The aim of the thesis is to gain knowledge concerning what pupils aged 8 and 9 need to learn to become familiar with negative numbers. The framework used in this research, variation theory, impliesthat students' problems in learning what was intended may have to do with the fact that some critical aspects of the studied object have not yet been discerned by the student. To get the pupils to understand the idea behind each critical aspect, carefully constructed examples were used. According to variation theory it is necessary to experience differences before you experience similarities. To answer the research question data was collected by using the learning study model. It is characterized by an iterative design where I as a researcher collaborate with teachers to try to find and orchestrate the critical aspects. The method is interventionist, which means that interventions are done in teaching. In the learning study I have cooperated with two primary school teachers and 64 pupils in four different classes. The data consists of video-recordings of lessons, pre- and posttests, interviews with pupils and notes from the meetings of the learning study group. When planning lessons as well as analyzing data, concepts relating to the theory of variation have been used as analytical tools. This thesis contributes to research by investigating in detail what aspects students need to differentiate in order to become familiar with negative numbers. The results show that the pupils needed not only to discern, but also to differentiate three different critical aspects: To differentiate the values of two negative numbers. To differentiate the function of the minuend versus the function of the subtrahend in a subtraction. To differentiate the minus sign for negative numbers versus the minus sign for subtraction.

Visual representations are frequently used in teaching as well as in social and science studiesin order to capture and explain complex and abstract phenomena. However, they can bechallenging, complex and difficult to understand. This study, which is carried out within theframework of a research project, aims to investigate how a visual representation can be usedin teaching in order to qualify elementary pupils’ comprehension about democracy, moreclosely by using a flowchart to describe the legislative process. Before this study wasconducted the research group analysed phenomenographically the conceptual understandingof a flowchart of the legislative process of students in the 5th, 8th and 10th grade andidentified the critical aspects of this phenomenon. The method chosen for this study wasorganizing and conducting a lesson based on those identified critical aspects of the flowchartof the legislative process. A pre- and post-test on it were written by the participants whoseresults have been analysed phenomenographically based on the outcome space of thephenomenographic analysis of the research group. The analysis of the pre- and post-testaimed to investigate the pupils’ conceptions of the phenomenon before and after the lessonwhich therefore indicated which of the two critical aspects were discerned during the lesson.Parts of the conducted lesson and discussions between the pupils were analysed, too. Thisanalysis intended to identify elements during the lesson where the critical aspects seemed tobe possible for the pupils to discern. The results of the study showed that although the lessondid not facilitate for the pupils to discern both critical aspects simultaneously, many of themqualified their conception of the phenomenon from before the lesson to after, by havingdiscerned one of the critical aspects. Thus, using visual representations by focusing ondiscerning critical aspects can facilitate a qualified comprehension/conception of complexphenomena.

This study has investigated how preschool teachers perceive the children's experience, actions and assimilation of information through chemistry experiments. Interviews were conducted with six teachers. The informants were given the same basic questions based on an interview guide with spontaneous follow-up questions mixed in. The interviews were recorded and transcribed. The transcribed data was then analyzed using a framework that has been adapted for the study where data has been divided, categorized and analyzed to identify consistent themes. The results show that teachers perceive that the children have varying prior knowledge, which affects the ability to understand chemistry experiments. Those children who do not have as much prior knowledge are considered to need to find more basic critical aspects of the phenomenon first. In the teachers' experiences of how children act in an experiment, the children are considered to have a high level of interest and the interest increases when the children see clear results. It is important that the children participate physically. Regarding the assimilation of the meaning of the chemistry experiments, documentation and reflection are highlighted as key factors. The teachers considered it important to have open-ended questions to encourage the children's own reflections. Recommendation for the organization is to think about using the right concepts in the work with chemistry, to continue to document and reflect on the various activities together with the children and to take advantage of and follow the children's interest. / Denna studie har undersökt hur förskollärare uppfattar barnens erfarande, agerande och tillgodogörande av information i samband med kemiexperiment. Semi-strukturerade intervjuer har genomförts med sex förskollärare. Intervjuerna har spelats in och sedan transkriberats. Insamlade data har analyserats med hjälp av Maguire och Delahunts (2017) ramverk som anpassats för studien där data delats upp, kategoriserats och analyserats för att identifiera genomgående teman. Resultatet visar att förskollärare uppfattar att barnen har varierande förkunskaper, vilket påverkar förmågan att förstå kemiexperimenten som utförs. De barn som inte har lika mycket förkunskaper anses behöva hitta mer grundläggande kritiska aspekter av fenomenet först. I förskollärarnas upplevelser kring hur barn agerar i ett experiment anses barnen ha högt intresse och intresset ökar när barnen ser tydliga resultat. Förskollärarna upplever att det är viktigt att barnen får delta fysiskt. Gällande tillgodogörande av meningen med kemiexperimenten lyfts dokumentation och reflektion fram som nyckelfaktorer. Förskollärarna påpekar att det är viktigt med öppna frågor för att uppmuntra egen reflektion hos barnen. Även den fria leken anses vara viktig för vidare bearbetning av informationen. Rekommendation för verksamheten är att tänka på att använda rätt begrepp i arbetet med kemi, att fortsätta dokumentera och reflektera kring de olika aktiviteterna tillsammans med barnen samt ta tillvara och följa barnens intresse.

Tidigare forskning har identifierat svårigheter och missuppfattningar kopplade till bokstavssymboler inom algebra. Vi väljer att avgränsa oss till obekanta tal eftersom många elever kan möta svårigheter när bokstäver introduceras i matematikundervisningen. Syftet med denna studie är att bidra med kunskap om hur undervisning om bokstavssymboler i tidigare skolår kan planeras och genomföras utifrån variationsteorin. Syftet besvaras med hjälp av följande frågeställningar: Vilka kritiska aspekter behöver elever i årkurs 1 urskilja för att utveckla förståelse för obekanta tal? Hur förändras elevernas kunskaper om obekanta tal mellan för- och eftertest? Hur kan undervisning planeras och iscensättas utifrån identifierade kritiska aspekter? Studiens teoretiska ramverk är variationsteorin, där centrala begrepp omfattar lärandeobjekt, kritiska aspekter och variationsmönster. Variationsmönster används för att hjälpa eleverna att urskilja kritiska aspekter av ett lärandeobjekt, vilket i denna studie är obekanta tal, genom att synliggöra dessa med olika variationer. Studiens metod inspirerades av Learning Study. Lektionerna genomfördes med två olika undervisningsdesigner som grundades på variationsteoretiska principer. Studiens data består av för- och eftertester och observationsanteckningar från två lektionstillfällen. Resultatet visar att fyra kritiska aspekter har identifierats till det specifika lärandeobjektet, vilka är urskilja att: en ekvation är en likhet som innehåller obekanta tal, det obekanta talet står för ett tal (värde), tidigare kunskaper inom aritmetiken kan användas vid ekvationslösning, till exempel talfakta eller inversa operationer och samma bokstavssymbol kan stå för olika värden beroende på ekvation. Resultatet visar även att elevernas kunskaper utvecklas mellan för- och eftertest, när undervisningarna planeras och iscensätts utifrån variationsteoretiska principer. Slutsatser som kan dras är att variationsteorin kan vara användbar för lärare när de ska identifiera kritiska aspekter och planera lektioner. Efter att ha genomfört två lektioner med årkurs 1 har vi kommit fram till att det är möjligt och fördelaktigt att introducera obekanta tal redan i tidig ålder. / Previous research has identified difficulties and misconceptions associated with letter symbols in algebra. A common difficulty can be interpreting and using unknown numbers. A misconception identified in a few studies is that letter symbols can be perceived to represent positions in the alphabet. We choose to focus on unknown numbers because many students may encounter difficulties when letter symbols are introduced in mathematics. The aim of this study is to investigate the teaching of letter symbols in grade 1. The aim is addressed through the following research questions: What critical aspects do students in grade 1 need to dicern in order to develop an understanding of unknown numbers? How do students' knowledge of unknown numbers change between pre- and post tests? How can teaching be planned and staged based on identified critical aspects?  The theoretical framework of the study is the variation theory of learning, where central concepts include learning object, critical aspects and patterns of variations. Patterns of variation are ursed to help students discern critical aspects of a learning object, which in this study are unknown numbers, by highlighting these through different variations. This method of study was inspired by Learning Study. Lessons was carried out with two different teaching designs, based on principles of variation theory. The study´s data include pre- and posttest and observation protocols from two conducted teaching sessions.  The results indicate that four critical aspects have been identified for the specific learning object, which are discerning that: an equation is an equality containing unknown quantities, the unknown quantity represents a number (value), prior knowledge in arithmetics can be used in equation solving, such as number facts or inverse operations, and the same letter symbol can represent different values depending on the equation. The results also show that students' knowledge evolve between pre- and post tests when teaching is planned and staged based on variation theoretical principles. It is shown that variation theory can be useful for teachers in identifying critical aspects and planning lessons. After conducting two lessons with grade 1, we have concluded that it is possible and beneficial to introduce unknown quantities at an early age.

Syftet med avhandlingen är att bidra till den ämnesdidaktiska kunskapsbasen för undervisning och lärande om tekniska system i grundskolan. Dagens teknikkomplexa samhälle är uppbyggt av system som vi människor interagerar med. Informations-, energi- och kommunikationssystem är exempel på tekniska system som vi kommer i kontakt med dagligen. Tekniska system ingår som en del av kunskapsinnehållet i den svenska grundskolans teknikämne, men forskning som rör teknikämnet visar att undervisning av tekniska system är begränsad och det råder osäkerhet kring vad lärande av tekniska system innebär. De övergripande frågeställningarna är: Hur uppfattar unga tekniska system? Hur kan ungas uppfattningar av tekniska system användas för att utveckla undervisningen om tekniska system? Vilken potential, att bidra till en ökad förståelse av teknik i dagens samhälle, har tekniska  system som kunskapsinnehåll i teknikämnet? Avhandlingen bygger på två studier som presenteras i fyra artiklar. I två av artiklarna fokuseras ungas uppfattningar av tekniska system och i två artiklar lyfts didaktiska dimensioner av tekniska system fram. En fenomenografisk ansats används för att kartlägga ungdomars olika sätt att erfara tekniska system genom empiriska undersökningar av kvalitativa skillnader i det kollektiva erfarandet av fenomenet. Resultatet av studierna indikerar att dimensioner av tekniska system och kritiska aspekter inom dessa dimensioner är avgörande för en utvecklad förståelse av tekniska system. Genom att lärare blir medvetna om ungas uppfattningar om tekniska system kan de med detta som utgångspunkt utveckla undervisningen. Inom den ämnesdidaktiska kunskapstraditionen ses den lärandes uppfattning av innehållet innan undervisningen startar som en betydelsefull aspekt av lärarens ämnesdidaktiska kompetens Det är därför viktigt att lärare är medvetna om dimensionerna och de kritiska aspekterna då de planerar och genomför undervisning för att kunna erbjuda kraftfulla sätt att lära om tekniska system. De didaktiska implikationerna, när det gäller tekniska system, lyfter fram aspekter av teknik som har betydelse för förståelsen av teknik även på ett mer generellt plan. Det handlar om kunskaper som aktiva medborgare i dagens teknikkomplexa samhälle behöver, så som insikter om krav på resurser, människans intentioner med och inblandning i tekniken samt hur teknikens struktur och organisation ser ut. Tekniska system som kunskapsinnehåll erbjuder en förståelse av teknik där viktiga medborgerliga aspekter som engagemang, konsekvensanalys och användaransvar kan synliggöras och problematiseras. / The purpose of this thesis is to contribute to the field of technology education research, specifically that which concerns teaching and learning about technological systems. Today's technologically complex society is made up of a variety of systems that humans interact with. Information, energy and communication are examples of technological systems with which we are involved daily. Education in technology prepares young people for participating as active citizens in a technologyintensive society and therefore includes technological systems as part of the knowledge content in the Swedish compulsory school subject of technology. Research related to the technology subject shows that the teaching of technological systems is limited and there is uncertainty about what the learning entails. The overall questions which this thesis intends to investigate are: How do young people experience technological systems? How can young people’s experiences of technological systems be used to develop the teaching of technological systems? What potential does knowledge about  technological systems have in contributing to a better understanding of technology in today’s society? The thesis is based on two studies presented in four articles. Two of the articles focus on young people’s experiences of technological systems and the other two highlight pedagogical dimensions of technological systems for teaching and learning. The studies take the perspective of  the learners’, using a phenomenographic approach, and investigate young people’s ways of experiencing technological systems. To start from the learners’ experience is an important aspect of the tradition of pedagogical research that concerns content specific knowledge. The phenomenographic approach offers empirical ways of investigating qualitative differences in the collective experience of the phenomenon and an opportunity to highlight what teaching should focus on to create learning opportunities. The main result of the studies consists of knowledge about dimensions of technological systems and critical aspects within those dimensions. Together they offer a perspective for teaching, providing possible starting points for teachers when they plan instruction. If teachers address their own and young people’s awareness of dimensions of variation, it could enable more powerful ways of learning about technological systems. The pedagogical implications in terms of technological systems also point to aspects that are relevant for understanding technology on a more general level, namely skills which active citizens in today’s technologically complex society must possess. Technological systems knowledge offers an understanding of technology in which key aspects of civil commitment, impact and user responsibility can be made visible and thus  problematized.

Illustrated pictures play an important role as mediators of mathematical concepts and relations in mathematical textbooks used in Swedish primary school classrooms. Theories of the meaning of semiotics for teaching and learning, as well as the Variation theory of learning, are used as a framework to investigate the critical aspects for developing multiplicative reasoning through visual presentations. In this study143 students in 3rd grade (age 9-10) participated in a test where multiplication is represented with illustrations, using both additive and multiplicative groupings. The students were also instructed to draw multiplication expressions with various visual supports. Students’ responses were analyzed by quantifying groupings based on the multiplication expressions students identify, as well as the extend of which they use additive and multiplicative visual representations and support in their own drawings. Results show indications of different properties of multiplication that can be presented in illustrations. How teacher knowledge can be used to identify the critical aspects for learning multiplication is here discussed, leading to suggestions regarding ways is which textbook pictures can be useful tools in teaching and learning.

Bråk beskrivs som ett av de mest komplexa ämnesområden inom matematik. Däremot har en del läromedel tonat ner bråkens betydelse samtidigt som forskning visar att läroböcker utgör en central del av lärares matematikundervisning. Syftet med denna studie är att ge en bild av vad tre läromedel läroböcker erbjuder elever att lära om bråk. Undersökningen har inspirerats av variationsteorin och utgår från fyra kritiska aspekter som utifrån tidigare forskning visat sig vara viktiga för elevers förståelse för bråk. Totalt har sex läroböcker analyserats med hjälp av en innehållsanalys, i syfte att identifiera vilka variationsmönster som används för att synliggöra fyra kritiska aspekter. Dessutom undersöktes vilka lärandeobjekt som förekommer i respektive lärobok. Av resultatet framgår en variation i vilken utsträckning de kritiska aspekterna behandlas och vilka variationsmönster som används för att öka elevers förståelse för de kritiska aspekterna. Framförallt ges elever möjlighet att lära om täljaren och nämnarens funktion genom generalisering som mönster av variation. I läromedlen kunde sammanlagt 19 unika lärandeobjekt identifieras. Däribland återfinns lärandeobjekt om såväl bråk som del av en helhet som bråk som del av antal. Även om inget av de analyserade läromedlen behandlar samtliga lärandeobjekt är det möjligt att utläsa en tydlig skillnad vad gäller såväl omfång som innehåll. Oavsett står det klart att en kombination av flera läromedel kan berika elevers möjligheter att lära bråk. För att en sådan variation ska vara möjlig krävs en stärkt skollag som säkerställer elevers rätt till läromedel. / Research shows that there are a number of reasons students struggle with fractions. On the other hand, some mathematics textbooks still reduce the importance of fraction, while research found mathematics textbooks as a central part of teachers’ math teaching. However, the aim of this study is to depict what three educational materials offer pupils to learn about fractions. The study has been inspired by the theory of variation and is based on four critical aspects that, based on previous research, have been found as critical for students’ understanding of fractions. A total of six mathematics textbooks have been analysed using a content analysis, in order to identify which patterns of variation are used to make the selected critical aspects visible. Furthermore, this study examined which learning objects that appear in each textbook. The results show a variation in the extent each critical aspect is treated and which variation patterns that are used to improve pupils understanding of selected critical aspects. A total of 19 unique learning objects could be identified in the mathematics textbooks. Due to the fact that none of the analysed teaching materials deals with all learning objects, it is possible to distinguish a variation of both volume and content. However, result from this study claims that mathematics textbooks combined would offer greater opportunities to learn obviously. If teachers should be able to overcome a various teaching material it is necessary to strengthen students’ rights to teaching material.

­­­­­­­­­­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.

Fingrar som verktyg är ett fenomen som individer både lärare, elever och vårdnadshavare runt om i världen har skilda åsikter om. Vissa ser fingrarna som ett positivt verktyg för elevers matematiska tänkande medan andra tvekar på fingrarnas tillgång. Syftet med studien är därför att ge bild av hur lärare uppfattar elevers användande av fingrar som verktyg i matematik. För att besvara syftet har studien tagit utgångspunkt i följande frågeställningar: på vilka skilda sätt uppfattar lärare i årskurs 1–6 elevers användande av fingrar som verktyg och vad karaktäriserar respektive uppfattning? Vilka kritiska aspekter kan identifieras relaterat till de skilda uppfattningarna? Studien tar sin utgångspunkt i den fenomenografiska ansatsen. I studien har fem lärare intervjuats för att få bild av hur de uppfattar användandet av fingrar som verktyg i matematik. Allt material har transkriberats och även analyserats för att kunna redogöra för ett tillförlitligt resultat. Resultatet presenteras genom ett utfallsrum som visar att samtliga lärare uppfattar fenomenet mestadels som något positivt, då en uppfattning är mer negativ. Dessa uppfattningar kan i sin tur delas in i fyra skilda uppfattningar: fingrar som ett hinder för elevers lärande, fingrar som ett konkret material, fingrar som ett stöd i matematik samt fingrar som en grund för aritmetisk förståelse. Av de fyra uppfattningarna gick det att urskilja tre kritiska aspekter, nämligen urskilja möjligheter, att fingrar är ett medierenade verktyg samt urskilja relationer/förståelse. / Fingers as a tool is a phenomenon that individuals such as teachers, pupils and caregivers around the world has different opinions about. Some consider that the fingers are a positive tool to pupils mathematical thinking, while others hesitate the fingers access. The purpose of this study is therefore to give a picture of how teacher perceive the use of fingers as a tool in mathematic. To answer the purpose, following questions have been used: In what ways perceive teacher in grade 1-6 pupils use of fingers as a tool and what characterizes each perception? Which critical aspects can be identified related to the different perceptions? The study is based on phenomenography. Five teachers have been interviewed to give a picture of how teacher perceive the use of fingers as a tool in mathematic. The material has been transcribed and analyzed to be able to account a reliable result. The result presented through an outcome space that shows each teacher perceives the phenomenon mostly as something positive, since one perception is more negative. These perceptions can be divided into four different perceptions: fingers as an obstacle for pupils learning, fingers as a concrete material, fingers as a support in mathematics and fingers as a foundation for arithmetic comprehension. Based on these four perceptions, it was possible to distinguish three critical aspects namely, distinguish possibility, that fingers are a mediating tool and distinguish relations/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.