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Relationer eller operationer - två sidor av samma mynt: Elevers utforskande av en del-helhetsmodell som redskap för att urskilja relationer i additiva strukturer
Stockholm University, Faculty of Humanities, Department of Teaching and Learning.ORCID iD: 0000-0003-4383-0613
2022 (Swedish)Licentiate thesis, comprehensive summary (Other academic)
Abstract [sv]

Syftet med denna licentiatuppsats är att pröva en specifik strukturell modell som stöd för elevers utforskande av relationer mellan tal i additiva strukturer även när negativa tal är inkluderade. En intention är att resultatet ska kvalificera undervisningen och utgöra ett stöd vid planering och genomförande av en undervisning avseende ekvationer med additiv struktur utifrån algebraisk undervisning som alternativ till att i första hand finna en lösning med stöd av regler och procedurer. Under studien kartlades, kategoriserades och beskrevs elevernas erfarande av fenomenet relationer mellan kvantiteter. I studien prövades även om, och i sådant fall, på vilka sätt en specifik strukturell modell användes av elever under arbetet med att utforska ekvationers struktur.

Elever från årskurserna 3, 8 och 9 deltog i semistrukturerade intervjuer samt i forskningslektioner. Tre grupper av lärare samt två forskare planerade, genomförde och reviderade forskningslektionerna baserat på ansatsen learning study.

Fenomenografi var den teoretiska ansatsen för de inledande semistrukturerade intervjuerna. Variationsteori och lärandeverksamhet var de teoretiska ramverken för forskningslektionerna där Davydovs program var en inspirationskälla.

För att uppnå syftet formulerades följande forskningsfrågor:

  1. Vilka skilda sätt att erfara fenomenet relationer mellan kvantiteter kan urskiljas i elevintervjuer?
  2. På vilka sätt använder elever en specifik strukturell modell för att utforska ekvationer?

Den första forskningsfrågan besvaras i Artikel 1 som visar att elever erfar relationer mellan kvantiteter som någonting som ska beräknas alternativt någonting som ska relateras. Den andra forskningsfrågan besvaras i Artikel 2 som visar att elever använde sig av den i studien prövade specifika strukturella modellen såsom ett formulär att fylla i alternativt som en lärandemodell och redskap för att identifiera del-helhetsstrukturen mellan tal i en ekvation samt för att välja lämplig operation att lösa ut det obekanta talet.    

                      Resultaten visar på möjligheten, men även utmaningen, att introducera en algebraisk undervisning med fokus på analys och teoretiska resonemang även för elever med erfarenheter från en alternativ bakgrund.

Abstract [en]

The aim of the licentiate thesis is to examined a specific structural model in order to support students' exploration of relationships between numbers in additive structures even when negative numbers is included. One intention is that the finding should qualify the teaching and constitute a support when plan and implement teaching regarding equations with additive structure based on algebraic teaching as an alternative to primarily finding a solution with support of rules and procedures. During the study, the students' experiences of the phenomenon relationships between quantities was examined, categorized and described. In the study it was also examined whether and, if so, in what ways a specific structural model was used by students during the work of exploring the structure of equations.

Students from grades 3, 8 and 9 participated in semi-structured interviews and in research lessons. Three groups of teachers and two researchers planned, conducted and revised the research lessons based on the learning study approach.

Phenomenography was the theoretical approach for the initial semi-structured interviews. Variation theory and learning activity were the theoretical frameworks for the research lessons where Davydov's curriculum was a source of inspiration.

In order to achieve the aim, the following research questions were formulated:

1) What different ways of experiencing the phenomenon of relationships between quantities can be discerned in student interviews?

2) In what ways do students use a specific structural model to explore equations?

The first research question is answered in Article 1, which shows that students experience relationships between quantities as something to be calculated or something to be related. The second research question is answered in Article 2, which shows that students used the specific structural model examined in the study as a form to fill in alternatively as a learning model and a tool to identify the part-whole structure between numbers in an equation and to choose appropriate operation to find the unknown number.

The findings show the possibility, but also the challenge, of introducing an algebraic teaching with a focus on analysis and theoretical reasoning also for students with experiences from an alternative background.

Place, publisher, year, edition, pages
Stockholm: Stockholms universitets förlag, 2022. , p. 107
Keywords [en]
learning model, part-whole structure, equations, negative numbers, learning activity, learning study
Keywords [sv]
lärandemodell, del-helhetsstruktur, lärandeverksamhet, tematisk analys, ekvationer
National Category
Didactics
Research subject
Mathematics Education
Identifiers
URN: urn:nbn:se:su:diva-211327ISBN: 978-91-8014-332-5 (print)OAI: oai:DiVA.org:su-211327DiVA, id: diva2:1713720
Presentation
2022-06-17, Stockholm, 11:00 (Swedish)
Opponent
Supervisors
Funder
Swedish Research CouncilAvailable from: 2022-11-29 Created: 2022-11-27 Last updated: 2022-11-29Bibliographically approved
List of papers
1. Relate before calculate: Students’ ways of experiencing relationships between quantities
Open this publication in new window or tab >>Relate before calculate: Students’ ways of experiencing relationships between quantities
2018 (English)In: Didactica Mathematicae, ISSN 0208-8916, Vol. 40, p. 5-33Article in journal (Refereed) Published
Abstract [en]

The aim of this article is to contribute with findings concerning students’ ways of experiencing general mathematical structures and, in particular, relationships in additive structures.When students discern relationships in additive structures, it may lead to positive consequences for students’ future ability to perform calculations in addition and subtraction tasks. In the study, semi-structured interviews were conducted with students in grades 3, 8, and 9. An illustration showing a set of different quantities was the starting point during the interviews, together with an opening question regarding how the diverse quantities could be equalised. After the students’ discussions, they were asked if this could be described mathematically using written symbols. The students’ expressions concerning the phenomenon “relationships between quantities” were analyzed using phenomenography as an analytical tool. According to phenomenography, there are a limited number of ways in which a phenomenon can be experienced. Further, it is not about exploring how many individuals hold a specific experience that is of interest. In the case of this article, it is about capturing qualitatively different ways of experiencing the phenomenon relationships between quantities. Despite no specific numbers being presented, many students attributed specific numbers and values when expressing relationships between quantities. The students expressed general mathematical structures only to a limited extent and, in those cases, mostly only after encouragement from the interviewer. Following the phenomenographical analysis, the students’ ways of experiencing “relationships between quantities” are: as something that has to be calculated, or as something that has to be related. The first of these was most common in all grades. In this study, one critical aspect was identified, namely, that quantities are related to each other, additively. Instead of introducing mathematics with a focus on answer-oriented tasks, it is essential to introduce mathematics based on general structures such as additive structures. Even if the students are not familiar with such a mathematical “culture”, it is worth it. This was confirmed in our study.

Keywords
general mathematical structures, relationships, part-whole, additive structures, quantities, phenomenography, ways of experiencing, critical aspect
National Category
Didactics
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-163382 (URN)10.14708/dm.v40i0.6431 (DOI)
Funder
Swedish Research Council
Available from: 2018-12-29 Created: 2018-12-29 Last updated: 2022-11-29Bibliographically approved
2. En modell som stöd för att utforska ekvationer: [A model to support exploring equations]
Open this publication in new window or tab >>En modell som stöd för att utforska ekvationer: [A model to support exploring equations]
2022 (Swedish)In: LUMAT: International Journal on Math, Science and Technology Education, E-ISSN 2323-7112, Vol. 10, no 1, p. 182-209Article in journal (Refereed) Published
Abstract [sv]

Syftet med artikeln är att lyfta fram om, och i sådant fall på vilka sätt, en specifik strukturell modell kan utgöra stöd när elever utforskar matematiska strukturer i ekvationer. Artikeln bygger på en empirisk forskningsstudie där elever utforskade matematiska strukturer med stöd av modellen, vilken är avsedd att visualisera strukturer. Lärare och forskare arbetade i en kollaborativ och intervenerande studie i iterativa processer. Sammantaget 149 elever från grundskolans årskurser 3, 8 och 9 deltog i filmade forskningslektioner utifrån forskningsansatsen learning study. Lektionerna designades med inspiration från ramverket lärandeverksamhet och eleverna utmanades i ett teoretiskt arbete. Analysen utfördes utifrån tematisk ansats och två kvalitativt skilda kärnteman identifierades: Formulär respektive Lärandemodell. I analysen framträdde att undervisningen behöver vara tillräckligt utmanande för att eleverna ska finna modellen meningsfull. Undervisningen behöver möjliggöra för eleverna att urskilja relationer mellan alla tal i en ekvation, där relationerna kan beskrivas som en del-helhetsstruktur.

Abstract [en]

The aim of the article is to highlight whether, and if so in what ways, a selected model can constitute support when students explore mathematical structures in equations. The article is based on an empirical research study where students explored mathematical structures with support by the model, which is intended to visualize structures. Teachers and researchers worked in a collaborative and interventional study in iterative processes. A total of 149 students from compulsory school grades 3, 8 and 9 participated in video recorded research lessons based on the research approach learning study. The lessons were designed with inspiration from the framework of learning activity and the students were challenged in a theoretical work. The analysis was performed on the basis of a thematic approach and two qualitatively different core themes were identified: Template respectively Learning model. In the analysis, it emerged that the teaching has to be challenging enough for the students to find the model meaningful. The teaching needs to enable students to discern relationships between all numbers in an equation, where the relationships can be described as a part-whole structure.

Keywords
learning model, equation, part-whole structure, learning activity, thematic analysis, lärandemodell, ekvation, del-helhetsstruktur, lärandeverksamhet, tematisk analys
National Category
Educational Sciences
Research subject
Didactic Science for Teachers and Teaching Professions; Didactics
Identifiers
urn:nbn:se:su:diva-207120 (URN)10.31129/LUMAT.10.1.1581 (DOI)2-s2.0-85140719544 (Scopus ID)
Available from: 2022-07-06 Created: 2022-07-06 Last updated: 2023-09-12Bibliographically approved

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Andersson Charlotta 2022(5162 kB)143 downloads
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