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Implementering av digitala verktyg har haft inverkan på hur lärare väljer att undervisa i matematikklassrummet de senaste åren. Därför som blivande matematiklärare blir det intressant att undersöka hur lärare ser på hur digitala verktyg påverkar den egna undervisningen och elevernas lärande samt undersöka lärares val av undervisningsmetoder i geometriämnet när användningen av digitala hjälpmedel står i fokus. För att samla in data genomfördes intervjuer av sju matematiklärare från tre olika skolar. De teoretiska ramverken som använts för analys i vår studie är SAMR-modellen (SAMR är en akronym som står för fyra engelska begrepp: Substitution, Augmentation, Modification och Redefinition) som beskriver hur digitala verktyg används i olika steg i undervisning och Drijvers orkestreringstyper för att identifiera olika undervisningsmetoder och beskriva deras kännetecken. Studien visade att alla intervjuade lärare använder olika digitala verktyg i sin geometriundervisning samt att de överlag är positiva till att använda verktygen. Lärarna beskriver att verktygen understödjer undervisningen utifrån de behov som uppstår i praktiken, till exempel behov av individanpassade uppgifter, olika redovisningsformer medmera. Resultatet av analysen visade att flera av de intervjuade lärare når steg tre i SAMR-modellen, vilket innebär att synen på elevernas lärande och undervisningen har börjat förändrats. Studien visade också att lärarna varierar och utnyttjar olika orkestreringstyper för att organisera sin undervisning för specifika lärandemål. / The aim of this thesis is to execute digital tools in mathematics teaching. The study focuses its attention on how teachers choose the type of the digital tools and carry out mathematics teaching, especially in the subject of geometry. As a future teacher, it will be interesting to investigate the teachers’ views concerning the effect of digitizing on their teaching and students’ learning, as well as the teacher's choice of teaching methods and the design of the teaching within that method, will depend on the lessons of geometry where the use of digital technology is the focus in the classroom. The empirical materials were collected through interviews with seven teachers from three different schools. The data were analyzed with a focus on the ways by which teachers employ digital tools to support and work with mathematical content. The theoretical frameworks in our study are the SAMR model (SAMR is an acronym that stands for four English concepts Substitution, Augmentation, Modification, and Redefinition) that describes how digital tools are used at different stages in teaching and Drijver's orchestration type to identify different teaching methods and describe their characteristics. The results based on the interviews show that most teachers meet requirements for reaching step three in the SAMR model. The study also shows an overview of different teaching methods in mathematics teaching where digital technology is used. The results also show that the teachers utilize several types of orchestration in the work with the subject of geometry in their teaching. In total, this study shows that teachers use up to five different types of orchestration. Generally, all teachers have positive opinions concerning the employment of digital technology in mathematics education, and the utilities of digitization that they spotlight include assisting and supporting students' mathematical knowledge as well as developing their understanding of geometric concepts.

The purpose of this study is to examine teachers, in the elementary grades 2 and 3, view on pupils ' understanding of regrouping in subtraction algorithms and the use of concrete materials to facilitate pupils' understanding of regrouping in subtraction algorithms. The study emanates from teachers who teach, or have taught, in regrouping in subtraction algorithms in the elementary grades of 2 and 3. With the use of a semi-structured interview, we were able to gain an understanding of teachers' opinions and experience about pupils' understanding of regrouping in subtraction algorithms and their use of concrete materials to facilitate pupils' understanding.  The teachers in our study agree on the fact that pupils can have difficulties with regrouping in subtraction algorithms. The study shows that the teachers´ opinions and experience indicate that concrete materials can facilitate pupils' understanding of mathematics, but they also show different opinions and experiences for whom the opportunities exist. Furthermore, the results of the study show that the teachers who participated in the study can also see limitations in using concrete materials in their mathematics teaching, such as the amount of material that is available, that the concrete material invites to play and more. Most of the teachers teach with concrete material for regrouping in subtraction algorithms to facilitate the pupils understanding and to provide clarity for the pupils, as seen from the results of the study. Furthermore, the results show that there are teachers who do not teach with concrete materials for regrouping in subtraction algorithms.

Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Mathematics anxiety is a personal characteristic which is widespread and continuing. It has a debilitating effect on mathematics performance and contributes to perceptions and attitudes that perpetuate a dislike for mathematics and a lack of confidence when dealing with mathematical problems. An investigation of relevant literature on mathematics anxiety identifies sources and symptoms and emphasises a need for a comprehensive approach to remediation. The historical development of an appropriate measuring instrument is documented and statistical evidence is used to create a mathematics anxiety rating scale suitable for measuring anxiety levels of secondary school pupils and student teachers. The extensive literature interest, research publications and remedial programmes emphasise the problem of mathematics anxiety and thus the need for a comprehensive approach to remediation. Mathematics teaching and curriculum design is expounded to provide the necessary direction to the alleviation of mathematics anxiety. General perspectives on curriculum design are discussed and a cyclical systems approach is recommended. Elements of this approach are detailed and are linked to important personal characteristics to add a humanistic and socio-cultural view of curriculum design in mathematics. The didactic viability of constructivism as an approach to mathematics curriculum design is investigated. Constructivism embodies a philosophy and a methodology which addresses the critical aspects influencing mathematics anxiety. Classroom topics and activities are reviewed in terms of a constructivist approach and the sources of mathematics anxiety are discussed from a constructivist perspective. A longitudinal case study of pupils during their five years at secondary school as well as a study involving student teachers was undertaken. Mathematics performance, perceptions, attitudes and levels of anxiety were investigated by means of tests, questionnaires, and mathematics anxiety rating scales. The statistical results of this research provide evidence to support a comprehensive approach to the remediation of mathematics anxiety. Constructivism is seen as the synthesis of critical aspects of teaching and curriculum development which will stem the perpetuation of mathematics anxiety. Constructivism provides the didactic approach to develop each individual's intellectual autonomy and mathematics power, by instilling a problem solving attitude and a self-confidence when doing mathematics. / Curriculum and Instructional Studies / D. Ed. (Didactics)

Nous visons à mieux comprendre, à travers la notion composite de rapport(s) au(x) savoir(s), la façon dont le sujet apprenant haïtien se construit dans l'espace scolaire comme membre de la société et s'y intègre. L’approche se base sur des données construites sur des singularités d’« histoires » scolaires et des statistiques issues d’études sociologiques. Elles sont analysées et interprétées dans un cadre théorique inspiré de travaux développés par l’équipe ESCOL (Charlot, Bautier, Rochex) ou par Beillerot dans une perspective clinique, ou culturelle comme Hayder. Les situations sociopolitiques et économiques rendent complexes les rapports à l’école et à ses objets : ainsi dans la société haïtienne, les enseignants éprouvent des difficultés à susciter l’intérêt pour les savoirs scolaires et les apprenants expriment des sentiments d’abandon à leur sort. La réussite à l’école ne leur suffit pas pour se projeter dans un avenir plus assuré qu’il ne l’est actuellement. Malgré tout, des familles se sacrifient pour que leurs enfants continuent à fréquenter l’école. Ici s’origine une idée de mieux comprendre ce que signifie, pour un jeune haïtien le fait d’aller à l'école, d'y travailler ou non, d'y apprendre et comprendre les savoirs proposés. Notre question centrale : de quel ordre sont les facteurs qui influencent la(es) rapport(s) au(x) savoir(s) des sujets apprenant dans l’espace scolaire haïtien ? Notre hypothèse principale : La mobilisation du sujet apprenant pour investir la(es) savoir(s) dépend de ses rapports à lui-même, à l'institution (à travers les interactions avec les enseignant(e)s et personnels de direction) et de ce que ces savoirs mobilisent en lui.Ces questions sont abordées à partir de l’organisation de l’enseignement dans différentes institutions scolaires prenant en compte entre autres : qualité d’enseignement, rapports à la culture scolaire, question du sens de l’école et des savoirs, etc.. Nous tentons de formuler un problème éducatif en Haïti à partir d’une lecture «en positif » des situations des sujets apprenant. Un des constats : les savoirs étant présentés en vrac, ceux qui n’ont pas une stratégie de « survie » se perdent dans l’indifférence de la société. / We predict to better understanding the Haitian learning experience through the notions of its relationships with knowledge and how these relationships are built within the school system environment as a member of society leading up to the integration. This approach is based on individual students’ stories and statistical facts issued from sociological studies. On one hand, these stories and facts are been analyzed and interpreted within theoretical and academic frameworks developed by ESCOL (Charlot, Bautier, Rochex). On the other hand, they have been analyzed by Beillerot from a clinical perspective, and analyzed by Haydwe from a cultural standpoint. The socio-politic and economic situations have contributed to the complexity of the relationships between the learners and school environments. Consequently, teachers have endured difficulties to motivate students to learn meanwhile students have expressed desire to drop out from school given that academic success has not been sufficient to ensure a sustainable future to them. Nevertheless, Haitian parents, in spite of limited resources continue to invest in the education of their children. Herein, begin the idea of understanding what it means for young Haitians to attend school, to work or not, to learn and understand what they are being taught in school.Our central question: What are the influential factors within the relationships between the learners and Haitian school system? Our hypothesis: The motivation of a student to learn depends on intrinsic motivation; and the interaction between him or her and the school institutions trough interaction with teachers and administrative personnel.These issues are raised from the organization of the educative system through several institutions in assessment of the quality of instruction, relation with the school culture, questions on the importance of education, etc. We try to formulate the problem of the Haitian school system from a “positive model” of the learners’ circumstance. One of the findings, since knowledge is presented in bulk, those who do not have a "survival" strategy, are lost in the society frightening indifference.

Made available in DSpace on 2016-04-27T16:33:02Z (GMT). No. of bitstreams: 1 IvaneteSantos.pdf: 5495028 bytes, checksum: ed603c90b37829789f56bd1856e7a18f (MD5) Previous issue date: 2006-02-23 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / This study examines the production of Edward Lee Thorndike related to the teaching of Arithmetic s, Algebra, and Geometry, which had important implications in the restructuring of the teaching of Mathematics in the United States, beginning in the first decades of the Twentieth Century. The hypothesis adopted here was that the author came up with a new approach for the teaching of Mathematics in the United States in that period. The publications by Thorndike that were analyzed here were the ones related to the contents that deal with Mathematics, Psychology, and Education, with the objective of identifying the new pedagogical pattern that was developed for the teaching of Mathematics, as well as the singularity of his scientific production, considering the changes and debates that were occurring in the United States in regards to the teaching of Mathematics in the first decades of the Twentieth Century. As for what concerns Psychology, the concepts he adopted in the production of students manuals designed for the elementary and high school levels were analyzed. Also investigated were the different means that were used by Thorndike to work with students, teachers, psychologists, and educators that taught Mathematics. It was found that the manuals The Thorndike Arithmetics and The Thorndike Algebra are the most systematic expression of the author s efforts in order to apply the laws and principles of the Psychology of Learning and the Experimental Education that were developed by him, which placed him in the movement of the modernization of the teaching of Mathematics within and without the United States. / Neste trabalho, é examinada a produção de Edward Lee Thorndike relativa ao ensino de Aritmética, Álgebra e Geometria que teve significativas implicações nas reformas do ensino de Matemática nos Estados Unidos, a partir das primeiras décadas do século XX. A hipótese adotada foi que autor conformou um novo padrão para o ensino de Matemática norte-americano nesse período. Foram analisadas as publicações de Thorndike relativas aos conteúdos matemáticos, à Psicologia e à Educação, com o objetivo de identificar o novo padrão pedagógico que conformou para o ensino de Matemática, bem como a singularidade da sua produção científica, considerando as transformações e debates que estavam ocorrendo, nos Estados Unidos, em relação ao ensino de Matemática nas primeiras décadas do século XX. No que se refere à Psicologia, foram verificados os conceitos que mobilizou para produzir manuais destinados ao aluno da elementary school e da high school. Foram investigados, também, os diferentes meios utilizados por Thorndike para se dirigir a alunos, professores, psicólogos e educadores matemáticos. Constatou-se que os manuais The Thorndike Arithmetics e The Thorndike Algebra são a expressão mais sistemática do esforço empreendido pelo autor para aplicar leis e princípios da Psicologia da Aprendizagem e da Educação Experimental por ele mesmo elaborados, que o inscrevem no movimento de modernização do ensino de Matemática dentro e fora dos Estados Unidos.

Made available in DSpace on 2016-04-27T16:57:18Z (GMT). No. of bitstreams: 1 ALEX SANDRO MARQUES.pdf: 7139616 bytes, checksum: f96912783b8a80ca4b7918ade2f51542 (MD5) Previous issue date: 2005-11-04 / The purpose of our search, to verify how mathematics was organized at junior high school in 1950 s, it s located between two fundamental moments in brazilian mathematics education history: the born of mathematics as a subject, in 1929, and the advent of Modern Mathematics Movement (MMM), in the beginning of 1960 s decade. Inicially, to analyse this question, we worked out on a synthesis about the educational reforms Francisco Campos and Gustavo Capanema, that are responsable for the criation of mathematics as a subject and junior high school, respectivily. After that, based on the new educational legislation, the Ministry Edict of 1951 we verified the estabilishment of Minimum Programs to be carried out obrigatorily, motivating a worry about the possibility of that among the teachers. On this context, we analysed the Annals of the 1st National Congress of Mathematics Teaching in High School, that happened in Salvador in 1955, witch counted on the participation of the teacher and author of the school books Osvaldo Sangiorgi, one of the protagonists of MMM in Brazil, and we verified that the discussion epicenter wasn t around the programs but had been dislocated to the quastion of weekly number of this subject classes. With the purpouse to get traces of the school daily we analysed the collection of the most representative brazilian school books of pre-modern times, that authorized us to certificate the subject estability on this decade. That way, we conclued that the pre-modern times didn t represent an ideal setting for a revolution on this subject; on the contrary, were years of agreement among the teachers about the methods and contents to be taught, were years of estability / O objeto de nosso trabalho, a verificação de como estava organizada a matemática escolar do ginásio nos anos 1950, está situada entre dois momentos fundamentais da história da educação matemática brasileira: o nascimento da disciplina matemática, em 1929, e o advento do Movimento da Matemática Moderna (MMM), no início do decênio de 1960. Inicialmente, para análise desta questão, elaboramos uma síntese sobre as reformas educacionais Francisco Campos e Gustavo Capanema, responsáveis pela criação da disciplina matemática e do ginásio, respectivamente. Em seguida, partindo da nova legislação educacional, a Portaria Ministerial de 1951, verificamos o estabelecimento de Programas Mínimos a serem cumpridos obrigatoriamente, motivando uma preocupação entre os professores em relação à viabilidade de tal medida. Nesse contexto, analisamos os Anais do I Congresso Nacional de Ensino da Matemática no Curso Secundário realizado em Salvador no ano de 1955, que contou com a participação do professor e autor de livros didáticos Osvaldo Sangiorgi, um dos protagonistas do MMM no Brasil, e verificamos que o epicentro das discussões não estava em torno dos programas e havia se deslocado para a questão do número de aulas semanais para a disciplina matemática. Com a finalidade de obter indícios do cotidiano escolar analisamos as coleções de livros didáticos brasileiros mais representativas dos tempos pré-modernos, o que nos autorizou atestar sobre a estabilidade da disciplina nessa década. Assim, concluímos que os tempos pré-modernos não representavam um cenário ideal para uma revolução na disciplina; pelo contrário, foram anos de consenso entre os professores sobre métodos e conteúdos a serem ministrados, foram anos de estabilidade