In the Netherlands there is discussion about the best way to teach mathematics, especially in the case of primary school students. Being able to identify and understand pupils’ multiple problem solving strategies is one of the pillars of pedagogy. However, it is very demanding for teachers, since it requires to notice and analyze pupils’ mathematical thinking and to understanding their actions. The skill to notice and analyze a student’s mathematical thinking is usually not emphasized in Dutch primary school teacher training. It is important to find ways to help teacher-students to analyze student mathematical reasoning, and to learn to recognize the importance of such analysis. Sherin and van Es used the concept of video clubs to help teachers in US schools to notice and analyze their students’ mathematical thinking. In such video clubs, students jointly discuss their filmed lessons. This leads to the following research question:How can video clubs be used to teach students who are learning to become primary school teachers to analyze their pupils’ mathematical thinking and to learn to recognize the importance of such analysis?This paper describes a study that monitors a video club with four participants.
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Learning mathematical thinking and reasoning is a main goal in mathematical education. Instructional tasks have an important role in fostering this learning. We introduce a learning sequence to approach the topic of integrals in secondary education to support students mathematical reasoning while participating in collaborative dialogue about the integral-as-accumulation-function. This is based on the notion of accumulation in general and the notion of accumulative distance function in particular. Through a case-study methodology we investigate how this approach elicits 11th grade students’ mathematical thinking and reasoning. The results show that the integral-as-accumulation-function has potential, since the notions of accumulation and accumulative function can provide a strong intuition for mathematical reasoning and engage students in mathematical dialogue. Implications of these results for task design and further research are discussed.
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The implementation of the new mathematical knowledge base in Dutch teacher education institutes for primary education raises a need for curriculum development. Teacher educators have to raise student teachers’ subject matter knowledge to a higher level. In working on this aim teacher educators experience that student teachers often feel uncertain about their mathematical skills and are not very interested in formal and abstract mathematics. Student teachers prefer to focus on mathematical pedagogical content knowledge. This paper presents two design studies that try to tackle this problem. The first one targets the development of student teachers’ specialized content knowledge (SCK) and the second one focuses on their horizon content knowledge (HCK). Both studies target developing student teachers’ mathematical subject matter knowledge in the perspective of teaching mathematics in primary school. In the studies we established student teachers’ learning environments that kept them involved and motivated, even when they found the mathematics hard to do. Primarily, this attitude supported their mathematical growth, while it also developed their pedagogical skills and insight. INTRODUCTION
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A primary teacher needs mathematical problem solving ability. That is why Dutch student teachers have to show this ability in a nationwide mathematics test that contains many non-routine problems. Most student teachers prepare for this test by working on their own solving test-like problems. To what extent does these individual problem solving activities really contribute to their mathematical problem solving ability? Developing mathematical problem solving ability requires reflective mathematical behaviour. Student teachers need to mathematize and generalize problems and problem approaches, and evaluate heuristics and problem solving processes. This demands self-confidence, motivation, cognition and metacognition. To what extent do student teachers show reflective behaviour during mathematical self-study and how can we explain their study behaviour? In this study 97 student teachers from seven different teacher education institutes worked on ten non-routine problems. They were motivated because the test-like problems gave them an impression of the test and enabled them to investigate whether they were already prepared well enough. This study also shows that student teachers preparing for the test were not focused on developing their mathematical problem solving ability. They did not know that this was the goal to strive for and how to aim for it. They lacked self-confidence and knowledge to mathematize problems and problem approaches, and to evaluate the problem solving process. These results indicate that student teachers do hardly develop their mathematical problem solving ability in self-study situations. This leaves a question for future research: What do student teachers need to improve their mathematical self-study behaviour? EAPRIL Proceedings, November 29 – December 1, 2017, Hämeenlinna, Finland
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Explicit language objectives are included in the Swedish national curriculum for mathematics. The curriculum states that students should be given opportunities to develop the ability to formulate problems, use and analyse mathematical concepts and relationships between concepts, show and follow mathematical reasoning, and use mathematical expressions in discussions. Teachers’ competence forms a crucial link to bring an intended curriculum to a curriculum in action. This article investigates a professional development program, ‘Language in Mathematics’, within a national program for mathematics teachers in Sweden that aims at implementing the national curriculum into practice. Two specific aspects are examined: the selection of theoretical notions on language and mathematics and the choice of activities to relate selected theory to practice. From this examination, research on teacher learning in connection to professional development is proposed, which can contribute to a better understanding of teachers’ interpretation of integrated approaches to language and mathematics across national contexts.
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Citizens regularly search the Web to make informed decisions on daily life questions, like online purchases, but how they reason with the results is unknown. This reasoning involves engaging with data in ways that require statistical literacy, which is crucial for navigating contemporary data. However, many adults struggle to critically evaluate and interpret such data and make data-informed decisions. Existing literature provides limited insight into how citizens engage with web-sourced information. We investigated: How do adults reason statistically with web-search results to answer daily life questions? In this case study, we observed and interviewed three vocationally educated adults searching for products or mortgages. Unlike data producers, consumers handle pre-existing, often ambiguous data with unclear populations and no single dataset. Participants encountered unstructured (web links) and structured data (prices). We analysed their reasoning and the process of preparing data, which is part of data-ing. Key data-ing actions included judging relevance and trustworthiness of the data and using proxy variables when relevant data were missing (e.g., price for product quality). Participants’ statistical reasoning was mainly informal. For example, they reasoned about association but did not calculate a measure of it, nor assess underlying distributions. This study theoretically contributes to understanding data-ing and why contemporary data may necessitate updating the investigative cycle. As current education focuses mainly on producers’ tasks, we advocate including consumers’ tasks by using authentic contexts (e.g., music, environment, deferred payment) to promote data exploration, informal statistical reasoning, and critical web-search skills—including selecting and filtering information, identifying bias, and evaluating sources.
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An essential condition to use mathematics to solve problems is the ability to recognize, imagine and represent relations between quantities. In particular, covariational reasoning has been shown to be very challenging for students at all levels. The aim of the project Interactive Virtual Math (IVM) is to develop a visualization tool that supports students’ learning of covariation graphs. In this paper we present the initial development of the tool and we discuss its main features based on the results of one preliminary study and one exploratory study. The results suggest that the tool has potential to help students to engage in covariational reasoning by affording construction and explanation of different representations and comparison, relation and generalization of these ones. The results also point to the importance of developing tools that elicit and build upon students' self-productions
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Atherosclerosis is the development of lipid-laden plaques in arteries and is nowadays considered as an inflammatory disease. It has been shown that high doses of ionizing radiation, as used in radiotherapy, can increase the risk of development or progression of atherosclerosis. To elucidate the effects of radiation on atherosclerosis, we propose a mathematical model to describe radiation-promoted plaque evelopment. This model distinguishes itself from other models by combining plaque initiation and plaque growth, and by incorporating information from biological experiments. It is based on two consecutive processes: a probabilistic dose-dependent plaque initiation process, followed by deterministic plaque growth.
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In mathematics, sciences and economics, understanding and working with graphs are important skills. However, developing these skills has been shown to be a challenge in secondary and higher education as it involves high order thinking processes such as analysis, reflection and creativity. In this study, we present Interactive Virtual Math, a tool that supports the learning of a specific kind of graphs: dynamic graphs which represent the relation between at least two quantities that covary. The tool supports learners in visualizing abstract relations through enabling them to draw, move and modify graphs, and by combining graphs with other representations, especially interactive animations and textual explanations. This paper reports a design experiment about students’ learning graphs with this tool. Results show that students with difficulty in generating acceptable graphs improve their ability while working with the tool.
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Conducting a classroom dialogue for reasoning can be very challenging for teachers and, in particular for students-teachers. Research on classroom discussions provides examples of models that can be used to help teachers organize, analyse and conduct classroom discussions. One of these models is the five-practices framework (Stein, Engle, Smith, & Hughes, 2008). In this study we investigate how this framework has been applied in one course of mathematics' pedagogies for in-service student-teachers at the applied university of Amsterdam to support mathematics student-teachers in conducting classroom dialogue based on students reasoning. The study was conducted in the academic year 2017-18 and involved 15 in-service student-teachers and their teachers. The data concerns students written rapports to an individual assignment in which they were requested to use the five-practices in an hypothetical classroom discussion. The preliminary results confirm that the model can be useful for students-teachers to prepare themselves beforehand and to think about creating opportunities for dialogue to occur in the classroom. But, the practice of making connections during the classroom discussion remains misunderstood or superficially performed by the students teachers. These results suggest the need of a more fine grained description of the practice of connecting as ways to involve students-teachers in it.
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