This master thesis concerns progress of junior school pupils in understanding of equations. It describes benchmark problems and mathematical environments in which equations and equation systems occur. Occurrences of such exercises in text books for primary schools are also mapped here. The main part of this thesis is focused on comprehension of pupils' intellectual actions during solving the equations. Thanks to several experiments, it includes description of pupils' solutions of exercises. The result of the phenomenal analysis of pupils' solving processes is an overview of phenomena, which appeared during solving exercises.

Obwohl das Angebot von Feedback in Lehr-Lernsituationen eine wichtige instruktionale Maßnahme darstellt,um Lernende in ihrem Lernprozess zu unterstützen, wird dessen Potenzial in multimedialen Lernumgebungen nur selten optimal genutzt. Feedback, welches in aller Regel im Anschluss an die Bearbeitung von Lernaufgaben bereitgestellt wird und den Lerner über die Qualität seiner Aufgabenbearbeitung informiert, beschränkt sich häufig auf die Bereitstellung einer richtig/falsch Rückmeldung mit einem sich anschließenden, zusätzlichen Lösungsversuch bzw. der Präsentation der richtigen Lösung. Als prototypisch für diese Art der Feedbackgestaltung erweist sich das in mathematischer Grundschulsoftware implementierte Feedback. Der Lerner erhält weder Informationen zu dem Ort und der Art des Fehlers noch einen Hinweis auf die korrekte Lösungsstrategie. Vor diesem Hintergrund bestand ein zentrales Ziel der vorliegenden Arbeit in der Entwicklung von fehlerspezifischem informativem tutoriellem Feedback (ITF) für die schriftliche Subtraktion auf der Grundlage des heuristischen Feedbackmodells von Narciss (2004) und der aus diesem Modell abgeleiteten Gestaltungsprinzipien für ITF. Obwohl allgemein erwartet wird, dass informatives Feedback lernwirksamer ist als Feedback, welches nur die richtige Lösung präsentiert, konnte diese Annahme durch die Ergebnisse der empirischen Feedbackforschung bisher nicht bestätigt werden. So ist die Feedbackforschung durch eine inkonsistente Befundlage gekennzeichnet. Vor diesem empirischen Hintergrund bestand ein weiteres zentrales Ziel der vorliegenden Arbeit in der Überprüfung der Effektivität des entwickelten fehlerspezifischen ITFs hinsichtlich verschiedener Lern- und motivationaler Parameter. Fehlerspezifisches ITF zeichnet sich dadurch aus, dass bei fehlerhaften Aufgabenlösungen Informationen angeboten werden, die auf der Grundlage einer kognitiven Anforderungs- und Fehleranalyse als korrekturrelevant identifiziert worden waren. Diese korrekturrelevanten Informationen werden dem Lernenden ohne Bekanntgabe der korrekten Lösung präsentiert, so dass sich dieser dazu aufgefordert sieht, die bereitgestellten strategischen Informationen in einem erneuten Lösungsversuch unmittelbar anzuwenden. In zwei Studien mit Viertklässlern wurde dieses computergestützte ITF im Hinblick auf seine Lern- und motivationale Wirksamkeit evaluiert. In der ersten Studie (N = 30) erfolgte diese Evaluation gegen eine Feedbackbedingung, welche das üblicherweise in mathematischer Lernsoftware realisierte Knowledge of Response und Knowledge of Correct Response (KR-KCR) Feedback präsentierte. In der zweiten Studie (N = 105) wurden zwei zusätzliche Feedbackbedingungen "Knowledge about Mistake (KR-KM) und Knowledge on How to proceed (KR-KH)" aufgenommen. Diese sind in ihrem Informationsgehalt zwischen dem ITF und dem KR-KCR Feedbackalgorithmus anzusiedeln. Die Ergebnisse beider Studien bestätigen die postulierte Lern- und motivationale Wirksamkeit des fehlerspezifischen ITFs für die schriftliche Subtraktion. Besonders im Vergleich zu der wenig informativen KR-KCR Feedbackbedingung konnten in beiden Studien signifikante Wirksamkeitsunterschiede nachgewiesen werden. Auch im Vergleich zu den komplexeren Feedbackformen Knowledge about mistake (KR-KM) und Knowledge on how to proceed (KR-KH) konnten für die Schüler der ITF Bedingung bessere Leistungen beobachtet werden, auch wenn diese Unterschiede nur partiell signifikant wurden. Die Ergebnisse der vorliegenden Untersuchungen zeigen, dass es möglich ist, auf der Grundlage des von Narciss (2004) vorgestellten Feedbackmodells und der aus diesem Feedbackmodell abgeleiteten Gestaltungsprinzipien lern- und motivationsförderliches ITF zu entwickeln. So liefern die Ergebnisse beider Studien erste empirische Belege für die Annahme, dass informatives Feedback nicht nur den Lernprozess unterstützt, sondern auch motivationsförderlich wirkt. / Feedback is considered to be an important factor to promote learning and motivation with computer-based training tools. However the findings of studies on the effectiveness of feedback are rather inconsistent. One reason for these inconsistent findings might be that the implementation of feedback is more based on intuition than on well-founded design principles. Therefore the aims of the present work was (1) to develop theoretical well-founded elaborated feedback forms for a procedural skill like the written subtraction, (2) to implement them in an adaptive feedback algorithm that induces the mindful processing of feedback, and (3) to evaluate this elaborated feedback regarding his impact on learning and motivation. Using results from prior feedback research and from cognitive task and error analysis of written subtraction tasks, information relevant to the correction of typical systematic errors (e.g. location of error, source of error, type of error and hints to the correct solution strategy) were selected. This information was used to design different types of bug-related feedback messages. These different feedback messages were arranged in an adaptive bug-related algorithm presenting three levels of feedback with increasing informational value to support the learner in finding the correct solution on his own. Because the developed feedback allows assisted multiple response attempts for an item by providing relevant information for error correction, but no immediate Knowledge of Correct Response (KCR) feedback, and by encouraging the learner to apply the corrective information to a further attempt, this kind of feedback was called bug-related "informative tutoring feedback (ITF)". In two computer-based learning experiments cognitive and motivational effects of this bug-related ITF were evaluated with forth-grade students with difficulties in written subtraction. In the first study (N=30) the bug-related ITF was compared to the effects of a standard Knowledge of Result-Knowledge of Correct Response feedback (KR-KCR) algorithm. In the second study (N=105) two additional feedback algorithms - Knowledge about Mistake (KR-KM) und Knowledge on How to proceed (KR-KH) - were implemented. These both feedback algorithms range between the bug-related ITF and the simple KR-KCR feedback regarding to their amount of information. In both studies significant positive effects of the bug-related ITF compared to the KR-KCR feedback could be found. Regarding to the more informative KR-KM and KR-KH feedback a positive effect of the bug-related ITF could be showed sporadic. These results are discussed in detail.

L’utilisation du jeu pour faire apprendre des mathématiques aux élèves s’impose de plus en plus comme une pratique pédagogique de référence au niveau de l’institution scolaire. Ce terme générique recouvre cependant des mises en œuvre très différentes n’induisant pas le même vécu pour les élèves. Notre travail a pour objectif d’explorer certaines de ces pratiques et de déterminer les conditions pour que les élèves jouent et développent des compétences mathématiques au cours d’une même séquence d’apprentissage. Nous cherchons dans un premier temps à cerner les activités qui peuvent être des jeux en classe, sachant que le contexte scolaire influe fortement, et négativement, sur la possibilité pour les élèves de vivre de réels moments de jeux. Nous proposons une caractérisation de ces activités à partir de quatre outils conceptuels, le game (la structure de jeu), le play (l’attitude de jeu), le potentiel ludique et le potentiel d’apprentissage. Nous dégageons ensuite deux modalités principales d’utilisation du jeu en classe : la ludicisation d’une situation didactique ou l’exploitation d’un jeu existant. Faisant l’hypothèse que la seconde possibilité est plus à même d’induire une attitude de jeu chez les élèves, nous concevons une séquence d’apprentissage basée du jeu de go dans une classe de cycle 3 de l’école élémentaire pour en étudier les impacts en termes d’apprentissages mathématiques. Pour ce faire, nous présentons une méthode d’analyse des jeux qui, appliquée au jeu de go, nous permet de dégager deux potentiels d’apprentissages important : l’argumentation heuristique (Duval, 1992) en résolution de problèmes et l’appréhension séquentielle des figures (Duval, 1994) en géométrie dans le cadre des contraintes actuelles de l’institution scolaire. La séquence, d’une dizaine de séances, est mise en œuvre par des enseignants expérimentateurs non spécialistes du jeu de Go. Nous analysons ensuite la séquence effective de manière à déterminer si les élèves ont vécu des moments de jeu suivant leurs propres conceptions du jeu et si des apprentissages mathématiques ont émergé. Nous nous posons enfin la question de la mobilisation de ces compétences, construites dans le cadre du jeu de go, dans d’autres domaines mathématiques. A partir de cette expérimentation nous cherchons à dégager l’ensemble des conditions et des contraintes pour l’implémentation d’un jeu de société à l’école élémentaire. / Using the game to teach mathematics to pupils became increasingly an obvious standard pedagogical practice in the scholastic institution. This generic term also reflects very different implementations that do not result in the same experience according to pupils. The objective of our work is to explore some of these practices and to determine the conditions in order for the pupils to play and develop mathematical competence during the same learning sequence. In the first instance, we are trying to define the activities that can be in-class games, keeping in mind that the school environment strongly and negatively influences the ability to experience real playing periods. We suggest a characterisation of these activities from four conceptual tools: the game (the structure of the game), the play (the game attitude), the playful potential and the learning potential. Then, we will draw two main conditions of in-class use of games: the “gameifying” of a didactic situation or the exploitation of an existing game. Assuming that the second possibility is more likely to lead to a gaming attitude for pupils, we are designing a learning sequence based on the strategy board game “go” in a junior division classroom at an elementary school, in order to study mathematical learning impacts. To do so, we are introducing a method of game analysis that, applied to Go, can extract two important learning potentials: the heuristic argumentation (Duval, 1992) for problem-solving and the sequential apprehension of figures (Duval, 1994) for geometry. This analysis method is implemented as part of the current constraints of the scholastic institution. The sequence, made of ten sessions, is implemented by experimenter teachers who are not Go specialists. We will then analyse the actual sequence in order to determine if the pupils have experienced gaming periods according to their own game understanding and if mathematical learning has emerged. We ultimately ask the question: how these skills, built as part of the Go game, have been deployed in other mathematical fields? From this experimentation, we want to clear the conditions and constraints of a boarding game implementation at the elementary school.

One of the difficulties that Colombian students face in order to access higher education lies in their low performance in primary and secondary education levels, in areas like writing and mathematics. This is demonstrated, for example, in the results reported from their exams on international (Report PISA ), national (Saber 11, ICFES ), and local (UN admissions exam) scales. The admissions exam for UN allows incoming students to be identified who need to strengthen their performance in areas of which UN offers leveling tests. In this context, I was invited by the DNIA , the DNPPr , and the Academic Vice-Rector of the UN, in order to design of a pedagogical proposal mediated by the use of ICT to support these leveling courses. The pedagogical proposal revolved around the implementation of an open educational practice, a type of “open teaching” (Chiappe, 2012), in order to promote the “empowerment of learners as co-producers on the road towards lifelong learning,” in other words, self-regulated or autonomous (Ehlers & Conole, 2010; Ehlers, 2011). In this document I describe the theoretical underpinnings and criteria that guided the design of each one of the components of this type of educational practice, namely: (a) a virtual site for the interaction between students, professors, and content in order to achieve the construction of a “Virtual Learning Community”; (b) the norms and rules of behavior to regulate the social relations of that community; (c) the instruments utilized in the interactive activities; and (d) the rules that establish the division of tasks in the same activity (Engeström, 1996; Cole & Engeström, 2001; Diaz Barriga, 2003; Onrubia, 2005; Coll, et al., 2008). The central axis of these components refers to the construction of a virtual learning community; therefore, the main objective of the project was to promote this construction. Faced with the potential offered by virtual learning communities, I wondered what and how does people, who decide to participate in them, learn. Some constructivist conceptions of human learning suggest that in VLC people participate to learn, learn by interacting with others, learn to understand and manifest understanding by performing cognitive performances such as: explaining, arguing, generalizing, applying, representing in a novel way, build, model, etc. Based on these conceptions, I set out to identify the potential of the VLC that I designed to strengthen the mathematical learning competencies of its participants. Thus, I analyzed the messages added by the participants in the virtual forums to characterize the processes of: a) participation, following the elements proposed by Hrastinski, (2008; 2009); b) interaction, from the community of inquiry approach (Garrison et al., 2010); c) mathematical learning, based on the basic standards of mathematical competences (MEN, 2006) and the ontosemiotic approach (Godino, 2017). I developed this research project using a mixed approach: the technique of content analysis and a combination of qualitative and quantitative methods. The proposed research design constitutes a significant contribution to the analysis of the mathematical learning achieved by the VLC participants, given its approach from different perspectives. Upon application, the following was revealed: On the one hand, the majority managed to: meet the criteria that characterize a participation in high quality virtual forums; weave an optimal network structure for interaction; show, in an authentic way, the skills and difficulties they have in communicating their ideas and learning with others; demonstrate good performance to correctly deal with mathematical concepts and to apply algorithms. Within the framework of this research, the importance of having expert moderators in guiding the discussions, from the side, was stressed in order to get VLC participants to develop high-level thoughts. Computing tools are also required to monitor and evaluate the discussion process in real time. In addition, it is essential to achieve the design of situations-generating problems of discussion and the implementation of an evaluation process that incorporates various sources and perspectives, to identify in a concrete and effective way the types of learning achieved. According to the above, it can be affirmed that the VLCs offer a high potential to strengthen the mathematical learning process of their participants, since in the debate zones you can easily observe the types of thoughts that they activate and put into operation when trying to find a solution to mathematical situations. This highlights the mathematical competences that each participant dominates and those that are difficult to understand, which makes it possible to develop targeted support strategies by teachers and moderators. The analysis of the mathematical learning of participants in virtual communities has not been sufficiently explored; In some cases, it has been limited to quantitatively reviewing the aggregate contributions in the discussion spaces, but a qualitative investigation of the discursive processes that demonstrate the concrete learning achievements has not been proposed, the combination of these two forms of analysis is one of the greatest contributions of this thesis.:Contents I. Introduction 13 a) How did I learn to learn? 13 b) How was my doctoral research project born? 17 c) Characteristics and purposes of the project 21 Theoretical Considerations 25 1 Chapter I: Human learning in community 25 1.1 Human learning from the constructivist perspective 25 1.2 The scope of learning 27 1.3 School learning from constructivism 29 1.4 Mathematical learning 31 1.5 Educational practices based on constructivism 39 1.6 How to define the notion of community? Contexts and debates 40 1.7 A definition of virtual learning community 45 1.8 Theoretical references for the design of virtual learning environments 53 Practical design considerations 61 2 Chapter II: Design Process of the Virtual Learning Community of Basic Mathematics 61 2.1 Academic activities carried out in the pre-project execution stage 61 2.2 First pilot test of the basic math virtual learning community design 64 2.2.1 Pedagogical Knowledge (PK) 65 2.2.2 Content Knowledge (CK) 65 2.2.3 Pedagogical Content Knowledge (PCK) 65 2.2.4 Technological knowledge (TK) 68 2.2.5 Technological Pedagogical Knowledge (TPK), Technological Content Knowledge (TCK) and Typology of uses of ICT 68 2.2.6 Rules for participation and feedback 73 2.2.7 Results obtained in the first pilot test offered by the basic math virtual learning community 76 2.3 Second pilot test of basic math virtual learning community design 76 2.3.1 Content Knowledge (CK) 77 2.3.2 Pedagogical Knowledge (PK) 77 2.3.3 Pedagogical Content Knowledge (PCK) 78 2.3.4 Technological knowledge (TK) 78 2.3.5 Technological Pedagogical Knowledge (TPK), Technological Content Knowledge (TCK) and Typology of uses of ICT 79 2.3.6 Norms or rules for participation and feedback 86 2.3.7 Results obtained in the second pilot test of the basic math virtual learning community 86 Methodological considerations 88 3 Chapter III: Final version of the virtual learning community offered 88 3.1 Characteristics of the website that hosted the virtual learning community 88 3.2 Human team that participated in the design process of the VLC, in its final version. 94 3.3 Activities that characterized the implementation of the Virtual Learning Community 95 3.3.1 Who was invited to participate and in what context was this invitation made? 95 3.3.2 What evaluation agreements were generated? 96 3.3.3 What academic units were offered? 96 3.3.4 What activities were carried out to promote the effective participation of students in the virtual learning community? 96 3.4 Global map of the Virtual Learning Community offered. Final version. 98 3.4.1 Pedagogical conceptions that support open educational practices, of the 'open teaching' type 99 3.4.2 Connections between technological, pedagogical and content knowledge. 104 3.4.3 Key elements for the design of Virtual Learning Communities (VLC). 111 3.4.4 Dynamics that characterized the VLC offered, in its final version. 113 4 Chapter IV: Research Design 116 4.1 Mixed approach on educational research 116 4.2 Research questions 120 4.3 Analysis of online participation 120 4.4 Analysis of online interaction 127 4.5 Analysis of mathematical learning 132 Empirical Findings and Discussion 148 5 Chapter V: Research data and process of analysis 148 5.1 Collection and selection of data 148 5.2 Structural Analysis 149 5.2.1 First research question about the online participation 149 5.3 Content Analysis 154 5.3.1 Second research question about the online interaction 154 5.3.2 Third research question about the mathematical learning 165 5.3.3 Analysis based on the basic standards of mathematical competences 166 5.3.4 Analysis based on the ontosemiotic approach 169 6 Chapter VI: Results of data analysis and discussion of results 176 6.1 Structural analysis results 176 6.1.1 Online participation 176 6.2 Results of content analysis 194 6.2.1 Online interaction 194 6.2.2 Mathematical Learning 221 6.3 Review of results and conclusions 285 6.3.1 Results and discussion 285 6.3.2 Conclusions 301 7 References 307 8 Annexes 325

Many high school math teachers are not trained for teaching students with mild disabilities. Math curriculum is not typically developed with the needs of students with mild disabilities in mind. Teachers may not be aware of the unique barriers for students with mild disabilities, and strategies to help them master mathematics. The purpose of this study was to identify research-based techniques used by general education and special education teachers for teaching math to high school students with mild disabilities. The study also investigated teachers’ perceptions of and willingness to implement specific strategies to teach math skills to students with mild disabilities. Academic research articles were reviewed to identify strategies. A survey was taken by nine high school general education and five special education teachers who teach math. The results showed a reasonable degree of knowledge, experience, and positive perceptions of evidence-based strategies, especially co-teaching. There were significant differences between the responses of teachers with math teaching licenses compared to those with special education licenses. Based on the existing literature and the survey results, a handbook of resources was created for teachers of secondary math classes to support learning for students with mild disabilities.

The overall aim of this thesis was to explore the role of different assessment tools and training regimens in assessment and remediation for children with special educational needs in school. A central purpose of assessment explored was that it should inform remediation, teaching and instruction. The concepts of working memory, complex executive function and metacognitive strategy training for children with special educational needs were specifically explored in relation to this purpose of assessment. Complex executive function refers to planning and metacognitive ability, that many children with special educational needs struggle with, and which they are expected to handle in learning during school day. Of particular interest in the thesis was the contrast between working memory and complex executive function and how these concepts inform assessment and remediation practices. In this context, special attention was given to mathematical learning difficulties. The thesis was based on four studies (I‑IV). Study I explored the prevalence of different assessment tools, and dilemmas and challenges as perceived by assessment professionals, teachers and parents, in the work with children with special educational needs in Europe. In Study II, a metacognitive strategy training framework was developed as a training regimen, guided by research on complex executive function, and applied on working memory training. Effects of working memory training were compared between the two training regimens, with and without metacognitive strategy training, and also the overall effect of working memory training on cognitive functioning and the school related skills of reading, writing and arithmetic. In Study III, different types of measures of working memory and their predictive capacities in relation to mathematics achievement in national curriculum assessments were explored, as well as the effects of working memory training on mathematics achievement. In Study IV the role of working memory and complex executive function in identifying risk for mathematical learning difficulties in children with special educational needs was explored. The results from Study I suggested that assessment and remediation practices can contribute to a deficiency‑oriented outlook on children with special educational needs. In contrast parents and teachers in Sweden also reported that assessment could help them to better understand the needs of the child. Results from studies II-IV showed that only the use of a metacognitive strategy training regimen targeting complex executive function resulted in improvements following working memory training. The results also indicated that working memory training strongly predicted mathematical performance in national curriculum assessments of mathematics in school, and that a more complex change measure of working memory was a better predictor than simple working memory measures in this regard. Finally, the results also showed that complex executive function, defined as planning ability, was a better predictor than simple working memory in the assessment of risk for mathematical learning difficulties. The results from the studies were discussed in relation to the purpose of assessment to inform remediation, teaching and instruction for children with special educational needs. It was concluded that, in addition to working memory, as complex executive function – planning and metacognitive ability - seems to be an important cognitive function related to learning, this should be addressed both in the assessment of children with special educational needs as well as in the remediation when designing training regimens and interventions for children with special educational needs in general, and children at risk for mathematic learning difficulties in particular. It was also highlighted that in remediation, the role of the teacher as a mediator of metacognition and complex executive function seems vital. / Det övergripande syftet med avhandlingen var att undersöka den roll som olika utredningsverktyg och begrepp spelar i utformandet av utrednings‑ och stödinsatser för barn i behov av stöd i skolan. Ett centralt syfte med utredningar som utforskades är att de ska bidra med kunskaper vid utformande av stödinsatser och undervisning. I relation till detta syfte utforskades specifikt begreppen arbetsminne, komplex exekutiv funktion och metakognitiv strategiträning för barn i behov av stöd. Komplex exekutiv funktion syftar till planerings- och metakognitiv förmåga, något som många barn i behov av stöd upplevs ha svårigheter med, och som de förväntas kunna hantera i lärandet i skolans vardag. Av särskilt intresse i avhandlingen var kontrasten mellan arbetsminne och komplex exekutiv funktion och hur dessa begrepp bidrar till en förståelse i utformandet av utrednings- och stödinsatser. I detta sammanhang uppmärksammades särskilt matematiksvårigheter hos barn i behov av stöd. Avhandlingen bygger på fyra studier (I-IV). I Studie I undersöktes förekomsten av olika utredningsverktyg, samt utredares, lärares och föräldrars uppfattningar av dilemman och utmaningar i arbetet kring barn i behov av stöd, i Europa. I Studie II utformades ett koncept för metakognitiv strategiträning med utgångspunkt från forskning kring komplex exekutiv funktion och tillämpades i arbetsminnesträning. Effekten av arbetsminnesträning med och utan metakognitiv strategiträning jämfördes, liksom effekten av arbetsminnesträning på kognitiva funktioner och skolrelaterade färdigheter inom läsning, skrivning och aritmetik. I Studie III undersöktes olika mått på arbetsminne, och deras prediktiva kapacitet i relation till matematisk förmåga mätt genom nationella prov i matematik, samt effekten av arbetsminnesträning på matematisk förmåga. I Studie IV undersöktes vilken roll arbetsminne och komplex exekutiv funktion har i identifiering av barn i behov av stöd i riskzon för matematiksvårigheter. Resultaten från Studie I visade att utrednings- och stödinsatser kunde bidra till att förstärka ett brist-orienterat synsätt på barn i behov av stöd. I kontrast till detta, delgav lärare och föräldrar i Sverige att utredningar kunde hjälpa dem att förstå barnets behov på ett bättre sätt. Resultaten från studie II-IV visade att enbart den metakognitiva träningen, fokuserad på komplex exekutiv funktion, bidrog till förbättringar efter arbetsminnesträning. Resultatet indikerade också att arbetsminnesträning predicerar matematisk prestation i nationella prov i matematik, och att ett mer komplext arbetsminnesmått var en bättre prediktor än enklare arbetsminnesmått. Slutligen visade resultaten också att komplex exekutiv funktion i form av planeringsförmåga var en bättre prediktor än enkelt arbetsminnesmått vid utredning av risk för matematiksvårigheter. Resultaten från studierna diskuterades i relation till syftet med utredning: att bidra med kunskaper vid utformande av extra anpassningar, särskilt stöd och undervisning för barn i behov av stöd. Eftersom komplex exekutiv funktion –planerings- och metakognitiv förmåga - verkar vara en viktig kognitiv funktion i lärandet, bör man ta hänsyn till detta både i utrednings- och stödinsatser kring barn i behov av stöd generellt, och särskilt kring barn i riskzon för matematiksvårigheter. Det belystes också att i stödinsatser är lärarens roll som mediator av metakognition och komplex exekutiv funktion viktig. / <p>Vid tidpunkten för disputationen var följande delarbeten opublicerade: delarbete 3 (inskickat), delarbete 4 (inskickat)</p><p>At the time of the doctoral defence the following papers were unpublished: paper 3 (submitted), paper 4 (submitted)</p>

<p>Syftet med detta examensarbete är att undersöka undervisning och lärande under en matematiklektion som präglas av traditionell skolmatematik. Metoden för undersökningen var en deltagande observation av en matematiklektion i åk 3 på gymnasiet. Med hjälp av begreppen matematikens lärandeobjekt, matematiska resurser, eleven som lärande aktör och sociomatematiska normer har jag tolkat de resultat som genererats från observationen. Två slutsatser som kan dras av undersökningen är att eleverna stimuleras till att bli oberoende lärande aktörer i undervisningen av traditionell skolmatematik samt att det i första hand är läraren som synliggör potentiella matematiska resurser för eleverna. Medvetenheten om elevernas användande av matematiska resurser skulle kunna påverka elevernas lärande genom att läraren synliggör matematiska resurser på ett mer medvetet sätt.</p> / <p>The aim with this degree project is to examine teaching and learning during a mathlesson characterized by traditional school mathematics. The method of the study was aparticipant observation of a mathematics lesson in year 3 in upper secondary school. Using the concepts of mathematical learning objects, mathematical resources, and pupil as an active learner in combination with socio-mathematical norms, I have interpreted the results generated from the observation. Two main conclusions can be drawn from the study. Firstly, pupils are encouraged to become independent as active learners in the teaching of traditional school mathematics. Secondly, it is primarily the teacher who makes potential mathematical resources visible and available for the pupils. With an increasing awareness of pupils’ use of mathematical resources, teachers can affect pupils’ learning by making potential mathematical resources explicit in a more conscious way.</p>

Syftet med detta examensarbete är att undersöka undervisning och lärande under en matematiklektion som präglas av traditionell skolmatematik. Metoden för undersökningen var en deltagande observation av en matematiklektion i åk 3 på gymnasiet. Med hjälp av begreppen matematikens lärandeobjekt, matematiska resurser, eleven som lärande aktör och sociomatematiska normer har jag tolkat de resultat som genererats från observationen. Två slutsatser som kan dras av undersökningen är att eleverna stimuleras till att bli oberoende lärande aktörer i undervisningen av traditionell skolmatematik samt att det i första hand är läraren som synliggör potentiella matematiska resurser för eleverna. Medvetenheten om elevernas användande av matematiska resurser skulle kunna påverka elevernas lärande genom att läraren synliggör matematiska resurser på ett mer medvetet sätt. / The aim with this degree project is to examine teaching and learning during a mathlesson characterized by traditional school mathematics. The method of the study was aparticipant observation of a mathematics lesson in year 3 in upper secondary school. Using the concepts of mathematical learning objects, mathematical resources, and pupil as an active learner in combination with socio-mathematical norms, I have interpreted the results generated from the observation. Two main conclusions can be drawn from the study. Firstly, pupils are encouraged to become independent as active learners in the teaching of traditional school mathematics. Secondly, it is primarily the teacher who makes potential mathematical resources visible and available for the pupils. With an increasing awareness of pupils’ use of mathematical resources, teachers can affect pupils’ learning by making potential mathematical resources explicit in a more conscious way.

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Obwohl das Angebot von Feedback in Lehr-Lernsituationen eine wichtige instruktionale Maßnahme darstellt,um Lernende in ihrem Lernprozess zu unterstützen, wird dessen Potenzial in multimedialen Lernumgebungen nur selten optimal genutzt. Feedback, welches in aller Regel im Anschluss an die Bearbeitung von Lernaufgaben bereitgestellt wird und den Lerner über die Qualität seiner Aufgabenbearbeitung informiert, beschränkt sich häufig auf die Bereitstellung einer richtig/falsch Rückmeldung mit einem sich anschließenden, zusätzlichen Lösungsversuch bzw. der Präsentation der richtigen Lösung. Als prototypisch für diese Art der Feedbackgestaltung erweist sich das in mathematischer Grundschulsoftware implementierte Feedback. Der Lerner erhält weder Informationen zu dem Ort und der Art des Fehlers noch einen Hinweis auf die korrekte Lösungsstrategie. Vor diesem Hintergrund bestand ein zentrales Ziel der vorliegenden Arbeit in der Entwicklung von fehlerspezifischem informativem tutoriellem Feedback (ITF) für die schriftliche Subtraktion auf der Grundlage des heuristischen Feedbackmodells von Narciss (2004) und der aus diesem Modell abgeleiteten Gestaltungsprinzipien für ITF. Obwohl allgemein erwartet wird, dass informatives Feedback lernwirksamer ist als Feedback, welches nur die richtige Lösung präsentiert, konnte diese Annahme durch die Ergebnisse der empirischen Feedbackforschung bisher nicht bestätigt werden. So ist die Feedbackforschung durch eine inkonsistente Befundlage gekennzeichnet. Vor diesem empirischen Hintergrund bestand ein weiteres zentrales Ziel der vorliegenden Arbeit in der Überprüfung der Effektivität des entwickelten fehlerspezifischen ITFs hinsichtlich verschiedener Lern- und motivationaler Parameter. Fehlerspezifisches ITF zeichnet sich dadurch aus, dass bei fehlerhaften Aufgabenlösungen Informationen angeboten werden, die auf der Grundlage einer kognitiven Anforderungs- und Fehleranalyse als korrekturrelevant identifiziert worden waren. Diese korrekturrelevanten Informationen werden dem Lernenden ohne Bekanntgabe der korrekten Lösung präsentiert, so dass sich dieser dazu aufgefordert sieht, die bereitgestellten strategischen Informationen in einem erneuten Lösungsversuch unmittelbar anzuwenden. In zwei Studien mit Viertklässlern wurde dieses computergestützte ITF im Hinblick auf seine Lern- und motivationale Wirksamkeit evaluiert. In der ersten Studie (N = 30) erfolgte diese Evaluation gegen eine Feedbackbedingung, welche das üblicherweise in mathematischer Lernsoftware realisierte Knowledge of Response und Knowledge of Correct Response (KR-KCR) Feedback präsentierte. In der zweiten Studie (N = 105) wurden zwei zusätzliche Feedbackbedingungen &amp;quot;Knowledge about Mistake (KR-KM) und Knowledge on How to proceed (KR-KH)&amp;quot; aufgenommen. Diese sind in ihrem Informationsgehalt zwischen dem ITF und dem KR-KCR Feedbackalgorithmus anzusiedeln. Die Ergebnisse beider Studien bestätigen die postulierte Lern- und motivationale Wirksamkeit des fehlerspezifischen ITFs für die schriftliche Subtraktion. Besonders im Vergleich zu der wenig informativen KR-KCR Feedbackbedingung konnten in beiden Studien signifikante Wirksamkeitsunterschiede nachgewiesen werden. Auch im Vergleich zu den komplexeren Feedbackformen Knowledge about mistake (KR-KM) und Knowledge on how to proceed (KR-KH) konnten für die Schüler der ITF Bedingung bessere Leistungen beobachtet werden, auch wenn diese Unterschiede nur partiell signifikant wurden. Die Ergebnisse der vorliegenden Untersuchungen zeigen, dass es möglich ist, auf der Grundlage des von Narciss (2004) vorgestellten Feedbackmodells und der aus diesem Feedbackmodell abgeleiteten Gestaltungsprinzipien lern- und motivationsförderliches ITF zu entwickeln. So liefern die Ergebnisse beider Studien erste empirische Belege für die Annahme, dass informatives Feedback nicht nur den Lernprozess unterstützt, sondern auch motivationsförderlich wirkt. / Feedback is considered to be an important factor to promote learning and motivation with computer-based training tools. However the findings of studies on the effectiveness of feedback are rather inconsistent. One reason for these inconsistent findings might be that the implementation of feedback is more based on intuition than on well-founded design principles. Therefore the aims of the present work was (1) to develop theoretical well-founded elaborated feedback forms for a procedural skill like the written subtraction, (2) to implement them in an adaptive feedback algorithm that induces the mindful processing of feedback, and (3) to evaluate this elaborated feedback regarding his impact on learning and motivation. Using results from prior feedback research and from cognitive task and error analysis of written subtraction tasks, information relevant to the correction of typical systematic errors (e.g. location of error, source of error, type of error and hints to the correct solution strategy) were selected. This information was used to design different types of bug-related feedback messages. These different feedback messages were arranged in an adaptive bug-related algorithm presenting three levels of feedback with increasing informational value to support the learner in finding the correct solution on his own. Because the developed feedback allows assisted multiple response attempts for an item by providing relevant information for error correction, but no immediate Knowledge of Correct Response (KCR) feedback, and by encouraging the learner to apply the corrective information to a further attempt, this kind of feedback was called bug-related &amp;quot;informative tutoring feedback (ITF)&amp;quot;. In two computer-based learning experiments cognitive and motivational effects of this bug-related ITF were evaluated with forth-grade students with difficulties in written subtraction. In the first study (N=30) the bug-related ITF was compared to the effects of a standard Knowledge of Result-Knowledge of Correct Response feedback (KR-KCR) algorithm. In the second study (N=105) two additional feedback algorithms - Knowledge about Mistake (KR-KM) und Knowledge on How to proceed (KR-KH) - were implemented. These both feedback algorithms range between the bug-related ITF and the simple KR-KCR feedback regarding to their amount of information. In both studies significant positive effects of the bug-related ITF compared to the KR-KCR feedback could be found. Regarding to the more informative KR-KM and KR-KH feedback a positive effect of the bug-related ITF could be showed sporadic. These results are discussed in detail.