Applying realistic mathematics education in Vietnam teaching middle school geometry Institut für Mathematik Lehrstuhl für Didaktik der Mathematik Titel der Dissertation Applying Realistic Mathematics.
CHARACTERISTICS OF MATHEMATICS EDUCATION IN VIETNAM
B RIEF INTRODUCTION TO EDUCATIONAL REFORMS IN V IETNAM AFTER 1945
This section provides a concise overview of three significant education reforms in Vietnam following its independence in September 1945 Table 1.1 summarizes the starting dates, primary motivations, and scope of each reform, drawing on insights from Pham Minh Hac (2002, pp 65-70) and Bui Minh Hien.
Table 1.1: The first three Educational Reforms in Vietnam
1 st 1950 Extremely primitive education Liberated areas
2 nd 1956 Necessary unification of general education after the restoration of peace in the Northern Vietnam
3 rd 1979 Necessary unification of general education after the war
Following Vietnam's independence in September 1945, the education system faced significant challenges, with over 95% of the population being illiterate and only about 3% attending school (Pham Minh Hac, 2002) By July 1950, a new educational reform project was initiated, particularly in liberated areas, leading to the establishment of a nine-year educational system This system was divided into three levels: Level I (grades 1-4), Level II (grades 5-7), and Level III (grades 8-9) Notably, students in Hanoi exhibited a greater enthusiasm for learning mathematics compared to their peers in Munich (Helmke et al., 2003).
After the implementation of educational reforms in Vietnam, the curricula in French-occupied territories remained similar to those used before 1945 (Pham Minh Hac, 2002; Bui Minh Hien, 2005) Following the restoration of peace in Northern Vietnam in 1954, two distinct educational systems emerged: one based on a nine-year reform and another with a twelve-year duration, reflecting the pre-French occupation structure (Pham Minh Hac).
2002, p 67) Therefore, one urgent task at that time was to unify the education system (Pham Minh Hac, 2002, p 67; Bui Minh Hien, 2005, pp 153-154)
In May 1956, the Ministry of Education implemented the second educational reform project, which established a general educational system comprising 10 grades This system was organized into three levels: Level I for grades 1 to 4, Level II for grades 5 to 7, and Level III for grades 8 to 10 (Pham Minh Hac, 2002; Bui Minh Hien, 2005).
After the unification of Vietnam in April 1975, the country faced the challenge of integrating two distinct educational systems: the 11-year system from Northern Vietnam and the 12-year system from Southern Vietnam, a legacy of the wars (Pham Minh Hac, 2002, p 68; Bui Minh Hien, 2005, pp 181-182) Recognizing the need for a unified education system, the Vietnamese government acknowledged that despite significant achievements from the second educational reform initiated in 1956, the education sector failed to keep pace with societal, scientific, and technological advancements, hindering the nation's postwar reconstruction efforts (Pham Minh Hac, 2002, p 68) Consequently, the government launched the third educational reform in 1979 to address these shortcomings and create a cohesive educational framework for the country.
1981-1982, textbooks for grade 1 called ‘reformed’ textbooks were originally used (Pham Minh Hac, 2002, pp 69-70; Bui Minh Hien, 2005, p 183) Subsequently, the
In the 1992-1993 school year, Vietnam implemented 'reformed' textbooks for grade 12, successfully unifying its general educational system into a comprehensive 12-year curriculum spanning grades 1 to 12 This initiative ensured that all schools across the country utilized a standardized set of curricula and textbooks, promoting consistency in education (Pham Minh Hac, 2002; Bui Minh Hien, 2005).
The existing curricula and textbooks have played a crucial role in Vietnam's educational development, yet they have also exposed several shortcomings Consequently, there is a significant demand for a new series of curricula and textbooks tailored for Vietnamese schools Since the early 2000s, this new series has been piloted and progressively implemented in educational institutions.
G ENERAL EDUCATION IN V IETNAM
This section briefly introduces general education (primary, middle and high school education) in the Vietnamese education system
The Vietnamese national education system, as outlined by the Vietnam National Assembly (1998, chapter 1, article 6; 2005, chapter 1, article 4), encompasses various levels including pre-school, general, vocational, and higher education General education consists of primary, middle, and high school, as depicted in figure 1.1 (Vietnam National Assembly, 1998, chapter 2, article 22; 2005, chapter 2, article 26) For a comprehensive overview of the national education system, refer to the Ministry of Education and Training (MoET, n.d a).
Vietnam's education system features a standardized nationwide curriculum and set of textbooks Notably, between the early and late 1990s, there were three distinct series of high school textbooks for certain subjects, developed by different authors from Hanoi University of Education, Ho Chi Minh City’s University of Pedagogy, and the National Institute for Educational Sciences (NIES).
2000, these series were corrected, edited and unified to make a unique series called
Primary education (Grades 1 to 5, ages
Middle school education (Grades 6 to
High school education (Grades 10 to
The Ministry of Education and Training (MoET) attempted an educational experiment from 1992 to 1997, dividing high school students into three streams: Natural Sciences (Stream A), Technique (Stream B), and Social Sciences (Stream C) However, this initiative ultimately failed due to a lack of interest, particularly in the Technique Stream Recently, the curriculum has been revamped, introducing a cohesive series of textbooks for both primary and middle schools Starting in the 2006-2007 school year, grade-ten students have been categorized into streams, with plans to implement this division for grades eleven and twelve in the subsequent school years.
In the 2006-2007 school year, Vietnamese grade-ten students predominantly selected the Basic Stream, with approximately 73% opting for this pathway, while only about 6% chose the Social Stream, highlighting a significant preference among students for the Basic Stream over the Social Stream (Kim Dung, 2006; Vietnam News).
C HARACTERISTICS OF MATHEMATICS EDUCATION IN THE ‘ REFORMED ’ PERIOD
1.3.1 School mathematics curriculum and textbooks
This section outlines key features of the school mathematics curriculum and textbooks during the reformed period from 1979 to the early 2000s It begins with a brief overview of the mathematics curriculum and textbooks, followed by an exploration of their specific characteristics Finally, it presents and discusses additional notable features of these educational resources.
1.3.1.1 Overview of mathematical curriculum and textbooks
The Vietnamese national mathematical curriculum is established by the Ministry of Education and Training (MoET), which serves as the foundation for the development of experimental mathematics textbooks These textbooks undergo testing in various schools over several years and are revised before their official implementation in all educational institutions All school textbooks in Vietnam are published by the Educational Publishing House Each mathematics textbook is organized into chapters that encompass lessons, typically featuring formal definitions and theorems.
Recently, the Ministry of Education and Training (MoET) has implemented a new policy focusing on high-level regulations and formulas This policy has led to the inclusion of probability, analysis, and analytic geometry in high school textbooks, a practice that has been ongoing since the early 1990s.
Despite students residing in diverse regions, including urban, rural, and remote areas, schools typically use a single series of mathematical textbooks Research conducted by the National Institute for Educational Sciences (NIES) from 1998 to 2000 on third and fifth-grade students across fourteen primary schools in five provinces reveals significant disparities in academic achievement among Vietnamese students based on their geographical location.
Significant disparities in student achievement exist not only between provinces but also among schools within the same province Notably, grade 3 students in Hanoi, the capital, demonstrate higher levels of proficiency in mathematics and reading comprehension in Vietnamese compared to grade 5 students from four more rural provinces, highlighting the educational inequalities present in the country (The World Bank).
The Primary Education for Disadvantaged Children Project highlights the disparities in educational opportunities and quality across the country, despite efforts by the Ministry of Education and Training (MoET) to bridge this gap.
The use of a standardized set of textbooks in Vietnam offers certain advantages, but it also presents notable disadvantages One key issue is the lack of choice for both students and teachers, who are required to utilize a specific series Additionally, these common textbooks must cater to varying levels of student ability Significant disparities exist in the academic performance of Vietnamese students from urban, rural, and remote areas, with urban students benefiting from superior teaching and learning conditions compared to their counterparts in less accessible regions.
Different countries utilize various sets of mathematics textbooks for education For instance, South Korea offers around ten different mathematics textbook series for secondary schools, while only one series is used for primary schools Japan has six distinct series of mathematics textbooks for middle schools Historically, China had a singular series of curricula and textbooks, but it has since adopted multiple textbook series.
Recently, there has been a debate regarding the unique school textbook set in Vietnam, as noted in section 3 of chapter 3 Since 1998, various educational approaches have been observed globally In the Netherlands, teachers have access to about six series of mathematics textbooks and the flexibility to either select a textbook or design their own curriculum Similarly, Germany's 16 states each maintain their own education systems, allowing teachers to choose from a list of state-approved mathematics textbooks or create their own materials In the United States, several prominent middle school mathematics textbook series are utilized, including Mathematics in Context, Math Thematics, Connected Mathematics, MathScape, and Pathways to Algebra and Geometry.
A study conducted by Project 2061 evaluated thirteen middle school mathematics textbooks in the United States, revealing that only four of these textbooks met satisfactory standards (Dekker & Querelle, 2001; Kulm, Roseman & Treistman, 1999).
1.3.1.2 Rigorous characteristic of mathematical curriculum and textbooks
Comparing the mathematical curriculum and textbooks of Vietnam with those of other countries presents challenges, primarily in selecting an appropriate country for comparison and establishing relevant criteria This article focuses on examining the correlation between the volume of content in mathematics textbooks and the amount of instructional time available to teachers in schools.
Mathematics teachers face a significant challenge due to the extensive content in textbooks and the limited time available for instruction Many educators express concerns about insufficient teaching time, leading them to prioritize content delivery over effective teaching methods This situation highlights the need for a balance between curriculum demands and practical teaching strategies to enhance student learning in mathematics.
12 Prof Dr Celia Hoyles was awarded the first (2003) Hans Freudenthal Medal of International Commission on Mathematical Instruction (ICME) for her outstanding contribution in the technology and mathematics education domain
13 This set of textbook which is based on RME is created by researchers at the Freudeuthal Institute in the Netherlands
14 Dr Nguyen Thi Quy is the Vice-Director of the Institute for Educational Research, Ho Chi Minh City
A study by Do Dinh Hoan highlights that Vietnamese primary students spend less time in school compared to their peers in other countries To address this issue, he recommends implementing a two-shift system, allowing students to learn both in the morning and afternoon (Do Dinh Hoan, 2003, p 14) Additionally, research by Nguyen Thi Quy (2004) emphasizes the need for all elementary students to attend school during both shifts, citing a disparity between the volume of knowledge to be taught and the limited time available for instruction.
Many middle and high school students, particularly in rural areas, face challenges that prevent them from attending school full-time Some of these students are required to assist their parents with household chores and agricultural work, which often leads to them attending school in shifts This situation highlights the need for educational reforms to accommodate the unique circumstances of these learners (Do Dinh Hoan, 2003; Nguyen Thi Quy, 2004; Huynh Cong Minh, 2017).
In addition to regular school classes, students often participate in extra lessons to keep pace with their curriculum and prepare for exams, a cost typically borne by their parents This trend has become prevalent even among primary school students in larger cities, leaving them with minimal time for self-study or leisure activities Mathematics is the most common subject for which students seek extra lessons The increasing reliance on supplementary education has emerged as a significant social issue, drawing the attention of educators and the community alike.
G ENERAL IDEAS TO IMPROVE MATHEMATICS EDUCATION
Vietnamese mathematics education faces significant challenges that require complex, long-term solutions The Ministry of Education and Training (MoET) is actively implementing key initiatives such as the Renovation of the Curriculum and Teaching Methods, the Training of Information Technology (IT) officers, and the integration of IT into school education Additional projects include the Primary Education Project, Primary Teacher Development Project, Lower Education Project, and Lower Education Teacher Development Project, all aimed at enhancing the quality of school education This article will explore various strategies for improving mathematics education in Vietnam.
The transition from the 'reformed' series of curricula and textbooks to new educational materials marks a significant shift in mathematics education This section will examine the key changes in the primary, middle, and high school mathematics curricula and textbooks, highlighting the differences between the previous and current resources.
According to Do Dat (2000, p 6), the Primary Mathematics Curriculum for 2000 includes the following modifications:
• Increasing mathematical application in reality;
• Adjusting time for teaching natural numbers, fractions, decimal and percentage;
• Introducing more geometrical shapes such as cylinder and sphere
A new series of textbooks aligned with the Middle School Mathematics Curriculum introduced by the Ministry of Education and Training (MoET) on January 24, 2002, has recently been developed Many mathematics educators in Vietnam consider these new resources to be significantly better than the previous 'reformed' curriculum and textbooks.
Some of the principles for building the middle school mathematics curriculum are:
The new primary curricula remain rigorous, as indicated by the Ministry of Education and Training (MoET), which recommends a 15% reduction despite their recent implementation.
• do not pay excessive attention to structural and precise characteristics of the system of mathematical contents in the curriculum;
• do not present results with pure theoretical meaning and long complicated proofs which are not suitable for majority of students to the curriculum;
• allow students to practice and exercise calculation skills and applying mathematical knowledge in life and other school disciplines
The new series of middle school mathematics textbooks offers several advantages, as highlighted by Nguyen Minh Phuong (2001), Ton Than (2000, 2003), Tran Phuong Dung (2003), Pham Gia Duc (2003), and Vu Huu Binh (2004) These benefits include enhanced clarity in content presentation, improved alignment with educational standards, and increased engagement for students, ultimately fostering a more effective learning environment.
Textbooks are being revised to minimize inappropriate mathematical content by relocating unsuitable concepts, theorems, and exercises to higher grade levels or eliminating them altogether Additionally, complex proofs for certain theorems are omitted, and abstract definitions are simplified to enhance student comprehension.
Textbooks now offer a diverse range of exercises designed to enhance student skills in areas such as calculation, figure drawing, characteristic prediction, reasoning, and proof creation These exercises come in various formats, including word problems, gap-fill activities, true-false questions, multiple-choice queries, crossword puzzles, and error identification in provided solutions This marks a shift from the traditional focus on conventional word problems seen in earlier textbooks.
• The new series emphasizes problem situations and self-study
• While the old set of textbooks did not strongly incorporate mathematics application, the new one places special importance on mathematics application in real life and other school disciplines 29
• The new series also hopefully attracts students’ interest by providing them interesting stories about history of mathematics
• The new curriculum covers calculator usage in mathematical calculation and reality
While the new mathematics curriculum and textbooks emphasize the importance of mathematical application, they still follow a traditional approach by teaching formal mathematics before applying it to problem-solving For a more in-depth analysis, refer to section 2 of chapter 3.
In middle school geometry, the traditional axiom-based approach is shifted towards alternative visual methods and reasoning Rather than being a purely deductive science that systematically builds theorems through mathematical proofs, this educational phase allows students to measure, observe, and experiment, leading to conclusions without the need for formal proofs Consequently, the emphasis on geometric proofs is significantly diminished in grades 6 and 7.
Through activities such as drawing and observing shapes, measuring segments and angles, and manipulating paper through folding and cutting, students can engage in discussions and make predictions about geometric properties before formally learning about theorems and their proofs.
Since 2000, a revised set of high school mathematics textbooks, known as the "2000 corrected unified high school" series, has been in use Recently, a new high school mathematics curriculum was approved, leading to the development and gradual replacement of this series with a newly tested set of textbooks.
In comparison with the ‘reformed’ textbooks (see section 1.3.1, this chapter), the ‘2000 corrected unified high school’ ones have the following modifications (Nguyen Huy Doan, 2000):
• Reducing theoretical knowledge and increasing practical knowledge (while more abstract, complicated, theoretical knowledge is significantly reduced in this new curriculum, more applicable practical knowledge includes);
• Unifying mathematical signs and terms in textbooks;
• Removing extremely difficult exercises and using moderated number of exercises
A new series of high school mathematics textbooks has recently been tested and is now being gradually implemented in schools These new textbooks feature significant changes compared to the previously 'reformed' curriculum and materials, as outlined by the Ministry of Education and Training (MoET).
• Adjusting structure of mathematics contents;
30 Typically, there is a unique series of mathematics textbooks in Vietnamese school There was an exception for high school because there were three series of mathematics textbooks available from the
To enhance the understanding of mathematics, it is essential to minimize theoretical complexities and focus on practical applications This approach involves omitting intricate theoretical concepts, substituting formal mathematical definitions with intuitive explanations, and avoiding the presentation of complex formal proofs By prioritizing application over theory, learners can grasp mathematical principles more effectively.
• Changing ways of presenting mathematical knowledge in textbooks (unlike the
‘reformed’ mathematics textbooks (see section 1.3.1.5 of this chapter), the new mathematics textbooks often offer students some tasks before mathematics definitions, theorems, rules and formulae are formally presented);
• Introduction to the use of calculators in teaching and learning mathematics;
• Introduction to the use of tests in examination;
• Emphasizing and encouraging students’ self-study
Because of the fact of underdeveloped teaching methods (section 1.3.2, this chapter), a reform of teaching methods is urgently required to remedy mathematics education in Vietnam (Nguyen Thi Quy, 2004)
The Ministry of Education and Training (MoET) emphasizes that middle school mathematics teaching should activate student engagement and focus on developing self-study, discovery, and problem-solving skills This approach aims to foster active, independent, and creative characteristics in students Consequently, teachers are expected to act as designers, organizers, guides, and controllers, while students are encouraged to take charge of their own learning and practice to enhance their personal development and meet the demands of future employment.
According to MoET (2002 b), general orientation for teaching high school mathematics is:
• Paying attention to activeness, initiative and ability of self-study for students;
• Using advantages of each teaching methods and paying attention to using problem solving approach;
• Providing students with the necessary knowledge and skills for real life
Recently, an assessment reform in Vietnamese school has become a necessary task Here are some orientations for this reform:
• The MoET considered assessment as a part of the current mathematics curriculum In this curriculum, the MoET suggests some changes of assessment (MoET, 2002 a, b & c)
• More research of assessment in mathematics is required (Le Thi Thanh Thao, 2004)
• Assessments and examinations are necessary to research and reform: develop students’ creativeness and reduce memorization (Hoang Thi Tuyet, 2004)
• Conduct other types of assessment and experiment use of tests for examinations (Hoang Thi Tuyet, 2004; Nguyen Quang Trung, 2004).
R ESEARCH QUESTION
The urgent need for reform in mathematics education in Vietnam highlights the importance of identifying effective teaching and learning approaches Given the current underdeveloped methods in this field, Realistic Mathematics Education (RME) emerges as a promising strategy to address these challenges However, for RME to be successfully integrated into Vietnamese mathematics education, it requires careful expansion and thoughtful application.
This dissertation focuses on the general question: how can RME enrich teaching and learning mathematics in Vietnamese school?
This dissertation focuses on refining the research question to align with the author's expertise in training middle and high school mathematics teachers Given the novelty of Realistic Mathematics Education (RME) for curriculum developers and educators in Vietnam, particular attention is directed towards middle school education Despite improvements in mathematics teaching and learning, existing weaknesses persist, highlighting the need to explore RME's potential in addressing these challenges Therefore, the central research question of this dissertation is established.
How can RME be used as a potential teaching and learning approach which can help mathematics education in middle school overcome its disadvantages?
1.5.2.1 Sub-question 1 (concerning a grade and a mathematics strand)
In Vietnam, the complexity of mathematics education escalates from primary to high school, with an emphasis on formal mathematics in curricula and textbooks Geometry, particularly formal deductive geometry, poses significant challenges for middle school students This dissertation specifically addresses the topic of middle school geometry, noting that geometrical proofs are introduced in grade 7, with an increasing number of proofs presented in grades 8 and 9.
It is worth considering how RME can be applied in teaching and learning grade-seven geometry The first sub-question is:
How can RME be applied in teaching and learning grade-seven geometry in Vietnamese school?
1.5.2.2 Sub-question 2 (considering difficulties teachers and students may meet)
Teaching and learning styles in Vietnam significantly differ from those in Western countries, especially in the Netherlands Therefore, research on the implementation of Realistic Mathematics Education (RME) in Vietnam must consider these cultural and educational distinctions.
What difficulties do teachers and students meet while RME is applied in teaching and learning middle school geometry?
1.5.2.3 Sub-question 3 (concerning potentials of RME)
RME appears to be particularly effective for guiding the reform of mathematics teaching methods in Vietnam However, it is essential to explore the potential impacts of RME on Vietnamese mathematics education Therefore, the focus of Sub-question 3 should be on investigating these effects.
What is the potential of RME, and how can this potential help mathematics education in Vietnam overcome its shortcomings?
1.5.2.4 Sub-question 4 (considering possible proposals for applying RME)
In Vietnam, teaching styles tend to be rigid and conventional, which can create challenges for both teachers and students when implementing Realistic Mathematics Education (RME) To address these difficulties, it is essential to explore potential solutions and suggestions that can help educators effectively navigate these obstacles.
What and how proposals should be made so that RME can be applied in teaching and learning in Vietnam?
1.5.2.5 Sub-question 5 (concerning frequency use of RME in teaching and learning in Vietnam)
In Vietnam, educational regulations impose strict guidelines on lesson duration and scheduling This creates a conflict between the extensive content found in mathematics textbooks and the limited time available for both teachers and students in the classroom Furthermore, a standardized set of textbooks is recognized as the official teaching material within the school system.
One question is posed: How often should RME be implemented in teaching and learning in middle school?
REALISTIC MATHEMATICS EDUCATION
O VERVIEW OF RME HISTORY
In this section, the Wiskobas project is discussed In the Netherlands, the innovative Wiskobas project inaugurated the period of elementary mathematics education in which RME was formulated and developed
Launched in the Netherlands in 1968, the Wiskobas project aimed to innovate elementary mathematics education by reforming teacher training This initiative focused on developing a new curriculum for elementary mathematics, with researchers analyzing various trends in mathematics education both domestically and internationally, including the Arithmetical (Mechanistical), Structural, and Empirical trends.
Dutch Arithmetic Education (Treffers, 1987, pp 14-17) Apart from the necessary
“pre-institutional stage” (1968-1971), the Wiskobas project has three important periods: “an exploratory phase” (1971-1973), “an integration phase” (1973-1975) and
“spin-off, further development and research” (1975-1977) (Treffers, 1987, pp 11-13)
The Dutch elementary mathematics education remained unaffected by the "New Math" approach, as highlighted by Van den Heuvel-Panhuizen (2000) The foundational principles that evolved into the Realistic Mathematics Education (RME) approach emerged in the Netherlands during the 1970s, primarily rooted in Freudenthal's philosophy regarding mathematics and its teaching.
2000, p 3) RME also incorporates elements from the aforementioned educational approaches Moreover, it takes advantages from these approaches (Treffers, 1987, pp 14-18)
For over thirty years, the Realistic Mathematics Education (RME) approach has been primarily developed by mathematics educators at the Freudenthal Institute of Utrecht University and various research institutions in the Netherlands Currently, approximately 75% of elementary schools in the Netherlands utilize RME-based textbooks, highlighting its widespread adoption (Treffers, 1991, p 11) Despite being three decades old, RME continues to evolve through ongoing research and development (Van den Heuvel-Panhuizen, 1998 & 2000, p 3) Numerous dissertations and research projects from the Freudenthal Institute and other Dutch institutions have contributed to the advancement of RME, as detailed in the following paragraphs.
The Hewet project (1981-1985) created the Mathematics A curriculum, tailored for high school students intending to pursue humanities or social sciences in university De Lange's dissertation (1987) provides a comprehensive analysis of the historical context, development, content structure, theoretical framework, teaching methods, and assessment strategies associated with Mathematics A.
Gravemeijer’s dissertation (1994) entitled, “Developing Realistic Mathematics Education” thoroughly analyzes and discusses “instructional design as a learning process”, “an instruction-theoretical reflection on the use of manipulatives”,
“mediating between concrete and abstract”, “educational development and developmental research in mathematics education” and “implementation and effect of realistic curricula”
Van den Heuvel's dissertation (1996) thoroughly discusses and analyzes the development of RME assessment strategies, including the initial phase, current practices, and written assessments, along with an in-depth exploration of the MORE project.
The principles of Realistic Mathematics Education (RME) have been utilized in the development of the Mathematics in Context series, a prominent collection of middle school mathematics textbooks in the United States (Romberg, 2001; Meyer et al., 2001) While the majority of research surrounding RME concentrates on K-12 education, there is a growing interest in its application to undergraduate mathematics teaching and learning (Rasmussen & King, 2000; Kwon, 2002; Ju & Kwon, 2004).
S OME BASIC IDEAS OF F REUDENTHAL FOR RME
This section explores key concepts of Freudenthal's Realistic Mathematics Education (RME) theory, focusing on mathematics as a human activity, guided reinvention, and didactical phenomenology Freudenthal's ideas are interconnected and stem from his inquiry into the purpose of teaching mathematics for practical use, as highlighted in his lecture and first article in Educational Studies in Mathematics.
Freudenthal (1991, p 6) emphasizes the crucial role of common sense in education, particularly in mathematics instruction, highlighting how it is frequently overlooked by the natural sciences, such as physics, chemistry, and astronomy, along with their teaching principles.
In education, it is advisable to begin with common sense ideas instead of dismissing them as outdated This approach is reinforced by the natural progression observed in the development of mathematics.
Freudenthal discusses the poor relationship between classroom and school experience and life experience; however, education should emphasize real life experience, argues Freudenthal (1991, pp 4-6)
Below, Freudenthal (1991, p 9) discusses strategies to repair this relationship:
To evolve into true mathematics and facilitate progress, common sense must be systematized and organized Everyday experiences merge into foundational rules, like the commutativity of addition, which then elevate our understanding and form the basis for more advanced mathematical concepts This creates a vast hierarchy of knowledge, driven by a remarkable interplay of ideas.
According to Freudenthal (1991, p 9), mathematics is the oldest science and
Freudenthal (1991, p 11) argues that mathematics is distinct from other sciences, indicating that its teaching and learning should prioritize the dynamic relationship between form and content He emphasizes the importance of respecting this interplay within the educational process.
2.2.1.2 Mathematics as a ready-made product and mathematics as a human activity Freudenthal discusses two different approaches to mathematics The first approach considers mathematics as a ready-made product, and the second one regards mathematics as an activity
Freudenthal highlights the concept of mathematics as a human activity, distinguishing it from the static nature of mathematics found in books and minds He asserts that mathematical activity encompasses a wide range of outputs, including propositions, theorems, proofs, definitions, notations, and their presentation in both print and thought This perspective broadens the understanding of what constitutes mathematics beyond mere written symbols.
Freudenthal (1991) considers mathematizing to be one of main characteristics of mathematical activity Details of mathematizing are discussed in section 2.3.1 of this chapter
In mathematics education, Freudenthal strongly objects to what he called anti- didactical inversion: teaching mathematics by beginning with ready-made mathematics
(Freudenthal, 1973, p 106, 1983, p ix; Gravemeijer & Terwel 2000, p 780) Instead, Freudenthal believes that mathematics should be taught as an activity
Freudenthal emphasizes the significance of guided reinvention in learning, describing inventions as essential steps in the learning process, highlighted by the "re" in reinvention He notes that the term "guided" refers to the instructional environment that supports this learning journey.
What I have called re-invention, is often known as discovery or re-discovery I have also used these terms a few times, and it would not really matter which are used […] Perhaps the term “invention” was chosen because students are expected to find something which is new and un-known to them but well-known to the instructor He also explains why he prefers the notion of ‘guided reinvention’ to other notions such as problem solving, discovery learning, heuristics and genetic method (Freudenthal, 1991, pp 45-48)
31 The adjective ‘guided’ here might refer to guide from not only teachers and learning materials but also other peers (students) (Freudenthal, 1991, p 47)
Students should not be expected to replicate the historical learning process of mathematics but should have the opportunity to reinvent it with the guidance of their teachers and appropriate learning materials (Freudenthal, 1991) Freudenthal outlines key principles of "guided reinvention," emphasizing the importance of helping students abstract, schematize, and formalize their understanding of reality To effectively guide this process, he identifies five essential tenets: selecting learning situations relevant to the learner's experience, providing tools for deeper mathematical understanding, fostering interactive instruction, encouraging student-generated production, and integrating various learning strands (Freudenthal, 1991).
Freudenthal explains that the way in which mathematics is published and presented is different from the way in which it is invented (Freudenthal, 1983) From this he suggests that:
Young learners have the right to experience a learning process that reflects the evolution of human knowledge This should not be a simplistic summary of past learning, nor should we expect them to begin exactly where previous generations concluded.
Brousseau (1998) supports Freudenthal's perspective by developing the Theory of Didactical Situations in Mathematics, which examines the interactions among mathematicians, teachers, and students He notes that mathematicians present knowledge in a decontextualized and impersonal manner, while teachers work to recontextualize and personalize this knowledge by creating meaningful learning situations for their students Following these experiences, students are guided by their teachers to redepersonalize and redecontextualize the knowledge they have acquired.
32 Reality here means what is experienced real to the students
In 2003, Professor Guy Brousseau received the inaugural Felix Klein Medal from the International Commission for Mathematics Instruction (ICMI) for his significant contributions to mathematics education and his dedication to teacher training His influential Theory of Didactical Situation in Mathematics was introduced in Vietnam during the 1990s, as noted by Nguyen Ba Kim in 2002.
Freudenthal (1983, p ix) discusses the notions of phenomenology and didactical phenomenology as follows:
The phenomenology of a mathematical concept involves exploring its relationship to the phenomena it was designed to address and its evolution throughout human learning This approach, known as didactical phenomenology, highlights key areas where educators can guide young learners in their understanding of mathematics, revealing pathways for deeper engagement with the subject.
Treffers argues that phenomenology is not a novel concept, as it has been utilized to identify appropriate mathematical applications within various instructional methods However, he highlights a crucial distinction in Freudenthal's perspective, emphasizing the concept of didactical phenomenology.
M EANING OF ‘ REALISTIC ’ IN RME
The term 'realistic' in Realistic Mathematics Education (RME) stems from Treffers's classification of four approaches to mathematics education, which are evaluated through the concepts of horizontal and vertical mathematization This article first clarifies the definitions of mathematization, as well as horizontal and vertical mathematization It then examines four distinct trends in mathematics education: mechanistic, structuralistic, empiristic, and realistic approaches Lastly, the discussion highlights the importance of authenticity as a key characteristic in these educational frameworks.
34 Treffers (1987, p 246) confirms that “such a didactical phenomenology is of course not new” It
As discussed previously (section 2.2.1.2 of this chapter), Freudenthal considers mathematizing primarily as an activity (Freudenthal, 1991, p 30; see also Gravemeijer,
1994, p 82) Freudenthal (1991, p 31) explains that “[…] the origin of the term mathematising as an analogue to axiomatising, formalising, schematising.” He also discusses aspects of mathematizing (Freudenthal, 1991, pp 35-36) Gravemeijer (1994, p 83) explains that:
[…] mathematizing mainly involves generalizing and formalizing Formalizing embraces modelling, symbolizing, schematizing and defining, and generalizing is to be understood in a reflective sense
De Lange defines mathematizing as “an organizing and structuring activity according to which acquired knowledge and skills are used to discover unknown regularities, relations and structures”(1987, p 43)
Horizontal and vertical mathematization are concepts that differentiate between "transforming a problem field into a mathematical problem" and "processing within the mathematical system" (Treffers, 1987, p 247) Treffers acknowledges that distinguishing between these two types of mathematization may not always be straightforward.
The distinction between horizontal and vertical components is somewhat artificial, as they can be significantly interrelated, a notion supported by De Lange (1987) and Freudenthal (1991).
De Lange, building on the work of Treffers and Goffree, identifies two key components of mathematization: horizontal and vertical The horizontal component involves converting a real-world problem into a mathematically formulated problem, while the vertical component focuses on the mathematical manipulation and refinement of that problem once it has been transformed into mathematical terms (De Lange, 1987, p 43).
At first, Freudenthal was not willingly to accept Treffers’s distinction between horizontal and vertical mathematization:
For a considerable time, I hesitated to embrace this distinction, worried that it might undermine the theoretical equivalence of both activities and their practical status My concerns stemmed from frequent disappointments with mathematicians focused on education who often oversimplified these complexities.
The 35 real-world problems should not be taken literally; instead, they should be approached by considering both their vertical components and the educational methods used in mathematics instruction, which often limit the focus to horizontal aspects.
Freudenthal eventually approves of this distinction “because of its consequences for mathematics education, and in particular, for characterising educational styles.”
(1991, p 41) Freudenthal (1991, pp 41-42) distinguishes horizontal and vertical mathematization as follows:
Horizontal mathematisation transitions from the lived experience to the realm of symbols, where individuals engage in actions and emotions In contrast, vertical mathematisation involves the mechanical, reflective, and comprehensive manipulation of these symbols.
Freudenthal also confirms the equal roles of horizontal and vertical mathematization and their presence in all mathematical activity levels (Van den Heuvel-Panhuizen, 1996, p 11; Nguyen Thanh Thuy, 2005, p 26)
De Lange (1987, p 43) enumerates some activities containing strong horizontal components:
• identifying the specific mathematics in a general context;
• formulating and visualizing a problem in different ways;
• recognizing isomorphic aspects in different problems;
• transferring a real world problem to a mathematical problem;
• transferring a real world problem to a known mathematical model
He also refers to some activity containing strong vertical components:
2.3.2 Different approaches to mathematics education
The term 'realistic' in mathematics education was introduced by Treffers, who identified four distinct approaches: mechanistic, structuralist, empiricist, and realistic This classification is based on the criteria of horizontal and vertical mathematization, as discussed in the works of De Lange (1987), Treffers (1987), and Freudenthal (1991).
De Lange describes the mechanistic approach as follows:
In the mechanistic approach mathematics is a system of rules The rules are given to the students, they verify and apply them to problems similar to previous examples
The mechanistic approach to education inadequately addresses key elements such as application, methodology, structure, interrelatedness, and insight (De Lange, 1987) Similar critiques are echoed by Streefland (1991), highlighting the limitations of this perspective Freudenthal (1991) further critiques this approach by likening individuals to "computer-like instruments," emphasizing its shortcomings Both horizontal and vertical mathematization are notably weak within this framework, underscoring the need for a more holistic educational methodology.
The structuralist approach considers mathematics as “an organized, closed deductive system” (De Lange, 1987, p 93) Hence, this approach emphasizes mathematical structures in school In the 1960s and 1970s, this approach, labeled the
‘New Mathematics’, widely influenced mathematics education (De Lange, 1987, p
The article critically examines the impacts and criticisms of the structuralist approach to mathematics education, highlighting the excessive emphasis on vertical mathematization while neglecting horizontal mathematization Key analyses are referenced from De Lange (1987), Freudenthal (1991), and Streefland (1991).
The empiricist approach used mainly in Great Britain (Streefland, 1991, p 22; see also Freudenthal, 1991, p 135) is described as follows:
Learners gain valuable experiences from their surroundings, yet they often lack the encouragement to organize and analyze these experiences This limitation hinders their ability to overcome environmental barriers and broaden their understanding of the world beyond their immediate reality.
36 ‘Training for examination’ approach in Vietnam is quite similar to the mechanistic approach
In the empiricist approach, horizontal mathematization is emphasized, but vertical mathematization is weak
On the contrary, the realistic approach fully incorporates both vertical and horizontal mathematization
The following table is often used to illustrate the differences of the four mentioned mathematics education approaches under the criterion of horizontal and vertical mathematization (De Lange, 1987, p 101; Treffers, 1987, p 251; Freudenthal,
Figure 2.1: Approaches in mathematics education
Jahnke (2001) emphasizes the importance of authenticity in productive exercises for mathematics lessons, stating that a problem situation is considered authentic when learners recognize its relevance and engage with it He advocates for the inclusion of authentic tasks in mathematics education to enhance learning experiences (Jahnke, 2001, p 7; see also Jahnke, 2005) However, it is important to note that authenticity is not a mandatory feature within the framework of Realistic Mathematics Education (RME).
The term 'realistic' can be somewhat deceptive, as it is frequently misinterpreted both within the Netherlands and beyond Contrary to popular belief, the realistic approach does not solely emphasize reality or authenticity, as highlighted by Van den Heuvel-Panhuizen (2000, p 4).
In their 1990 work, Gravemeijer, Van den Heuvel-Panhuizen, and Streefland emphasize that 'realistic' mathematics education involves linking mathematical concepts to real-world contexts while also enabling learners to build their own mathematical understanding.
The term 'realistic,' as defined by Van den Heuvel-Panhuizen (2003), originates from the Dutch verb 'zich realiseren,' meaning 'to imagine.' She emphasizes that 'realistic' pertains more to the goal of providing students with problem situations they can envision, rather than a strict definition of realism itself.
T ENETS ( PRINCIPLES ) OF RME
The development of Realistic Mathematics Education (RME) is guided by five key principles: the use of real-life contexts, the application of models, the encouragement of students' own construction and production, the interactive nature of learning, and the integration of various learning strands These principles are derived from the combination of Van Hiele’s levels of mathematical understanding, Freudenthal’s didactical phenomenology, and Treffers’s concept of progressive mathematization.
The author of this dissertation aligns with Gravemeijer, suggesting that the phrase "these will never become real life problems" should be revised to "in some cases, these will not become real life problems" for greater accuracy.
Borasi defines context as “a situation in which the problem is embedded” (Van den Heuvel-Panhuizen, 1996, p.118) Traditional mathematics textbooks often lack contextualized problems, presenting them only in brief introductions or as end-of-section story problems (Meyer et al., 2001, p 522) Consequently, students face challenges when tackling contextual problems, as they must first translate these into non-contextual forms before attempting to solve them (Meyer et al., 2001, p 522).
According to Gravemeijer and Doorman (1999, p 111), context problems are
The issues encountered in problem situations are tangibly relevant to the student, encompassing both real-world contexts and abstract mathematical challenges This duality highlights the importance of addressing not only practical problems but also "pure mathematical" ones, as discussed by Gravemeijer and Doorman (1999) and Van den Heuvel-Panhuizen (2000).
In the realm of Realistic Mathematics Education (RME), the concept of context problems aligns closely with the idea of problem situations as defined by Nguyen Ba Kim and Vu Duong Thuy (1997) They outline three essential conditions that characterize these problem situations.
• It contains a problem which students do not know any algorithm to solve this problem;
• Students understand the problem’s relevance; and
Despite lacking an immediate solution or a specific algorithm, students possess relevant knowledge and skills related to the problem at hand They maintain a belief that with dedicated effort and perseverance, they can ultimately find a resolution to the challenge they face.
Freudenthal discusses how to use contexts properly when discussing mathematics application:
In traditional mathematical instruction, the approach often follows a pattern of didactical inversion, where mathematical concepts are introduced first, and their real-world applications are presented later This method contrasts with starting from concrete problems and exploring them through mathematical means, limiting the connection between mathematics and its practical uses.
In RME, “[…] the use of realistic contexts became one of the determining characteristics” (Van den Heuvel-Panhuizen, 2003, p 9), and “[…] context problems play a role from the start onwards.” (Gravemeijer & Doorman, 1999, p 111)
De Lange identifies three levels of context use in mathematics: the third order, which is the most significant for introducing and developing mathematical models, the second order, which is important for helping students organize and structure relevant mathematics to tackle real-world problems, and the first order, often found in traditional textbooks, where mathematical operations are embedded in contexts requiring only a simple transition to a mathematical problem (De Lange, 1987, pp 76-77).
Based on De Lange’s three levels of context use, Meyer et al (2001, p 523) point out five different roles of context in teaching and learning mathematics which are often interactive:
• motivating students to explore new mathematics;
• offering students a chance to apply mathematics;
• serving as a source of new mathematics;
• suggesting a source of solution strategy;
• providing an anchor for mathematical understanding
In 1975, Freudenthal introduced initial concepts of models at a broader didactical level within Realistic Mathematics Education (RME), distinguishing them from traditional mathematical models (Van den Heuvel-Panhuizen, 2003, p 15) Streefland expanded on these ideas in 1985 by developing the concepts of "model of" and "model for" within a micro-didactic context (Van den Heuvel-Panhuizen, 2003, p 15).
A model serves as a crucial intermediary that simplifies complex realities or theories, making them accessible for formal mathematical analysis The term "mathematical model" can be misleading, as it implies a direct application of mathematics to real-world environments, which is only accurate when mathematics is closely linked to those environments Emphasizing the model's role is essential, as many overlook its necessity Frequently, mathematical formulas are misapplied like recipes in complex situations without a proper intermediary model to validate their use.
Freudenthal (1991) discusses the development of mathematical models in problem-solving contexts, as elaborated by Streefland (cited in Van den Heuvel-Panhuizen, 2003) Initially, a context-based model emerges from a specific problem situation, referred to as a "model of." Subsequently, this model evolves into a more generalized framework, known as a "model for," which can be applied to various situations beyond the initial context This transition serves as a bridge between informal and formal knowledge In practice, students first engage with a problem situation, using their informal knowledge to devise context-specific strategies They then refine these strategies into broader approaches that can address not only the original problem but also similar challenges in different contexts.
Gravemeijer (1994) critiques the information processing approach that relies on formal mathematics as a foundation, utilizing didactical models to illustrate this knowledge He argues that while these models may appear concrete, the underlying mathematical concepts often remain abstract for students, hindering their ability to gain true mathematical insight This "top-down" instructional method directs learning from formal to informal mathematics, which Gravemeijer suggests is ineffective in fostering a deeper understanding of the subject.
Consequently, he stresses that in order to overcome the mentioned weakness,
Gravemeijer (1994) emphasizes that the development of abstract mathematical knowledge should begin with informal knowledge and real-life situations This approach advocates for a reverse progression in mathematics instruction, where students evolve from informal understanding to formal concepts through self-developed models He refers to this as an "alternative" or "bottom-up" approach, highlighting the importance of starting with practical knowledge to enhance formal mathematical learning.
Gravemeijer (1994, p 101) discerns four levels in RME, including situations, model of, model for and formal mathematics by using the following figure:
Figure 2.2: Self-developed models in RME Gravemeijer (1994, p 101) describes the levels in more general terms:
• the level of the situations, where domain specific, situational knowledge and strategies are used within the context of the situation (mainly out of school situations);
• a referential level, where models and strategies refer to the situation which is sketched in the problem (mostly posed in a school setting);
• a general level, where a mathematical focus on strategies dominates the reference to the context;
• the level of formal arithmetic, where one works with conventional procedures and notations
Later Gravemeijer uses the notion of emergent models instead of self- developed models (Gravemeijer & Doorman, 1999; Gravemeijer, 2002 & 2004) He explains that
The term "emergent" describes the dual nature of the process in Realistic Mathematics Education (RME), highlighting how models develop and how these models facilitate the emergence of formal mathematical understanding.
2.4.2.3 “Didactical modeling”, “mathematical modeling” and “emergent modeling” Gravemeijer elaborates three different types of modeling in mathematics education:
“didactical modeling”, “mathematical modeling” and “emergent modeling” (Gravemeijer, 2004, pp 97-99) He also confirms that each modeling assumes a certain role in mathematics education (Gravemeijer, 2004, p 97)
The limitations of didactical models are highlighted in section 2.4.2.2 According to Gravemeijer (2004), "mathematical modeling" treats the mathematical model and the situation being modeled as distinct entities (p 97) He proposes that emergent modeling can act as a precursor to the formal knowledge modeling of situations, bridging the gap between the two concepts (p 97) This relationship between emergent modeling and mathematical modeling can be visually represented in a figure.
Figure 2.3: “Emergent modeling” and “mathematical modeling”
2.4.3 The students’ own productions and constructions
S OME EXAMPLES
Numerous examples about RME can be found in De Lange (1987), Treffers (1987), Streefland (1991), Gravemeijer (1994), Van den Heuvel-Panhuizen (1996), Bakker
In a study featuring Vietnamese middle school mathematics teachers, the example of "T-shirts and Sodas" is utilized to illustrate problem-solving strategies This problem is specifically designed for middle school students, encouraging them to explore various informal strategies to arrive at a solution.
Freudenthal highlighted the significant connections between mathematics and various disciplines, particularly emphasizing the strong relationship between mathematics and physics, as well as its relevance to chemistry and biology.
Physics relies on mathematics as a supportive tool, yet it also emerges from the experiential reality that informs the subject matter and structure of mathematics This relationship highlights the interconnectedness of the two disciplines.
It is unfortunate that physics education often relies on simplified mathematics, resulting in students learning physics without a strong mathematical foundation This approach leads to a disconnect between mathematics and physics, preventing students from understanding the crucial role that mathematics plays in the physical sciences.
Freudenthal (1973) advocates for an approach where mathematics serves as a central discipline that integrates and organizes content from other subjects, rather than merely coordinating with them This concept can be categorized into internal and external relations An example of internal relations is observed in students aged 8 and 9, who, having only learned multiplication up to 10, develop informal strategies to tackle long division In contrast, an example of external relations is the use of an empty number line, which aids students in mastering addition and subtraction up to 100.
In this section, an example of developing long division is described This example is presented following Gravemeijer (1994, pp 83-84)
Children age 8 or 9 who have learned only multiplication up until number 10 are given the following problem:
Tonight 81 parents will be visiting our school Six parents can be seated at each table How many tables do we need?
Then the teacher gives the students some suggestion by drawing the following figures:
Figure 2.4: Guided figures (Gravemeijer, 1994, p 84) The students can find different solutions to the problem:
Some draw tables for 6 persons until they get enough tables for 81 persons Then they count a number of necessary tables:
They can count 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78 and 84 Then they count a number of tables: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14
Some students start by using the multiplication 10 × 6 = 60 Then they use repeated addition or multiplication:
Figure 2.6: Using multiplication and repeated addition
One student starts by using the multiplication 6 × 6 = 36 Then this student doubles 36, adds 6 and adds 6 to find a number of necessary tables
After that, students are encouraged to compare their solutions Most of them agree that the solution starting by multiplication 10 × 6 = 60 is reasonable
Finally, a similar problem is given to the students:
One pot serves seven cups of coffee; each parent gets one cup How many pots of coffee must be brewed for the 81 parents?
At this time, most of the student use “ten times” solution to deal with this problem, although their teacher has not told them to use it
Van Galen and Feijs (1991) explore the diverse problem-solving approaches of third-grade students during a lesson enhanced by videodisc technology in the Netherlands In their findings, they highlight how Anita, one of the students, utilizes drawing as a key strategy to articulate her solutions effectively.
(14) tables with chairs, but without numbers; Fatiha starts by drawing 4 tables (without chairs and numbers), then she uses numbers instead of tables; Osman starts by drawing
Noura begins her calculations with two multiplications: 10 × 6 = 60 and 4 × 6 = 24 She then lists a series of additions: 6 added together multiple times, culminating in 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 3 = 81 This process illustrates her method of transitioning from tables with chairs and numbers to simply tables with numbers, ultimately progressing from 60 to 63 and reaching 81 (Van Galen & Feijs, 1991).
2.5.2 Empty number line for additions and subtraction up to 100
Gravemeijer gives three reasons related to “phenomenological analysis of number”,
“informal solution procedures” and “level-raising qualities” for using the empty number line to add and subtract up to 100 (Gravemeijer, 1994, pp 123-125)
Moreover, he also identified a disadvantage of using full number line which tempts the students to use “primitive counting strategies” when they dealt with the subtraction (Gravemeijer, 1994, pp 123-125)
Gravemeijer proposes an effective method for introducing the empty number line in mathematics education Initially, students are presented with a contextual problem to engage their thinking Following this, they work with a bead string, which helps them visualize numerical relationships Finally, the empty number line is introduced as a model that represents the bead string, reinforcing their understanding of the concept.
Figure 2.7: Making numbers of the bead string (Gravemeijer, 1994, p 125)
Figure 2.8: Modeling a bead string solution with an empty number line (Gravemeijer, 1994, p
Gravemeijer (1994, p 125) explains that “[…] there are no marks on the number line, the student places the marks that he or she chooses.”
Students can utilize the empty number line to explore various methods for solving addition and subtraction problems up to 100 For example, when tackling the addition problem 27 + 38, different strategies can be illustrated through the figure provided, showcasing the diverse approaches students take in their calculations.
Figure 2.9: Different strategies for addition 27 +38 (Gravemeijer, 1994, p 120)