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  • 1. Breen, Sinéad
    et al.
    Larson, Niclas
    O'Shea, Ann
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    A study of students’ concept images of inverse functions in Ireland and Sweden2017In: Nordisk matematikkdidaktikk, NOMAD: [Nordic Studies in Mathematics Education], ISSN 1104-2176, Vol. 22, no 4, p. 85-102Article in journal (Refereed)
    Abstract [en]

    In this paper we focus on first-year university students’ conceptions of inverse function. We present results from two projects, conducted in Ireland and Sweden respectively. In both countries, data were collected through questionnaires, as well as through student interviews in Sweden. We draw on the notion of concept image and describe the components of students’ evoked concept images. The students’ responses involved e.g. ”reflection”, ”reverse”, and concrete ”examples”, while just a few students gave explanations relating to the definition of inverse functions. We found that the conceptions of inverses as reflections and reverse processes are important and relatively independent of local factors, and the data seemed to suggest that a ”reverse” conception is linked to an appreciation of injectivity more than a ”reflection” conception.

  • 2.
    Bälter, Olle
    et al.
    KTH Royal Institute of Technology.
    Cleveland-Innes, Martha
    Athabasca University.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Scheja, Max
    Stockholm University, Faculty of Social Sciences, Department of Education.
    Svedin, Maria
    KTH Royal Institute of Technology.
    Student approaches to learning in relation to online course completion2013In: Canadian Journal of Higher Education, ISSN 0316-1218, Vol. 43, no 3, p. 1-18Article in journal (Refereed)
    Abstract [en]

    This study investigates the relationship between approaches to studying and course completion in two online preparatory university courses in mathematics and computer programming. The students participating in the two courses are alike in age, gender, and approaches to learning. Four hundred and ninety-three students participating in these courses answered the short version of the Approaches and Study Skills Inventory for Students (ASSIST). Results show that students demonstrating a deep approach to learning in either course are more likely to complete. In the mathematics  course, a combination of deep and strategic approaches correlates positively with course completion. In the programming course, students who demonstrate a surface approach are less likely to complete. These results are in line with the intentions of the course designers, but they also suggest ways to improve these courses. Furthermore, the study demonstrates that ASSIST can be used to evaluate course design.

  • 3.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Discerning multiplicative and additive reasoning in co-variation problems2015In: Proceeding of ICMI STUDY 23: primary mathematics study on whole numers / [ed] Xuhua Sun, Berinderjeet Kaur, Jarmila Novotná, Macao: University of Macao , 2015, p. 559-566Conference paper (Refereed)
    Abstract [en]

    In this study of arithmetical reasoning, which extends earlier work, we explore what properties students, when working in pairs, discern in additive and multiplicative co-variation problems that help them to distinguish between problem types. Results showed that pairs who solved each problem appropriately discerned mathematically significant properties such as speed, starting time and distance. Pairs who over-used additive reasoning focused on the distance difference without considering speed. While speed is considered to be a difficult quantity, here it seems to help students distinguish between multiplicative and additive situations.

  • 4.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andrews, Paul
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Students’ conceptualisation of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbersManuscript (preprint) (Other academic)
    Abstract [en]

    Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students’ multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication.

  • 5.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andrews, Paul
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Students' conceptualisations of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers2017In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 48, p. 1-13Article in journal (Refereed)
    Abstract [en]

    Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students' multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication.

  • 6.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andrews, Paul
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    The ambiguous role of equal groups in students’ understanding of distributivityManuscript (preprint) (Other academic)
    Abstract [en]

    Distributivity is considered to be essential for multiplicative understanding but difficult to learn. The difficulties may arise as overgeneralisations of addition strategies. Rectangular models emphasise the two-dimensionality of multiplication, separating it from addition and are suggested to support understanding of distributivity better compared to equal groups. Coincidently, studies report of students’ understanding of distributivity based on equal groups, leaving no consensus on equal groups’ suitability for understanding distributivity. In this paper we investigate how students can exploit equal groups to understand distributivity, by analysis of two students’ reasoning when they successfully explain distributivity construing the multiplication as heaps of sticks and bags of coins. The role of equal groups with respect to multiplicative understanding of distributivity is discussed in relation to previous ambiguous findings and to the extension of multiplication beyond integers, in which the equal groups model may be inappropriate.

  • 7.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Algorithmic and formal contextualisations: Exploring students’ understanding of threshold concepts in calculus2009In: Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education: In search for theories in mathematics education / [ed] M. Tzekaki, M. Kaldrimidou & H. Sakonidis, Thessaloniki: PME , 2009, p. 1-447-1-447Conference paper (Refereed)
  • 8.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Algorithmic contexts, intuitive ideas and formal reasoning: Exploring students’ understanding of concepts in calculus2011In: Voices on learning and instruction in mathematics / [ed] Jonas Emanuelsson [och 5 andra], Göteborg: Nationellt centrum för matematikutbildning (NCM), 2011, 1:1, p. 301-313Chapter in book (Refereed)
    Abstract [en]

    Two case studies were carried out to explore students’ understanding of concepts in calculus. The first study explored engineering students’ understanding of limit and integral. The second study involved students within a mathematics programme, working on a challenging task including the concepts function and derivative, requiring proof by induction. Drawing on a theory of contextualisation data were analysed within a constructivist research framework following the principles of intentional analysis.  The results reveal that the students in the mathematics programme expressed their understanding in a formal context where also intuitive ideas played an important role. The engineering students expressed their understanding in an algorithmic context, in which procedural knowledge was predominant. The studies display how students may have a potential for developing a formal understanding of a mathematical concept previously viewed within an algorithmic context.

  • 9.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Exploring students’ experiences of understanding the mathematical concept of function2011In: Education for a Global Networked Society, Proceedings EARLI conference 2011, 30 aug-3 sep, Exeter, England, 2011, p. 20-21Conference paper (Refereed)
    Abstract [en]

    This study explores the nature of university students' experiences of understanding the threshold concept ‘function'. Fifteen teacher students taking an introductory course in mathematics were at the beginning of the course asked to reflect in writing on the meaning of ‘function'. Subsequent interviews explored in greater detail individual students' conceptual understandings of this concept. At the end of the course, students were again asked to reflect in writing on the meaning of function and then to elaborate further on those written reflections in a qualitative research interview. In a previous study of students' understandings of limit and integral an intentional analysis of the interview transcripts underscored the crucial interplay between understandings developed within an algorithmic and conceptual context respectively. The present study seeks to develop this contextual framework for analysing students' understanding in mathematics by linking the analysis of students' emerging understandings of function to ongoing research on threshold concepts in higher education and more broadly to research on students' experiences of understanding disciplinary ways of thinking and practising.

  • 10.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Förståelse genom tröskelbegrepp2010Other (Other (popular science, discussion, etc.))
    Abstract [sv]

     I elevers strävan att utveckla förståelse inom ett område blir vissa begrepp mer avgörande än andra. Dessa begrepp kan benämnas tröskelbegrepp. De fungerar som en portal till ett tidigare onåbart och i början problematiskt sätt att tänka om någonting. De har en potential att öppna för förståelse av hela området. Exempel på tröskelbegrepp är funktion, gränsvärde, derivata och integral. I min avhandling har jag undersökt studenters begreppsuppfattningar för dessa begrepp. Resultaten visar att studenterna tolkar den matematik de möter på olika sätt, och att studenterna utnyttjar en dynamik mellan olika aspekter av sina begreppsuppfattningar. Denna dynamik utgör en viktig del i utvecklingen av förståelsen av tröskelbegreppen. Även i grundskolans matematik finns tröskelbegrepp, t.ex. bråkbegreppet. Kunskaper om dessa begrepp kan hjälpa oss att skapa goda lärsituationer för eleverna.

  • 11.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Grepp om begrepp – en tröskel?2012Conference paper (Other (popular science, discussion, etc.))
  • 12.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Grepp om begrepp – en tröskel?2011Other (Other (popular science, discussion, etc.))
    Abstract [sv]

    I elevers strävan att utveckla förståelse inom ett område blir vissa begrepp mer avgörande än andra. Dessa begrepp kan benämnas tröskelbegrepp. De fungerar som en portal till ett tidigare onåbart och i början problematiskt sätt att tänka om någonting. De har en potential att öppna för förståelse av hela området. Exempel på tröskelbegrepp är funktion, gränsvärde, derivata och integral. I min avhandling har jag undersökt studenters begreppsuppfattningar för dessa begrepp. Resultaten visar att studenterna tolkar den matematik de möter på olika sätt, och att studenterna utnyttjar en dynamik mellan olika aspekter av sina begreppsuppfattningar. Denna dynamik utgör en viktig del i utvecklingen av förståelsen av tröskelbegreppen. Även i grundskolans matematik finns tröskelbegrepp, t.ex. bråkbegreppet. Kunskaper om dessa begrepp kan hjälpa oss att skapa goda lärsituationer för eleverna.

  • 13.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    The threshold concept of a function:  A case study of a student’s development of her understanding2012Conference paper (Refereed)
  • 14.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Threshold concepts: A framework for research in university mathematics education2011Conference paper (Refereed)
    Abstract [en]

    A threshold concept is a ‘portal’ or a ‘conceptual gateway’ that leads to a previously inaccessible, and initially troublesome, way of thinking about something. A new way of understanding may thus emerge – a transformed view of the subject. The framework of threshold concepts has been used for some years in research of teaching and learning in higher education in several subjects but there are only few articles in mathematics education using this framework. The aim of this paper is to introduce threshold concepts into mathematics education. The result of searching papers in mathematics education using threshold concepts is presented. The need for more research using this framework to create a meeting point for mathematicians and educationalists and to improve students’ learning is pointed out.

  • 15.
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    University students' contextualisations of threshold concepts in calculus2010In: Mathematics and mathematics education: Cultural and social dimensions: Proceedings of MADIF7, The seventh mathematics education research seminar, Stockholm, 26-27 January 2010 / [ed] Christer Bergsten, Eva Jablonka, Tine Wedege, Linköping: Svensk förening för matematikdidaktisk forskning , 2010Conference paper (Refereed)
    Abstract [en]

    Two empirical studies were carried out to explore students’ understandings of threshold concepts in calculus. The studies investigated engineering and mathematics students’ understandings of the concepts of function, limit, derivative and integral. The engineering students expressed their understanding in an algorithmic context, in which procedural knowledge was predominant. The students in the mathematics programme expressed their understanding in a formal context where also intuitive ideas played an important role. The results also point out the importance of contextual shifts in the development of conceptual understanding of function, limit, derivative and integral as threshold concepts.

  • 16.
    Pettersson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Larson, Niclas
    Seminar groups as part of first-semester mathematics teaching: What did the students learn?2018In: Proceedings of INDRUM 2018 Second conference of the International Network for Didactic Research in University Mathematics : April 5-7, 2018, Kristiansand, Norway / [ed] Viviane Durand-Guerrier, Reinhard Hochmuth, Simon Goodchild, Ninni Marie Hogstad, Kristiansand: University of Agder and INDRUM , 2018, p. 368-369, article id 174463Conference paper (Refereed)
    Abstract [en]

    In this study, the learning encouraged by teaching activities in a small-group setting was investigated through the analysis of students’ responses to survey and interview questions. The results indicate that the students perceived an increased ability to communicate mathematics in written form, but to a lesser extent developed their ability to discuss mathematics and build conceptual understanding.

  • 17.
    Pettersson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Scheja, Max
    Stockholm University, Faculty of Social Sciences, Department of Education.
    Algorithmic contexts and learning potentiality: A case study of students’ understanding of calculus2008In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 39, no 6, p. 767-784Article in journal (Refereed)
    Abstract [en]

    The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.

  • 18.
    Pettersson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Scheja, Max
    Stockholm University, Faculty of Social Sciences, Department of Education.
    Prospective mathematics teachers’ development of understanding of the threshold concept of a function2012Conference paper (Refereed)
  • 19.
    Pettersson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Stadler, Erika
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Tambour, Torbjörn
    Stockholm University, Faculty of Science, Department of Mathematics.
    Transformation of students’ discourse on the threshold concept of function2013In: Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (CERME 8, February 6 - 10, 2013) / [ed] Behiye Ubuz, Çiğdem Haser, Maria Alessandra Mariotti, Ankara: Middle East Technical University and ERME , 2013, p. 2406-2415Conference paper (Refereed)
    Abstract [en]

    In this study of university students’ discourse on the threshold concept of function, the aim was to examine what kind of changes of their mathematical discourse could be observed during a study year. Data was collected through interviews, questionnaires and observations during the students’ first year of mathematics courses and was analyzed using the commognitive framework of Sfard. The result shows substantial differences in the students’ discourse. The transformation of their discourse resulted in broader registers of words, mediators, narratives and routines. Students not transforming their discourse showed an unbalanced use of visual mediators and fewer connections between the interrelated features of the discourse.

  • 20.
    Pettersson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Svedin, Maria
    Stockholm University, Faculty of Science, Numerical Analysis and Computer Science (NADA). KTH Royal Institute of Technology, Sweden.
    Scheja, Max
    Stockholm University, Faculty of Social Sciences, Department of Education.
    Bälter, Olle
    Stockholm University, Faculty of Science, Numerical Analysis and Computer Science (NADA). KTH Royal Institute of Technology, Sweden.
    Approaches to studying in first-year engineering: comparison between inventory scores and students' descriptions of their approaches through interviews2018In: Higher Education, ISSN 0018-1560, E-ISSN 1573-174X, Vol. 75, no 5, p. 827-838Article in journal (Refereed)
    Abstract [en]

    This combined interview and survey study explored the relationship between interview data and data from an inventory describing engineering students' ratings of their approaches to studying. Using the 18-item Approaches and Study Skills Inventory for Students (ASSIST) students were asked to rate their approaches to studying in relation to particular statements. A subsample of nine first-year engineering students participated in subsequent interviews exploring their experiences of studying and learning. The students' views were examined and interpreted into inventory scores which were compared to the students' actual ratings. The interviews confirmed the scales measured in the inventory and provided illustrations to them. While students who were extreme in either approach were easier to interpret, others provided a good example of the complex combination of approaches that can exhibit itself in one individual. The study illustrates how combined data sets can contribute to achieve a holistic understanding of student learning in its context.

  • 21. Ryve, Andreas
    et al.
    Nilsson, Per
    Pettersson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Analyzing effective communication in mathematics group work: The role of visual mediators and technical terms2013In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 82, no 3, p. 497-514Article in journal (Refereed)
    Abstract [en]

    Analyzing and designing productive group work and effective communication constitute ongoing research interests in mathematics education. In this article we contribute to this research by using and developing a newly introduced analytical approach for examining effective communication within group work in mathematics education. By using data from 12 to 13-year old students playing a dice game as well as from a group of university students working with a proof by induction, the article shows how the link between visual mediators and technical terms is crucial in students' attempts to communicate effectively. The critical evaluation of visual mediators and technical terms, and of links between them, is useful for researchers interested in analyzing effective communication and designing environments providing opportunities for students to learn mathematics.

1 - 21 of 21
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