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  • 1.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Connections for learning multiplication2015In: Proceedings from Symposium Elementary Mathematics Education 15, (SEMT´15) Developing mathematical language and reasoning / [ed] Novotná, Jarmila & Moraová, Hana, Prague, 2015, p. 202-211Conference paper (Refereed)
    Abstract [en]

    This paper presents an analytical tool designed to capture the connections students make between three components essential to the learning of multiplication: the calculative act, the arithmetical properties and multiplicative representations. Connections are viewed as building blocks for learning and conceptual understanding and their components are central for multiplication. The analytical tool is presented alongside examples of students’ connections shown as calculations or statements. By examples from one student’s work, it is demonstrated what can be inferred from the exploration of these connections. Finally other important connections concerning understanding of multiplication not included in the analytical tool are discussed.

  • 2.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Finding Erik and Alva: uncovering students who reason additively when multiplying2016In: Nordisk matematikkdidaktikk, ISSN 1104-2176, Vol. 21, no 2, p. 69-88Article in journal (Refereed)
    Abstract [en]

    This article presents a study in which grade 5 students' responses to multiplicative comparison problems, a well-known method for distinguishing additive reasoning from multiplicative, are compared to their reasoning when calculating uncontextualised multiplicative tasks. Despite recognising the multiplicative structure of multiplicative comparison problems a significant proportion of students calculated multiplicative problems additively. Therefore, multiplicative comparison problems are insufficiant on their own as indicators of multiplicative reasoning.

  • 3.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Mulitplicative thinking in relation to commutativity and forms of representation2013In: Tasks and tools in elementary mathematics / [ed] Jarmila Novotná and Hana Moraová, Prague: Charles University, Faculty of Education , 2013, p. 179-187Conference paper (Refereed)
    Abstract [en]

    Multiplicative thinking and commutativity in multiplication have proved to be difficult for many children to learn and to use. Different representations of multiplicative situations may enhance the possibilities for children to understand multiplicative properties. In this paper 24 5th grade students’ use of multiplicative thinking is related to their understanding of commutativity and their choice of form of representation. The study discloses that students who use multiplicative thinking show understanding of commutativity to a greater extent as well as ability to use multiple forms of representation of multiplication compared to students who use additive thinking.

  • 4.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Multiplicative thinking in relation to multiplication of two two-digit numbers2013In: Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education: Mathematics learning across the life  an / [ed] Lindmeier, Anke M. & Heinze, Aiso, Kiel: Breitschuh & Kock GmbH , 2013, p. 237-237Conference paper (Refereed)
  • 5.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Sixth grade students' explanations and justifications of distributivity2015In: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer, Nada Vondrová, 2015, p. 295-301Conference paper (Refereed)
    Abstract [en]

    Equal groups and rectangular arrays are examples of multiplicative situations that have different qualities related to students' understanding of the distributive and the commutative properties. These properties are, inter alia, important for flexible mental calculations. In order to design effective instruction we need to investigate how students construct understanding of these properties. In this study sixth grade students were invited to reason with a peer about calculation strategies for multiplication with the goal of explaining and justifying distributivity. Their discussions demonstrate that the representation of multiplication as equal groups helps them to explain and justify distributivity. At the same time this representation hinders their efficient use of commutativity.

  • 6.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Structured classifications for subtraction?: A study of mathematics education literature2012Conference paper (Other academic)
  • 7.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Students' understandings of multiplication2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Multiplicative reasoning permeates many mathematical topics, for example fractions and functions. Hence there is consensus on the importance of acquiring multiplicative reasoning. Multiplication is typically introduced as repeated addition, but when it is extended to include multi-digits and decimals a more general view of multiplication is required.

    There are conflicting reports in previous research concerning students’ understandings of multiplication. For example, repeated addition has been suggested both to support students’ understanding of calculations and as a hindrance to students’ conceptualisation of the two-dimensionality of multiplication. The relative difficulty of commutativity and distributivity is also debated, and there is a possible conflict in how multiplicative reasoning is described and assessed. These inconsistencies are addressed in a study with the aim of understanding more about students’ understandings of multiplication when it is expanded to comprise multi-digits and decimals.

    Understanding is perceived as connections between representations of different types of knowledge, linked together by reasoning. Especially connections between three components of multiplication were investigated; models for multiplication, calculations and arithmetical properties. Explicit reasoning made the connections observable and externalised mental representations.

    Twenty-two students were recurrently interviewed during five semesters in grades five to seven to find answers to the overarching research question: What do students’ responses to different forms of multiplicative tasks in the domain of multi-digits and decimals reveal about their understandings of multiplication? The students were invited to solve different forms of tasks during clinical interviews, both individually and in pairs. The tasks involved story telling to given multiplications, explicit explanations of multiplication, calculation problems including explanations and justifications for the calculations and evaluation of suggested calculation strategies. Additionally the students were given written word problems to solve.

    The students’ understandings of multiplication were robustly rooted in repeated addition or equally sized groups. This was beneficial for their understandings of calculations and distributivity, but hindered them from fluent use of commutativity and to conceptualise decimal multiplication. The robustness of their views might be explained by the introduction to multiplication, which typically is by repeated addition and modelled by equally sized groups. The robustness is discussed in relation to previous research and the dilemma that more general models for multiplication, such as rectangular area, are harder to conceptualise than models that are only susceptible to natural numbers.

    The study indicated that to evaluate and explain others’ calculation strategies elicited more reasoning and deeper mathematical thinking compared to evaluating and explaining calculations conducted by the students themselves. Furthermore, the different forms of tasks revealed various lines of reasoning and to get a richly composed picture of students’ multiplicative reasoning and understandings of multiplication, a wide variety of forms of tasks is suggested.

  • 8.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Subtraktion2011In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 38, no 4Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    Vad är egentligen subtraktion? Vad behöver en lärare veta om subtraktion och subtraktionsundervisning? Om elevers förståelse av subtraktion och om elevers vanliga missuppfattningar? Om hur subtraktion kan användas? Finns det forskning om subtraktion som kan hjälpa lärare att planera sin undervisning mer effektivt? Den här artikeln kommer inte att ge svar på alla ovanstående frågor, men förhoppningsvis belysa att det finns en hel del kunskap om subtraktion som kan vara användbara utgångspunkter för lärares planering av undervisning.

  • 9.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Subtraktionsberäkningar2012In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, no 1, p. 21-28Article in journal (Other (popular science, discussion, etc.))
  • 10.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Varför ska man "göra olika"?: En litteraturstudie om beräkningsstrategier för subtraktion2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [sv]

    Denna studie är en litteraturstudie där kurslitteratur i matematikdidaktik har undersökts med avseende på hur beräkningsstrategier för subtraktion beskrivs. Kurslitteraturen användes vid Stockholms universitet läsåret 2008/09 i kurser som riktar sig mot matematikundervisning i tidiga skolår. Beskrivna beräkningsstrategier i den undersökta litteraturen kategoriseras i en matris över olika beräkningsstrategier. Matrisen bygger på forskning om vilka beräkningsstrategier som elever använder då de utför beräkningar i huvudet. Det är både strategier som eleverna fått lära sig i skolan och strategier som de själva har utvecklat. Studien har även analyserat vilka ord som författarna till den undersökta litteraturen använder för beräkningsstrategi.

    Resultatet av studien gällande beskrivningar av beräkningsstrategier har analyserats ur ett variationsteoretiskt perspektiv. Den undersökta litteraturen ger läsaren olika möjligheter att uppfatta en variation av vilka beräkningsstrategier som finns. Studien visar att ingen enskild bok eller artikel ger en fullständig och strukturerad bild av vilka beräkningsstrategier för subtraktion som elever använder. Resultatet av författarnas ordval för beräkningsstrategier sätts i relation till de ordval som samma litteratur uppvisar för situationer inom räknesätten. Studien visar att ordvalen för beräkningsstrategier varierar och överlappar ordvalen för situationer.

    Resultaten diskuteras i förhållande till vilka möjligheter lärarstuderande ges att erfara strukturer av hur olika beräkningsstrategier kan kategoriseras. Det förs även en diskussion om vad det kan innebära för de studerande att de ord som används för beräkningsstrategier och situationer varierar och sammanfaller. Studien föreslår gemensamma och entydiga termer för såväl begreppet beräkningsstrategi, som för olika beräkningsstrategier samt en tydlig struktur för att kategorisera olika beräkningsstrategier.

  • 11.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    What about subtraction: Can research enhance teaching and learning in elementary school?2011In: Proceedings: the mathematical knowledge needed for teaching in elementary schools : Prague, the Czech Republic, Charles University, Faculty of Education, August 21-26, 2011 : proceedings / [ed] Jarmila Novotná and Hana Moraová, Prague: Charles University, Faculty of Education , 2011, p. 381-382Conference paper (Other academic)
  • 12.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Andrews, Paul
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Ridderlind, Inger
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Mathematical games can make a difference: An intervention for children at risk2019In: / [ed] Jarmila Novotná, Hana Moraová, Prague: Charles University , 2019Conference paper (Refereed)
    Abstract [en]

    Children in public care are at risk not achieving academically to their potential. To support this group, Letterbox Club is an intervention program sending parcels with e.g. mathematical games to the children hoping this would increase their engagement to, and skills in, mathematics. We report of the effects of such an intervention where the LBC members were compared to their peers by pre- and post-test design. The over-all test results demonstrated promising effects, but no significant differences. However, certain tasks, especially subtraction tasks, stood out as LBC members had significant lower scores in the pre-test. There were also tasks where the LBC members improved significantly better than their peers. These promising results call for more studies of the effects of mathematical development by sending mathematical games to children in care.

  • 13.
    Larsson, Kerstin
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Larson, Niclas
    Matematiska institutionen, Linköpings universitet.
    Räkning - en kul historia2011In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 38, no 2, p. 48-52Article in journal (Other (popular science, discussion, etc.))
    Abstract [sv]

    Att arbeta med att synliggöra olika talsystem öppnar möjligheter att beakta ett (kultur)historiskt perspektiv på matematiken. Författarna ger konkreta exempel på hur en medveten variation av skilda talsystem i undervisningen också bidrar till elevers förståelse av vårt tiobassystem.

  • 14.
    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.

  • 15.
    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.

  • 16.
    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.

  • 17.
    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.

  • 18. Levenson, E.
    et al.
    Barkai, R.
    Larsson, Kerstin
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Functions of explanations: Israeli and swedish elementary school curriculum documents2013In: Tasks and tools in elementary mathematics / [ed] Jarmila Novotná and Hana Moraová, Prague: Charles University, Faculty of Education , 2013, p. 188-195Conference paper (Refereed)
    Abstract [en]

    Explanations are an integral part of mathematics education in primary school. This paper investigates some of the possible functions of explanations according to curriculum documents in Israel and Sweden and provides a way of classifying those functions. Findings indicated that explanations may have several various functions depending on the context in which they are requested or given.

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