Free play is a term that often common in educational activities. The definition of the term is difficult, but it is about the children themselves decides the scope, content and process in their play. On an after-school, this may mean that it is the children are given the opportunity to freely choose their own activity in the leisure constraints. Free play can be an alternative to other, more controlled activities where adults control framework. The purpose of this study is to investigate the variations in how some children perceive and experience the phenomenon of free play in the after-school and how the children perceive the teacher´s role in the free game. The study empirical data is limited to a class, in a department in an after-school center in the inner suburbs of Stockholm. School centers and schools are integrated with each other and operations are in close cordinations and cooperation in common areas. Study based in two qualitative semi-structured focus group interviews with three children in each other group. To analyze the material used a phenomenographic analysis model. The result show that children feel that it is imortant to have free play in the school center. The children experience a degree of freedom, because you yourself may determine the content of their play. Pedagogue role in the free game perceive children as a role, where the adults keep order and ensure that the children needs are met and to avoid conflict. Educators will alto be assisting with activities to inspire creativity. Free play is also linked to emotions such as desire and curiosity and to friends enriches free play greatly.

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According to the recent Statistics South Africa (2007), there are 668 176 children with disability between 6 – 23 years old of which 337 660 are males and 330 516 females in the Amathole District of the Eastern Cape. This number consists of schoolchildren with visual, auditory, physical, mental, multiple and non- specified types of disabilities. However, there are limited research studies on the challenges around opportunities for children to participate in variety of sporting activities within the school settings. The purpose of this study was to provide a more accurate picture of the barriers against inclusion of students with disabilities in sports, among the identified Special Schools in the Amathole district, Eastern Cape. Data collection involved interviews, questionnaire and participant observation methods. Results showed that 87 percent of the students with disabilities participated in one sport or the other with male student participating more (92.2 percent than the female participants (86.7 percent). Overall, the participation rate was 86.7 percent. Further analysis of data revealed that 44 percent female and 41.5 percent male students agreed strongly that sporting activities are beneficial to their health. 7.1 percent of the sample indicated that their disability prevented them from sports participation. The reported main barriers towards sports participation include; lack of trained personnel; lack of adequate funding; lack of cooperation from parents and guardians and inadequate facilities and equipment. Although students with disabilities were using the limited resources at their disposal to participate in sports, they need support to sustain their efforts.

Considering that geometry is taught according to certain principles that do not encourage creativity, I have decided to employ the mixed methods philosophical framework applying the concurrent transformative design in the form of an exploratory case study. The case study to (i) explore and design a model that influences learning using polygon pieces and mathematics dictionary in the teaching and learning of geometry to grade 8 learners; (ii) investigate if the measurement of angles and sides of polygons using polygon pieces assisted by mathematics dictionary promote learners’ comprehension of geometry and (iii) investigate how mathematics teachers should use polygon pieces along with mathematics dictionary to teach properties of triangles in order to promote learners’ conceptual understanding. Drawing from my research findings a model has been developed from the use of polygon pieces and mathematics dictionary. The model use of mathematics dictionary in teaching and learning geometry is to develop learners’ mathematics vocabulary and terminology proficiency. Polygon pieces are to enhance the comprehension of geometric concepts. The quantitative data emerged from marked scripts of the diagnostic and post-intervention tests, the daily reflective tests and intervention activities were analysed as percentages and presented in line and bar graphs. Qualitative data obtained from observation notes and transcribed interviews were analysed in three forms: thematically, constant comparison and keywords in context. These findings support other research regarding the importance of using physical manipulatives with mathematics dictionary in teaching and learning geometry. They align with other findings that stress that manipulatives are critical facilitating tools for the development of mathematics concepts. The investigations led into the designing of a teaching model for the topic under study for the benefit of the mathematics community in the teaching and learning of geometry, focusing on properties of triangles. The model developed during this study adds to the relatively sparse teaching models but growing theoretical foundation of the field of mathematics. / Mathematics Education / Ph. D. (Mathematics Education)

CAPES / Apresentamos neste trabalho alguns paradoxos lógico-matemáticos, como o paradoxo de Galileu, e também alguns paradoxos geométricos, como os paradoxos de Curry, de Hooper e de Banach-Tarski. Empregamos os paradoxos de Curry e de Hooper para motivar o estudo de conceitos de Geometria e de Teoria dos Números, tais como área, semelhança de triângulos, o Teorema de Pitágoras, razões trigonométricas no triângulo retângulo, o coeficiente angular da reta e a sequência de Fibonacci, e organizamos atividades lúdicas para a sala de aula no Ensino Fundamental e no Ensino Médio. / We present in this work some logical-mathematical paradoxes, as Galileo's paradox, and also some geometric paradoxes, such as Curry's paradox, Hooper's paradox and the Banach-Tarski paradox. We employ the Curry and Hooper paradoxes to motivate the study of concepts of Geometry and Number Theory, such as area, triangle similarity, Pythagorean Theorem, trigonometric ratios in the right triangle, angular coefficient of the line, and Fibonacci sequence, and we organize recreation activities for the classroom in Elementary and High School.