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Flexibility in knowing school mathematics in the contexts of a Swedish and an Indian school class

Dash, Ingrid LU (2009)
Abstract (Swedish)
Popular Abstract in Swedish

En av de viktigaste forskningsfrågorna inom matematikutbildning handlar om förståelse. Det huvudsakliga syftet med den här avhandlingen har varit att nå insikt i olika förståelseformer i skolmatematik och former av lärandeidentiteter i två lärandemiljöer. Avhandlingens specifika fokus var flexibla sätt att urskilja delar och avgränsa helheter och undersöka hur lärande individer förstår del-helhetsförhållanden vid problemlösning i matematik.

Den teoretiska delen innehåller tolkningar av perspektiv på förståelse från fenomenografi, variationsteori och från socialkonstruktionistiska teoretiska antaganden. Det empiriska materialet samlades in från en skolklass i södra Sverige och från en... (More)
Popular Abstract in Swedish

En av de viktigaste forskningsfrågorna inom matematikutbildning handlar om förståelse. Det huvudsakliga syftet med den här avhandlingen har varit att nå insikt i olika förståelseformer i skolmatematik och former av lärandeidentiteter i två lärandemiljöer. Avhandlingens specifika fokus var flexibla sätt att urskilja delar och avgränsa helheter och undersöka hur lärande individer förstår del-helhetsförhållanden vid problemlösning i matematik.

Den teoretiska delen innehåller tolkningar av perspektiv på förståelse från fenomenografi, variationsteori och från socialkonstruktionistiska teoretiska antaganden. Det empiriska materialet samlades in från en skolklass i södra Sverige och från en skolklass i Orissa, centralöstra Indien.

Meningen som den lärande gav uttryck för, i verbala uttryck eller med hjälp av matematik, under intervjuer och observationer, analyserades med kontextuell analysmetod.

Huvudresultatet består av tre förståelseformer i matematik: Associativt flexibelt erfarande, kompositionellt flexibelt erfarande och kontextuellt flexibelt erfarande.



Relationen mellan dessa förståelseformer representerar en tydlig skillnad i hur lärande individer använder variation när de hanterar ett kunskapsobjekt, hur djup förståelsen i matematik är, och vilken lärandeidentitet man konstituerat.

Associativt flexibelt erfarande uppvisas när förståelse och meningsskapande sker på slumpmässiga sätt och fokus ligger på slumpmässigt urskilda aspekter som är associativt relaterade.

Kompositionellt flexibelt erfarande engagerar den lärande i att nå förståelse med fokus på kompositioner, som t ex tal-förhållanden och matematiska samband.

Inom förståelseformen kontextuellt flexibelt erfarande ligger fokus på kontexten, ur vilken mening söks till det matematiska. Denna förståelseform uppvisade en högre differentierad förståelse för den matematiska kontextens logik och innehåll, jämfört med de två andra förståelseformerna.



Variationen inom kategorierna var relaterad till hur den lärande uppfattade lärandekontexten och vad situationen kräver. Den dominerande förståelseformen var den kompositionella. Den mest framträdande formen av lärandeidentitet var, inom den svenska studien, samtidigt självständig och kollaborativ, och dessutom samtidigt kreativ och produktiv. I den indiska studien var den dominerande formen av lärandeidentitet autonom och hängiven.



Ett av huvudresultaten pekar på att det är viktigt att ge elever i skolan matematikuppgifter som innehåller möjligheter för att förstå variation och som bereder tillfällen för medskapande av mening (authorship). Ett sådant fokus får konsekvenser för undervisningen. Läraren bör regelbundet observera och utvärdera elevens förståelse och användning av variationsmöjligheter. (Less)
Abstract
A central question in mathematics education research concerns understanding. The main objective of the present thesis has been to obtain insights into flexible modes of knowing in school mathematics in two school class contexts, and how these relate to modes of being a learner in these contexts, with specific focus on learners’ flexible ways of discerning parts and delimiting wholes, and how they understand part- and whole-relationships while doing mathematics. The theoretical exploration of knowing school mathematics was informed by perspectives from phenomenography and variation theory, as well as constructionist theoretical standpoints. Empirical material was collected from a school class in Southern Sweden and a school class in Orissa,... (More)
A central question in mathematics education research concerns understanding. The main objective of the present thesis has been to obtain insights into flexible modes of knowing in school mathematics in two school class contexts, and how these relate to modes of being a learner in these contexts, with specific focus on learners’ flexible ways of discerning parts and delimiting wholes, and how they understand part- and whole-relationships while doing mathematics. The theoretical exploration of knowing school mathematics was informed by perspectives from phenomenography and variation theory, as well as constructionist theoretical standpoints. Empirical material was collected from a school class in Southern Sweden and a school class in Orissa, Central-Eastern India. The meaning the learners expressed during interviews and observations, verbally or with the help of mathematics, was analysed using contextual analysis. In line with methods in phenomenographic research, the main results of the thesis are different categories of description. Three modes of knowing emerged from the empirical material. These were: associative flexible experiencing; compositional flexible experiencing and contextual flexible experiencing.



These modes of knowing feature distinct differences: in the depth of understanding mathematics, in how learners use variation when dealing with an object of knowledge, and in learner identity. The associative mode of knowing involved the learner in arbitrary ways of making sense of the material s/he was working with, with a focus on arbitrarily discerned aspects in chains of associations. The compositional mode of knowing meant that the learner made an effort to understand, keeping a focus on compositions, such as number-relations or formulas. Finally, the contextual mode of knowing engaged the learners in ways of understanding the context from which critical aspects were to be discerned. The contexts gave meanings to the content. The knowledge about the context, mathematical and also reality-based, gave meaning to the theoretical constructs. The logic of the mathematical context and content was understood in a more differentiated way than within the two other modes of knowing. In all parts of empirical material, the compositional flexible mode of knowing predominated. The dominant mode of being a learner in the Swedish school class context was simultaneously independent and collaborative, as well as creative and productive. In the Indian school class context, the dominating mode of being a learner was autonomous and committed.



A major finding is that in mathematics education there is a need to give pupils tasks containing possibilities both for experiencing variation and for authorship. This also demands of the teacher to observe and evaluate the individual pupil’s understanding and use of the possibilities offered. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Prof. Wistedt, Inger, Stockholms universitet
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Mathematics education, compulsory school, Sweden, Orissa, India, flexibility in knowing, modes of knowing, authorship, agency, phenomenography, contextual analysis, intercultural perspective
defense location
Palaestras nedre hörsal
defense date
2009-01-30 10:00
ISBN
978-91-628-7366-0
language
English
LU publication?
yes
id
8305fe51-fd72-4380-b5da-91a967599ace (old id 1274817)
date added to LUP
2008-12-31 13:08:21
date last changed
2016-09-19 08:45:18
@phdthesis{8305fe51-fd72-4380-b5da-91a967599ace,
  abstract     = {A central question in mathematics education research concerns understanding. The main objective of the present thesis has been to obtain insights into flexible modes of knowing in school mathematics in two school class contexts, and how these relate to modes of being a learner in these contexts, with specific focus on learners’ flexible ways of discerning parts and delimiting wholes, and how they understand part- and whole-relationships while doing mathematics. The theoretical exploration of knowing school mathematics was informed by perspectives from phenomenography and variation theory, as well as constructionist theoretical standpoints. Empirical material was collected from a school class in Southern Sweden and a school class in Orissa, Central-Eastern India. The meaning the learners expressed during interviews and observations, verbally or with the help of mathematics, was analysed using contextual analysis. In line with methods in phenomenographic research, the main results of the thesis are different categories of description. Three modes of knowing emerged from the empirical material. These were: associative flexible experiencing; compositional flexible experiencing and contextual flexible experiencing.<br/><br>
<br/><br>
These modes of knowing feature distinct differences: in the depth of understanding mathematics, in how learners use variation when dealing with an object of knowledge, and in learner identity. The associative mode of knowing involved the learner in arbitrary ways of making sense of the material s/he was working with, with a focus on arbitrarily discerned aspects in chains of associations. The compositional mode of knowing meant that the learner made an effort to understand, keeping a focus on compositions, such as number-relations or formulas. Finally, the contextual mode of knowing engaged the learners in ways of understanding the context from which critical aspects were to be discerned. The contexts gave meanings to the content. The knowledge about the context, mathematical and also reality-based, gave meaning to the theoretical constructs. The logic of the mathematical context and content was understood in a more differentiated way than within the two other modes of knowing. In all parts of empirical material, the compositional flexible mode of knowing predominated. The dominant mode of being a learner in the Swedish school class context was simultaneously independent and collaborative, as well as creative and productive. In the Indian school class context, the dominating mode of being a learner was autonomous and committed. <br/><br>
<br/><br>
A major finding is that in mathematics education there is a need to give pupils tasks containing possibilities both for experiencing variation and for authorship. This also demands of the teacher to observe and evaluate the individual pupil’s understanding and use of the possibilities offered.},
  author       = {Dash, Ingrid},
  isbn         = {978-91-628-7366-0},
  keyword      = {Mathematics education,compulsory school,Sweden,Orissa,India,flexibility in knowing,modes of knowing,authorship,agency,phenomenography,contextual analysis,intercultural perspective},
  language     = {eng},
  school       = {Lund University},
  title        = {Flexibility in knowing school mathematics in the contexts of a Swedish and an Indian school class},
  year         = {2009},
}