Magister Educationis - MEd / In South Africa it is widely known that most learners struggle with mathematics. The results for mathematics are poor. The department of basic education offers a number of intervention programmes to assist learners in mathematics but the problem still persists. Algebra is the most basic and important topic in mathematics as it becomes an element in almost all the other topics in mathematics curriculum. Algebraic fractions in particular are a challenge for most leaners. Research shows that learners commit a number of errors when they work with algebraic fractions. The study investigated the implementation of an intentional teaching model into grade 9 mathematics learners’ ways of working with rational algebraic fractions. An intentional teaching model is a teaching strategy which emphasizes teaching intentions or teaching objectives are brought to the fore during a lesson, the model emphasizes the use of spiral revision and assessment for learning. Ways of working in this study refers to the way in which learners deal with algebraic fractions when they simplify them including the errors they commit from the misconceptions they have about aspects of working with fractions. The study was conducted in a group of grade 9 mathematics learners at Gugulethu High school, which is located in Guguletu, a township in the Western Cape Province of South Africa. The study is premised on a qualitative research paradigm which focuses on studying situations in their natural settings and applying an interpretive perspective. Data was collected by means of observation and video recording of lessons while learners were engaged in working with algebraic fractions. Learners’ written work was analysed as part of the data collection. The results of the study show that leaners commit a number of errors when they manipulate algebraic fractions. Among other errors are: (i) Cancellation errors which had the highest frequency of occurrence (ii) Defractionalisation (iii) No recognition of the common factor and (iv)Exponential laws error. It was found that the learners’ ways of working with algebraic fractions are mostly characterised by their misunderstanding of exponential laws and difficulty in working with fractions needing the use of factorisation to simplify and find the lowest or highest common denominator during addition or subtraction. The results of the study also reveal that learners struggle to articulate extensively or in detail what they are actually doing as they simplify rational algebraic fraction.

Magister Educationis - MEd / This research investigated how grade 7 learners dealt with introductory aspects of integers when they are introduced through a temperature model. In particular, the study analysed the effect of an intentional teaching strategy on learners' engagement with integers. The idea of combining an intentional teaching strategy with the introduction of integers in grade 7 learners using a temperature model is what makes this study unique. A qualitative study was adopted. Data was collected by means of audio and video and also by means of learners' completed worksheets. The results of the study indicate that the majority of learners could recognise, compare and order integers. It is recommended that the application of intentional teaching with a temperature model is a viable strategy to introduce grade 7 learners to integers.

While there are numerous models to practice instructional design, few instructional models to teach instructional design have been documented. This dissertation documents the development of a reflexive teaching model for the teaching of instructional design (ID) in a graduate education program. The model supports co-participatory learning of instructional design and mutual examination of one's learning and participation by both instructor and students. A design and development framework is used to describe the design decisions, model implementation, and evaluation of the model across six deliveries, or case studies, of a master's level instructional design course from 1994-1998. Design decisions included course sequence, learning tasks, instructional materials, and assessment rubric. Model implementation described student responses to instruction and instructor efforts to assist learners. The model was evaluated in terms of student performance on instructional design projects, student perceptions of their learning, and instructor responsivity to learner needs. The model's development was summarized in terms of changes in design decisions, model implementation, and model evaluation over the six cases. A discussion of the reflexive model is presented using Joyce and Weil's (1996) conceptual approach, describing the model's social system, syntax, participants' reaction, support system, plus the model's instructional and nurturant effects. Four categories of conclusions address improvements to the instructional approach, conditions that promote successful use of the model, impact of the model on student and teacher learning, and conditions conducive to efficient model development. Limitations of the study, future research options, and the implications of the model for ID instruction, the ID process, and teacher inquiry are discussed. / Ph. D.

Magister Educationis - MEd / This research has employed an intentional teaching model to investigate the algebraic reasoning of grade 9 learners from a low socio-economic background. It has also sought to study how learners engage with algebra to make generalizations and to scrutinize any misconceptions deriving from the experience. They looked for patterns, paid attention to aspects of the patterns that are important and then generalized from familiar to unfamiliar situations. Algebraic reasoning underpins all mathematical thinking including arithmetic because it allows us to explore the structure of mathematics. This study is based on the Curriculum and Assessment Policy Statement which states that learners are expected to investigate patterns to establish the relationships between variables, as well as represent and analyse the change of patterns. The study also had a huge emphasis on algebra. According to Mphuthi & Machaba (2016): “Algebraic expressions form part of the senior phase CAPS curriculum in South Africa. A substantial amount of time is allocated to this section on evaluating expressions and simplifications of algebraic expressions in grade 7-9.” The study is premised on a qualitative research paradigm and a design-based research methodology for data collection. A set of tasks based on algebraic patterns and generalizations was given to an opportunistic sample of 20 grade 9 learners in a school in Delft, a low socio-economic suburb about 30 kilometres from Cape Town in the Western Cape Province of South Africa. Three weeks after completing the tasks, learners were interviewed to identify their reasoning and how they felt about the tasks. The results of the study show that the majority of the learners struggled with tasks especially when asked what the rules they could derive from the patterns. Learners did not seem to understand what they were doing because they were unable to articulate the given tasks in words and did not have knowledge of concepts like the perimeter.

Though, most of online courses conducted in the pass were asynchronous oriented, due to the recent advance of information technology on the Internet bandwidth and the IP-based conferencing system, conducting synchronous online courses are now feasible. There are no longer obstacles for promoting e-learning in terms of limited computing power, short memory capacity and not enough network bandwidth. The new challenges have shifted to how to conduct a high qualify online course by combing synchronous and asynchronous learning into a blended mode. The feature of asynchronous-oriented instruction and the text only interaction mode in the passed e-learning are very inconvenient and inefficient. Because Chinese input for many senior teachers are difficult and using text express the meaning of content and answer questions would result bad performance of demonstrating instructors¡¦ professional. The advanced Learning Management System nowadays can support IP-based video-conferencing and document sharing for teachers and students to do online synchronous face to face instruction and interactions; the question is most teachers are not aware of this kind of potential and what kind of teaching models can be adopted in the online synchronous classroom. The purpose of this thesis is to explore different teaching models and their best practices by action research, such that the results can serve as good references for teachers. This research environment is the course named e-Learning Theory & Practice conduced for many semesters on NSYSU Cyber University.

I detta arbete används begreppet ”kvalitativt god matematikundervisning” i syfte att ringa in matematikundervisning som är framgångsrik och leder till meningsfullt lärande i matematik. Mot bakgrund av att svenska skolelevers resultatnedgång i matematik väcker frågor kring vad kvalitativt god matematikundervisning är, syftar denna studie till att: beskriva vad som kan anses utgöra kvalitativt god undervisning i matematik och försöka överföra beskrivningen till en för lärare användbar undervisningsmodell. För studiens genomförande har forskningsfrågorna varit: Vad kännetecknar kvalitativt god matematikundervisning? och vilka roller och ansvar innebär kvalitativt god matematikundervisning för läraren och eleverna? En metaanalytisk inspiration ligger till grund för en metodologi baserad på systematisk översyn av aktuell svensk matematikdidaktisk forskning. I urvalet ingår sju doktorsavhandlingar och en vetenskaplig rapport publicerade mellan åren 2006 och 2013. Granskningen utav de åtta forskningspublikationerna visar att kvalitativt god matematikundervisning kännetecknas utav en kommunikativt vital undervisningsmiljö där gemensamma och utforskande aktiviteter används för att etablera en konceptuell förståelse i matematik. I kvalitativt god matematikundervisning behandlas öppna och matematiskt rika problem som kräver tolkningsförmåga och tränar elever i att matematisera sitt språk och sitt tänkande. Läraren bör eftersträva insyn i hur elevernas uppfattningar och matematiska förståelser är beskaffade genom hela undervisningsförloppet för att bemöta dem med relevant återkoppling. Läraren arbetar som en ciceron och erbjuder elever möjligheter till lärande genom att bjuda in till möten med matematiska lärandeobjekt. Eleverna är aktiva och delaktiga i diskussioner och återkopplingar genom undervisningen och fungerar därmed både som sina egna, varandras och lärarens resurser. Matematikundervisningens olika kvalitetsaspekter sammanfattas i en modell med viss potential att fungera även som lektionsmall. Modellens och resultatens giltighet diskuteras mot tidigare forskning vilket åskådliggör likheter med faktorer för framgångsrik undervisning överlag, ej enbart i matematik.

This paper aims to present a model of teaching and learning mathematics in China. The model is “Two Basic”, basic knowledge and basic ability. Also, the paper will analyze some of the background of the model and why it is efficient in mathematics education. The model is described by a framework of “slab” and based on a model of learning cycle, allow students to work with mathematical thinking. Though the model looks of demonstration and practice looks very traditional, the explanation behind allows us to understand why Chinese students achieved well in many international studies in mathematics. The innovation of the model is the teacher intervention during the learning process. Such interventions include repeated practice, and working on group of selected related questions so that abstraction of the learning process is possible and student can link up mathematical expression and process. Examples used in class are included and the model can be applied in teaching advanced mathematics, which is usually not the case in some many other existing theories or framework.

A presente Tese está inserida no contexto do ensino de Administração, com o desafio de relacionar entre si os conceitos acerca da aprendizagem, do ensino, da Administração de Empresas, da Tecnologia da Informação, do construtivismo e dos ambientes de aprendizagem. Foi tomado como objetivo o de se realizar um estudo original, elaborando uma nova proposta para o ensino e a aprendizagem, com base nos pressupostos do construtivismo, a partir da elaboração de um Modelo Construtivista para o Ensino de Administração. O tema escolhido como ponto de partida foi o da necessidade do entendimento de como os estudantes constroem o seu conhecimento, conforme o que é proposto na teoria da Epistemologia Genética (PIAGET, 1978). O conhecimento se dá através da ação deste indivíduo sobre o objeto sobre o qual ele está aprendendo. Os novos conhecimentos são construídos com base nas estruturas que estes já possuíam previamente, sendo os conhecimentos retirados das ações refletidas sobre os objetos observáveis com os quais o sujeito interage. A ação do sujeito, ao interagir com o objeto em estudo, dá a este indivíduo a oportunidade de atribuir significados àquilo sobre o que ele estiver aprendendo. Na educação, há variados modelos que empregam o construtivismo. Dentre eles, há os: do “design instrucional”; da “cognição situada”; da “instrução ancorada”; da “aprendizagem vivencial”; e o do “aprendizado baseado em problemas”. Além disto, há alguns ambientes que utilizam os pressupostos construtivistas, como os seguintes: “ambientes complexos de aprendizagem”; “ambientes para novas aprendizagens”; e “sistemas de gerenciamento da aprendizagem”. Os atuais modelos de ensino, em Administração, apresentam-se limitados para os novos conhecimentos que são exigidos dos profissionais, dentre eles a autonomia do estudante, a sua capacidade para a tomada de decisões e a socialização. A metodologia utilizada na tese aqui apresentada foi a da “pesquisa-ação”, sendo esta formada pelas etapas: exploratória; principal; de ação; e de avaliação do modelo. O que se propôs na presente Tese foi um Modelo Construtivista para o Ensino de Administração, composto por uma metodologia de ensino e por um ambiente de aprendizagem, ambos elaborados com base nos pressupostos da Epistemologia Genética. O modelo foi formado por “Contextos de Aprendizagem” e por “Situações de Aprendizagem” nos quais os estudantes resolviam problemas empresariais reais, tendo que utilizar as suas concepções prévias, discutir em grupos, tomar decisões e elaborar relatórios para a resolução das situações. Os resultados da presente pesquisa indicam que o modelo proposto atende às seguintes necessidades: elaborar atividades de resolução de problemas; fazer com que os materiais pedagógicos sejam elaborados com base na contextualização; apoiar a construção do conhecimento do estudante; apresentar condições para a realização de atividades contextualizadas, aplicáveis à realidade; de estas serem resolvidas em grupos; e de que tais tarefas sejam realizadas através de problemas que reflitam situações concretas, encontradas no mundo real. A tecnologia, os materiais pedagógicos e o ambiente físico facilitam o processo de construção do conhecimento, tendo sido planejados juntamente com os demais elementos. / This Doctoral Thesis is in Business teaching context, with the challenge of relating the concepts of learning, teaching, Business Administration, Information Technology, constructivism and learning environments. It was defined as objective to realize an original study, building a new purpose for teaching and learning, based on constructivism assumptions, through the development of a Constructivist Business Teaching Model. The theme chosen as a start point was the need of understanding how students construct their knowledge according to the assumptions of Genetic Epistemology theory (PIAGET, 1978). Knowledge creation occurs through the action of these individuals over the object that is being learned. New knowledge is constructed based on structures that have already been constructed, being the knowledge withdrawn reflected action over observable objects they interact with. The individual action, interacting with the object being studied, gives him the opportunity of making sense about the subjects being learned. In Education, there is a variety if models that apply constructivism. Among them, there are: “instructional design”; “situated cognition”; “anchored instruction”; “experiential learning”; and “problem based learning”. Therefore, there are some environments that use the constructivism assumptions: “complex teaching learning environments”; “new learning environments”; and “learning management systems”. The current teaching models, in Business area, are restricted regarding new learning requirements by professionals, like student autonomy, his capability of decision making and socialization. The methodology used in the Thesis was the “action research”, constituted by these phases: exploration; principal; action; and model evaluation. The purpose of this Thesis was to present a Constructivist Business Teaching Model, composed by a teaching methodology and a learning environment, both of them created with basis on Genetic Epistemology assumptions. The model was composed by “Learning Contexts” and “Learning Situations” in which students solve real business problems, having to use their previous conceptions, discuss in groups, make decisions and elaborate reports to solve the situations. The results of this research indicate that the proposed model is in accordance with the following needs: elaboration of problem solving activities; develop the pedagogical material based on the contexts; support the knowledge construction of students; to present conditions for contextualized activities, applicable to reality; having these activities solved in groups; the tasks has to be solved through problems that reflect concrete situations, found in real world. The technology, the pedagogical materials and the physical environment enable the knowledge construction process, having being planned with other elements.

This thesis explores the phenomenon of model making. The first part is devoted to the history of models in the world and in the Czech Republic. First, it is generally described the history from prehistoric times to the present. Further the thesis is focused on selected models and institutions in which models can be found. The second part describes selected publications and periodicals dealing with this topic. The third part deals with the relationship between model making and art education, or relationship model making and teaching at the elementary school. There are described children's views on theoretical models and outline projects whose the main theme are models. The work aims to inform teachers primarily about using of models in teaching.

A presente Tese está inserida no contexto do ensino de Administração, com o desafio de relacionar entre si os conceitos acerca da aprendizagem, do ensino, da Administração de Empresas, da Tecnologia da Informação, do construtivismo e dos ambientes de aprendizagem. Foi tomado como objetivo o de se realizar um estudo original, elaborando uma nova proposta para o ensino e a aprendizagem, com base nos pressupostos do construtivismo, a partir da elaboração de um Modelo Construtivista para o Ensino de Administração. O tema escolhido como ponto de partida foi o da necessidade do entendimento de como os estudantes constroem o seu conhecimento, conforme o que é proposto na teoria da Epistemologia Genética (PIAGET, 1978). O conhecimento se dá através da ação deste indivíduo sobre o objeto sobre o qual ele está aprendendo. Os novos conhecimentos são construídos com base nas estruturas que estes já possuíam previamente, sendo os conhecimentos retirados das ações refletidas sobre os objetos observáveis com os quais o sujeito interage. A ação do sujeito, ao interagir com o objeto em estudo, dá a este indivíduo a oportunidade de atribuir significados àquilo sobre o que ele estiver aprendendo. Na educação, há variados modelos que empregam o construtivismo. Dentre eles, há os: do “design instrucional”; da “cognição situada”; da “instrução ancorada”; da “aprendizagem vivencial”; e o do “aprendizado baseado em problemas”. Além disto, há alguns ambientes que utilizam os pressupostos construtivistas, como os seguintes: “ambientes complexos de aprendizagem”; “ambientes para novas aprendizagens”; e “sistemas de gerenciamento da aprendizagem”. Os atuais modelos de ensino, em Administração, apresentam-se limitados para os novos conhecimentos que são exigidos dos profissionais, dentre eles a autonomia do estudante, a sua capacidade para a tomada de decisões e a socialização. A metodologia utilizada na tese aqui apresentada foi a da “pesquisa-ação”, sendo esta formada pelas etapas: exploratória; principal; de ação; e de avaliação do modelo. O que se propôs na presente Tese foi um Modelo Construtivista para o Ensino de Administração, composto por uma metodologia de ensino e por um ambiente de aprendizagem, ambos elaborados com base nos pressupostos da Epistemologia Genética. O modelo foi formado por “Contextos de Aprendizagem” e por “Situações de Aprendizagem” nos quais os estudantes resolviam problemas empresariais reais, tendo que utilizar as suas concepções prévias, discutir em grupos, tomar decisões e elaborar relatórios para a resolução das situações. Os resultados da presente pesquisa indicam que o modelo proposto atende às seguintes necessidades: elaborar atividades de resolução de problemas; fazer com que os materiais pedagógicos sejam elaborados com base na contextualização; apoiar a construção do conhecimento do estudante; apresentar condições para a realização de atividades contextualizadas, aplicáveis à realidade; de estas serem resolvidas em grupos; e de que tais tarefas sejam realizadas através de problemas que reflitam situações concretas, encontradas no mundo real. A tecnologia, os materiais pedagógicos e o ambiente físico facilitam o processo de construção do conhecimento, tendo sido planejados juntamente com os demais elementos. / This Doctoral Thesis is in Business teaching context, with the challenge of relating the concepts of learning, teaching, Business Administration, Information Technology, constructivism and learning environments. It was defined as objective to realize an original study, building a new purpose for teaching and learning, based on constructivism assumptions, through the development of a Constructivist Business Teaching Model. The theme chosen as a start point was the need of understanding how students construct their knowledge according to the assumptions of Genetic Epistemology theory (PIAGET, 1978). Knowledge creation occurs through the action of these individuals over the object that is being learned. New knowledge is constructed based on structures that have already been constructed, being the knowledge withdrawn reflected action over observable objects they interact with. The individual action, interacting with the object being studied, gives him the opportunity of making sense about the subjects being learned. In Education, there is a variety if models that apply constructivism. Among them, there are: “instructional design”; “situated cognition”; “anchored instruction”; “experiential learning”; and “problem based learning”. Therefore, there are some environments that use the constructivism assumptions: “complex teaching learning environments”; “new learning environments”; and “learning management systems”. The current teaching models, in Business area, are restricted regarding new learning requirements by professionals, like student autonomy, his capability of decision making and socialization. The methodology used in the Thesis was the “action research”, constituted by these phases: exploration; principal; action; and model evaluation. The purpose of this Thesis was to present a Constructivist Business Teaching Model, composed by a teaching methodology and a learning environment, both of them created with basis on Genetic Epistemology assumptions. The model was composed by “Learning Contexts” and “Learning Situations” in which students solve real business problems, having to use their previous conceptions, discuss in groups, make decisions and elaborate reports to solve the situations. The results of this research indicate that the proposed model is in accordance with the following needs: elaboration of problem solving activities; develop the pedagogical material based on the contexts; support the knowledge construction of students; to present conditions for contextualized activities, applicable to reality; having these activities solved in groups; the tasks has to be solved through problems that reflect concrete situations, found in real world. The technology, the pedagogical materials and the physical environment enable the knowledge construction process, having being planned with other elements.