<p>Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them.</p><p>The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives.</p><p>The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection.</p><p>When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.</p>

Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them. The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives. The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection. When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.

Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them. The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives. The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection. When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression. / Ekvationer och funktioner har en viktig roll i olika matematiska moment, som exempelvis algebra, trigonometri, programmering och analys. Under gymnasiets matematikkurs B förväntas det att eleverna ska lära sig lösa andragradsekvationer och vad som kännetecknar en funktion samt att de ska kunna tolka och använda en andragradsfunktion. Trots det ökade intresset för medborgare med djupare matematiska kunskaper redovisas ständigt larmrapporter från landets tekniska högskolor och universitet om allt sämre matematikkunskaper hos de nyantagna studenterna. För att förstå elevernas problem med och i matematik behövs ökad kunskap om elevernas lärande i relation till vad det är i innehållet som behandlas i klassrummet. Syftet med denna studie är att analysera, söka förstå och förklara relationen mellan vad som framställs i matematiskt innehåll rörande andragradsekvationer och andragradsfunktioner i klassrumspraktiken och elevernas lärande av detsamma. Fokus ligger på relationen mellan det framställda och det lärda innehållet och inte på att analysera lärarnas uppfattningar eller deras kunskap i ämnet. Denna begränsning innebär att det är innehållet som är det centrala i min studie och som kommer att analyseras ur olika perspektiv. 45 elever och två lärare deltog i undersökningen. Data består av videoinspelade lektioner, lärarnas individuella genomgång, sekvenser när lärarna tillsammans med mig tittade på och diskuterade den individuella genomgången samt intervjuer med eleverna. Elevernas prov utgör en viktig del i samlandet av data. Det variationsteoretiska perspektivet ger mig teoretiska begrepp som fungerar som analysverktyg för att tolka det empiriska materialet i min studie. Tillämpningen av variationsteori har gjort det möjligt att analysera lärandet ur två perspektiv, nämligen vad som erbjuds och vad som erfars i ett innehåll. I det erbjudna lärandeobjektet har lärarnas undervisningshandlingar analyserats som uttryck för de aspekter, delar och helheter som eleverna erbjuds att urskilja samt deras relation till varandra. Det framställda innehållet i läromedlet har analyserats utifrån samma princip, det vill säga genom att identifiera fokuserade aspekter, delar och helheter samt deras relation till varandra. Därefter har analysen fokuserat på att identifiera de variationer som öppnas upp eller begränsas i lärarens och läromedlets framställning av objekten för lärande. På så sätt kunde de aspekter som är möjliga att urskilja utifrån framställningen av lärandeobjekten identifieras och relateras till mönster av variation. Elevernas erfarande har studerats som uttryck för de aspekter, delar och helheter som urskiljs när de löser olika uppgifter samt hur dessa aspekter relateras till varandra. De aspekter som blir urskiljda och sättet på vilket detta görs, har gjort det möjligt att identifiera vilka aspekter som är kritiska för elevernas lärande. Resultaten visar att komplexa dimensioner av variation öppnas upp i det innehåll som eleverna erbjuds. Det förefaller vara vad som här kallas för konvergenta variationer som leder till ett mer fullständigt lärande. Det är denna variation som gör det möjligt för eleverna att göra generaliseringar inom varje objekt för lärande (ekvationer och funktioner) och mellan dessa lärande objekt. Dessa generaliseringar kvarstår, trots att man arbetar med nya lärandeobjekt. Dessutom kan det konstateras att parametrar, den obekanta storheten i en ekvation och funktionens argument är kritiska aspekter i elevens lärande och att meningen med dem ändras flera gånger när lärare presenterar innehållet i klassrummet och när eleverna löser olika uppgifter. Vidare demonstreras att huruvida funktionens argument framträder i explicit eller implicit form kan ha avgörande betydelse för om läraren i sin framställning av lärandeobjekten och elever i sitt erfarande av dem skiljer eller inte skiljer en funktion från en ekvation.

Computer programming is a core area in computer science education that involves practical as well as conceptual learning goals. The literature in programming education reports however that novice students have great problems in their learning. These problems apply to concepts as well as to practise. The empirically based research presented in this thesis contributes to the body of knowledge on students' learning by investigating the relationship between conceptual and practical learning in novice student learning of programming. Previous research in programming education has focused either on students' practical or conceptual learning. The present research indicates however that students' problems with learning to program partly depend on a complex relationship and mutual dependence between the two. The most significant finding is that practise, in terms of activities at different levels of proficiency, and qualitatively different conceptual understandings, have dimensions of variation in common. An analytical model is suggested where the dimensions of variation relate both to concepts and activities. The implications of the model are several. With the dimensions of variation at the center of learning this implies that when students discern a dimension of variation, related conceptual understandings and the meaning embedded in related practises can be discerned. Activities as well as concepts can relate to more than one dimension. Activities at a higher level of proficiency, as well as qualitatively richer understandings of concepts, relate to more dimensions of variation. Concrete examples are given on how variation theory and patterns of variation can be applied in teaching programming. The results can be used by educators to help students discern dimensions of variation, and thus facilitate practical as well as conceptual learning.