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  • 1.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    A geometric evolution problem2002Inngår i: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 60, nr 1, s. 37-73Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A traditional approach to compression moulding of polymers involves the study of a generalized Hele-Shaw flow of a power-law fluid, and leads to the p-Poisson equation for the instantaneous pressure in the fluid. By studying the convex dual of an equivalent extremal problem, one may let the power-law index of the fluid tend to zero. The solution of the resulting extremal problem, referred to as the asymptotically dual problem, is known to have the property that the flow is always directed towards the closest point on the boundary. In this paper we use this property to introduce the concept of boundary velocity in the case of piecewise C2 domains with only convex corners, and we also give an explicit solution to the asymptotically dual problem in this case. This involves the study of certain topological properties of the ridge of planar domains.With use of the boundary velocity, we define a geometric evolution problem and the concept of classical solutions of it. We prove a uniqueness theorem and use a comparison principle to study the persistence of corners. We actually estimate "waiting times" for corners, in terms of geometric quantities of the initial domain.

  • 2.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    A geometric evolution problem arising in an asymptotic approach to compression moulding1998Licentiatavhandling, monografi (Annet vitenskapelig)
  • 3.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Conceptualizing reasoning-and-proving opportunities in textbook expositions: Cases from secondary calculus2017Inngår i: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1-5, 2017) / [ed] Dooley, T., & Gueudet, G., Dublin, Ireland: European Society for Research in Mathematics Education , 2017, s. 91-98Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Several recent textbook studies focus on opportunities to learn reasoning-and-proving. They typically investigate the extent to which justifications are general proofs and what opportunities exist for learning important elements of mathematical reasoning. In this paper, I discuss how a particular analytical framework for this might be refined. Based on an in-depth analysis of certain textbook passages in upper secondary calculus textbooks, I make an account for analytical issues encountered during this process and identify aspects of reasoning-and-proving in textbooks that might be missed unless the framework is refined. Among them there are characterizations of generality, use of different representations, logical and mathematical structure, and ordering of material and student activities. Finally, implications beyond textbook research are discussed.

    Fulltekst (pdf)
    Conceptualizing reasoning-and-proving opportunities in textbook expositions: Cases from secondary calculus
  • 4.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    On a generality framework for proving tasks2015Inngår i: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer; Nada Vondrova, Prague: Charles University in Prague, Faculty of Education and ERME , 2015, s. 86-92Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper I present an analytic framework for generality in textbook proving tasks that involve functions. The framework is discussed in relation to results obtained when analysing tasks in integral calculus. The results show that the frameworks’ categories are easily distinguishable if the functions are explicitly described. The results are also promising regarding the possibility to clarify differences between textbooks. The analysed sections exemplify that there is not necessarily a correlation between the number of general proving tasks and the opportunities for students to engage in reasoning about arbitrary functions. Limitations and possible refinements of the framework are also discussed.

  • 5.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Proof-related reasoning in upper secondary school: characteristics of Swedish and Finnish textbooksInngår i: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Despite the central role of proofs in mathematics, research often shows that school textbooks offer limited support for the teaching and learning of proof-related reasoning. This study contributes to this field of research by studying Swedish and Finnish upper secondary textbooks on logarithms and combinatorics. Justifications in expository sections are analysed and students' tasks are categorized according to the type and nature of reasoning they require. The findings imply that opportunities to learn proof-related reasoning are few, and are more oriented towards deductive reasoning in Finnish textbooks and towards empirical reasoning and conjecturing in Swedish textbooks. The results are discussed in relation to similar studies from both Scandinavian and United States contexts, and address future research and development of the theoretical framing of proof-related reasoning.

  • 6.
    Bergwall, Andreas
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Hemmi, Kirsti
    Åbo Akademi University, Åbo, Finland; Uppsala University, Uppsala, Sweden.
    The State of Proof in Finnish and Swedish Mathematics Textbooks: Capturing Differences in Approaches to Upper-Secondary Integral Calculus2017Inngår i: Mathematical Thinking and Learning, ISSN 1098-6065, E-ISSN 1532-7833, Vol. 19, nr 1, s. 1-18Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish text-books than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.

  • 7.
    Bergwall, Andreas
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Knutsson, Malin
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Students’ views on mathematical challenges in working as a teacher2014Inngår i: The Seventh Nordic Conference onMathematics Education, NORMA 14 Turku, June 3–6, 2014: PROGRAMME AND ABSTRACTS / [ed] Harry Silfverberg, 2014Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Many frameworks for teacher knowledge describe the work of a mathematics teacher as highly complex and therefore requires specialized competences (e.g. Rowland & Ruthven, 2011). At the same time, countries around the world have difficulties in recruiting highly qualified students to teacher education. Furthermore, teacher educators often express that prospective teachers question the amount and level of mathematics that they have to learn. Research on views about teaching often focuses on teachers’ views (Philipp, 2007). Less is known about students’ views about working as a teacher and how those might influence their choice of a profession. The focus of this study is on the views that students have on mathematical challenges associated with the work of a mathematics teacher, when compared to other occupations.

  • 8.
    Knutsson, Malin
    et al.
    Örebro universitet, Institutionen för naturvetenskap och teknik. Akademin för utbildning, kultur och kommunikation, Mälardalens högskola, Västerås, Sweden.
    Hemmi, Kirsti
    Akademin för utbildning, kultur och kommunikation, Mälardalens högskola, Västerås, Sweden.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik. Akademin för utbildning, kultur och kommunikation, Mälardalens högskola, Västerås, Sweden.
    Ryve, Andreas
    Umeå University, Umeå, Sweden.
    School-based mathematics teacher education in Sweden and Finland: characterizing mentor – prospective teacher discourse2013Inngår i: The Eighth Congress of the European Society for Research in Mathematics Education. Feb 6th - Feb 10th, 2013, Antalya, Turkey, 2013Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Despite many similarities between the neighbouring countries Sweden and Finland, prior studies indicate that conceptualizations and discourses about school-based teacher education are very different. In this paper we add to this picture of differences, and contribute to the research discourse about school-based teacher education, by identifying and characterizing aspects of mathematics teaching made relevant in review meetings between mentors and prospective primary teachers. While the Swedish discourse typically focuses on the students’ individual work with textbooks, connections to everyday experiences and teaching as individual supervision, the Finnish discourse emphasizes lesson aims, learning and progression in mathematics through formative assessment and differentiation according to pupils’ abilities

    Fulltekst (pdf)
    fulltext
  • 9.
    Ryve, Andreas
    et al.
    Mälardalen University, Västerås, Sweden.
    Nilsson, Per
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Palm, Torulf
    Umeå University, Umeå, Sweden.
    Van Steenbrugge, Hendrik
    Mälardalen University, Västerås, Sweden.
    Andersson, Catarina
    Umeå University, Umeå, Sweden.
    Bergwall, Andreas
    Örebro universitet, Institutionen för naturvetenskap och teknik.
    Boström, Erika
    Umeå University, Umeå, Sweden.
    Larsson, Maria
    Mälardalen University, Västerås, Sweden.
    Vingsle, Lotta
    Umeå University, Umeå, Sweden.
    Kartläggning av forskning om formativ bedömning, klassrumsundervisning och läromedel i matematik: Delrapport från skolforsk-projektet2015Rapport (Annet vitenskapelig)
    Abstract [en]

    The current project focuses on mathematics education, and is partitioned into three subprojects mapping research on formative assessment, classroom teaching, and curriculum programs in mathematics. The rationale for focusing on these three areas is that they are all highly relevant for understanding and improving Swedish mathematics education and students’ knowing of mathematics. Therefore, the aim of the project is to map research on formative assessment, classroom teaching, and curriculum programs in mathematics education.

    The methodology of the literature review has been inspired by Gough, Oliver, and Thomas (2013), and we have focused on the mapping on journal articles published on Web of Science (WoS).

    The results from the sample of articles on formative assessment show that strategies of formative assessment in mathematics are positively correlated to students’ performance in mathematics with medium and large effect sizes. However, based on the current mapping it is difficult to specify aspects of how the formative strategies are to be implemented in order to promote students’ knowing of mathematics.

    Despite the change in perspective of what constitutes knowledge in mathematics to also include reasoning, problem-solving and communication, the map shows that research is mainly focused on examining teaching methods and their effects on students’ skills in mathematics. A closer examination of the studies that do focus on teaching for supporting students in developing competencies like reasoning and problem-solving shows that connections between and comparison of students’ solutions, as well as teachers’ ways of asking questions to support students in explaining their solutions clearly and in detail, are important for students’ learning of these competencies.

    A central finding stemming from this review of curriculum programs is the complexity involved in how the programs can support teachers in establishing classroom practices. Curriculum resources and teacher resources, as well as other influencing factors, impact the quality of instruction, and studies have begun to point out how curriculum resources and teacher resources uniquely and jointly impact classroom practices. Multiple research articles have expressed the need for teacher support in implementing curriculum programs, by means of professional development, teacher education and support provided by the curriculum programs themselves. Interesting in this regard is the state of the research field concerning the design of educative curriculum programs, and how teachers make use of such support. Studies have proposed design approaches, regarding both the actual development of educative curriculum programs as well as how to use them in teacher education to support prospective teachers’ development of knowledge. Further, although research has revealed that it is important to prepare for teaching in certain ways, we found very little research that explicitly analyzed how teachers actually prepare for teaching a mathematics lesson.

    Limitations of the project include: (1) the lack of searching in potentially relevant databases, (2) the fact that a relatively small proportion of articles found in the search have been coded, (3) that we have not engaged in deep considerations as to whether and in what ways results from international research are relevant in the Swedish context, and (4) that we therefore have not been able to synthesize the results of the study. In relation to the Swedish context (Hemmi & Ryve, 2014; Boesen et al., 2014), international research (Hattie, 2009; Smith & Stein, 2011), and the current project’s findings, we recommend that Skolforskningsinstitutet focus on two aspects of great importance for developing students’ knowing of mathematics. First, Skolforskningsinstitutet should synthesize research that supports actors, such as teachers and principals, in acting within school practices. In the case of teachers, support is needed to engage them in actively anticipating students’ thinking, using curriculum programs effectively, introducing mathematical content, acting in group work, formatively assessing students’ learning, and orchestrating whole-class mathematical discussions. Secondly, actors within school practices need support not only in initiating and implementing developments but also in institutionalizing such developments. Skolforskningsinstitutet should specify the kind of support needed in order to ensure that material, routines, competences, and organizations become integral and permanent features of Swedish school practice. 

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