We investigated whether Early Algebra lessons that explicitly aimed to elicit mathematical discussions (Shift-Problem Lessons) invoke more and qualitatively better mathematical discussions and raise students’ mathematical levels more than conventional lessons in a small group setting. A quasi-experimental study (pre- and post-test, control group) was conducted in 6 seventh-grade classes (N =160). An analysis of the interaction processes of five student groups showed that more mathematical discussions occurred in the Shift-Problem condition. The quality of the mathematical discussions in the Shift-Problem condition was better compared to that in the Conventional Textbook condition, but there is still more room for improvement. A qualitative illustration of two typical mathematical discussions in the Shift-Problem condition are provided. Although students’ mathematical levels were raised a fair amount in both conditions, no differences between conditions were found. We concluded that Shift-Problem Lessons are powerful for eliciting mathematical discussions in seventh-grade Shift-Problem Early Algebra Lessons.
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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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This textbook is intended for a basic course in problem solving and program design needed by scientists and engineers using the TI-92. The TI-92 is an extremely powerful problem solving tool that can help you manage complicated problems quickly. We assume no prior knowledge of computers or programming, and for most of its material, high school algebra is sufficient mathematica background. It is advised that you have basic skills in using the TI-92. After the course you will become familiar with many of the programming commands and functions of the TI-92. The connection between good problem solving skills and an effective program design method, is used and applied consistently to most examples and problems in the text. We also introduce many of the programming commands and functions of the TI-92 needed to solve these problems. Each chapter ends with a number of practica problems that require analysis of programs as well as short programming exercises.
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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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Individuals with autism increasingly enroll in universities, but little is known about predictors for their success. This study developed predictive models for the academic success of autistic bachelor students (N=101) in comparison to students with other health conditions (N=2465) and students with no health conditions (N=25,077). We applied propensity score weighting to balance outcomes. The research showed that autistic students’ academic success was predictable, and these predictions were more accurate than predictions of their peers’ success. For first-year success, study choice issues were the most important predictors (parallel program and application timing). Issues with participation in pre-education (missingness of grades in pre-educational records) and delays at the beginning of autistic students’ studies (reflected in age) were the most influential predictors for the second-year success and delays in the second and final year of their bachelor’s program. In addition, academic performance (average grades) was the strongest predictor for degree completion in 3 years. These insights can enable universities to develop tailored support for autistic students. Using early warning signals from administrative data, institutions can lower dropout risk and increase degree completion for autistic students.
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Openbare les Prof. Dr. Paul Drijvers, 11 oktober 2018
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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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This paper explores the contributions of research to the field of adults learning mathematics (ALM) in the last twenty years. The results of the review of the literature on ALM show that the most cited studies that have been published in the last twenty years tend to focus on the field of numeracy to understand health data (such as understanding how to dose a medicine in a medical treatment). However, we know little about key aspects of how adults learn mathematics, what obstacles they encounter, and how they overcome them. This paper identifies the main gaps that ALM research faces in the coming years.
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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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Reading and writing is modelled in CSP using actions containing the symbols ? and !. These reading actions and writing actions are synchronous, and there is a one-to-one relationship between occurrences of pairs of these actions. In the CPA conference 2016, we introduced the half-synchronous alphabetised parallel operator X ⇓ Y , which disconnects the writing to and reading from a channel in time. We introduce in this paper an extension of X ⇓ Y , where the definition of X ⇓ Y is relaxed; the reading processes are divided into sets which are set-wise asynchronous, but intra-set-wise synchronous, giving full flexibility to the asynchronous writes and reads. Furthermore, we allow multiple writers to the same channel and we study the impact on a Vertex Removing Synchronised Product. The advantages we accomplish are that the extension of X ⇓ Y gives more flexibility by indexing the reading actions and allowing multiple write actions to the same channel. Furthermore, the extension of X ⇓Y reduces the end-to-end processing time of the processor or coprocessor in a distributed computing system. We show the effects of these advantages in a case study describing a Controlled Emergency Stop for a processor-coprocessor combination.
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