Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Teaching for the learning of additive part-whole relations: The power of variation and connections
Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, results from four empirical studies and a re-analysis are synthesized with what can constitute a structural approach to teaching and learning additive part-whole relations among learners aged four to eight years. In line with a structural approach to additive relations, the relations of parts and whole are in focus from the outset and are seen as the basis for addition and subtraction (Davydov 1982; Neuman, 1987). This approach was introduced by the researches in two intervention studies across different contexts. The researches collaborated with teachers in planning part-whole activities, teachers teaching them in their own settings, and then reflecting on them together with the research team. The empirical material consists of video-recorded lessons (Grade 3), small-group teaching (preschool) and individual video-recorded task-based learner interviews (with preschoolers). The teaching episodes and interviews were analyzed on a micro-level, using analytical tools and concepts from variation theory (Marton, 2015). To deepen the knowledge, a re-analysis was also conducted with the purpose of identifying qualitative differences in teachers’ enactments of mathematical ideas and principles associated with a structural approach to additive relations.

Looking at the articles and the re-analysis, the results suggest that, for learning, it matters which representations are offered to the children. Some representations seem to facilitate the discernment of the parts and whole, and their relations. The results suggest that it matters which examples are offered. A systematic sequence of examples has the potential to bring to the fore relations between different part-whole examples, which offer the children opportunity to learn mathematical principles such as commutativity. Furthermore, the results indicate that what is made possible to learn about additive part-whole relations is associated with what aspects are opened up as dimensions of variation (Marton, 2015). Foremost, though, the results reveal the importance of making connections to highlight number relations and key features associated with the structural approach to additive relations. The results suggest that how variation is offered, and whether and how the teacher explicitly (verbally and gesturally) draws attention to relations, ideas and aspects, is crucial for the learning of additive part-whole relations. Moreover, through the separate articles and the re-analysis, the outcomes indicate that the structural approach to additive part-whole relations and conjectures from variation theory are possible to implement in different contexts and for different ages of children.

Place, publisher, year, edition, pages
Jönköping: Jönköping University, School of Education and Communication , 2019. , p. 121
Series
Doktorsavhandlingar från Högskolan för lärande och kommunikation, ISSN 1652-7933 ; 038
Keywords [en]
variation, variation theory, connections in teaching, making connections, early numbers, teaching and learning additive relations, part-whole relation, part-whole relations of numbers, intervention study
National Category
Didactics Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-46899ISBN: 978-91-88339-28-7 (print)ISBN: 978-91-88339-29-4 (electronic)OAI: oai:DiVA.org:hj-46899DiVA, id: diva2:1372663
Public defence
2019-12-13, Hb116, School of Education and Communication, Jönköping, 13:00 (English)
Opponent
Supervisors
Available from: 2019-11-25 Created: 2019-11-25 Last updated: 2019-11-25Bibliographically approved
List of papers
1. Coding teaching for simultaneity and connections: Examining teachers’ part-whole additive relations instruction
Open this publication in new window or tab >>Coding teaching for simultaneity and connections: Examining teachers’ part-whole additive relations instruction
2016 (English)In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 93, no 3, p. 293-313Article in journal (Refereed) Published
Abstract [en]

In this article, we present a coding framework based on simultaneity and connections. The coding focuses on microlevel attention to three aspects of simultaneity and connections: between representations, within examples, and between examples. Criteria for coding that we viewed as mathematically important within part-whole additive relations instruction were developed. Teachers’ use of multiple representations within an example, attention to part-whole relations within examples, and relations between multiple examples were identified, with teachers’ linking actions in speech or gestures pointing to connections between these. In this article, the coding framework is detailed and exemplified in the context of a structural approach to part-whole teaching in six South African grade 3 lessons. The coding framework enabled us to see fine-grained differences in teachers’ handling of part-whole relations related to simultaneity of, and connections between, representations and examples as well as within examples. We went on to explore the associations between the simultaneity and connections seen through the coding framework in sections of teaching and students’ responses on worksheets following each teaching section.

Keywords
Additive relations, Coding framework, Connections, Primary mathematics, Simultaneity, South Africa, Variation theory
National Category
Didactics
Identifiers
urn:nbn:se:hj:diva-31993 (URN)10.1007/s10649-016-9700-0 (DOI)000386370900002 ()2-s2.0-84965019191 (Scopus ID)HLKSkolnäraIS (Local ID)HLKSkolnäraIS (Archive number)HLKSkolnäraIS (OAI)
Available from: 2016-10-17 Created: 2016-10-17 Last updated: 2020-02-14Bibliographically approved
2. Weaving in connections: Studying changes in early grades additive relations teaching
Open this publication in new window or tab >>Weaving in connections: Studying changes in early grades additive relations teaching
2018 (English)In: South African Journal of Childhood Education, ISSN 2223-7674, E-ISSN 2223-7682, Vol. 8, no 1, article id a540Article in journal (Refereed) Published
Abstract [en]

In this article, we present aspects of teaching that draw attention to connections – both within and between examples – in order to explore the potential objects of learning that are brought into being in the classroom space and thus what is made available to learn. Our focus is on exploring differences in teaching over time, in the context of learning study style development activity of additive relation problems in three Grade 3 classes in South Africa. In a context where highly-localised and fragmented instruction has been noted, this study reports on the nature and extent of changes in connections in instruction over time. The application of a coding framework focused on simultaneity and connections in teaching points to a richer range of structural relationships within examples, and more connecting work between examples in the second year in comparison to the first year.

Place, publisher, year, edition, pages
University of Johannesburg, 2018
Keywords
Connections; Example space; Additive relations; Primary mathematics; Part-part-whole teaching; Missing number problem teaching; Changes in teaching; Variation theory; South Africa
National Category
Didactics
Identifiers
urn:nbn:se:hj:diva-39015 (URN)10.4102/sajce.v8i1.540 (DOI)000430807400001 ()HLKPUFIS (Local ID)HLKPUFIS (Archive number)HLKPUFIS (OAI)
Available from: 2018-03-16 Created: 2018-03-16 Last updated: 2019-11-25Bibliographically approved
3. Different learning possibilities from the same activity: Swedish preschool teachers’ enactment of a number relation activity
Open this publication in new window or tab >>Different learning possibilities from the same activity: Swedish preschool teachers’ enactment of a number relation activity
2020 (English)In: Scandinavian Journal of Educational Research, ISSN 0031-3831, E-ISSN 1470-1170Article in journal (Refereed) Epub ahead of print
Abstract [en]

In this paper, differences in the implementation of a number activity called the snake game are studied. Nine Swedish preschool teachers worked in collaboration with a research team, enacting the same activity with their groups of 5-year-old children over a 3-month period. Variation theory forms the basis for the analysis of 67 videorecorded enactments. The results suggest that an activity such as the snake game can bring various aspects of numbers to the fore through differences in enactment. The activity became mathematically richer when the teacher compared children’s different finger patterns and used systematically varied examples of number relations. This study’s results contribute knowledge about characteristics of teaching that foregrounds numbers’ part-whole relations.

Place, publisher, year, edition, pages
Taylor & Francis, 2020
Keywords
Variation theory, mathematical activity, in-service preschool teachers, early numbers, part-whole relationship, part-whole relations, finger patterns
National Category
Didactics Mathematics
Identifiers
urn:nbn:se:hj:diva-46895 (URN)10.1080/00313831.2020.1739131 (DOI)000523149300001 ()2-s2.0-85082747980 (Scopus ID)GOA HLK 2020 (Local ID)GOA HLK 2020 (Archive number)GOA HLK 2020 (OAI)
Available from: 2019-11-25 Created: 2019-11-25 Last updated: 2020-05-15
4. Implementing a structural approach in preschool number activities: Principles of an intervention program reflected in learning
Open this publication in new window or tab >>Implementing a structural approach in preschool number activities: Principles of an intervention program reflected in learning
2020 (English)In: Article in journal (Refereed) Epub ahead of print
Abstract [en]

We report here on an intervention implementing a structural approach to arithmetic problem-solving in relation to learning outcomes among preschoolers. Using the fundamental principles of the variation theory of learning for developing the intervention and as an analytical framework, we discuss teaching and learning in commensurable terms. The research question is how teaching grounded on a structural approach and designed based on principles of variation theory is reflected in children’s learning of numbers. To answer this, three analyses were conducted, addressing: i) how the children’s ways of experiencing numbers changed after participating in the intervention, ii) how the theoretical ideas were afforded in the intervention program, and iii) synthesizing how the affordance was associated with the children’s arithmetic learning. One group of eight children participating in the intervention program was chosen for thorough analysis. Progression was observed in how the children changed their ways of experiencing numbers during the intervention that allowed them to enact more advanced arithmetic strategies, which was associated with the structural approach in teaching. The results also show how analysis focusing on aspects discerned in learning and aspects afforded in teaching provides a way of describing arithmetic learning with significant implications for teaching practices.

Place, publisher, year, edition, pages
Taylor & Francis, 2020
Keywords
Arithmetic skills, early childhood education, intervention, variation theory of learning
National Category
Didactics Mathematics
Identifiers
urn:nbn:se:hj:diva-46896 (URN)10.1080/10986065.2020.1756027 (DOI)
Note

Included in doctoral thesis in manuscript form with the title "Implementing a structural approach in preschool number activities: Effects of an intervention program".

Available from: 2019-11-25 Created: 2019-11-25 Last updated: 2020-05-15

Open Access in DiVA

Kappa(4477 kB)228 downloads
File information
File name FULLTEXT01.pdfFile size 4477 kBChecksum SHA-512
c9100dd6136c1b3bc6f21d8444a1420f88455221ec97e057177ffc9e84ba1c9502be3f284f7c416075cac53ca70f099dbcb1e0dcd880bc4ff6911fffaa587eae
Type fulltextMimetype application/pdf

Authority records BETA

Ekdahl, Anna-Lena

Search in DiVA

By author/editor
Ekdahl, Anna-Lena
By organisation
Mathematics Education Research
DidacticsMathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 228 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 555 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf