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Säfström, A. I. & Nagy, C. (2018). Embodied fractions: Conceptual difficulties in the light of grounding metaphors. In: Ewa Bergqvist, Magnus Österholm, Carina Granberg & Lovisa Sumpter (Ed.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education: . Paper presented at 42nd Conference of the International Group for the Psychology of Mathematics Education (PME42), July 3-8, 2018, Umeå, Sweden (pp. 287-287). Umeå, 5
Open this publication in new window or tab >>Embodied fractions: Conceptual difficulties in the light of grounding metaphors
2018 (English)In: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education / [ed] Ewa Bergqvist, Magnus Österholm, Carina Granberg & Lovisa Sumpter, Umeå, 2018, Vol. 5, p. 287-287Conference paper, Poster (with or without abstract) (Refereed)
Abstract [en]

Fractions and rational numbers are known to be hard to both teach and learn, as there are many conceptual difficulties concerning fractions. For example, pupils may interpret the entirety of a picture as the whole (Mack, 1990), or seeing a part as a fourth as long as the whole is divided in four parts, regardless of the size of the parts (Ball, 2007). A recent study has revealed additional difficulties: Seeing fractions as divisions may hinder pupils to recognise one of the parts as ¼, and claim that it is the partition that is ¼. The role of numerator and denominator can be mixed up, or the denominator may be seen as the remaining parts, resulting in a picture of 2 fifths to be named 2/3. Pupils can also claim that a fraction has a specific representation, for example that it should be the upper right fourth of a circle that should be shaded, in order for the picture to represent one fourth. One possible reason for misconceptions is stereotypical or restricted use of representations of rational numbers, especially area models (Zhang, Clements & Ellerton, 2015). However, if the number line is introduced, there is a risk that the difficulties are transferred to the new representation. In the recent study, some pupils saw the number line as a whole, and place one half at the centre, regardless of the part of the line visible.

In this study, we relate conceptual difficulties concerning fractions to Lakoff and Núñez (2000) four grounding metaphors for numbers, by analysing the underlying metaphors of visual models used by pupils when the difficulties manifest. The results give implications for the introduction of fractions in the early years of elementary school. Our poster will present how misconceptions can manifest in area models and on the number line, how these misconceptions are related to the metaphor implicitly used in the models, and suggested activities where metaphors aid the understanding of fractions.

Place, publisher, year, edition, pages
Umeå: , 2018
Keywords
bråk, representationer, metaforer
National Category
Didactics
Identifiers
urn:nbn:se:hh:diva-37543 (URN)
Conference
42nd Conference of the International Group for the Psychology of Mathematics Education (PME42), July 3-8, 2018, Umeå, Sweden
Available from: 2018-07-10 Created: 2018-07-10 Last updated: 2018-09-20Bibliographically approved
Nagy, C. (2018). Teaching about fractions in mathematics: Professional learning about progression with an action research approach. In: The 46th Annual Congress of the Nordic Educational Research Association: Educational Research: Boundaries, Breaches and Bridges: Abstract book. Paper presented at The 46st NERA Nordic Conference on Educational Research, Oslo, Norway, March 8-10, 2018 (pp. 556-556). Oslo
Open this publication in new window or tab >>Teaching about fractions in mathematics: Professional learning about progression with an action research approach
2018 (English)In: The 46th Annual Congress of the Nordic Educational Research Association: Educational Research: Boundaries, Breaches and Bridges: Abstract book, Oslo, 2018, p. 556-556Conference paper, Oral presentation with published abstract (Refereed)
Abstract [en]

Topic, aim and framework

Progression in mathematics instruction can be seen as a quality in teaching that entails gradually increased demands on the student (Säfström, 2017). If there is a lack of progression in teaching or the demands increase too much, this can reduce students’ opportunities to learn. An assumption for this study was that progression in teaching between school stages was important. Furthermore, teachers should take joint responsibility for learning within their teams to develop the quality in teaching (cf. Wennergren, 2016).

The approach of the study was based on action research. Four teachers from preschool to 9th grade (age 1-16) were invited to a temporary team, a community of practice. The overall aim of the study was to develop knowledge about teaching fractions when teachers used students’ understandings as a point of departure for their actions plans. A second aim was to illuminate what influences progression in their teaching. In addition, I intend to highlight the importance of the team’s negotiation of teaching fractions. Wenger’s dimensions of social learning were used as an analytical tool: joint enterprise, mutual engagement and shared repertoire.

Methodology/research design

The team of teachers used the four phases of action research: plan, act, observe and reflect, during their learning processes. The teachers’ learning sessions were videotaped and transcribed and this provided the main data that formed the basis of the results.

Findings

Four themes that described teachers’ negotiation of qualities in mathematics instruction were identified.

Interpreting students’ understandings: By analysing students’ understandings through videotaped instruction, the teachers interpreted students’ understandings and observed which kind of instruction enabled learning.

Basing instruction on students’ understandings: The teachers reified mathematical instructions based on students’ understandings, which entailed an expanded content.

Visualizing fractions: There was not always an automatic correspondence between visualizing fractions with everyday materials or manipulative materials and the conventions regarding fractions within mathematics, which affected students’ learning opportunities.

Ensuring students’ understanding: Even if the students seemed to be united about fractions, the teachers were uncertain how to ensure students’ understanding. The teachers were uncertain whether every student had an understanding of the concept and whether they were able to use their knowledge in another context.

Conclusions

When teachers, regardless of what stage was involved, reified similar instructions, it did not benefit students’ learning opportunities. In order to improve progression in teaching fractions, it was important that teachers succeeded in identifying students’ understandings and that the team negotiated different qualities in their community of practice. The shared repertoire (the pre-tests and the video recordings) formed the core of negotiating progression based on students’ understandings.

Relevance to Nordic educational research

This study contributes knowledge about diverse qualities in analysing students’ understandings by means of pre-tests or video recordings. The study also contributes knowledge about professional learning when participants have diverse experience.

Place, publisher, year, edition, pages
Oslo: , 2018
Keywords
action research, analytical tools, communities of practice, fraction, negotiations, professional learning, progression, school stages
National Category
Didactics
Identifiers
urn:nbn:se:hh:diva-37522 (URN)
Conference
The 46st NERA Nordic Conference on Educational Research, Oslo, Norway, March 8-10, 2018
Available from: 2018-07-07 Created: 2018-07-07 Last updated: 2018-07-09Bibliographically approved
Nagy, C. (2017). Fler bråk i matematikundervisningen: En aktionsforskningsstudie där lärare lär om progression. (Licentiate dissertation). Göteborg: Göteborgs universitet
Open this publication in new window or tab >>Fler bråk i matematikundervisningen: En aktionsforskningsstudie där lärare lär om progression
2017 (Swedish)Licentiate thesis, monograph (Other academic)
Alternative title[en]
Teaching about fractions in mathematics : Professional learning about progression with an action research approach
Abstract [en]

Few studies have a focus on progression in teaching and learning mathematics. An assumption for this study was that progression in teaching between school stages was important. The approach of the study was based on action research. Four teachers from preschool to 9th grade (age 1-16) were invited to a temporary team, a community of practice. The overall aim of the study was to develop knowledge about teaching fractions when teachers used students’ understandings as a point of departure for their action plans. A second aim was to illuminate what influences progression in their teaching.

The team of teachers used the four phases of action research: plan, act, observe and reflect, during their learning processes. The teachers’ learning sessions were videotaped and transcribed and this provided the main data that formed the basis of the results. Wenger’s dimensions of social learning were used as an analytical tool: joint enterprise, mutual engagement and shared repertoire.

Four themes that described teachers’ negotiation of qualities in mathematics instruction were identified: interpreting students’ understandings, basing instruction on students’ understandings, visualizing fractions and ensuring students’ understanding. When teachers, regardless of what stage was involved, reified similar instructions, it did not benefit students’ learning opportunities. In order to improve progression in teaching fractions, it was important that teachers succeeded in identifying students’ understandings and that the team negotiated different qualities in their community of practice. The shared repertoire (the pre-tests and the video recordings) formed the core of negotiating progression based on students’ understandings. The team showed a mutual engagement, with students’ learning as their joint enterprise. An implication of the study is that teachers from different educational stages can negotiate progression and improve it.

Place, publisher, year, edition, pages
Göteborg: Göteborgs universitet, 2017. p. 177
Keywords
action research, analytical tools, communities of practice, fraction, negotiations, professional learning, progression, school stages
National Category
Pedagogical Work
Identifiers
urn:nbn:se:hh:diva-36057 (URN)
Presentation
2017-11-17, O104, Högskolan i Halmstad, Halmstad, 13:00 (Swedish)
Opponent
Supervisors
Available from: 2018-01-12 Created: 2018-01-09 Last updated: 2018-01-12Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-2441-1341

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