 2005). The results of the study indicated that the students develop a sense for the multiplication and division of rational numbers, showing familiarity with different meanings and contexts of operations, flexibility in the use of proprieties of operations, critical ability in the analysis of processes and results, and capacity to use symbols and formal mathematical language with meaning.

#### *12.2.4 Algebra*

In the last ten years, algebra has attracted much attention in mathematical education research in Portugal. This investigation is mainly influenced by the ideas of Carpenter, Kaput, Kieran and Radford concerning the development of algebraic thinking. Nevertheless, influences deriving from RME have also been noted. For example, in an article for mathematic teachers, Ponte, Branco, and Matos (2008) analysed the role played by symbols in the development of students' algebraic thinking. The authors presented the perspective of Freudenthal (1983) concerning the teaching of algebra, indicating that "the symbols must mean something, at least initially, by analogy to what happened in the historical development of algebra" (Ponte et al., 2008, p. 90). They underline the importance of the process of progressive formalisation and also presented Freudenthal's perspective concerning algebraic language as a system framed by syntactic rules, which allow for the development of certain actions and highlights that the complexity of the algebraic language may originate incorrect interpretations from students.

In their study, Pereira and Saraiva (2013), proposed a learning and teaching model based on the notion of the parameter of a function to structure the mathematical reasoning of secondary school students in Grade 11. This model analyses the students' concepts, concerning relevance, cohesion and algebraic coherence, with teaching organised in three levels: reference operational, informal operational, and structural operational. For the authors, the concepts are structured at each level, representing contexts that promote the creation of meanings in a hierarchical logic. This work showed the influence of Gravemeijer's perspective of levels of students' mathematical activity (2005) in the construction of tasks in a teaching experiment, with the reference operational level matching the 'model of' level, the informal operational level matching the 'model to' level, and the structural operational level matching the formal level. According to Pereira and Saraiva, the results of their study suggest that this model is useful to structure tasks to promote students' thinking and to develop students' reasoning with mathematical concepts.

In an investigation carried out with Grade 4 students, intended to understand how to promote their relational thinking, Mestre and Oliveira (2013) focussed on the issue of the context and its connection to representations. They gave special attention to the way the teacher orchestrates the whole class discussion of a mathematical task and guides the systematisation of learning. For the authors, the tasks have an important role, with special emphasis on their contexts that must be significant to promote the development of students' relational thinking. Supported by Gravemeijer and Doorman (1999), the authors considered that contextualised problems constitute a source for mathematical activity, allowing the transition from informal to formal strategies. They also indicated that as students experiment with the process of reinventing mathematics through solving contextualised problems, they develop their mathematical knowledge and broaden their understanding of the real world. On this matter, the authors underlined the reflexive relation between the utilisation of contextualised problems and the apprehension of reality, arguing that these problems have roots in this reality and their solution helps students to broaden their own notion of reality. They concluded that the students used several representations, successfully presenting the values of the variables in the algebraic symbolic form.

The development of students' mathematical reasoning is an issue that has been studied in Portugal, mainly in studies related to the learning of algebra and also of rational numbers. Ponte, Mata-Pereira, and Henriques (2012), Mata-Pereira and Ponte (2013) and Ponte and Quaresma (2014a) regarded mathematical reasoning as the process of formulating inferences in a justified way, considering that this involves deductive, inductive and abductive aspects. In their view, justification is the central process of deductive thinking and generalisation is the central aspect of inductive and abductive reasoning. One of the main ideas of these studies is that reasoning is strictly connected to the representations used, which may assume a more formal or informal nature. In a study carried out with Grade 5 students, Ponte and Quaresma (2014a) presented a model that distinguishes between formal reasoning with and without understanding, where formal reasoning with understanding is based on informal reasoning, in a back and forth process, while formal reasoning without understanding relies essentially on memorised learning. This perspective on mathematical reasoning assumes that the big problem in teaching this subject is knowing how to make the progressive articulation between formal and informal reasoning processes and is supported by Gravemeijer's (2005) model of levels of mathematical activity.

In other research reports involving the teaching and learning of algebra, the influence of the RME authors is also visible. For example, Ponte (2005), in an article discussing the approach to algebra in the school curriculum, pointed to the role of 'real situations' in learning, making reference to the work of De Lange (1993). Pimenta and Saraiva (2013), in research aimed at the development of the algebraic thinking of Grade 4 and 5 students, referred to Freudenthal's notion of vertical mathematising (1973). And Silvestre and Ponte (2012), in a study of the development of the proportional reasoning of Grade 6 students, mentioned Gravemeijer's (2005) model of levels mathematical activity.

#### *12.2.5 Geometry*

In contrast to numbers and algebra, geometry is a mathematical topic that has attracted less attention from Portuguese mathematics education research. At an early stage, in the 1980s, the model of Van Hiele's (1984) levels of geometrical reasoning played an important role, especially in the work of Matos (1984), who studied the geometric thinking ability of prospective early-years teachers. However, in recent years, there is no record of references based on this model of geometric thinking. In today's Portuguese work on geometry the most visible influences are those of American authors such as Battista and Clements. However, there are also echoes of RME ideas, especially with regard to general perspectives on the teaching of geometry.

Pinheiro and Carreira (2013) discussed the development of geometric reasoning in the context of the use of Geogebra in a teaching experiment with Grade 7 students in order to know how they develop their understanding of the properties and relationships of geometric figures in studying triangles and quadrilaterals. In formulating their educational perspective, Pinheiro and Carreira presented the ideas of Freudenthal (1971, 1991) on the role of geometry in the school curriculum and on the teaching of geometry. Hence, the authors emphasised the role of geometry, given its importance to understand and organise spatial phenomena and they assumed that teaching should focus on the construction of conceptual models. They also valued the importance of manipulating physical materials in specific situations. Furthermore, they found that deductive reasoning should be promoted in accordance with students' maturity and that the most suitable way to learn geometry is "to allow the pupil to gradually become aware of their intuitive understanding of space" (Pinheiro & Carreira, 2013, p. 148). The results of this study showed that the sequence of tasks built and the way that the tasks were solved in the classroom helped to promote students' understanding of the mathematical concepts involved. They also found that the use of the dynamic geometry environment contributed to the development of students' spatial reasoning ability, and therefore of their geometric reasoning.

In another study, Mestrinho and Oliveira (2012) analysed how the use of the tangram may support understanding of the area concept in prospective early-years teachers, as part of a teacher education experiment in the second year of the programme. The authors referred to the idea of Freudenthal (1983), according to which the concept of area is much more complex than the concept of length, since the definition of an equivalence relation and of an order relation as well as the creation of a composition operation are much more complex for area. They also showed three perspectives on the concept of area referred to by Freudenthal (1983), namely 'equitable distribution' (situations in which it is necessary to divide a figure into equivalent parts), 'comparison and reproduction' (situations that involve the comparison of two parts of a surface or the reproduction of a certain amount of area with a different shape), and 'measurement' (situations involving filling a part of a surface with congruent figures, decomposition and recomposition operations or the use of general geometric relationships). The authors found that the use of the tangram as a resource promotes the development of basic ideas for understanding of area measurement, and allows to explore different approaches to this concept.

#### **12.3 Influences on Curriculum Documents**

Abrantes (1994) studied how an innovative curriculum developed by the project MAT789 5 influenced the ability and disposition of students to tackle problems involving relationships of mathematics with reality and the way they saw mathematics and mathematics learning. In this experimental curriculum, there are two clear influences of RME: the perspectives about how to conduct curriculum development and how to frame evaluation processes. On a small scale the work of Abrantes is similar to the Dutch approach to curriculum development through projects such as carried out at OW&OC,6 the predecessor of the Freudenthal Institute, or the curriculum development that was done in collaboration with SLO, Netherlands Institute for Curriculum Development. Characteristics of this curriculum development was that successive versions of materials were trialled, evaluated, and modified before being generalised. Abrantes (1994) referenced that regarding evaluation, the major influence came from the HEWET7 project (De Lange, 1987)—in which teaching materials were developed on various mathematical topics for pre-university education—and highlighted the concern that evaluation must generate learning situations, be consistent with aims and methodologies, have a positive nature, and occur in a climate of trust and clarity.

At the national level, the mathematics curriculum for basic education (ME, 2007) shows a clear influence of RME ideas, notably in the topic of numbers and operations. An important methodological guideline is the informal use of counting that evolves through replay and repetition to become structured knowledge. For example, the mathematics curriculum for basic education indicates that

the exploration of counting processes used by students associated with different possibilities to structure and relate numbers, contributes to the understanding of the first numerical relationships. These relationships are fundamental to understand the arithmetic operations and, besides, are a foundation for the development of number sense. (ME, 2007, p. 14)

5A project that developed and tested an innovative curriculum for students aged 12 to 15 years carried out by Paulo Abrantes, Eduardo Veloso, Leonor Santos, Paula Teixeira, and Margarida Silva.

6Onderzoek Wiskundeonderwijs en Onderwijs Computercentrum (Mathematics Education Reasearch and Educational Computer Centre).

7Herverkaveling Wiskunde I en II (Re-allotment Mathematics I and II); the HEWET project resulted in Mathematics A and Mathematics B, a new mathematics curriculum for the upper grades (age 16–18) of VWO, the pre-university level of secondary education.

This curriculum does not integrate the RME approach to written calculation in number operations, but it recognises the importance of delaying the introduction of the standard algorithms and stresses the idea that it is important to progressively develop more high-level abbreviated strategies. This document also recommends the use of the empty number line as a model that can be used alongside others. In addition, for teaching all topics, this curriculum stresses the importance of working from tasks posed in meaningful contexts, highlighting the importance of contexts that may engage students in asking questions, notice patterns, and lead them to use mathematical models and adequate representations.

#### **12.4 Conclusion**

As indicated in this chapter, many RME authors have influenced Portuguese mathematics education. The most salient are, in a first phase, Freudenthal, Treffers and Streefland, and, in more recent times, Gravemeijer, Van den Heuvel-Panhuizen, Fosnot and Dolk and Buys. Such influences can be seen in many mathematical fields, from numbers and operations to algebra and geometry, often mixed with influences from other mathematics education research programmes. The influences concern RME general ideas such as the perspective about representations and the notion of model and the levels of mathematical activity with attention to the progressive refinement of students' strategies from informal towards formal levels. There are also frequent references to the processes of vertical and horizontal mathematising and to the use of experientially real situations as a basis for learning.

The importance of carefully formulating the contexts of tasks as well as the articulation among them stands in studies related to numbers and operations. Tasks and the work that is done based on them in the classroom must favour the transformation of students' reasoning and mental calculation processes, from informal towards progressively more formal levels, and support the development of mathematical concepts. The importance of algebraic language as well as the process of progressive formalisation stands in studies regarding the teaching and learning of algebra. In the case of geometry, RME influences concern the importance of this topic in the curriculum and the didactical approach, underlining the role of the manipulation of materials, as well as the phenomenological analysis of concepts. There are several other fields in which Portuguese research comes close to RME ideas, such as the use of technology as a support for students' learning, organising teacher education with a strong connection to practice, and framing studies as design-based research; however, in these cases, the most quoted authors are usually from other approaches.

In several crosscutting topics, we also see RME influences. The mathematical tasks used in most recent research studies in Portugal strive to be framed in interesting contexts and to allow for a wide variety of students' solutions. The sequences of tasks constructed in these studies indicate possible learning routes, supporting a process of progressive mathematising, an important RME principle. Didactical phenomenology, another important RME idea, is also present in several research studies, in which a given mathematical topic is explored in depth, with great attention to everyday situations in which it can be traced. At another level, we may say that the pragmatic spirit of RME, and its concern for improving mathematics education by working in close connection with teachers and schools is also present in studies undertaken in our country. Globally, the work developed by Portuguese researchers using RME notions and tools has proven to be fruitful and underscores the value of the results and ideas of RME perspectives. In addition, these perspectives have been an important support for the development of research in mathematics education in Portugal, with positive effects on teachers' professional practices and, we believe, on students' learning.

#### **References**

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## **Chapter 13 Supporting Mathematical Learning Processes by Means of Mathematics Conferences and Mathematics Language Tools**

#### **Christoph Selter and Daniel Walter**

**Abstract** In recent decades, the instructional theory of Realistic Mathematics Education has exerted a powerful influence on mathematics education around the world. The idea of progressive mathematisation has gained international acceptance. In this chapter, we will illustrate the way in which we benefited from the idea of organising the teaching and learning of mathematics in keeping with this guiding principle. After some personal memories of the first author, we start by describing what we consider to be the central elements of the principle of progressive mathematisation. This is followed by a description of two methods, the mathematics conferences and mathematics language tools, for rendering the learning and teaching concepts entailed by the principle of progressive mathematisation—especially its vertical component—even more expedient and fruitful. The contribution concludes with an explanation of how we understand the term 'realistic' in Realistic Mathematics Education.

**Keywords** Progressive schematisation · Mathematics conferences · Language and mathematics · Individual learning processes · Co-operative learning · Mathematics language tools

### **13.1 The Santa Claus Problem**

It must have been at the end of 1983 that the first author—at the time studying to become a primary school teacher—became aware of Adri Treffers' paper "Fortschreitende Schematisierung – ein natürlicher Weg zur schriftlichen Multiplikation und Division im 3. und 4. Schuljahr" (Treffers, 1983). Taking the multiplication of large

C. Selter (B)

D. Walter

© The Author(s) 2020

Faculty for Mathematics/IEEM, TU Dortmund, Dortmund, Germany e-mail<EMAIL_ADDRESS>
Faculty for Mathematics, Münster University, Münster, Germany e-mail<EMAIL_ADDRESS>
M. van den Heuvel-Panhuizen (ed.), *International Reflections on the Netherlands Didactics of Mathematics*, ICME-13 Monographs, https://doi.org/10.1007/978-3-030-20223-1\_13

**Fig. 13.1** Solutions to the Santa Claus problem

numbers as an example, this paper describes how students can be motivated to apply their individual approaches and develop them further in a purposeful manner. The starting question in the paper is: "Santa Claus has his gifts distributed in the village by eight helpers. Each has 23 parcels. How many parcels do they have altogether?"

Treffers' paper describes how eight-year-old students solved this problem using strategies they had individually developed in various ways (Fig. 13.1). The strategies naturally differed in terms of elegance and efficiency (Treffers, 1993), and therewith either represent the stages observable in the development of a single child or the heterogeneity noticeable within a learning group of students. The 'Santa Claus' problem serves as a representative example for illustrating a possible starting point of the so-called 'principle of progressive mathematisation'.

Starting from problems like this one, the various solution strategies of the students are discussed, explained, and elaborated in interactions between the students or between the teacher and the students. The students can see how other students work and thereby assess the advantages and disadvantages of different strategies (Treffers, 1991). An individual student's (mental) actions are as vitally important as his or her interaction with other students. The illustrations included in the paper are taken from a lesson where the students were not shown how to solve this type of problem based on the principles of isolating difficulties and increasing complication—as was still widespread in use in the classrooms of that time—but instead by encouraging them to develop their own approaches and then also develop them further. To put it briefly: from inventions to conventions.