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Proficiency in basic mathematical skills is a topic of heated discussions in many countries. International comparative studies on mathematical skills such as PISA and TIMMS have lead to concerns about the mathematics curriculum especially in countries with a relatively low rating in the summary reports (the league-tables effect). In the Netherlands...
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... the Dutch system there is an early differentiation at the end of primary education (age 12). At the end of primary education, students choose for one of the three streams for sec- ondary education (Figure 1.1). These streams are VWO (pre-university education), HAVO (general education or pre-higher vocational education), and VMBO (pre-vocational sec- ondary education). The VWO stream, which represents about 15 % of the population, typically prepares students for university. Gymnasium is a VWO stream that offers stu- dents additional courses in the classical languages Latin and Greek. The HAVO stream (25 %) prepares students for higher vocational education. The VMBO stream is divided in four sub-streams, i.e. BBL (basic vocational programme), KBL (middle management vocational programme), GL (combined programme), and TL (theoretical programme). These streams represent 17 %, 16 %, 6 %, and 21 % of the total number of students in secondary education respectively. The majority (72 %) of the VMBO "certified" students continues their stud- ies in MBO (post-secondary vocational education and training). Only a small number of VMBO students (approximately 6% from GL and 12% from TL), continues in the HAVO stream (Van Esch and Neuvel, 2007). The selection for these streams is based on the advice of the teacher of grade 6 and the results of a nation-wide test, the CITO-test (e.g. Cito, 2010). ...
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... "Les mathématiciens n'étudient pas des objets, mais des relations entre les objets; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matière ne leur importe pas, la forme seule les intéresse.", (Poincaré, 1902) As Poincaré formulated in the above, mathematics is often called a science of structures. The importance of structure was beautifully exemplified by Sfard (1991). She shows that questions that are easily answered if the structure is known (Figure 7.2) are much harder if the same structure is presented as a set of four seemingly unrelated rules (Figure 7.1). This example shows also that it is almost impossible to reason about more sophisticated and generalizing tasks starting from those four rules. Just the recognition of the structure of a mathematical concept provides for deeper insight. Although recogni- tion of structure is a form of abstraction, it must not be confused with the use of abstract mathematical ...
Context 3
... analyzing these fragments we found that in each of these textbook series, fraction multiplication is strongly connected with informal strategies, contexts and models. Al- though textbooks differ, we distinguish four specific forms of fraction multiplication that characterize all of them ( Figure 6.1). Typical of those specific forms is that they are number- specific, that is to say, they depend on the type of fractions involved. Similar to the multiplication of whole numbers, the multiplication of a 'whole number times a proper fraction' is presented as 'repeated addition' 2 . In these types of tasks the whole number is typically small. In the various textbooks different models are chosen to represent this. Examples are the number line or natural predecessors thereof and contexts such as measuring ...
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Citations
... Este resultado coincide con estudios realizados en otros países, que revelan una tendencia por parte de los libros de texto a focalizar el proceso de enseñanza-aprendizaje de los problemas en la adquisición de habilidades y procedimientos algorítmicos (Bruin-Muurling, 2010;Li, Chen y An, 2009;Van Stiphout, 2011). ▪ En el estudio realizado por Wijaya et al. (2015) también se evidenció que la gran mayoría de los problemas presentados por los libros de texto de matemáticas en Indonesia aparecían en un contexto "puramente matemático". ...
... Por otro lado, con respecto al segundo objetivo específico planteado y de acuerdo con el papel que habitualmente desempeñan los problemas en los libros de texto de matemáticas, prevemos que una parte considerable de los problemas analizados estarán supeditados a la ejercitación de las operaciones aritméticas, más que al desarrollo del razonamiento (Bruin-Muurling, 2010;Castro y Ruiz, 2015;Li et al., 2009;Sánchez y Vicente, 2015;Säljö y Wyndhamn, 1987;Schoenfeld, 1989Schoenfeld, ,1991Sowder, 1988;Stern, 1992;Stigler, Fuson, Han y Kim, 1986;Van Stiphout, 2011;Vincent y Stacey, 2008;Verschaffel, 2012). ...
... En resumen, si tenemos en cuenta los resultados de la frecuencia de los dos tipos de situaciones analizados (33.80%) junto con los resultados de los estudios que han demostrado que el criterio adoptado por los libros de texto para agrupar los problemas es la última operación estudiada (Bruin-Muurling, 2010;Castro y Ruiz, 2015;Li et al., 2009;Sánchez y Vicente, 2015;Säljö y Wyndhamn, 1987;Schoenfeld, 1989Schoenfeld, ,1991Sowder, 1988;Stern, 1992;Stigler, Fuson, Han y Kim, 1986;Van Stiphout, 2011;Vincent y Stacey, 2008;Verschaffel, 2012), podemos concluir que son abundantes las ocasiones en las que la función del problema se supedita a la ejecución de la operación matemática. De todo ello damos cuenta en el siguiente epígrafe dedicado a la discusión. ...
Los libros de texto son el tipo de material curricular más empleado en la educación obligatoria. El objetivo de este trabajo es analizar el tratamiento que reciben los problemas aritméticos verbales de estructura aditiva en estos materiales. Se llevó a cabo una revisión de estudios recientes que analizan la estructura semántica de los problemas en los libros de texto de diferentes sistemas educativos. Se concluye que los libros de texto no constituyen, por sí solos, herramientas eficaces para abordar el proceso de enseñanza-aprendizaje de la resolución de problemas. Además, es necesario tener en cuenta las carencias de estos materiales curriculares y utilizar problemas matemáticos que abarquen todas las estructuras semánticas.
... , fractions(Bruin-Muurling, 2010), and algebra (Van Stiphout, 2011). Analysing the results of those three Ph.D. studies, Gravemei- jer, Bruin-Muurling, Kraemer and Van Stiphout ...
This chapter describes a socio-constructivist elaboration of Realistic Mathematics Education (RME) that emerged from my collaboration with Paul Cobb and Erna Yackel. It is argued that RME and socio-constructivism are compatible and complement each other. Socio-constructivism points to the critical role of the classroom culture, while RME offers a theory on supporting students in (re-)constructing mathematics. Furthermore, the role of symbols and models is discussed, which was considered problematic in constructivist circles, while being central in RME. The emergent modelling design heuristic is presented as a solution to this puzzle. Together, guided reinvention, didactical phenomenology, and emergent modelling, are combined to delineate RME as an instructional design theory. This is complemented by a discussion of pedagogical content tools as counter parts of the emergent modelling and guided reinvention design heuristics at the level of classroom instruction. Finally, research on student learning and enactment of RME in Dutch classrooms is discussed.
... The students, however, spontaneously started using splitting and reasoning methods, which they did not master. Bruin-Muurling (2010) investigated operations with fractions in primary and secondary school. She showed that students were struggling with fractions well into secondary school. ...
The gist of this article is that a shift is needed towards a mathematics curriculum in which teaching for understanding is the main objective. Before elaborating on what such a shift might entail, two compelling arguments for making such a change are presented. One is based on research, which showed that a one-sided emphasis on skills induced the teaching of isolated, topic-specific, skills-leading to a low level of proficiency. The other argument is grounded in the observation that the role of mathematics in society is changing and that, as a consequence thereof, the importance of mastering routine skills diminishes, while the need for mathematical understanding grows. On the basis of these two arguments, a shift is advocated, from an emphasis on skills, to an emphasis on understanding. This is connected to the thesis that deep mathematical understanding can only be achieved when students construct mathematical objects by reifying mathematical processes. For the domain of number, the idea of mathematical objects is linked to the notion of junctions in networks of number relations. The core of the article is an exploration of what mathematics education in the number domain might look like if it would be organized along the line of process-object transitions, and how number relations can be used for solving the kind of number tasks that are commonly solved with standard procedures. This exploration is closed with a sketch of a potential instructional sequence for addition and subtraction up to 100. Resumo. A essência deste artigo é que é necessária uma mudança para um currículo de matemática em que o objetivo principal é o ensino para a compreensão. Antes de discutir o que essa mudança pode acarretar, são apresentados dois argumentos convincentes para fundamentar essa mudança. Um é baseado na investigação, que mostra que a ênfase unilateral nas técnicas induz o ensino de técnicas isoladas, específicas de um tópico, levando a um baixo nível de proficiência. O outro Fostering process-object transitions… 7 Quadrante, Vol. 28, N.º 2, 2019 argumento fundamenta-se na observação de que o papel da Matemática na sociedade está a mudar e que, como consequência disso, a importância de dominar as técnicas rotineiras diminui, enquanto a necessidade de compreender a Matemática aumenta. Com base nestes dois argumentos, defende-se uma mudança, de uma ênfase nas técnicas rotineiras para uma ênfase na compreensão. Isto liga-se à tese de que uma compreensão matemática profunda só pode ser alcançada quando os alunos constroem objetos matemáticos reificando processos matemáticos. Para o domínio dos números e operações, a ideia de objetos matemáticos está ligada à noção de articulações em redes de relações numéricas. A parte central do artigo é uma exploração de como seria a educação matemática no domínio dos números e operações se fosse organizada ao longo da linha de transições de processo-objeto e de como as relações numéricas podem ser usadas para resolver o tipo de tarefas numéricas comummente resolvido com procedimentos padrão. Esta exploração termina com o esboço de uma potencial sequência de ensino para a adição e a subtração até 100.
... Student proficiency regarding fractions is often problematic. In Grade 5 in the Netherlands, for example, many students find it difficult to solve problems related to improper fractions and mixed numbers or addition and subtraction including fractions, even though instruction has been provided to them about these topics (Bruin-Muurling 2010). ...
In this study, we investigated which teacher characteristics influence student proficiency development regarding fractions in Grade 5 of Dutch primary education. At least three domains of research (i.e. perspectives) on effective teaching can be distinguished: studies focusing on teachers’ background characteristics, on their knowledge and conceptions regarding the subject they are teaching and on the domain-specific and general pedagogical characteristics of their teaching. In this study, effects of the three perspectives on student fraction proficiency were examined simultaneously using multilevel analyses. Findings revealed that teachers’ age and experience in the upper grades, their pedagogical content knowledge and the degree of student participation in their lessons had positive effects. Their subject matter knowledge, quality of their concept maps and the richness of the mathematics in their lessons had negative effects. Thus, effects were found pertaining to all three perspectives on teachers and their teaching we included in our study.
... There is a clear focus in introducing different conceptions of fractions grounded in contextual problems that could be considered real for students in the Dutch educational system (Bruin-Muurling, 2010;Keijzer, 2003;Keijzer & Terwel, 2001;Streefland, 1991). ...
... Other meanings of fractions such as partwhole, ratio and operator are also represented in the textbooks. However, detailed analysis of fractions in four primary and two secondary textbook series by Bruin-Muurling (2010) revealed that while fractions and fraction multiplication are firmly grounded in contexts, students' various informal strategies are often followed by teaching specific procedures. They contend that this approach does not support the development of more advanced understanding of fractions such as "the dual conceptualization of a fraction as a number and as an operation" (p. ...
This study examined the relationship between teachers' questioning techniques and students' strategies in solving contextual mathematical problems. This case study was undertaken with one pre-service teacher (and 22 Year 4 students) from Indonesia and one pre-service teacher (and 25 Year 4 students) in the Netherlands. Both pre-service teachers assigned the same problems to their students and these problems were novel for the students in both countries. The lessons were observed by the first author and video recorded for data analysis. Qualitative data analysis was undertaken through within-case and cross-case analysis. The findings suggest that the contextual problems, the way pre-service teachers prompt students' thinking, and the curriculum context were highly influential in the way the students solved the problems.
... In 2006-2009 a comprehensive research project was carried out on the proficiency in the domain of fractions, concerning students in grade 6 through 9 (Bruin-Muurling, 2010). For this purpose a series of tests was constructed to measure both procedural and conceptual aspects of proficiency in the fraction domain. ...
This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three cases, the disappointing results appeared to be caused by a deviation from the original intentions of the reform, resulting from the textbooks' focus on individual tasks. It is suggested that this “task propensity”, together with a lack of attention for more advanced conceptual mathematical goals, constitutes a general barrier for mathematics education reform. This observation transcends the realm of textbooks, since more advanced conceptual mathematical understandings are underexposed as curriculum goals. It is argued that to foster successful reform, a conscious effort is needed to counteract task propensity and promote more advanced conceptual mathematical understandings as curriculum goals.
... Student proficiency regarding fractions is often problematic. In Grade 5 in the Netherlands, for example, many students find it difficult to solve problems related to improper fractions and mixed numbers or addition and subtraction including fractions, even though instruction has been provided to them about these topics (Bruin-Muurling 2010). ...
... , fractions(Bruin-Muurling, 2010), and algebra (Van Stiphout, 2011). Analysing the results of those three Ph.D. studies, Gravemei- jer, Bruin-Muurling, Kraemer and Van Stiphout ...
The gist of this article is that a shift is needed towards a mathematics curriculum in which teaching for understanding is the main objective. Before elaborating on what such a shift might entail, two compelling arguments for making such a change are presented. One is based on research, which showed that a one-sided emphasis on skills induced the teaching of isolated, topic-specific, skills-leading to a low level of proficiency. The other argument is grounded in the observation that the role of mathematics in society is changing and that, as a consequence thereof, the importance of mastering routine skills diminishes, while the need for mathematical understanding grows. On the basis of these two arguments, a shift is advocated, from an emphasis on skills, to an emphasis on understanding. This is connected to the thesis that deep mathematical understanding can only be achieved when students construct mathematical objects by reifying mathematical processes. For the domain of number, the idea of mathematical objects is linked to the notion of junctions in networks of number relations. The core of the article is an exploration of what mathematics education in the number domain might look like if it would be organized along the line of process-object transitions, and how number relations can be used for solving the kind of number tasks that are commonly solved with standard procedures. This exploration is closed with a sketch of a potential instructional sequence for addition and subtraction up to 100. Resumo. A essência deste artigo é que é necessária uma mudança para um currículo de matemática em que o objetivo principal é o ensino para a compreensão. Antes de discutir o que essa mudança pode acarretar, são apresentados dois argumentos convincentes para fundamentar essa mudança. Um é baseado na investigação, que mostra que a ênfase unilateral nas técnicas induz o ensino de técnicas isoladas, específicas de um tópico, levando a um baixo nível de proficiência. O outro Fostering process-object transitions… 7 Quadrante, Vol. 28, N.º 2, 2019 argumento fundamenta-se na observação de que o papel da Matemática na sociedade está a mudar e que, como consequência disso, a importância de dominar as técnicas rotineiras diminui, enquanto a necessidade de compreender a Matemática aumenta. Com base nestes dois argumentos, defende-se uma mudança, de uma ênfase nas técnicas rotineiras para uma ênfase na compreensão. Isto liga-se à tese de que uma compreensão matemática profunda só pode ser alcançada quando os alunos constroem objetos matemáticos reificando processos matemáticos. Para o domínio dos números e operações, a ideia de objetos matemáticos está ligada à noção de articulações em redes de relações numéricas. A parte central do artigo é uma exploração de como seria a educação matemática no domínio dos números e operações se fosse organizada ao longo da linha de transições de processo-objeto e de como as relações numéricas podem ser usadas para resolver o tipo de tarefas numéricas comummente resolvido com procedimentos padrão. Esta exploração termina com o esboço de uma potencial sequência de ensino para a adição e a subtração até 100.
Mathematics as a compulsory school subject was introduced in the Netherlands in the first decades of the 19th century. While in the beginning there was some involvement of Dutch academic mathematicians, later on their engagement with mathematics teaching was only marginal. That changed in the second half of the 20th century. Hans Freudenthal, professor of mathematics in Utrecht, became deeply involved in mathematics teaching. He became the first director of the IOWO, the Institute for the Development of Mathematics Education, that dominated Dutch mathematics teaching from the 1970s on. In the 1960s, under the influence of New Math, other mathematicians had already played a role in the modernisation of the teaching of mathematics, but from the 1970s on, their role became minimal again. In the first decade of the 21st century the dominance of the ideas of Realistic Mathematics Education elicited protests from mathematics departments at several universities. This criticism induced fierce and often heated debates. At the moment, these discussions have calmed down and it seems that a new understanding between the worlds of school and university mathematics is growing.
