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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Torpedo is a digital learning environment for developing mathematical problem-solving ability through self-study for pre-service teachers in primary teacher education. To achieve this, Torpedo supports and challenges pre-service teachers’ reflection during and after solving non-routine mathematics problems. To investigate the feasibility of the Torpedo approach, 271 pre-service teachers used Torpedo during one month in a pilot study. They used and evaluated Torpedo’s reflective elements differently. The results varied from pre-service teachers who experienced that reflection really contributed to the development of their problem-solving ability, to pre-service teachers who hardly reflected. The last group consisted of those who found the problems too difficult to reflect upon and those who used Torpedo to prepare for the National Mathematics Test and preferred to do so by drill and practice. As a conclusion, the study provides clues for improving Torpedo so that it invites more reflective self-study behaviour. For pre-service teachers who consider reflection valueless, however, self-study in a digital learning environment may be insufficient to change this attitude.
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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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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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The aim of this study is to contribute to the body of knowledge on the use of contextual mathematical problems. Word problems are a predominant genre in mathematics classrooms in assessing students’ ability to solve problems from everyday life. Research on word problems, however, reveals a range of difficulties in their use in mathematics education. In our research we took an alternative approach: we designed image-rich numeracy problems as alternatives for word problems. A set of word problems was modified by systematically replacing the descriptive representation of the problem situation by a more depictive representation and an instrument was designed to measure the effect of this modification on students’ performance. The instrument can measure the effect of this alternative approach in a randomized controlled trial. In order to use the instrument at scale, we made this instrument also usable as a diagnostic test for an upcoming nationwide examination on numeracy. In this article we explain and discuss the design of the instrument and the validation of its intended uses.
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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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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 this paper we describe values in mathematics education in the Netherlands, in a context where both government and teachers are merely focused on test results. At the same time, newly proposed core goals emphasise mathematics education’s socializing function. These new core goals also focus on mathematical attitudes and relating mathematics and the world outside school. This paper describes an exploration to come to a model that can be used to both clarify the role of values in mathematics education and can be used to design education that aligns with this value-oriented character. We present the development of this model as it evolved in discussions with experts such as teacher educators, mathematics specialists, and primary school teachers. It developed from a linear model, via an adapted version making it too complex, to a model that is more focused on the relation between the main elements. The final model represents relations between ‘mathematics’, ‘values’, and ‘the world’ and names the connections between these aspects ‘mathematical literacy’, ‘citizenship’, and ‘value-based mathematics’. By developing this model, we experienced that its use shifted from a descriptive model, via a model that helps communicating about value based mathematics to a mental model. The final model can be used for future design research, where the first step will be establishing the present status of values in mathematics education.
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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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