Introduction: The aim of this study is to investigate whether the corona pandemic has influenced preschool teachers' prerequisites to teach in preschool settings. We want to examine preschool teachers' experience of teaching under the influence of a pandemic and how that experience compares to teaching before the pandemic. By teaching we mean planning what to teach, how to teach and if the preschool teachers indeed have been able to implement what they planned to teach. Method: We study this with a phenomenographic research approach where the participating preschool teachers' experiences and thoughts are the main focus and where the material therefore is analyzed with a hermeneutic approach. This approach combined with the restrictions and recommendations of the pandemic, has affected our choice of method. Considering the pandemic, the data was gathered using qualitative questionnaires via email, with open questions where the preschool teachers’ have had the opportunity to answer the questions using their own words. Theory: The result has been analyzed from two perspectives; one is a theory about teaching from variation. The idea is that knowledge is learned through variations within the learning subject and the outer world combined. This theory is called variation theory. The other perspective is called developmental pedagogy which includes the teachers’ ability to see children's potential and relating the development of that potential to their own role as teachers, with an emphasis on the influence they themselves have on the learning situation. Results: The result from preschool teachers that have been participating in this study points to negative consequences due to an extensive absence of pedagogues', high level of stress amongst the preschool teachers’ and a change of priority from teaching to caring. In addition, the teachers also feel that the pandemic has limited their communication with the principal. On the other hand, teaching has been possible, but through new strategies and changed approaches. Conclusion: The participants felt an uncertainty about children ́s and pedagogues’ presence, reduced support from the principals and an increased feeling of stress. This has led to reductions and pauses in the planned education. Further, the teachers have also experienced a development in their teaching strategy and a positive impact on teaching in relation to the outdoor environment.

Denna artikel presenterar en Learning Study som innehåller tre forskningslektioner.Syftet med studien var att analysera på vilket sätt variationsmönster inverkar på enelevgrupps lärande och förståelse för vad som påverkar blodomloppet. Studiensdesign baserades på Variationsteorin och undersökningsgruppen utgjordes av 45elever i årskurs sex. Resultatet visade att alla elever utvecklade sina kunskaper omvad som påverkar blodomloppet. Genom att åskådliggöra blodcirkulationen som ettsammanbundet system och hur en väska transporteras i ett helt system gentemot ettsystem med hål i fick eleverna se att oavsett var i systemet brottet är påverkas alla delarna. / This Learning Study consisting three research lessons with the aim to analyse inwhat way patterns of variation have impact on pupils’ learning about what affectsblood circulation. Design was based on Variation theory. Data consisted 45 pupilsgrade sixth. All students deepened their knowledge about what affects bloodcirculation. To show blood circulation as an interconnected system and how a liquidis transported in a interconnected system compared to a system with a preformationstudents could see that no matter where the fracture is all parts are affected.

Syftet med denna studie är att undersöka vilka svårigheter som finns för lärande inom multiplikation i årskurs 3. För att besvara studiens syfte tillämpades en diagnos samt semistrukturerade intervjuer som metod. Diagnosen genomfördes i två klasser i årskurs 3 och två pedagoger intervjuades för vidare undersökning. För att tydliggöra hur olika räkneuppgifter samt svårigheter hänger samman med olika fenomen tillämpades variationsteorin. Med utgångspunkt i variationsteorin samt begreppen kontrast, separation, generalisering och fusion kategoriseras det insamlade materialet utifrån analysverktyget meningskoncentrering. Resultatet visar att eleverna har svårigheter med följande: upptäcka likheter kring ett lärandeobjekt, skapa förståelse för de enskilda kritiska aspekterna av lärandeobjektet, se skillnad mellan två värden samt att välja en lämplig strategi. Vad gäller pedagogernas uppfattningar och beskrivningar av elevernas lösningsstrategier lyfts följande aspekter fram: stress och slarvfel, bristande förmåga att hitta ett samband mellan multiplikation och addition, bristfällig läs- och begreppsförståelse samt svårigheter att tillämpa en fungerande strategi. För att främja elevernas utveckling inom multiplikation påvisar resultatet vikten av att pedagoger i sin framtida undervisning arbetar med konkret material, didaktiska verktyg såsom areaformen och tallinje, visualisering av multiplikation genom addition, begreppsförståelse och samtal i grupp för att utöka matematikspråket. / The purpose of this study is to investigate the difficulties that exist for learning in multiplication in year 3. To answer the purpose of the study, a diagnosis and semi-structured interviews were used as a method. The diagnosis was performed in two classes in year 3 and two educators were interviewed for further investigation. In order to clarify how different arithmetic tasks and difficulties are related to different phenomena, the theory of variation was applied. Based on the theory of variation and the concepts of contrast, separation, generalization and fusion, the collected material is categorized on the basis of the analysis tool meaning concentration. The results show that the students have difficulties with the following: discover similarities around a learning object, create an understanding of the individual critical aspects of the learning object, see the difference between two values and choose an appropriate strategy. Regarding the educators 'perceptions and descriptions of the students' solution strategies, the following aspects are highlighted: stress and carelessness, lack of ability to find a connection between multiplication and addition, inadequate reading and comprehension and difficulties in applying a working strategy. To promote students' development in multiplication, the results show the importance of educators in their future teaching working with concrete material, didactic tools such as area form and number line, visualization of multiplication through addition,concept understanding and group discussions to expand the language of mathematics.

Syftet med detta arbete är att utveckla ett ramverk för analys av variationsmönster som förekommer inom decimala talsystemet i matematikläromedel, för att därefter gå över till att använda detta ramverk för analys av hur tre vanligt förekommande läromedel i Sverige behandlar det decimala talsystemet i årskurs 3. Studien utgår därmed från följande frågeställning: Hur kan kritiska aspekter inom området decimala talsystemet, som framkommer i tidigare forskning, användas för läromedelsanalys? På vilka sätt behandlar det tre utvalda läromedlen för årskurs 3 de kritiska aspekterna i det decimala talsystemet? Studiens fokus är det matematiska innehåll som eleven möter i läroboken vid inlärning av det decimala talsystemet och den variation som finns i de kritiska aspekterna som forskningen visar på. Forskningen visar nämligen att det är vanligt förekommande att elever har problem med inlärning av det decimala talsystemet och att detta kan leda till problem senare under deras skolgång. Metoden som används är en kvalitativ läromedelsanalys där tre läromedel för årskurs 3 analyserades med syftet att undersöka vilka variationsmönster som förekommer i de kritiska aspekterna som forskningen lyfter. De kritiska aspekter som granskades var platsvärde, gruppering, språk och variationsmönster. Analysen visar att varitionsmönsters förekomst skiljer sig mycket mellan de olika böckerna. De kritiska aspekterna förekom i alla läromedlen men i olika omfattning. Ett av lärmedlen hade ett mer framstående variationsmönster, särskilt inom området platsvärde. Utifrån denna studie är det rimligt att dra slutsatsen att det inte går att förlita sig enbart på läromedlen i undervisning av det decimala talsystemet, samt att läraren bör kritiskt granska dessa före användning. Det vill säga användandet av enbart läromedel kan utgöra problem vid lärandet av det decimala talsystemet och lärare behöver bli mer medvetna om sitt ansvar när det kommer till val av läromedel. / The purpose of this work is to develop a framework for analysis of variation patterns occurring within the decimal number system in mathematic textbooks, and then move on to using this framework for analysis of how three commonly used textbooks, in Sweden deal with the decimal number system in Year 3. This purpose is achieved by answering the following questions. In what way can critical aspects concerning the field of the decimal number system, which emerge from previous research, be used for textbook analysis? In what ways do the three selected textbooks for Year 3 deal with the critical aspects of the decimal number system? The focus of the study is the mathematical content that students encounter in textbooks when learning the decimal number system and the variation that exists in the critical aspects revealed by research. Research shows that it is common for students to have problems with learning the decimal number system and that this can lead to issues later in their schooling. Qualitative analysis was used to analyze three teaching materials for Year 3 with the aim of examining which variation patterns occur in the critical aspects highlighted by the research. The critical aspects examined were site value, grouping, and language. The analysis showed that the occurrence of variation patterns varies greatly among different books. The critical aspects occurred in all the teaching materials, but to varying degrees. One of the teaching aids had more prominent variation patterns, especially concerning place value. From this study, it can be concluded that it is not possible to rely solely on textbooks when teaching the decimal number system, and teachers should critically examine teaching aids before using them. Relying on textbooks alone can be a problem in teaching the decimal number system, and teachers need to become more aware of their responsibilities when choosing textbooks in mathematics.

­­­­­­­­­­I flera decennier har likhetstecknets betydande roll i matematikämnet studerats av en rad olika forskare. Där det tydligt framgår att den operativa uppfattningen av likhetstecknet dominerar i klassrummen, därav syftar vår studie till att undersöka hur elevers uppfattning och förståelse av likhetstecknets relationella funktion förändras om olika delar av undervisningsinnehållet varieras. Syftet kommer besvaras genom frågeställningarna; Hur visar sig elevernas förståelse för likhetstecknets funktion i förtestet, hur påverkar variationen av undervisningsinnehållet elevernas förståelse av likhetstecknet och vilken uppfattning av likhetstecknet visas av eleverna i eftertestet efter en variation av undervisningsinnehållet. Studien har sin utgångspunkt i variationsteorin som fokuserar på vad som ska läras och hur lärandet kan möjliggöras genom teorins centrala begrepp, kritiska aspekter och variationsmönster. För att uppfylla studiens syfte och besvara frågeställningarna kommer designbaserad forskning användas som forskningsmetod, i form av en lektionsserie på tre lektioner som baserats på ett förtest.  Resultatet visar att variation av undervisningsinnehållet kan med hjälp av variationsmönster; kontrastering, generalisering och fusion, synliggöra och urskilja de kritiska aspekterna för att kunna utveckla elevernas förståelse av likhetstecknet. / For several decades, the significant role of the equals sign in the subject of mathematics has been studied by several different researchers, where the operative conception of the equal sign dominates in the classroom. Therefore, our study aims to investigate how students' conception and understanding of the equal sign relational function changes if different parts of the teaching content are varied. In accordance with this aim, the following research questions are addressed in the present study; What is the students' understanding of the function of the equal sign in the pre-test, how does the variation of the teaching content affect the students' understanding of the equal sign and what perception of the equal sign is shown by the students in the post-test after a variation of the teaching content. The study has its starting point in variation theory, which focuses on what is to be learned and how learning can be made possible through the theory's central concepts, critical aspects, and patterns of variation. To fulfil the purpose of the study and answer the questions, design-based research will be used as a research method, in the form of a series of lessons (3) based on a pre-test.  The results show variation of the teaching content can be by means of patterns of variation; contrasting, generalizing, and fusion, making visible and distinguishing the critical aspects to develop the students' understanding of the equal sign.

Syftet med studien är att fördjupa kunskapen om hur eleverna introduceras till likhetstecknet och vilka svårigheter som ligger till grund för att eleverna brister i sin förståelse kring likhetstecknet. Studien utgår från två forskningsfrågor. I studien används en kvalitativ ansats i form av semistrukturerade intervjuer. I resultatet framkommer det att eleverna i viss mån introduceras till symbolerna för större än och mindre än samtidigt, dock är det få som även får kunskap om olikhetstecknet. Svårigheterna hos eleverna är att de utgår från en operationell förståelse för tecknet, att ett resultat ska räknas fram. Denna svårighet beror på bristande förståelse för likhetstecknet vilket kan påverka eleverna när de introduceras för ekvationer. Vår slutsats är att för att motverka detta bör lärarna arbeta med att utveckla den relationella förståelsen genom varierande räknehändelser och bör även introducera alla relevanta symboler samtidigt som större än, mindre än och olikhetstecknet samt undvika symboler för de olika räknesätt under denna introduktion för att undvika att eleverna skapar en operationell förståelse. / The purpose of the study is to deepen the knowledge about how students are introduced to the equal sign and what difficulties underlie the students' lack of understanding of the equal sign. The study is based on two research questions. In the study uses a qualitative approach in the form of semi-structured interviews. In the result appears that the students are to some extent introduced to the symbols for greater than and less than at the same time, however, few also gain knowledge of the sign of dissimilarity. The difficulty of the students is that they start from a operational understanding of the sign, that a result should be calculated. This difficulty is due to a lack of understanding of the equal sign, which can affect students when introduced to equations. Our conclusion is that to counteract this, teachers should work on developing the relational understanding through varying counting events and should also introduce all relevant symbols while being greater than, less than and the sign of difference and avoiding symbols for the different counting methods during this introduction to avoid students creating an operational understanding.

The objective of this study was to gain knowledge concerning what up-per secondary students need to learn to be able to construct historical explanations while working with historical sources. The study also explored how teaching could be designed to enhance student’s capabilities. The history didactical framework took its departure in the British Historical Thinking-tradition and the methodology was based on Learning Study, an iterative research method for analysis and enhancement of teaching and learning. Empirical data originates from one Learning Study undertaken in an upper secondary school performed in cooperation with three teachers teaching three different classes. The bulk of data consists of student interviews, video recordings from research lessons and essay texts performed by the students before and after research lessons. Variation Theory was used to analyse teaching and learning with the purpose to identify critical aspects of what characterizes the capabilities and what the students needed to learn to reason about causation, interpret and use historical sources. The historical content was framed on the late 19th century phenomenon known as The Scramble for Africa where the students were supposed to reason on the causes for thescramble while working with a variety of primary sources. The methodological design of the research-lessons was based on a combination of principles from Variation Theory and Learning Activity with the ambition to form an educational practice that would allow students to discern the critical aspects and advance their learning. The results show that the students who participated in this study needed to distinguish the following critical aspects to be able to construct Historical Explanations when working with sources: Cause To distinguish temporal aspects of historical change not to confuse causes and consequences. To distinguish that historical change are caused by societal structures and intervention by historical actors and do not occur by itself. To distinguish that historical change predominantly have their origin in several causes and cannot be reduced to single causes. To distinguish that historical explanations need support from evidence. Interpret and Use Historical Sources To asses sources critically rather than view them as neutral information. To contextualize sources to avoid presentism and literal interpretations. To corroborate sources to be able to validate claims and present different perspectives in a historical explanation. To see that values and limitations of sources depend on the historical questions we seek an answer to. To distinguish the difference between authentic sources and reliable claims.

Elevers uppfattning av likhetstecknet har länge diskuterats inom forskning och ansetts varakritiskt, då förståelsen för tecknets relationella innebörd ofta saknas. Syftet med denna textär att undersöka hur likhetstecknet framställs i svenska läromedel, och diskutera hur det kantänkas påverka elevers uppfattning av symbolen. Detta genom att besvara frågeställningarna;på vilket sätt uppmuntrar läroböckerna i matematik till en relationell förståelse avlikhetstecknet och på vilket sätt uppmuntrar läroböckerna i matematik till en operationellförståelse av likhetstecknet?Studien grundar sig i variationsteorins idéer och resultatet har analyserats kopplat tillvariationsmönster och kritiska aspekter. Kopplingar har även gjorts mellan traditionellaekvationer och en operationell syn på likhetstecknet, samt icke-traditionella ekvationer ochen relationell syn på tecknet. Metoden som använts är en kvantitativ innehållsanalys där sexolika svenska läromedel i matematik, riktade till årskurs 1, analyserats för att få syn på hur devarierar framställningen av likhetstecknet. Resultatet pekar på bristande variation i form avkontrast, generalisering och fusion, då det i störst grad är traditionella ekvationer som fårutrymme i böckerna. Detta kan tänkas bidra till att elever utvecklar en operationell syn pålikhetstecknet snarare än en relationell, vilket i längden kan försvåra övergången frånaritmetik till algebra i matematikundervisningen. / Students' perception of the equals sign has for a long time been debated in research andconsidered critical, as the understanding of the relational meaning of the equals sign is oftenlacking. The purpose of this text is to examine how the equals sign is presented in Swedishtextbooks, and to discuss how it might affect students' perception of the symbol. This is doneby addressing the following questions: in what way do mathematics textbooks encourage arelational understanding of the equals sign, and in what way do they encourage anoperational understanding of the equals sign?The study is based on the ideas of variation theory, and the results are analyzed in relation topatterns of variation and critical aspects. Connections have also been made betweentraditional equations and an operational view of the equals sign, as well as non-traditionalequations and a relational view of the sign. The method used is a quantitative contentanalysis where three different Swedish mathematics textbooks, aimed at grade 1, are analyzedto observe how they vary in the presentation of the equals sign. The results indicate a lack ofvariation in terms of contrast, generalization, and fusion, as the textbooks predominantlyfocus on traditional equations. This may contribute to students developing an operationalview of the equals sign rather than a relational one, which could ultimately complicate thetransition from arithmetic to algebra in mathematics education.

Två forskargrupper utvecklade, oberoende av varandra, två olika teoretiska ramverk som bygger på användandet av variation och invarians. Det ena ramverket, variationsteorin, grundas i att se skillnader genom variation. Det andra ramverket, bianshi, grundas i att se likheter genom variation. Syftet med denna studie är att undersöka om bianshi kan användas i en cyklisk interventionsstudie för att utveckla matematikundervisning. Syftet uppfylls genom att besvara följande forskningsfrågor: hur påverkas planeringsprocessen när bianshi används som teoretiskt ramverk (istället för variationsteorin) i en interventionsstudie med komponenter från learning study? och hur ställer sig bianshi som teoretiskt ramverk i förhållande till variationsteorin i en interventionsstudie med komponenter från learning study? Studien baseras på två cykliska interventionsstudier med komponenter från learning study, en med variationsteorin som utgångspunkt och en med bianshi som utgångspunkt. Variationsteorin har tidigare visat sig vara väl fungerande i sådana arrangemang. För att kunna använda bianshi i en interventionsstudie har en egen tolkning gjorts av process och undervisning. I de två interventionsstudierna deltog tio respektive tolv elever i årkurs 2. Det matematiska innehållet som behandlas i studien är likhetstecknets betydelse. Resultatet visar att bianshi skulle kunna användas i en interventionsstudie med komponenter från learning study på ett likvärdigt sätt som variationsteorin. Det framkom även en hypotes om att variationsteorin och bianshi skulle kunna användas som ett integrerat ramverk i en interventionsstudie. / Two groups of researchers, independent from each other, developed two different theoretical frameworks describing the use of variation and invariance. One framework, the variation theory, is based on seeing differences through variation. The other framework, bianshi, is based on seeing similarities through variation. The aim of this study is to investigate if bianshi can be used in a cyclical intervention study to develop mathematics teaching. The aim is fulfilled by answering the following research questions: how is the planning process affected when bianshi is used as a theoretical framework (instead of variation theory) in an intervention study with components from learning study? and how does bianshi work as a theoretical framework compared to variation theory in an intervention study with components from learning study? The study is based on two cyclical intervention studies with components from learning study, one using variation theory and one using bianshi. Variation theory has previously been shown to be well-functioning in such arrangements. To be able to use bianshi in an intervention study, we have made an interpretation of the process for analysis and lesson planning. In the two intervention studies, ten and twelve students respectively in grade 2 participated. The mathematical content addressed in the study is the meaning of the equal sign. The results show that bianshi could be used in an intervention study with components from learning study just as effectively as the variation theory. A hypothesis also emerged suggesting that the variation theory and bianshi could be used as an integrated framework in an intervention study.

In this study Learning study has been carried out as a model for skills development focusing teaching and learning processes for music teachers within the Community School of Arts. The model rests on variation theory to improve teaching and learning. The design of a Learning study consists of three video filmed lessons interwoven with continious monitoring and planning through collagiate dialogues. Work with Learning study starts from the choice of a certain aspect of knowledge, the learning object, and the identification of several critical aspects based on the pupil's current level of understanding. In the teaching situation, the teacher provides variation patterns to support the pupil’s learning.. The design of the model in this study is modified with a view to a better fit with the subject of music, and the context of teaching in the School of Arts. Starting from the cultural-psychological perspective on learning and a phenomenographic inspired method, this study aims to describe the participating teachers’ learning through their Learning study. The teachers have experienced a development of knowledge in themselves as well as in their pupils. They have assimilated a language for reflecting on the process of teaching. Reflected experience has made the didactic interaction in the teaching process visible. The collegiate dialogue has contributed to the development of more variation in the tools used for teaching. Learning study is therefore found be a useful tool in the development of music teaching in the context of Community School of Arts when enough time, continuity and a qualified coach are provided.