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Students' understandings of multiplication
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
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.
Series
Doctoral thesis from the department of mathematics and science education, 14
Keyword [en]
Multiplication, students’ understanding, connections, multiplicative reasoning, models for multiplication, calculations, arithmetical properties
National Category
Educational Sciences
Research subject
Mathematics Education
Identifiers
URN: urn:nbn:se:su:diva-134768ISBN: 978-91-7649-515-5ISBN: 978-91-7649-516-2OAI: oai:DiVA.org:su-134768DiVA: diva2:1038458
Public defence
2016-12-12, sal G, Arrheniuslaboratorierna, Svante Arrhenius väg 20 C, Stockholm, 09:00 (English)
Opponent
Supervisors
Note

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
List of papers
1. Sixth grade students' explanations and justifications of distributivity
Open this publication in new window or tab >>Sixth grade students' explanations and justifications of distributivity
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.

Keyword
Multiplicative reasoning, distributivity, commutativity, equal groups
National Category
Educational Sciences
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-131830 (URN)978-80-7290-844-8 (ISBN)
Conference
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
2. Finding Erik and Alva: uncovering students who reason additively when multiplying
Open this publication in new window or tab >>Finding Erik and Alva: uncovering students who reason additively when multiplying
2016 (English)In: Nordisk matematikkdidaktikk, ISSN 1104-2176, Vol. 21, no 2, 69-88 p.Article in journal (Refereed) Published
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.

Keyword
Multiplication, repeated addition, multiplicative reasoning, multiplicative comparison, calculations
National Category
Educational Sciences
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-131829 (URN)
Available from: 2016-06-29 Created: 2016-06-29 Last updated: 2016-10-24Bibliographically approved
3. The ambiguous role of equal groups in students’ understanding of distributivity
Open this publication in new window or tab >>The ambiguous role of equal groups in students’ understanding of distributivity
(English)Manuscript (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.

Keyword
Distributivity, equal groups, rectangular array
National Category
Educational Sciences
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-134771 (URN)
Available from: 2016-10-18 Created: 2016-10-18 Last updated: 2016-10-24Bibliographically approved
4. Students’ conceptualisation of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers
Open this publication in new window or tab >>Students’ conceptualisation of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers
(English)Manuscript (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.

Keyword
Models for multiplication, calculations, arithmetical properties, connections, multi-digit numbers, decimal numbers
National Category
Educational Sciences
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-134772 (URN)
Available from: 2016-10-18 Created: 2016-10-18 Last updated: 2016-10-24Bibliographically approved

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