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  • 1. Alfredsson, Lena
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Erixon, Patirk
    Heikne, Hans
    Matematik 5000, kurs 52013Book (Other (popular science, discussion, etc.))
  • 2. Alfredsson, Lena
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Erixon, Patrik
    Heikne, Hans
    Matematik 5000, kurs 1b2011Book (Other (popular science, discussion, etc.))
  • 3. Alfredsson, Lena
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Erixon, Patrik
    Heikne, Hans
    Matematik 5000, kurs 2c2012Book (Other (popular science, discussion, etc.))
  • 4. Alfredsson, Lena
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Erixon, Patrik
    Heikne, Hans
    Matematik 5000: kurs 3c, läromedel 2012Book (Other (popular science, discussion, etc.))
  • 5. Alfredsson, Lena
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Erixon, Patrik
    Heikne, Hans
    Matematik 5000, kurs 42013Book (Other (popular science, discussion, etc.))
  • 6.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    A new look at E.G. Björling and the Cauchy sum theorem2007In: Archive for History of Exact Sciences, ISSN 0003-9519, E-ISSN 1432-0657, Vol. 61, no 5, p. 519-535Article in journal (Refereed)
    Abstract [en]

    We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

  • 7.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    A Study of the Development of Concepts in Mathematics2004Licentiate thesis, comprehensive summary (Other academic)
  • 8.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Ambiguities of fundamental concepts in mathematical analysis during the mid 19th century2012In: Foundations of Science, ISSN 1233-1821, E-ISSN 1572-8471, Vol. 17, no 4, p. 301-320Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke's (Hist Math 20(3):265-284, 1993) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Bjorling's general view of real- and complexvalued functions. We argue that Bjorling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is how Bjorling interprets Cauchy's definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Bjorling (Kongl Vetens Akad Forh Stockholm 166-228, 1852) we consider Cauchy's theorem on power series expansions of complex valued functions. We investigate Bjorling's, Cauchy's and the Belgian mathematician Lamarle's different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy's theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Bjorling, Cauchy and Lamarle.

  • 9.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Development of school algebra - a comparison between the 1980 and 2011 Swedish mathematics curricula2019In: Proceedings of the Eighth European Summer University on History and Epistemology in Mathematics Education ESU 8 / [ed] U.T Jankvist, T. Tzanakis & B. Smestad, Oslo: OsloMet , 2019, p. 711-725Conference paper (Refereed)
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  • 10.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    E.G Björling's version of the Cauchy sum theorem2004In: HPM 2004 & ESU 4, Uppsala / [ed] F. Furinghetti, S. Kaijser & T. Tzanakis, Uppsala: Uppsala universitet, 2004, p. 48-52Conference paper (Refereed)
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  • 11.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    E.G. Björling's work with convergence of infinite function series in view of prose and calculus2015In: Proceedings of the 7th European Summer University of History and Epistemology of Mathematics Education (ESU 7). 14-18 July, Aarhus University, Denmark / [ed] E. Barbin, U.T Jankvist & T. Hoff Kjeldsen, 2015, p. 807-820Conference paper (Refereed)
  • 12.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Malmsten's proof of the integral theorem2002In: Study the masters: The Abel-Fauvel conference in Kristianssand, Norway. Proceedings of a Nordic pre-conference to ICMI 10 and the HPM satellite in 2004 / [ed] O. B. Bekken & R. Mosvold, Göteborg: Nationellt centrum för matematikutbildning (NCM), 2002, p. 135-145Conference paper (Refereed)
  • 13.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Studies in the Conceptual Development of Mathematical Analysis2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This dissertation deals with the development of mathematical concepts from a historical and didactical perspective. In particular, the development of concepts in mathematical analysis during the 19th century is considered. The thesis consists of a summary and three papers. In the first paper we investigate the Swedish mathematician E.G. Björling's contribution to uniform convergence in connection with Cauchy's sum theorem from 1821. In connection to Björling's convergence theory we discuss some modern interpretations of Cauchy's expression x=1/n. We also consider Björling's convergence conditions in view of Grattan-Guinness distinction between history and heritage. In the second paper we study visualizations in mathematics from historical and didactical perspectives. We consider some historical debates regarding the role of intuition and visual thinking in mathematics. We also consider the problem of what a visualization in mathematics can achieve in learning situations. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. In the third paper we consider Cauchy's theorem on power series expansions of complex valued functions on the basis of a paper written by E.G. Björling in 1852. We discuss Björling's, Lamarle's and Cauchy's different conditions for expanding a complex valued function in a power seris. In the third paper we also discuss the problem of the ambiguites of fundamental concpets that existed during the mid-19th century. We argue that Cauchy's and Lamarle's proofs of Cauchy's theorem on power series expansions of complex valued functions are correct on the basis of their own definitions of the fundamental concepts involved.

     

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  • 14.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Teaching traditions in Swedish school algebra - a project description2015In: "Dig where you stand" 3, Proceedings of the Third International Conference on the History of Mathematics Education, September 25-28, 2013, at Department of Education, Uppsala University, Sweden / [ed] Bjarnadottir, K.,Furinghetti, F., Prytz, J. & Schubring, G., Uppsala: Uppsala universitet, 2015, p. 81-92Conference paper (Refereed)
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  • 15.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Tolka visualiseringar2010In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 4, p. 29-33Article in journal (Other (popular science, discussion, etc.))
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  • 16.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Visualizations and intuitive reasoning in mathematics2012In: The Montana Mathematics Enthusiast, ISSN 1551-3440, E-ISSN 1551-3440, ISSN 1551-3440, Vol. 9, no 1&2, p. 1-18Article in journal (Refereed)
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  • 17.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Hemmi, Kirsti
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education. Åbo Akademi.
    Madej, Lars
    Åbo Akademi.
    Teoretiska och praktiska perspektiv på generaliserad aritmetik2018In: Perspectives on professional development of mathematics teachers. Proceedings of MADIF11: the eleventh research seminar of the Swedish Society for Research of Mathematics Education / [ed] J. Häggström, Y. Liljekvist, J. Bergman Ärlebäck, M. Fahlgren & O. Olande, Göteborg: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2018, p. 27-36Conference paper (Refereed)
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  • 18.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Hemmi, Kirsti
    Åbo Akademi University.
    Madej, Lars
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Röj-Lindberg, Ann-Sofi
    Åbo Akademi University.
    Towards research-based teaching of algebra - analyzing excpected student progression in the Swedish curriculum grades 1-92016Conference paper (Refereed)
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  • 19.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Kallem, Nicholas
    USA.
    Sriraman, Bharath
    The University of Montana.
    The didactical nature of some lesser known historical examples in mathematics2011In: Crossroads in the History of Mathematics and Mathematics Education / [ed] Bharath Sriraman, Charlotte, North Carolina: Information Age Publishing , 2011, p. 91-114Chapter in book (Other academic)
  • 20.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Madej, Lars
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Generaliserad aritmetik - en bro mellan aritmetik och algebra2017In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 4, p. 3-8Article in journal (Other academic)
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  • 21.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Madej, Lars
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Hemmi, Kirsti
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education. Åbo Akademi University, Åbo, Finland.
    Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks2019In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 24, no 1, p. 27-49Article in journal (Refereed)
    Abstract [en]

    In search of the reasons for Swedish students’ low achievement in algebra in inter-national and national evaluations, we investigate how the development of algebraic thinking is addressed in the Swedish national mathematics curriculum and two widely used mathematics textbook series for grades 1–6 in Sweden. The analytical tool used is based on the classification of ”big ideas” which research has shown as important for developing pupils’ algebraic understanding in early school grades. The results show that functional thinking, expressions, and equations are well represented topics both in the curriculum and the textbooks; however generalized arithmetic is a topic that is poorly developed in both the curriculum and the textbooks.

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  • 22.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Pejlare, Johanna
    On the relations between historical epistemology and students’ conceptual developments in mathematics2015In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 89, no 2, p. 251-265Article in journal (Refereed)
  • 23.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Pejlare, Johanna
    The role of Swedish school algebra in a historical perspective2019In: NERA 2019 Education in a globalized world, Uppsala, 2019, p. 279-280Conference paper (Refereed)
    Abstract [en]

    Solid knowledge of algebra is vital to manage mathematics at upper secondary and university level. Traditionally, algebra has been postponed until adolescence because of former assumptions that young children would not be cognitively capable of thinking algebraically. However, recent research reveals that it is possible and even beneficial to start working with algebra already in early grades (Blanton et al., 2015). This has influenced school mathematics where many countries have revised their syllabuses in order to incorporate algebra in primary school.In Sweden algebra is a part of mathematics that causes pupils major difficulties. In international evaluations, Swedish pupils have performed below the international average in algebra since the 1960s. Although there have been various attempts to improve school algebra teaching the results in algebra have not improved. The overall purpose of the present study is to contribute to the international research field regarding the complex issue of implementing algebra in early school mathematics by investigating the Swedish case. More specifically, we examine how algebra is traditionally treated in the last five Swedish syllabuses for grades 1–9 from 1962, 1969, 1980, 1994, and 2011. The study is part of a broader research project aiming at characterizing Swedish school algebra on both formulation and realization arenas (Hemmi et al., 2018). The project is theoretically embedded in Bernstein’s theory about classification and framing of educational knowledge.In order to characterize the algebraic content as well as to investigate what role algebra plays in school mathematics we have conducted a qualitative content analysis where Blanton et al.’s (2015) five big ideas of algebra have been applied as an analytical tool. The big ideas are: Expressions and equations, Generalized arithmetic, Functional thinking, Variables, and Proportional reasoning.The results show both similarities and differences between the syllabuses. For instance, in the 1980 syllabus algebra represents a very small part of the mathematical content, especially compared with the 2011 syllabus where algebra is emphasized already from earlier grades. All five syllabuses emphasize the importance of everyday mathematics and the practical use of mathematics in contexts relevant for the students. However, there are differences regarding which role algebra plays in everyday life. The 1980 syllabus states that algebra is less important in everyday life and students only need a “certain orientation” of algebra, which is probably a reaction to the great focus on abstract mathematics of “New math” in the 1969 syllabus. In the 2011 syllabus everyday life appears frequently within the algebraic content. A common feature of all five syllabuses is the weak emphasis on the big idea generalized arithmetic.

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  • 24.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Pejlare, Johanna
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Visualizations in mathematics2008In: Erkenntnis, ISSN 0165-0106, E-ISSN 1572-8420, Vol. 68, no 3, p. 345-358Article in journal (Refereed)
  • 25. Bråting, Kajsa
    et al.
    Sollervall, Håkan
    Stadler, Erika
    Algebra för lärare2017Book (Other academic)
  • 26.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Sollervall, Håkan
    Stadler, Erika
    Geometri för lärare2013Book (Other (popular science, discussion, etc.))
  • 27.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Sollervall, Håkan
    Stadler, Erika
    Sannolikhet och statistik för lärare2019Book (Other academic)
  • 28.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Arts, Department of Philosophy, Logic and Metaphysics.
    Om matematiska begrepp: en filosofisk undersökning med tillämpningar2005In: Filosofisk Tidskrift, ISSN 0348-7482, Vol. 26, no 4, p. 11-17Article in journal (Other academic)
  • 29.
    Bråting, Kajsa
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Österman, Tove
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Arts, Department of Philosophy, Logic and Metaphysics.
    John Dewey and mathematics education in Sweden2017In: "Dig where you stand" 4. Proceedings of the Fourth International Conference of the History of Mathematics Education, Roma: Edizioni Nuova Cultura , 2017, p. 61-72Conference paper (Refereed)
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  • 30.
    Dyrvold, Anneli
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Exploring teaching traditions in mathematics2019In: NERA 2019 Education in a globalized world, 6 - 8 March, 2019, Uppsala, Sweden, Abstact book 2019-03-06, Uppsala University, 2019, p. 269-270Conference paper (Refereed)
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  • 31.
    Hemmi, Kirsti
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Lepik, Madis
    Curricular approaches to algebra in Estonia, Finland and Sweden – a comparative study2020In: Mathematical Thinking and Learning, ISSN 1098-6065, E-ISSN 1532-7833Article in journal (Refereed)
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  • 32.
    Hemmi, Kirsti
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Lepik, Madis
    Madej, Lars
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Smedlund, Joakim
    Introduction to early algebra in Estonia, Finland and Sweden - some distinctive features identified in textbooks for Grades 1-32019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. / [ed] U.T Jankvist, T. Tzanakis & B. Smestad, Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute and ERME , 2019, p. 2039-2046Conference paper (Refereed)
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  • 33.
    Kilhamn, Cecilia
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education. Uppsala University.
    Algebraic thinking in the shadow of programming2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education / [ed] U.T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis, Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University , 2019, p. 566-573Conference paper (Refereed)
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  • 34. Pejlare, Johanna
    et al.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education.
    Writing the history of mathematics: Interpretations of the mathematics of the past and its relation to the mathematics of today2019In: Mathematics of the Arts and Sciences / [ed] B. Sriraman, Springer, 2019Chapter in book (Refereed)
  • 35.
    Österman, Tove
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Arts, Department of Philosophy, Logic and Metaphysics.
    Bråting, Kajsa
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Educational Sciences, Department of Education. Uppsala Univ, Postgrad Studies, Uppsala, Sweden.
    Dewey and mathematical practice: revisiting the distinction between procedural and conceptual knowledge2019In: Journal of Curriculum Studies, ISSN 0022-0272, E-ISSN 1366-5839, Vol. 51, no 4, p. 457-470Article in journal (Refereed)
    Abstract [en]

    We identify a recent trend in school mathematics as well as in some of the research literature in mathematics education: an emphasis on the practical uses of mathematics and an increased emphasis on verbalizations as opposed to numerical and computational skills. With tools provided by John Dewey, an early advocate of contextual and practical knowledge, we analyse the common research framework for discussing mathematical knowledge in terms of the procedural and the conceptual. We argue that procedural and conceptual knowledge should not be seen as opposites, and that the tendency to treat them as such might be avoided by emphasising the notion of operational skill. We argue that this is important in order for the students to gain both the contextual knowledge and the computational skill entailed in mathematical knowledge.

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