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Sixth grade students' explanations and justifications of distributivity
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
2015 (English)In: Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education / [ed] Konrad Krainer, Nada Vondrová, 2015, 295-301 p.Conference 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.

Place, publisher, year, edition, pages
2015. 295-301 p.
Keyword [en]
Multiplicative reasoning, distributivity, commutativity, equal groups
National Category
Educational Sciences
Research subject
Mathematics Education
URN: urn:nbn:se:su:diva-131830ISBN: 978-80-7290-844-8OAI: diva2:944487
CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Prague, Czech Republic, February 2015
Available from: 2016-06-29 Created: 2016-06-29 Last updated: 2016-11-21Bibliographically approved
In thesis
1. Students' understandings of multiplication
Open this publication in new window or tab >>Students' understandings of multiplication
2016 (English)Doctoral 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.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics and Science Education, Stockholm University, 2016. 88 p.
Doctoral thesis from the department of mathematics and science education, 14
Multiplication, students’ understanding, connections, multiplicative reasoning, models for multiplication, calculations, arithmetical properties
National Category
Educational Sciences
Research subject
Mathematics Education
urn:nbn:se:su:diva-134768 (URN)978-91-7649-515-5 (ISBN)978-91-7649-516-2 (ISBN)
Public defence
2016-12-12, sal G, Arrheniuslaboratorierna, Svante Arrhenius väg 20 C, Stockholm, 09:00 (English)

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2016-11-17 Created: 2016-10-18 Last updated: 2016-10-28Bibliographically approved

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