To reach for abstraction is a major but challenging goal in mathematics education: teachers struggle with finding ways how to foster abstraction in their classes. To shed light on this issue for the case of geometry education, we align theoretical perspectives on embodied learning and abstraction with practical perspectives from in-service teachers. We focus on the teaching and learning of realistic geometry, not only because this domain is apt for sensori-motor action investigations, but also because abstraction in realistic geometry is under-researched in relation to other domains of mathematics, and teachers’ knowledge of geometry and confidence in teaching it lag behind. The following research question will be addressed: how can a theoretical embodied perspective on abstraction in geometry education in the higher grades of primary school inform current teacher practices? To answer this question, we carried out a literature study and an interview study with in-service teachers (n = 6). As a result of the literature study, we consider embodied abstraction in geometry as a process of reflecting on, describing, explaining, and structuring of sensory-motor actions in the experienced world through developing and using mathematical artifacts. The results from the interview study show that teachers are potentially prepared for using aspects of embodied learning (e.g., manipulatives), but are not aware of the different aspects of enactment that may invite students’ abstraction. We conclude that theories on embodiment and abstraction do not suffice to foster students’ abstraction process in geometry. Instead, teachers’ knowledge of embodied abstraction in geometry and how to foster this grows with experience in enactment, and with the discovery that cognition emerges to serve action.
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In this chapter, I look back at the implementation of W12-16, a major reform of mathematics education in the lower grades of general secondary education and pre-vocational secondary education in the Netherlands including all students aged 12–16. The nationwide implementation of W12-16 started in 1990 and envisioned a major change in what and how mathematics was taught and learned. The content was broadened from algebra and geometry to algebra, geometry and measurement, numeracy, and data processing and statistics. The learning trajectories and the instruction theory were based on the ideas of Realistic Mathematics Education (RME): the primary processes used in the classroom were to be guided re-invention and problem solving. ‘Ensuring usability’ in the title of this chapter refers to the aim of the content being useful and understandable for all students, but also to the involvement of all relevant stakeholders in the implementation project, including teachers, students, parents, editors, curriculum and assessment developers, teacher educators, publishers, media and policy makers. Finally, I reflect on the current state of affairs more than 20 years after the nationwide introduction. The main questions to be asked are: Have the goals been reached? Was the implementation successful?
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Nowadays, digital tools for mathematics education are sophisticated and widely available. These tools offer important opportunities, but also come with constraints. Some tools are hard to tailor by teachers, educational designers and researchers; their functionality has to be taken for granted. Other tools offer many possible educational applications, which require didactical choices. In both cases, one may experience a tension between a teacher’s didactical goals and the tool’s affordances. From the perspective of Realistic Mathematics Education (RME), this challenge concerns both guided reinvention and didactical phenomenology. In this chapter, this dialectic relationship will be addressed through the description of two particular cases of using digital tools in Dutch mathematics education: the introduction of the graphing calculator (GC), and the evolution of the online Digital Mathematics Environment (DME). From these two case descriptions, my conclusion is that students need to develop new techniques for using digital tools; techniques that interact with conceptual understanding. For teachers, it is important to be able to tailor the digital tool to their didactical intentions. From the perspective of RME, I conclude that its match with using digital technology is not self-evident. Guided reinvention may be challenged by the rigid character of the tools, and the phenomena that form the point of departure of the learning of mathematics may change in a technology-rich classroom.
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Numeracy and mathematics education in vocational education is under pressure to keep up with the rapid changes in the workplace due to developments in workplace mathematics and the ubiquitous availability of technological tools. Vocational education is a large stream in education for 12- to 20-years-olds in the Netherlands and the numeracy and mathematics curriculum is on the brink of a reform. To assess what is known from research on numeracy in vocational education, we are in the process of conducting a systematic review of the international scientific literature of the past five years to get an overview of the recent developments and to answer research questions on the developments in vocational educational practices. The work is still in progress. We will present preliminary and global results. We see vocational education from the perspective of (young) adults learning mathematics.
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In this chapter, we discuss the education of secondary school mathematics teachers in the Netherlands. There are different routes for qualifying as a secondary school mathematics teacher. These routes target different student teacher populations, ranging from those who have just graduated from high school to those who have already pursued a career outside education or working teachers who want to qualify for teaching in higher grades. After discussing the complex structure this leads to, we focus on the aspects that these different routes have in common. We point out typical characteristics of Dutch school mathematics and discuss the aims and challenges in teacher education that result from this. We give examples of different approaches used in Dutch teacher education, which we link to a particular model for designing vocational and professional learning environments.We end the chapter with a reflection on the current situation.
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Blended learning, a teaching format in which face-to-face and online learning is integrated, nowadays is an important development in education. Little is known, however, about its affordances for teacher education, and for domain specific didactical courses in particular. To investigate this topic, we carried out a design research project in which teacher educators engaged in a co-design process of developing and field-testing open online learning units for mathematics and science didactics. The preliminary results concern descriptions of the work processes by the design teams, of design heuristics, and of typical ways of collaborating. These findings are illustrated for the case of two of the designed online units on statistics didactics and mathematical thinking, respectively.
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Integrated curricula seem promising for the increase of attention on science and technology in primary education. A clear picture of the advantages and disadvantages of integration efforts could help curriculum innovation. This review has focussed on integrated curricula in primary education from 1994 to 2011. The integrated curricula were categorized according to a taxonomy of integration types synthesized from the literature. The characteristics that we deemed important were related to learning outcomes and success/fail factors. A focus group was formed to facilitate the process of analysis and to test tentative conclusions. We concluded that the levels in our taxonomy were linked to (a) student knowledge and skills, the enthusiasm generated among students and teachers, and the teacher commitment that was generated; and (b) the teacher commitment needed, the duration of the innovation effort, the volume and comprehensiveness of required teacher professional development, the necessary teacher support, and the effort needed to overcome tensions with standard curricula. Almost all projects were effective in increasing the time spent on science at school. Our model resolves Czerniac’s definition problem of integrating curricula in a productive manner, and it forms a practical basis for decision-making by making clear what is needed and what output can be expected when plans are being formulated to implement integrated education.
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In ancient times, music education was an essential part of mathematics education: in the Quadrivium mathematics, geometry, music and astronomy formed a four-unit whole. In China, among others, Confucius emphasized the importance of music education and the connection between poetry, law and music.
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This work draws on the Programme for the International Assessment of Adult Competencies (PIAAC) survey. Last year a first review was conducted on the PIAAC Numeracy Framework (Tout. et al., 2017). In 2018 and 2019 the framework for the second cycle of PIAAC will be developed. This second cycle of the PIAAC survey aims to update the data about the numeracy skills of adults in different countries around the World (Hoogland, Díez-Palomar, Maguire, 2019). The objective of this paper is to highlight some relevant findings from literature on the concept numeracy in order to discuss a potential enrichment of the PIAAC Numeracy Assessment Framework (NAF).
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We are well into the 21st century now and the urgency for lifelong learning is growing especially regarding numeracy. There are major societal and policy pressures on education to prepare citizens for a complex and technologized society, in literature referred to as “21st century skills” (Voogt & ParejaRoblin, 2012), “global competences” (OECD, 2016a) or “the 4th industrial revolution” (Schwab, 2016). International research has demonstrated the economic and social value of literacy and numeracy knowledge and skills (Hanushek and Wöbmann, 2012; Grotlüschen, et al. 2016). With respect to numeracy (and/or mathematics) education, we explore the implications of these pressures to the mathematical demands at individuals living and working in modern life, and what is expected from numeracy education as society moves further into the 21st century. New means of communication and types of services have changed the way individuals interact with governments, institutions, services and each other, and social and economic transformations have in turn, changed the nature of the demand for skills as well.
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