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Relate before calculate: Students’ ways of experiencing relationships between quantities
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Stockholm University, Faculty of Humanities, Department of Humanities and Social Sciences Education.ORCID iD: 0000-0003-0764-5728
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.

Place, publisher, year, edition, pages
2018. Vol. 40, p. 5-33
Keywords [en]
general mathematical structures, relationships, part-whole, additive structures, quantities, phenomenography, ways of experiencing, critical aspect
National Category
Didactics
Research subject
Mathematics Education
Identifiers
URN: urn:nbn:se:su:diva-163382DOI: 10.14708/dm.v40i0.6431OAI: oai:DiVA.org:su-163382DiVA, id: diva2:1274294
Funder
Swedish Research CouncilAvailable from: 2018-12-29 Created: 2018-12-29 Last updated: 2022-11-29Bibliographically approved
In thesis
1. Relationer eller operationer - två sidor av samma mynt: Elevers utforskande av en del-helhetsmodell som redskap för att urskilja relationer i additiva strukturer
Open this publication in new window or tab >>Relationer eller operationer - två sidor av samma mynt: Elevers utforskande av en del-helhetsmodell som redskap för att urskilja relationer i additiva strukturer
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
learning model, part-whole structure, equations, negative numbers, learning activity, learning study, lärandemodell, del-helhetsstruktur, lärandeverksamhet, tematisk analys, ekvationer
National Category
Didactics
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-211327 (URN)978-91-8014-332-5 (ISBN)
Presentation
2022-06-17, Stockholm, 11:00 (Swedish)
Opponent
Supervisors
Funder
Swedish Research Council
Available from: 2022-11-29 Created: 2022-11-27 Last updated: 2022-11-29Bibliographically approved
2. Kritiska aspekter för att urskilja en del-helhetsstruktur: Ett undervisningsutvecklande forskningsprojekt i grundskolan
Open this publication in new window or tab >>Kritiska aspekter för att urskilja en del-helhetsstruktur: Ett undervisningsutvecklande forskningsprojekt i grundskolan
2022 (Swedish)Licentiate thesis, comprehensive summary (Other academic)
Abstract [sv]

Syftet med licentiatuppsatsen är att bidra med kunskap om vilka aspekter som kan vara nödvändiga för elever att urskilja för att identifiera och analysera en del-helhetsstruktur och i förlängningen bemästra ekvationer. I forskningsprojektet medverkade lärare och forskare i ett kollaborativt och intervenerande arbete. Elever i årskurserna 3, 8 och 9 deltog i semistrukturerade intervjuer och forskningslektioner för att producera data. Forskningslektionerna genomfördes med learning study som forskningsansats, vilken erbjuder iterativa processer. De teoretiska ramverken utgjordes av fenomenografi, variationsteori och lärandeverksamhet. Ansatsen fenomenografi utgjorde ett analysverktyg när intervjuerna analyserades. Variationsteori och lärandeverksamhet utgjorde kompletterande ramverk när forskningslektioner designades och analyserades. Resultatet i Artikel 1 påvisar en kritisk aspekt, vilken formulerades som: två kvantiteter tillsammans (två delar) bildar en tredje kvantitet (en helhet) med samma ”värde” som de två delarna tillsammans. Resultatet i Artikel 2 påvisar fem kritiska aspekter, vilka formulerades som (1) det är en relation mellan alla tal i en ekvation, (2) två delar tillsammans är ekvivalenta med en helhet med samma värde, (3) vad som utgör en helhet respektive delar, (4) samma relation kan formuleras på fyra olika sätt och (5) helheten kan anta ett lägre värde än delarna. En slutsats är att det kan vara en fördel redan för unga elever att delta i undervisning med utgångspunkt i early algebra och lärandeverksamhet där generella matematiska strukturer fokuseras. En ytterligare slutsats är att det kan vara gynnsamt att delta i teoretiska resonemang där ekvationer med negativa tal inkluderade utforskas med stöd av en lärandemodell.

Abstract [en]

The aim of the licentiate thesis is to contribute with knowledge concerning which aspects may be necessary for students to discern in order to identify and analyze a part-whole structure and in the long term master equations. In the research project, teachers and researchers participated in a collaborative and intervening work. Students in grades 3, 8 and 9 participated in semi-structured interviews and research lessons in order to produce data. The research lessons were conducted with learning study as research approach, which offers iterative processes.The theoretical frameworks consisted of phenomenography, variation theory and learning activity. The phenomenography approach was an analysis tool when the interviews were analyzed. Variation theory and learning activity constituted complementary frameworks when research lessons were designed and analyzed. The finding in Article 1 demonstrates a critical aspect, which was formulated as two quantities together (two parts) build up a third quantity (the whole) with the same “value” as the two parts together. The finding in Article 2 demonstrates five critical aspects, which wereformulated as (1) there is a relationship between all numbers in an equation, (2) two parts together are equivalent to a whole with the same value, (3) what constitutes a whole and parts, (4) the same relationship can be formulated in four different ways and (5) the whole can assume a lower value than the parts. One conclusion is that it may be an advantage already for young students to participate in teaching based on early algebra and learning activity where general mathematical structures are focused. A furtherconclusion is that it may be beneficial to participate in theoretical reasoning where equations with negative numbers included are explored with support by a learning model. 

Place, publisher, year, edition, pages
Stockholm: Stockholms universitet, 2022. p. 135
Keywords
critical aspects, relationships between quantities and between numbers, negative numbers, early algebra, learning activity, learning study, kritiska aspekter, relationer mellan kvantiteter samt mellan tal, negativa tal, early algebra, lärandeverksamhet, learning study
National Category
Didactics
Research subject
Mathematics Education
Identifiers
urn:nbn:se:su:diva-211326 (URN)978-91-8014-333-2 (ISBN)
Presentation
2022-06-22, Stockholm, 10:00 (Swedish)
Opponent
Supervisors
Funder
Swedish Research Council
Available from: 2022-11-29 Created: 2022-11-27 Last updated: 2022-11-29Bibliographically approved

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Tuominen, JaneAndersson, CharlottaBjörklund Boistrup, LisaEriksson, Inger

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