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  • 1.
    Abdulhamid, Lawan
    et al.
    University of the Witwatersrand, Johannesburg, South Africa.
    Venkat, Hamsa
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research. University of the Witwatersrand, Johannesburg, South Africa.
    Primary mathematics teachers’ responses to students’ offers: An ‘elaboration’ framework2018In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 51, p. 80-94Article in journal (Refereed)
    Abstract [en]

    Responding constructively ‘in-the moment’ to student offers is described as a critical, and yet difficult, aspect of skilful and responsive teaching. South African evidence points to limited evaluation of student offers in schools serving poor communities. In this paper, we present and discuss an ‘elaboration’ framework emerging from a grounded analysis of data drawn from video recordings of 18 mathematics lessons prepared and conducted by four in-service primary school teachers in South Africa. This analysis led to a categorization of the situations in which teacher responses to student offers occurred, and the nature and range of these responses. Three response situations are identified within the framework: breakdown, sophistication, and individuation/collectivization, with a range of response (and non-response) categories in each situation. Literature on responsive feedback is drawn in to explore hierarchies and relationships between the emergent categories within situations of elaboration. The elaboration framework provides a tool for lesson observation, and a model for thinking about developments in responsive teaching.

  • 2.
    Abramsson, Matilda
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers förståelse av likhetstecknet: En studie i årskurs 32016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of the study is to explore the understanding of the equal sign and how the teaching about the equal sign among third grade students can be varied to be as effective as possible.  The aim will be answered trough the questions: what patterns of variation can the studied students meet and what critical aspects have the students identified. Patterns of variation means that what is critical in the teaching should be varied to become visible. Critical aspects is what students need to identify to understand what should be learned. The foundation of the study is the Variation Theory, where patterns of variation and critical aspects are central concepts.

    The observations were accomplished during a third grade lesson and six students were selected for interviews about the equal sign. The result of the study shows that the students met six critical aspects during the lesson. For every critical aspect there were one or several patterns of variation that was exposed to the students. The result also states that the students who were interviewed have a relational and instrumental understanding of the equal sign. The students also have understanding of a critical aspect that they did not meet in the observed lesson, namely that all numbers have to enter in a task. Four out of six students have understanding of the critical aspect that there should be equivalence in a chain of similarities. The result also show that the students understanding of the equal sign is not dependent of that they meet patterns of variation in the teaching, but that they meet the critical aspects somehow.

  • 3.
    Andersson, Frida
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Hur matematikläroböcker presenterar räknelagar och räkneregler2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In mathematics education textbooks to a large extent determine what is offered for students to be learnt. With this in mind, in this study, five Swedish textbooks series is reviewed in a latent and manifest content analysis approach where both quantitative and qualitative data is presented. The result of the quantitative data indicate that only a few textbooks series mentions the associative and distributive law in explicit manners. The result of the qualitative data shows that the basic laws of arithmetic is often described in other contexts. Many examples in the textbooks makes generalizations that may lead to limited understanding of the basic laws and rules of arithmetic.

  • 4.
    Andersson, Julia
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    För- och nackdelar med olika undervisningsmaterial: En intervjustudie om undervisningsmaterial vid introduktionen av subtraktion2016Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of this study is to look into what teaching material seven teachers choose to use during the introduction of subtraction.The study has been inspired by life-world phenomenology, and has been conducted through semi-structured interviews, where the qualitative datasets later have been analyzed through different themes. The results show that all of the teachers agreed that concrete and laboratory materials where to prefer when introducing subtraction. The textbooks were used at a later stage to help the pupils consolidate their knowledge at a more abstract level.

  • 5.
    Axelsson, Johanna
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Geometriska mönster i Favorit matematik: En läromedelsgranskning av Favorit matematik i årskurserna 1-32017Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Algebra in mathematics is so much more than just equations and calculations. Through the primary grades in school, geometric patterns are a part of the Swedish curriculum. By working with patterns we prepare the students for the more difficult algebra to come, and they learn to express themselves generally. Many studies show the benefits with the work with patterns in the primary grades, for example students learn how to see connections and how to generalise.

     

    This study is a teaching material study of the material Favorit matematik in the primary grades. The aim is to contribute knowledge about which mathematical abilities students are given chance to develop by working with patterns in Favorit matematik. Focus is also to study what type of pattern, of repeating patterns and growing patterns, that are more processed in the material. The method that has been used is a chart where the patterns and the abilities were written into.

     

    The result shows flaws when it comes to patterns in Favorit matematik. The teacher has to work with this along with the material to give the students chance to explore every mathematical ability. Nor is it shown that patterns are something within the algebra, it is mostly linked to other areas from the curriculum. Mostly the exercises were about completing a pattern that was repeating, and therefore the majority of the exercises was connected to the first ability, which is about solving problems and evaluate your strategies and methods. The result also shows that growing geometric patterns is not presented as much as repeating patterns.

  • 6.
    Bjenning, Caroline
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Levenby, Jessica
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Sociomatematiska normer: Skuggan i matematikklassrummet2016Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Det valda ämnesområdet för litteraturstudien är sociomatematiska normer, eftersom dessa är oundvikliga och förekommer i alla matematikklassrum. Forskningen kring ämnet är begränsad trots att sociomatematiska normer är en del av lärares och elevers vardag. Syftet med litteraturstudien är att belysa sociomatematiska normer som förekommer i den dagliga verksamheten i skolan samt deras påverkan. Utgångspunkten i studien är både ur ett lärar- och ur ett elevperspektiv. Följande frågeställningar behandlas för att uppnå syftet med studien:

     

    • Hur skapas sociomatematiska normer av lärare och elever i matematikklassrummet?
    • På vilket sätt kan olika sociomatematiska normer samt en förändring av dessa påverka elevers lärande?

     

    Tillvägagångssättet i studien är informationssökning i olika söktjänster. Den funna analyserade litteraturen är vetenskapligt granskad och riktar sig mot grundskolan. Vidare består litteraturen av både internationell och nationell forskning. Resultatet i studien visar att sociomatematiska normer skapas genom interaktion mellan elever och mellan elever och lärare. Genom ett sådant skapande av dessa normer uppstår ett flertal lärandemöjligheter för såväl lärare som elever. Sociomatematiska normer är unika för varje elevgrupp och matematikklassrum. Elevers lärande påverkas av sociomatematiska normer och för att påverkan ska vara positiv krävs acceptans av och förståelse för dessa normer från både lärare och elever.  Även vid inkludering och exkludering av normer i matematikklassrummet påverkas elevers lärande.

     

    Sociomatematiska normer är ett komplext ämne och forskning visar att lärare har bristfällig kunskap kring ämnesområdet. Följaktligen finns ett stort behov av fortsatt forskning och en utveckling av kunskaper kring sociomatematiska normer. 

  • 7.
    Burman, Maja
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Clemborn, Frida
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    PCK: – En litteraturstudie om pedagogical content knowledge samt hur det kopplas till lärare och matematikundervisningen2017Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
  • 8.
    Clemborn, Frida
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    ”Allt ligger ju på läraren hur en lektion blir”: En kvalitativ studie om lärares planeringsförmåga av matematikundervisning i grundskolans årskurs F-32018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Att arbeta som lärare kräver att du erhåller vissa kunskaper. Du ska ha tillräckliga ämneskunskaper och pedagogiska kunskaper. Du ska även ha kunskaper om hur planering, utförande och utvärdering av undervisningen sker. Att skapa en lektionsplanering och integrera alla nödvändiga delar av en undervisning är komplext. Därför utvecklade Shulman (1987) sex processdelar undervisningen följer.

    Frågan är om dagens lärare använder dessa kunskaper. Syftet är därför att bidra med förståelse om vilka kunskaper lärare besitter samt hur planering av undervisning sker. Detta genom att låta lärarna planera en undervisning inom ämnet volym. Studien genomfördes genom kvalitativa intervjuer med fem stycken lärare. Resultatet av studien visar att alla lärarna följde processdelarna. Det var emellertid mycket som skiljde sig mellan lärarna. Forskning säger att erfarenhet, utbildning och fortbildning har betydelse för lärares planeringsförmåga. Resultatet av studien visar att detta i viss utsträckning har betydelse och att lärares förmågor är olika. Dock visar egen analys att det mer beror på att alla lärare är olika individer.

  • 9.
    Debreceni, Hanna
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers uppfattningar av mönster: En kvalitativ studie om hur elever i årskurs 1 uppfattar upprepade och växande geometriska mönster2017Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The work of patterns, through its content of structures and relationships, is an important part in the area of algebra. Teaching pupils in the lower grades about mathematical patterns, helps them to understand the more complex algebra. The study is a qualitative interview study inspired by phenomenography, where pupils in grade 1 were asked to solve a number of tasks related to repeating as well as growing geometric patterns. Based on this method, the study aims to contribute knowledge about how pupils in the younger ages perceive mathematical patterns. In connection with the purpose of this study it also contributes to the understanding about what the critical aspects connected to the two types of patterns could be. Once knowledge of the above-mentioned parts is available, teachers can design their education in a way that benefits pupils' learning.

    The result shows a number of critical aspects associated with the understanding of repeating and growing patterns. It may be critical to identify the repeating part in a repeated pattern and to continue a repeated pattern according to the same structure. Furthermore, it may be critical to distinguish a growing structure and a regular numerical connection in the growing patterns. Discovering a general relationship in a growing geometric pattern is another critical aspect that appears in the study.

    Pupils perceive patterns in many different ways, which teachers should keep in mind when planning and conducting teaching. There are important aspects that should be made visible in the education so that pupils can develop an understanding of repeating and growing patterns.

  • 10.
    Dlamini, Emmanuel
    et al.
    Wits University.
    Venkat, Hamsa
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research. Wits School of Education, University of the Witwatersrand, Johannesburg, South Africa.
    Askew, Mike
    Monash University, Melbourne.
    A modeling and model approach: performance on multiplication2015In: Proceeding of the 23rd annual meeting of the Southern African association for research in mathematics, science and technology / [ed] D. Huillet, 2015, p. 61-68Conference paper (Refereed)
  • 11.
    Ekdahl, Anna-Lena
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Grade 3 and 4 students' different ways of discerning Mathematical patterns2013In: Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education: Vol.5 (p.224). Kiel, Germany:PME / [ed] Lindmeier, A. M. & Heinze, A. (Eds.), 2013, p. 224-Conference paper (Refereed)
  • 12.
    Ekdahl, Anna-Lena
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Björklund, C.
    Runesson Kempe, Ulla
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Finger Patterns as means to experience numbers' part-part-whole relations.2018In: Proceedings of the 42nd Conference of the International Group for Psychology of Mathematics education / [ed] E. Bergqvist, M. Österholm, C. Granberg & L. Sumpter, Umeå: PME , 2018, Vol. 5, p. 42-42Conference paper (Other academic)
  • 13.
    Ekdahl, Anna-Lena
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Kullberg, Angelika
    University of Gothenburg, Sweden.
    Different learning possibilities in preschool mathematics from the same task2018Conference paper (Refereed)
    Abstract [en]

    In this paper one specific task in a series of tasks focusing on part-part-whole relations of the ten first natural numbers and finger patterns for structure number relations, is presented and discussed. The tasks were designed, planned and enacted in an intervention program conducted in Swedish preschool during an eight-month period. In the program nine preschool teachers worked in close collaboration with the research team in planning how to enact tasks with their 5-year-old children, in an iterative process. The specific task, called the ‘snake game’, consists of five or ten beads on a string, some of the beads where to be hidden and the children would find out the hidden part by using structured finger patterns. The task was designed in accordance with the variation theory assumptions that certain aspects need to be discerned as dimensions of variations. The aim of the paper is to examine which dimensions of variations that were opened up by the teachers and what was made possible for the children to learn from the enactment of the ‘snake game task’. The data set includes 67 video observations from the teacher’s enactment of the task. The results suggest that what seems to be a ‘non-complex task’ (five/ ten beads on a string) offers rich mathematical experiences and has potential to bring fore important aspects of numbers and number relations. However, depending on which dimensions of variations that were opened up reveals different learning possibilities.

  • 14.
    Ekdahl, Anna-Lena
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Lindgren, Helen
    Högskolan i Borås.
    Gemensamt fokus på förskolebarns taluppfattning och aritmetiska förmågor: Ett samverkansprojekt där teori och praktik flätas samman2018Conference paper (Refereed)
    Abstract [sv]

    I forskningsprojektet FASETT[1] flätas teori och praktik samman. Projektet syftar till att generera kunskap om barns tidiga taluppfattning, utifrån delvis andra perspektiv än de dominerande inom fältet, och se hur en pedagogisk verksamhet i samverkan med forskare kan bidra till barns utveckling av aritmetikfärdigheter.

    I projektet arbetade nio förskollärare och 65 femåringar på fem förskolenheter i ett tätt samarbete med en grupp forskare under en åttamånadersperiod. Deltagarna träffades kontinuerligt, diskuterade och fördjupade sig i aritmetiken med fokus på aktiviteter gällande de tio första talens del-del-helhetsrelationer. Aktiviteterna var till viss del välkända, men bearbetade utifrån tidigare forskningsresultat och variationsteorin (Neuman, 1987; Marton, 2015). Utgångspunkten och reflektionerna vid gruppträffarna var intervjuer av barnens olika sätt att uppfatta tal och lösa enklare aritmetikproblem samt lärarnas iscensättande av de planerade aktiviteterna. Genom att gemensamt diskutera de filmade aktiviteterna kunde aktiviteterna förfinas och förskollärarnas didaktik utvecklas för att möta barnens behov. 

    Pågående analyser visar att designen av projektet möjliggjort för förskollärare och forskare att i kollaboration implementera ett alternativt teoretiskt underbyggt sätt att utveckla barns taluppfattning och förmåga att   lösa enklare aritmetikproblem genom att exempelvis använda sig av fingrarna som redskap för att strukturera talrelationer och inte enbart räkna ’ett till ett’.

    Analysen av de barnintervjuer som gjordes innan aktiviteterna introduceras och de barnintervjuer som gjordes efter forskningsprojektets slut visar att förskollärarnas målorienterade processer med största sannolikhet haft effekter på barnens aritmetiska förmågor. De preliminära resultaten indikerar att valet av att fokusera på ett fåtal aktiviteter möjliggjorde för en djupare reflektion kring teoretiska antaganden och vad barn behöver få syn på för att lära sig om tal och talrelationer.

    [1] ”Förmågan Att Sinnligt Erfara de Tio första Talen som nödvändig grund för aritmetiska färdigheter”, finansierat av Vetenskapsrådet 2015-2018

  • 15.
    Ekdahl, Anna-Lena
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Teachers’ responses to incorrect answers on missing number problems in South Africa2015In: The twenty-third ICMI Study: Primary mathematics study on whole numbers: Proceedings / [ed] Xuhua Sun, Berinderjeet Kaur and Jarmila Novotná, 2015, p. 431-439Conference paper (Refereed)
    Abstract [en]

    This paper examines differences in how three Grade 3 South African teachers responded to students’ incorrect answers in whole class teaching of the part-whole relationship in additive missing number problems. Nine video recorded lessons, taught by three teachers, were analysed, with attention paid to teaching episodes containing incorrect students’ answers. The variation theoretical analyses indicated differences in the ways teachers responded to incorrect answers. We argue that different ways of responding to incorrect answers may provide different learning possibilities.

  • 16.
    Ekdahl, Anna-Lena
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Venkat, Hamsa
    Jönköping University, School of Education and Communication. University of Witwatersrand, Johannesburg, South Africa.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Coding teaching for simultaneity and connections: Examining teachers’ part-whole additive relations instruction2016In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 93, no 3, p. 293-313Article in journal (Refereed)
    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.

  • 17.
    Engström, Sofie
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Positionssystemet: Elevers möjligheter att förstå positionssystemet i årskurs 62016Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Swedish students are showing a continuous deterioration of knowledge about the concept of mathematical place-value and by looking at a closer analysis, made by the TIMSS, it was made even more visible that the concept of place-value, which is associated with our positional system, is very central to students' understanding of the number concept. The PISA - report further demonstrates that it is possible to discern the deteriorating knowledge of students' perceptions of digits’ place-values in calculations of standard algorithms. The study is carried out because I want to explore as well as contribute to the research on the opportunities students are given to understand the positional system.

    The study aims to create knowledge on students' opportunity to understand the positional system in the sixth grade. The questions the study seeks to answer are the following:

     What are the different patterns of variation the students meet during the lesson about the positional system?

     Which are the critical aspects that can be discerned when listening to the students' descriptions of the positional system?

    To get answers to these questions, and thus fulfill the purpose of the study, different teachings have been observed and students have been interviewed. The theory of this work is variation theory. This theory was chosen because it makes it possible to understand many different ways to treat a selected subject of matter as well as to relate it to students' opportunities to learn.

    Results of the study show that students encounter several different parts of the pattern of variation in the teaching about the positional system, including contrast, generalization and separation. By taking in the students' descriptions of the positional system, three critical aspects could be discerned. These were the place-value together with the significance of zero in multi-digit numbers, numbers properties and lastly number sence. The conclusion is that students have different opportunities to develop an understanding of the positional system in the sixth grade. The possibilities are different depending on which critical aspects that are distinguished by the students in the education of the positional system.

  • 18.
    Erixson, Lea
    et al.
    Ribbaskolan, Gränna.
    Frostfeldt Gustavsson, Karin
    Ribbaskolan, Gränna.
    Kerekes, Klara
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research. Linköpings universitet.
    Lundberg, Birgitta
    Ribbaskolan, Gränna.
    Att se det som inte syns – om talföljder i årskurs 3 och 42013In: Forskning om undervisning och lärande, ISSN 2000-9674, no 10, p. 64-81Article in journal (Refereed)
    Abstract [sv]

    Internationell forskning och undersökningar visar att elever har svårt att lära sig algebra i allmänhet samt att konstruera och beskriva talföljder i synnerhet. Undervisningen, som fokuserar på olika undervisningsmetoder i stället för på det som krävs för att lära, anses vara en av de viktigaste orsakerna till detta.

    Studiens syfte är att, utifrån ett variationsteoretiskt perspektiv, studera det som är kritiskt för elever i årskurs 3 och 4 när de ska lära sig att konstruera och beskriva vad som kännetecknar olika talföljder. I artikeln beskrivs de identifierade kritiska aspekterna och hur dessa gjordes synliga i undervisningen genom variation. Learning study användes som metod.

    Resultatet visar att eleverna utvecklade förmågan att beskriva talföljder när det i undervisningen gavs möjlighet för dem att urskilja sambandet mellan talen och talens inbördes förhållande till varandra, urskilja helheten, förstå att det finns ett system mellan talen som kan varieras i oändlighet och upptäcka att talföljder kan byggas upp på olika sätt. Detta benämns i studien som kritiska aspekter.

  • 19.
    Faag, Julia
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Faag, Louise
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Programmering som matematisk lärandemiljö: Kan programmering bidra till att utveckla matematiska förmågor?2017Student paper other, 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Finns det matematik i programmering? Den svenska regeringen och Skolverket verkar ha den uppfattningen. Det finns dock åsikter som pekar på motsatsen.

    På grund av samhällets ökade digitalisering gav regeringen Skolverket i uppdrag att ta fram ett förslag på Nationella it- strategier för skolväsendet (2016). Förslaget godtogs av regeringen den 9 mars 2017. Avsikten med förslaget är att utveckla elevers digitala kompetens genom att bland annat införa programmering i läroplanen, främst i kursplanen för matematik. Den vetenskapliga grunden för förslaget har däremot ifrågasatts. Denna litteraturstudie syftar därför till att undersöka förhållandet mellan programmering och matematik gällande vilka förmågor som kan utvecklas. De förmågor som programmering i forskning beskrivs kunna utveckla, jämförs med de fem förmågorna i kursplanen för matematik. Studiens slutsats är att forskning visar att programmering kan utveckla flera förmågor som kan anses vara matematiska, exempelvis problemlösningsförmåga, kreativ förmåga och samarbetsförmåga.

  • 20.
    Fenelius, Beatrice
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Undervisning om tals del-helhetsrelationer: En variationsteoretisk studie i förskoleklass2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The following thesis is an intervention study based on arithmetic teaching and part-whole relationships of numbers. Some researchers argue that part-whole relationships are a basis for further arithmetic skills, while other re-searchers believe that it is more important that students develop counting abilities. The pur-pose of this study is to investigate how teach-ing of part-whole relationships in addition and subtraction can be designed from a variation theory perspective. The object of learning for the teaching sequence is to identify an un-known number of a part-whole relationship, in the number range 0-10. The study was conducted with 13 students in preschool class through a pre-test, a teaching sequence of three lessons and a post-test. The teaching sequence and the tests were vide-otaped to provide a basis for analysis. The les-sons were designed to show different patterns of variation such as contrast, separation, gen-eralization and fusion so that the students would be able to discern critical aspects. For example, "Part-whole bars" and commuta-tivity were used in the lessons, in order to highlight the critical aspects. The critical as-pects that emerged in the result were to discern that the two parts fit in and together is as much as the whole, be able to use two numbers in a relation between the whole and the parts to find the third number, be able to use the whole and one part to find the second part and to be able to use knowledge about previous part-whole relationships to find the unknown number in new part-whole relationships.

  • 21.
    Flarup, Andrea
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Hur elever i årskurs 1 förstår likhetstecknet2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This study aims to analyze the understanding of the equal sign among students in first grade and to identify critical aspects in their descriptions of the equal sign. Data were collected from a classroom observation.  Also interviews with five students from the observed class were made.  The students were interviewed about their understanding of the equal sign and about their perceptions of mathematical tasks with focus on the equal sign.  There are elements in the study of Phenomenography and Variation Theory. The results of the study show that students have both relational and instrumental understanding of the equal sign. The result also show that different students have identified different critical aspects of the equal sign during their descriptions. The conclusion is that students’ ability to identify a critical aspect depends on the structure of the task.

  • 22.
    Flarup, Andrea
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Abramsson, Matilda
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Ett plus ett blir två: Introduktion av likhetstecknet i förskoleklass och årskurs 12015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 23.
    Fredriksson, Amanda
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Ek, Josefin
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Van Hiele´s teori: En litteraturstudie om elevers lärande och geometriundervisning utifrån van Hiele´s teori2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Ämnesområdet för litteraturstudien är matematik, mer specificerat mot området geometri, gällande elevers lärande om geometriska figurer utifrån van Hiele´s teori. Det finns flera problemområden inom matematik, ett av dem är elevers svårigheter att benämna geometriska figurer och dess egenskaper med korrekt terminologi. Elevers matematiska språk och erfarenheter anses idag vara influerat av vardagligt språk, exempelvis benämns fyrhörning frekvent som fyrkant. Därför är syftet med vår litteraturstudie att klargöra relationen mellan undervisning gällande geometriska figurer och elevers lärande, utifrån van Hiele´s teori. Genomförande av denna litteraturstudie har gjorts genom analys av vetenskapliga publikationer i form av doktorsavhandlingar, forskningsartiklar och en antologi. Publikationerna som använts har hittats i databaserna ERIC och Google Scholar. Analys har gjorts med hjälp av en analysmall för att synliggöra likheter och skillnader som framgick mellan publikationerna. Urvalet som gjorts har baserats på våra frågeställningar. Genom denna litteraturstudie har vi konstaterat att van Hiele´s teori består av fem tankenivåer. Varje nivå uppnås successivt genom en stegvis progression. Progressionen har sin utgångspunkt i det konkreta och strävar mot det abstrakta. Inom van Hiele´s teori har språket en väsentlig roll och lärandet sker i en social kontext. Laboration och konkret material används som medel för att nå nästkommande nivå. En god begreppsförståelse är utvecklad när elever har nått abstraktion, vilket gör att konkret material inte behöver tillämpas mer. Vår slutsats är, inom geometriundervisning måste det finnas en progression från konkret till abstrakt, för att elever ska kunna utveckla god begreppsförståelse. Det finns möjligheter att tillämpa van Hiele´s teori i praktiken, eftersom den kan användas som både undervisningsmetod med hjälp av stöttande faser och som verktyg för bedömning. Van Hiele´s teori kan ge både elever och lärare möjligheter att utveckla matematiska kunskaper inom området geometri och därför anser vi van Hiele´s teori som relevant inför vårt kommande yrke som lärare.

  • 24.
    Garancz, Gabriel
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Lundman, Jessica
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Att testa förståelse för vektorbegreppet i gymnasiets matematikkurs2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Some of the changes made in 2011 in the curriculum of secondary schools was that vectors were added to the central content to the plan of the subject Mathematics 1c. This study deals with the design of exercises that test students' understanding of the vector concept, where the selection in the study includes high school students who have read or are reading the course Mathematics 1c within the period of the thesis. The method used was semi-structured interviews where exercises acted as questions and the student had to answer attendant questions, this to observe the students' thoughts, speculations and hindrances.

    The responses that students gave to the exercises created a wide range of different answers, a space of different responses. This space of different responses became available choices to the exercises in the final test, that the study have brought forth. The test has thus created well-tried exercises that were created with a variety of starting points. The starting points in the task of creating the exercises took inspiration and support from the high school curriculum as well as the plan of the subject Mathematics 1c alongside research about opportunities and hindrances in the understanding of the vector concept. During the creation as well as during the collection of data, changes has been made to the test and we have observed the hindrances and thoughts that the students had, which is available in this thesis.

  • 25.
    Gol Mohammadi, Nazanin
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Zhang, Ying
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Matematik på svenska och kinesiska: en komparativ studie mellan svenska och kinesiska läroböcker inom området andragradsekvationer2011Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med den här studien är att 1) jämföra upplägget av läromedel i Sverige och i Kina (Shanghai), 2) jämföra de lösningsmetoder som presenteras i läromedlen samt 3) undersöka hur läromedlen präglats av den matematiska kulturen i respektive land. Vår studie bygger på APOS-teorin (kompletterat med PCK), en teori som bland annat beskriver processen av matematikinlärning i olika steg. Studien har vidare gått till så att vi gjort en textanalys och studerat de valda läromedlen, Matematik år 8 (Liu, 2006) för Kina och Exponent B Röd (Gennow m.fl., 2004) för Sverige, för att hitta såväl likheter som skillnader mellan läromedlen. Resultatet av studien sammanfattas till att läromedlen till största delen tar upp samma metoder inom det studerade området. Den upptäckta skillnaden ligger i de övningsuppgifter som presenterats. Exponent B Röd (Gennow m.fl., 2004) innehåller betydligt fler övningsuppgifter som dessutom i många fall har en tillämpande karaktär, medan uppgifterna i Matematik år 8 (Liu, 2006) innehåller färre uppgifter, som oftast är abstrakta och aritmetiska. Detta har vi tolkat som ett resultat av den rådande kulturen i respektive land. Den kinesiska skolan har en starkare betoning på prov och har därför läroboken som en förberedelse inför kommande provtillfällen. Den svenska skolan däremot använder termer som förståelse och lustfyllt lärande, vilket medför att även läroboken försöker spegla detta genom vardagsnära kopplingar, främst i form av övningsuppgifter.

  • 26.
    Gunnarsdotter, Ylva
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    IKT i matematikundervisningen: Didaktiska forskningsresultat om hur IKT används samt hur det påverkar elever och lärare.2016Independent thesis Basic level (university diploma), 10 credits / 15 HE creditsStudent thesis
  • 27.
    Gunnarsson, Elsa
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Problemlösning med sju- och åttaåringar: En fenomenografiskt inspirerad studie av elevers olika lösningsstrategier av ett matematiskt problem2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Problemlösning genomsyrar hela läroplanen och är en viktig del av matematik-undervisningen i skolan (Skolverket, 2011a). Att lösa problem kommer naturligt för barn och det är lärarens uppgift att ta vara på den förmågan och hjälpa elever att bli effektiva problemlösare. Förmågan att lösa problem är en viktig kunskap som varje elev har fördel av att kunna (Lester, 1996). Studiens syfte är att undersöka variationen av problemlösnings-strategier som elever använder samt undersöka hur eleverna resonerar när de löser ett problem. 39 elever från två olika skolor i England och Sverige fick lösa ett matematiskt problem och sedan intervjuades 12 av dem med olika lösningsstrategier. Resultatet visade att eleverna använde sig av fyra olika kategorier av lösningsstrategier. De olika kategorierna var: lösningsstrategi genom addition, lösningsstrategi genom addition och subtraktion, lösningsstrategi genom att gissa och resonera, och lösningsstrategi genom att söka mönster. Det fanns även en grupp elever som inte hade någon utläsbar lösningsstrategi. Slutsatsen av studien är att elever behöver explicit undervisning i problemlösning för att till fullo kunna behärska den.

    Problem solving permeates the Swedish national curriculum and it is an important part of mathematics education (Skolverket, 2011a). To solve problems comes naturally to children and it is the teacher’s task to harvest this ability and help pupils to be effective problem solvers. The ability to solve problems is an important knowledge and if known provides an advantage in life (Lester, 1996). The purpose of this study is to investigate the variation of problem solving strategies that pupil use and to investigate their mathematical reasoning while solving a mathematical problem. 39 pupils from two different schools in England and Sweden got to solve a mathematical problem and then 12 of them, which had different solution strategies, were selected for an interview. The result showed that the pupils used four categories or solving strategies. The categories were: finding a solution though addition, finding a solution though both addition and subtraction, finding a solution though guessing and reasoning and finding a solution though seeking patterns. There was also one group of pupils who did not have a distinguishable solution strategy. The conclusion of this study is that pupils need explicit teaching about problem solving to be able to fully master it.

  • 28.
    Gunnarsson, Robert
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Arithmetic expressions with multiple operations - How to solve it?2016In: Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Volume 2, 2016, Vol. 1, p. 298-Conference paper (Refereed)
  • 29.
    Gunnarsson, Robert
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Ekdahl, Anna-Lena
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Landén, Josefine
    Jönköping University, School of Education and Communication.
    Tegnefur, Jenny
    Jönköping University, School of Education and Communication.
    Students’ strategies to continue geometric number sequences2016Conference paper (Refereed)
    Abstract [en]

    Number sequences can be useful tools for teaching generalization, functions, or variables, for instance. Consequently, there are many studies that have studied students’ perception of number sequences and the strategies used to continue those sequences. However, a large part of the studies have been using arithmetic or quadratic number sequences. In this paper we present a study of students’ strategies to continue non-contextualized geometric number sequences. Interview data from 18 students in grades 9 to 12(age 15-19) (in Sweden) was analysed. Five qualitatively different strategies have been discerned in the data. These strategies are not completely overlapping the strategies previously described in literature.

  • 30.
    Gunnarsson, Robert
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Hernell, Bernt
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Sönnerhed, Wang Wei
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    On the use of emphasizing brackets when learning precedence rules2012In: Evaluation and comparison of mathematical achievement: Dimensions and perspectives: Proceedings of Madif 8 / [ed] C Bergsten, E Jablonka, M Raman, Linköping: Svensk förening för matematikdidaktisk forskning (SMDF) , 2012, p. 209-210Conference paper (Refereed)
  • 31.
    Gunnarsson, Robert
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Hernell, Bernt
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Sönnerhed, Wang Wei
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Useless brackets in arithmetic expressions with mixed operations2012In: Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education / [ed] T.Y. Tso, The International Group for the Psychology of mathematics Education , 2012, p. 2-275-2-282Conference paper (Refereed)
    Abstract [en]

    There can be different intentions with brackets in mathematical expressions. It has previously been suggested that mathematically useless brackets can be educationally useful when learning the order of operations in expressions with mixed operations. This paper reports how students (12-13 years) deal with the implicit mental conflict between brackets as a necessary part of the order of operations and brackets to emphasize precedence. The students taking part in this quasi-experimental study were instructed on the order of operations, but were also indirectly exposed to different use of brackets. It is concluded that emphasizing brackets impede the transfer from a left-to-right computation strategy to the use of precedence rules.

  • 32.
    Gunnarsson, Robert
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Karlsson, Annasara
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Brackets and the structure sense2014In: Development of Mathematics Teaching: Design, Scale, Effects: Proceedings of MADIF 9, The Ninth Swedish Mathematics Research Seminar / [ed] O. Helenius, A. Engström, T. Meaney, P. Nilsson, E. Norén, J. Sayers & M. Österholm., Linköping: Svensk förening för MatematikDidaktisk Forskning - SMDF, 2014, p. 47-55Conference paper (Refereed)
    Abstract [en]

    Brackets are essential structure elements in mathematics expressions. However, students have shown to have scattered understanding of the concept of brackets and how they are used in mathematical expressions. In this paper we present data that illustrate students’ perceptions of the word “brackets” and how these perceptions influence their use of brackets in numerical expressions. Based on our data we argue that the teaching of the concept of brackets also need to describe brackets as ordered pairs where each symbol has a unique counterpart and that insertion of brackets can, but does not have to, modify the structure of an expression.

  • 33.
    Gunnarsson, Robert
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Wei Sönnerhed, Wang
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Hernell, Bernt
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?2016In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 92, no 1, p. 91-105Article in journal (Refereed)
    Abstract [en]

    The hypothesis that mathematically superfluous brackets can be useful when teaching the rules for the order of operations is challenged. The idea of the hypothesis is that with brackets it is possible to emphasize the order priority of one operation over another. An experiment was conducted where expressions with mixed operations were studied, focusing specifically on expressions of the type a ± (b × c) with brackets emphasizing the multiplication compared to expressions of the type a ± b × c without such brackets. Data were collected from pen and paper tests, before and after brief (about 7 min) instructions, of 169 Swedish students in year 6 and 7 (aged 12 to 13). The data do not seem to support the use of brackets to detach the middle number (b) from the first operation (±) in a ± b × c type of expressions.

  • 34.
    Hérard, Jenny
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Karlsson, Sofi
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Appar i matematikundervisningen: En litteraturstudie om appars användning i de lägre åldrarna2017Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Allt fler skolor satsar idag på att införa digital teknik som ett verktyg i undervisningen, vilket innebär att elevers tillgång till en dator eller surfplatta har ökat. Även om den digitala tekniken har potential att förbättra lärandet, om den används på ett medvetet sätt, sker användandet av den här teknik relativt sparsamt i dagens matematikundervisningen. Den digitala tekniken har således god potential att påverka lärandet positivt men det är viktigt att införandet av tekniken görs med ett tydligt definierat syfte av det som har för avsikt att läras ut och hur det ska ske. Det är en utmaning, inte minst då forskning om hur appar kan användas i matematikundervisningen är begränsad.

     

    Syftet med litteraturstudien är att bidra med kunskap om vilken funktion appar kan fylla för matematikundervisningen i de lägre åldrarna. Litteraturstudien har en kvalitativ ansats där innehållet i 9 tidskriftsartiklar och 2 konferensbidrag har granskats och tolkats. Vid materialanalysen arbetades olika teman fram som sedan användes för att presentera resultatet.

     

    Resultatet visade att användning av appar kan bidra till att höja nivån av motivation och engagemang hos elever. Det framgick även att appar erbjöd en del unika möjligheter som kan vara svåra att uppnå med traditionell undervisning. Till de här hör möjlighet till omedelbar feedback och vägledning, individanpassning och användning av flera sinnen. Slutsatsen är att användning av appar kan ha potential att stärka elevers lärande. Lärare har en viktig roll i att kunna avgöra när en app är ett lämpligt verktyg för att uppnå lektionsmålet eller när traditionell undervisning lämpar sig bättre.

  • 35.
    Jakobsson, Jonas
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Svärd, Jennie
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Utomhuspedagogik som arbetsmetod: i matematikundervisning för elever i årskurs 4-62018Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med litteraturstudien var att utifrån ämnesdidaktisk forskning i matematik belysa hur utomhuspedagogik som arbetsmetod kan påverka elevers lärande i matematik i årskurs 4–6. Avsikten var att kartlägga hur utomhuspedagogik kan bidra till elevers lärande, vilka förmågor elever kan utveckla i matematik med hjälp av arbetsmetoden samt vilka hinder som kan påverka användningen av utomhuspedagogik i matematikundervisningen. Vidare var avsikten att genomföra en litteraturanalys över likheter och skillnader mellan de inkluderade vetenskapliga texterna. De bestod av två forskningsrapporter och flertalet tidskriftsartiklar, vilka resultatet grundades på. När litteraturanalysen var genomförd föll innehållet ut i kategorierna: matematik utomhus, kombinera utomhusmiljön med inomhusmiljön, elevers lust till att lära, utomhuspedagogikens inverkan på minnet och hinder med arbetsmetoden. Resultat som beskrivs i forskningen beträffande elevers lärande i matematik visar att utomhuspedagogik som arbetsmetod i kombination med traditionell klassrumsundervisning kan stärka elevers begrepps- och kommunikationsförmåga. Variationen av undervisning utomhus och inomhus synliggör även matematikämnets roll i vardagen för elever. Resultatet visar även på positiva effekter gällande elevers minne, motivation, lust till att lära och kognitiva förmåga. I resultatet beskrivs även att väder, ljudnivå, tid, lärares oro och självförtroende kan utgöra hinder för användande av utomhuspedagogik som arbetsmetod i matematikundervisning. Slutsatsen utifrån litteraturstudiens resultat är att utomhuspedagogik i kombination med traditionell klassrumsundervisning påverkar och motiverar elevers matematiklärande positivt.

  • 36.
    Johansson, Jesper
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Undervisning i elementär algebra med generella symboler: eller hur jag blev kompis med μ2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Studiens syfte är att testa, utveckla och jämföra elevers förmåga att lösa förstagradsekvationer samt förenkla algebraiska uttryck där den obekanta variabeln är av, för eleverna, familjär natur, x alternativt y, respektive en godtycklig symbol, till exempel λ eller σ. Deltagarna i studien har gjort ett inledande oförberett test varefter de under höstterminen i årskurs 1 har fått undervisning i ämnet under 32 timmar. Vid terminens slut har eleverna genomfört ett avslutande och likvärdigt test. Totalt har 87 elever på gymnasieskolans teknikprogram deltagit i studien. Materialet i rapporten kommer från elever som började på Ållebergsgymnasiet i Falköping 2011 och 2012.

    Här visas med 99 procentig och 72 procentig säkerhet att lösning av förstagradsekvationer och förenkling av algebraiska uttryck upplevs som svårare när de genomförs med en generell symbol, som λ eller σ, jämfört med en känd dito, som x, då elever kommer direkt från grundskolan. Det visas även med 85 procentig och 75 procentig säkerhet att lösningsfrekvensen för uppgifterna är högre då symbolen x används.

    Efter 32 timmars extraundervisning i symbolhanterande algebra, där de introducerats för det grekiska alfabetet, så gick lösningsfrekvenserna på likartade uppgifter upp med 30 – 200 % och hamnade i området 0,85 – 0,95. Skillnaden i upplevd svårighetsgrad och lösningsfrekvens försvann inom ekvationslösningen men kvarstod för förenklingsuppgifter.

    Studiens resultat visar på en linjär korrelation mellan resultat på det inledande diagnostiska provet och betyg från grundskolan. Elever med betyget G presterar med mer än 90 procentig säkerhet sämre än elever med betyget VG eller MVG. En linjär korrelation mellan lösningsfrekvens och upplevd svårighetsgrad kunde befästas vid såväl det inledande som avslutande testet.

    Resultaten har jämförts med resultat från en liknande studie som gjordes kring 2000. Vi fann att eleverna i denna studie högst troligt sämre förberedda från grundskolan då de börjar gymnasiet men efter en termin är de minst lika bra eller bättre. 

  • 37.
    Johansson, Lovisa
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Tre gånger så många eller Tre gånger fler: En kvantitativ studie kring hur elever tolkar vanliga aritmetiska jämförelser2016Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Mathematics should, as far as possible, be expressed unambiguously – there should be no doubt what is meant with a mathematical expression. There is no reason why arithmetic comparisons should be treated differently. However, there are suggestions that arithmetic comparisons could mean different things, depending on how they are presented.The aim of this study is to investigate how students in the grades 4-6 (age 10-13) interpret arithmetic comparisons in mathematical text assignments. Arithmetic comparisons refer to as proportional relations which describe a relationship, for example between two quantities. Do students separate comparisons as three times as many and three times more? 

    How do students interpret the comparisons double and half? 

    Do the interpretations differ between boys and girls? 

    Do the interpretations differ between students with Swedish as first language and students with Swedish as second language?

    The study is based on a survey in which 188 students participated. The analysis shows that the absolute majority of students in the study interpret times as many and times more as synonyms. Most students also make the generally accepted interpretation of the comparisons double and half.

  • 38.
    Johansson, Lovisa
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Jonsson, Josefine
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Structure sense: Ett matematikdidaktiskt begrepp som håller på att formas2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Structure sense är ett begrepp som nyligen har börjat användas inom matematikdidaktik föratt beskriva elevers förståelse för matematiska strukturer. Syftet med det härexamensarbetet är att utforska begreppet structure sense genom att jämföra olikadefinitioner av begreppet. Vidare är syftet också att jämföra structure sense med tre andraliknande begrepp: symbol sense, personal structure och awareness of mathematical pattern and structure. Litteraturstudien bygger på 13 vetenskapliga publikationer som samlades in ochanalyserades. Samtliga publikationer är skrivna av internationella forskare. I resultatetpresenteras och jämförs structure sense utifrån fyra olika definitioner. De olikadefinitionerna av begreppet är riktade mot olika nivåer av matematik men gemensamt förbeskrivningarna är att structure sense uppfattas som en eller flera förmågor som innefattarett mångfacetterat sätt att uppfatta matematisk struktur.

  • 39.
    Jonsson, Isabell
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers olika erfarenheter kring subtraktionstecknet: En kvalitativ studie om nio elevers olika användande och förståelse av subtraktionstecknet2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of this study is to investigate how nine pupils in schoolyear 5 experiences the subtraction sign. The study will present and reveal the pupils’ different perceptions and difficulties regarding the subtraction sign. This study has been performed through qualitative interviews by nine pupils describing how they comprehend the subtraction sign in different contexts. The theoretical point of view in this study is life-world phenomenology. When analyzing the study’s datasets, a number of themes has been formed that later are addressed in the result. The result of this study shows that the pupils considers the subtraction sign as to take something away or to decrease a number. The result also shows that the pupils considers subtraction to be difficult, and errors often occurs because of multiple steps during the calculations. Furthermore, the result reveal that the pupils are not aware of their choice regarding methods and strategies when making calculations in subtraction.

  • 40.
    Jonsson, Josefine
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Att strukturera och beräkna matematiska uttryck: En studie om hur elever i årskurs 5 hanterar utvecklade aritmetiska uttryck2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Some of the difficulties students experience in algebra can be due to lack of understanding of the structure in mathematical expressions. Structure, in this context, refers to how a mathematical entity consists of its parts, and how these parts are related to each other. Previous studies also indicate that students’ difficulties in algebra devolve upon a lack of arithmetical knowledge. In arithmetic, students can manage by using informal methods, while algebraic activities require a greater awareness of mathematical structures. It has therefore been argued that students’ difficulties with algebraic expressions are caused by a lack of knowledge of the structure in arithmetic expressions. 

    The purpose of this study is to investigate how 5th grade students calculate and structure longer arithmetic expressions, meaning numerical expressions with several operations, for example, 5 · 6 + 4 · 2 · 3. This study covers numerical expressions with three or four operations. 

    The study includes 116 students from three different schools. The analysis is based on data from solutions of tasks on a written worksheet. The worksheet consisted of ten arithmetic calculation assignments that the students worked with individually. The analysis of the data revealed different approaches that students used to structure and calculate the arithmetic expressions, particularly four methods were used in several tasks. Through the different approaches that students used to calculate mathematical expressions, different ways to create structure could be discovered. Many students based their calculations on the surface structure of an expression and only a few students seemed to be able to identify the hidden structure of an expression. 

  • 41.
    Josefsson, Elin
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    "Ser ni nu att det betyder samma sak?": En observationsstudie av lärares arbete med representationsformer i matematikundervisningen.2017Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Students experience problems regarding representations in mathematics. Therefore the purpose of this study is to examine how teachers’ work with different representations and how they clarify the connections between representations. The study is based on the principles of variation theory. The data, consisting of observations of six different teachers whole class introductions, have been analysed using Ekdahl, Venkat and Runessons theoretical framework (2016). The study shows that teachers use the representation called “written symbols” most frequent and that the representation vary depending of the subject. The result also shows that teachers use gestures and their verbal expression to clarify the relationship between representations. The conclusion is that teachers use different representations in teaching, and that teachers tend to use different linking actions in mathematics to make the connection between representations visible. 

  • 42.
    Josefsson, Elin
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Källberg, Paulina
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers och lärares möjligheter samt hinder i arbetet med representationsformer i matematikundervisningen2016Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Under vår verksamhetsförlagda utbildning uppmärksammade vi att eleverna i årskurs 4-6 uppvisade kunskapsluckor i användandet av olika representationsformer. Den representationsform som eleverna föredrog att arbeta med var symboler. Det väcktes ett intresse att undersöka varför eleverna i huvudsak valde att använda en representationsform vid beräkningar. Litteraturstudiens syfte är dels att öka förståelse för elevers och lärares arbete med olika representationsformer, dels att belysa på vilka sätt de olika formerna av representationer kan vara till nytta för elever i årskurserna 4-6 i matematikundervisningen. Studien fokuserar även på att synliggöra eventuella problem som kan uppstå vid användandet av representationsformer och hur lärare på ett effektivt sätt kan motivera elever att använda olika representationsformer. Arbetet baseras på vetenskapliga artiklar samt litteratur som hittats via informationssökningar. Resultatet av litteraturstudien visar att med hjälp av representationsformer kan elever öka sin matematiska förståelse samt utveckla sitt matematiska tänkande. Forskningen redovisade att elever ofta upplever svårigheter gällande växlingen mellan representationsformer men poängterade även att lärare upplever svårigheter med representationsformer på grund av bristande kunskap. Vidare visar resultatet att läraren har en viktig uppgift i att skapa och upprätthålla en positiv inställning till arbetet med representationer. Centralt i matematikundervisningen måste också vara att kommunikation mellan elev-lärare och elev-elev kring representationsformer sker för att gemensamt utveckla och befästa kunskaperna.

  • 43.
    Jönsson, Lukas
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Matematik "på riktigt": En kvalitativ studie av lärares uppfattningar om relationen mellan verkligheten och matematikundervisningen2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Mathematics has been used for several thousands of years. Even if the mathematics progresses there is one thing that consist: we still use it in the real world. To work with the connection between the mathematics and the reality is one of the mathematics teachers’ tasks. Previous research is pointing out that the education of mathematics in lower years should aim to prepare the students for higher education studies and an increasingly more demanding society where mathematical knowledge, among them mathematical modeling, is crucial to achieve a deeper connection between the real life and the mathematics. Since the mathematics is so important on several levels and teachers need to be able to motivate their students to work with the subjects that are situated in schools, does the teachers’ own view and attitude towards real-life connected mathematics play a big part.

    The study emanates from the teaching theory Realistic Mathematics Education and aims to contribute with knowledge about how mathematic teachers, who teaches students aged 9-13, can perceive the reality linked to their teaching. This is done by six semi-structured interviews with teachers who are active in teaching mathematics for students aged 9-13. To answer the aim of the study, it emanates from four questions:

    • What do teachers regard as the role of mathematic education in school?

    • How do mathematic teachers use the reality in their teaching?

    • How do mathematic teachers design a task linked to the reality, for their students?

    • How do mathematic teachers reason about a mathematic modeling-task?

    The results shows that teachers are positive towards the use of a real-life connected education in mathematics and that they believe it benefits the students. Several of the teachers in the study associated the term real-life connected teaching in mathematics to larger tasks or projects with the students and could see difficulties in finding the time to plan and execute said activities. When working with tasks that are connected to the reality, the teachers focus more on the process than the final answer. The discussion around the tasks is also something that is also important according the teachers

  • 44.
    Karlsson, Annasara
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Gunnarsson, Robert
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Students' perceptions of brackets2013In: Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, vol. 5 / [ed] A.M. Lindmeier & A. Heinze, Kiel, Germany: IGPME , 2013, p. 85-Conference paper (Refereed)
  • 45.
    Knutsmark, Matilda
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Multiplikativt tänkande: Olika strategier för beräkningar av uppgifter inom multiplikation och division2016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This study focuses on multiplicative thinking among pupils in grade 3. Multiplicative thinking is abstract (Clark & Kamii, 1996), involving applying strategies to solve multiplication and division tasks. The purpose of this study is to examine how pupils use different strategies within multiplicative thinking for multiplication and division. This study was inspired by Grounded Theory. From this theory, a number of semi-structured interviews, observations and analysis of data were made. Eight pupils participated in the interviews, after an initial pilot study. The collected material was based on the pupils' solutions of tasks in multiplication and division, notes from observations of the pupils' solutions and audiotaped interviews. The results show that almost every pupil uses an additive strategy in their solutions of multiplication and division tasks. It also show that only four out of eight pupils could show understanding 0f the connection between the two basic arithmetic operations. From the results, the pupils showed different strategies and solutions within multiplicative thinking, even though they have had the same mathematic education. 

  • 46. Kullberg, Angelika
    et al.
    Mårtensson, Pernilla
    Jönköping University, School of Education and Communication. Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Nilsson, Pernilla
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Vikström, Anna
    Learning study and teachers' change of practice2014Conference paper (Refereed)
  • 47.
    Kullberg, Angelika
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research. Göteborgs Universitet.
    Mårtensson, Pernilla
    Jönköping University, School of Education and Communication, HLK, School Based Research.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Exploration and search for the external and internal horizon of the object of learning2013In: Book of Abstracts: 5th Biennial Conference EARLI 2013, 2013, p. 7-8Conference paper (Other academic)
    Abstract [en]

    Within the phenomenographic tradition the object of learning depicts the capability that is to be learned by the learner (Marton &Booth, 1997). The object of learning can be defined by its critical aspects, since they are seen as necessary for the learner to discern in order to learn. The aim of this paper is to discuss the nature of the object of learning by investigating how its meaning can change as it is explored by teachers. We analyzed seven recorded meetings in which four teachers and a researcher discussed the nature of the object of learning while they were planning, analyzing and revising a lesson. We found that the meaning of the critical aspects identified changed for the teachers due to the discussion and analysis of the lessons and thereby the meaning of the object of learning changed also. From at first being defined, they later become refined and specified as the teachers acquired deeper understanding of the object of learning. Distinctions were made to separate out what was of significance for the object of learning and what is not (the objects external horizon). Furthermore, an exploration by the teachers was made of how different aspects relate to each other (the objects internal horizon). The findings indicate that qualitative differences in teachers’ experience of the object of learning emerge through the collaborative investigation.

  • 48.
    Kullberg, Angelika
    et al.
    University of Gothenburg.
    Mårtensson, Pernilla
    University of Gothenburg.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research. University of Gothenburg.
    Exploring teachers' investigation of the object of learning: An analysis of A Learning Study about division2012Conference paper (Refereed)
    Abstract [en]

    This paper investigates how teachers explore the object of learning in a learning study in mathematics. The object of learning depicts the capability that is learned by the learner. For each object of learning there are critical aspects that the learner needs to discern. The aim of the paper is to describe the meanings that the critical aspects have for the teachers at different stages in the learning study process. The study is a part of a lager study in Sweden investigating teachers’ learning from learning studies (LGK-project). In this paper we report on the analysis of seven collaborative meetings, with four teachers and a researcher, from the point of view of how the critical aspects and object of learning are discussed using the framework of variation theory. The object of learning was that students in the 7th grade would understand that in a division, with a denominator between 0 and 1, the quotient becomes larger than the numerator. The study shows that the meaning of the critical aspects, identified by the teachers, changes for the teachers due to the discussion and analysis of the lessons. From at first being defined, they later become refined and more explicit as the teachers get deeper understanding of the object of learning. Furthermore, student learning is enhanced, most likely, by the changes made in the teaching due to the teachers’ deeper understanding of the object of learning.

  • 49.
    Kullberg, Angelika
    et al.
    Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Göteborg, Sweden.
    Mårtensson, Pernilla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    What is to be learned? Teachers' collective inquiry into the object of learning2016In: Scandinavian Journal of Educational Research, ISSN 0031-3831, E-ISSN 1470-1170, Vol. 60, no 3, p. 309-322Article in journal (Refereed)
    Abstract [en]

    Within the phenomenographic research tradition, the object of learning depicts the capability that is to be learned by the learner. It has been argued that the object of learning cannot be fully known in advance since what is to be learned depends on the learners as well as on the content taught. The object of learning and its nature needs to be explored. In this paper, we analyze how a group of teachers collaboratively investigated an object of learning when they planned, enacted, analysed, and revised a mathematical task. We describe distinctions made by the group in the inquiry into teaching and learning, and how delimitations and distinctions made transformed the teaching and meaning of the object of learning.

  • 50. Kullberg, Angelika
    et al.
    Runesson, Ulla
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Examples with variation. Teachers’ choice and use of mathematical examples2015Conference paper (Refereed)
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