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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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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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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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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Presentation at the ALM28 Conference: Numeracy and Vulnerability, 5-7 july, Universität Hamburg, Germany.
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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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This paper reports on an experiment comparing students’ results on image-rich numeracy problems and on equivalent word problems. Given the well reported problematic nature of word problems, the hypothesis is that students score better on image-rich numeracy problems than on comparable word problems. To test the hypothesis a randomized controlled trial was conducted with 31,842 students from primary, secondary, and vocational education. The trial consisted of 21 numeracy problems in two versions: word problems and image-rich problems. The hypothesis was confirmed for the problems used in this experiment. With the insights gained we intend to improve the assessment of students’ abilities in solving quantitative problems from daily life. Numeracy, word problem, image-rich problem, randomized controlled trial, assessment
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Leerkrachten die hun leerlingen willen ontwikkelen tot reken-wiskundige probleemoplossers moeten zelf ook goede probleemoplossers zijn. Om toekomstige leraren basisonderwijs, aan dit doel te laten werken is het van belang dat ze non-routine reken-wiskundige problemen oplossen en bovendien daarbij reflecteren op hun rekenwerk.
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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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One of the core elements of the Programme for the International Assessment of Adult Competencies (PIAAC) is a survey of adult skills. The survey measures adults’ proficiency in literacy, numeracy, and problem solving in technology-rich environments. Furthermore, the survey gathers data on how adults use their skills at home, at work and in the wider community. The second cycle of PIAAC will take place in 2021 and 2022. Preparations have started by the Numeracy Expert Group in reviewing the numeracy framework used in the first cycle and designing items which will be used to measure the numerate capabilities and numerate behavior of adults.
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