Syftet med studien är att kartlägga talbaserade och sifferbaserade räknemetoder i subtraktion genom en läromedelsanalys av läromedlet Koll på matematik för årskurs 1-3. Studien ämnar till att undersöka i vilken ordning räknemetoderna introduceras samt att urskilja hur undervisningen av dessa utformas utifrån läromedlet. Studiens teoretiska perspektiv, hypothetical learning trajectory, kopplas samman med Skotts analysmodell för att kartlägga räknemetoderna i läromedlet. Resultatet visar att undervisningen av skriftliga räknemetoder i subtraktion utifrån läromedlet Koll på matematik sker på olika sätt beroende på årskurs, men liknande mönster går att urskilja. Inledande gemensamma samtal, stöttning, illustrationer, samt konkret material förekommer i alla årskurser men en övergripande regression av kvantitet sker i stigande årskurser. Det framgår även att talbaserade metoder är de som introduceras först och sifferbaserade metoder introduceras tidigast i elevbok 2B. En diskussion förs innan slutsatsen utmynnar i att tidigare forskning påvisar att talbaserade metoder bör introduceras tidigare än sifferbaserade metoder, vilket också är den ordning läromedlet introducerar räknemetoderna.

Integration is a core concept of calculus. As such, significant work has been done on understanding how students come to reason about integrals, including both the definite integral and the accumulation function. A path towards understanding the accumulation function first, then the definite integral as a single point on the accumulation function has been presented in the literature. However, there seems to be an accessible path that begins first with understanding the definite integral through an Adding Up Pieces (AUP) perspective and extending that understanding to the accumulation function. This study provides a viable hypothetical learning trajectory (HLT) for beginning instruction with an AUP perspective of the definite integral and extending this understanding to accumulation functions. This HLT was implemented in a small-scale teaching experiment that provides empirical data for the type of student reasoning that can occur through the various learning activities. The HLT also appears to be a promising springboard into developing the Fundamental Theorem of Calculus. Additionally, this study offers a systematic framework for understanding the process- and object-level thinking that occurs at different layers of integration.

Lesson study is a form of professional development for teachers adopted in recent years from Japan. Introducing lesson study to U.S. teachers and researchers has been the focus of most of the literature on this subject. Much of the literature outlines how lesson study works and describes its essential features. One of the features of lesson study is anticipating student responses, also known as anticipating student thinking. Anticipating student responses is passingly described in lesson study literature. This research was conducted to understand what it means to anticipate student responses for preservice mathematics teachers in a lesson study group. Lesson study literature indicates that anticipating student responses is to anticipate conceptual development from the students' perspective, and the purpose is to be prepared to have meaningful discussions and questions to enable students to develop the understanding. Anticipating student responses is highly related to the hypothetical learning trajectory described by Simon (1995), the self directed anticipative learning model described by Christensen and Hooker (2000) and the expert blind spot discussed by Nathan and Petrosino (2003). While their work does not stem from lesson study, they add theoretical perspective to the idea of anticipating student responses. Their work indicates that anticipating student responses is difficult, valuable, that one gets better at it through experience, and that it is very useful in refining lessons. Participants were enrolled in the mathematics education methods class of a large private university in the U.S. A characterization of anticipating student responses was developed as the participants met in group meetings to create a lesson. They anticipated student responses in ways that facilitated lesson planning and task design. Participants did not anticipate student responses toward students' conceptual development. This research reports five particular ways that anticipating student responses was used as a tool to define and refine the lesson so that it ran smoothly toward lesson goals. These ways are related to: goals, tasks and materials, procedural mathematical reasoning, successful student efforts, and emotional responses. It is believed that anticipating student responses towards task design is a necessary precursor to anticipating student responses toward students' conceptual development.

There is an overemphasis on procedures and manipulation of symbols in calculus and not enough emphasis on conceptual understanding of the subject. Specifically, students struggle to understand and correctly apply concepts in calculus such as the chain rule, implicit differentiation, and related rates. Students can learn mathematics more deeply when they make connections between different mathematical ideas. I have hypothesized that students can make powerful connections between the chain rule, implicit differentiation, and related rates through the mathematical concept of nested multivariation. Based on this hypothesis, I created a hypothetical learning trajectory (HLT) rooted in nested multivariation for students to develop an understanding of these three concepts. In this study, I explore my HLT through a small-scale teaching experiment with individual first-semester calculus students using tasks based on the HLT.Based on the teaching experiment, nested multivariational reasoning proved to be critical in understanding how the variables within a function composition change together and in developing intuition and understanding for the multiplicative nature of the chain rule. Later, nested multivariational reasoning was mostly important in recognizing the existence of a nested relationship and the need to use the chain rule in differentiation. Overall, through the HLT, students gained a connected and conceptual understanding for the chain rule, implicit differentiation, and related rates. I also discuss how the HLT might be adjusted and improved for future use.

The function concept is one of the most important concepts in the learning of mathematics (Dubinsky & Harel, 1992), yet it is considered by many researchers to be one of the least understood and most difficult concepts to master in the learning of high school mathematics (Eisenberg, 1992, Sfard, 1992). To this end, problems concerning its teaching and learning are often confronted (Mann, 2000) and few teachers know how learners come to understand functions (Yoon, 2007). As a result, most teachers teach functions using the conventional approach which starts by stating definitions followed by examples and then a few applications. The nature of this approach has not encouraged teachers to engage learners and their ways of reasoning in knowledge construction and adequately addressing their difficulties. The purpose of this study was to use design research to improve the teaching and learning of functions at grade 11 level. This was achieved by adapting design cycles of Wademan’s (2005) Generic Design Research model in which each cycle comprised different iterative APOS (Action, Process, Object, Schema) analysis, design, development and implementation of hypothetical learning trajectories (HLTs). I started by interrogating twelve grade 11 learners of a particular rural high school on the June 2011 mathematics paper 1 examination they had written to determine the APOS theory conception level each learner was operating at, and their difficulties. Learners’ difficulties from initial interviews and literature were grouped under the function definition and representation. I then designed instruction based on HLTs embedded with Realistic Mathematics Education (RME) activities and two separate tasks on the definition and representation as a form of intervention to help learners move up from their initial conception levels to the next and to overcome their difficulties. After each design cycle I interviewed learners based on the task for a particular concept and learners’ responses were analysed using APOS theory and used to design further instruction to help learners approximate the schema level of understanding concepts related to functions. The major findings of this study were that the use of learners’ conceptions and RME activities in designing instruction helped learners to progress smoothly through APOS theory conception levels though they did not fully reach the intended schema level. In addition, design research cycles and their HLTs implemented in a constructivist environment enabled learners to collectively derive working definitions of the function concept and to improve their conceptual understanding of the process of switching from a graph to an equation. Another contribution of this study has been a deeper understanding of the extent to which design research can be used to improve learners’ understanding of functions and an addition of some insights to the teaching and learning of functions. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Science, Mathematics and Technology Education / unrestricted

Made available in DSpace on 2016-04-27T16:57:00Z (GMT). No. of bitstreams: 1 Jose Manoel Vitolo.pdf: 939821 bytes, checksum: 0434aa524d750001b5fe4f4e1b7980c2 (MD5) Previous issue date: 2010-10-25 / Secretaria da Educação do Estado de São Paulo / This work has as its intents: to investigate the construction, discussion and evaluation of a teaching planning that gives an introductory notion to the idea of function, inside a constructivist learning perspective; to investigate how the researches in the Mathematics Education field contribute to the teaching organization and to analyze Mathematics teachers performance before a teaching proposal with this characteristic. The theoretic foundation is based on the formula proposed by Simon (1995), about Hypothetical Learning Trajectory (HLT). The study achieved has qualitative aspect and had the involvement of two Mathematics teachers of a public school, in the state of Sao Paulo, and their work with seventy-seven students shared in two groups, all of them in their first year of High School. The data came through interviews, questionnaires and observations. Although the HTL has been worked out with tasks that include problems resolution, technology, interdisciplinary approaches and application in day-to-day situations, the HLT per se does not guarantee learning under a constructivist perspective, without the constant support of the teacher in the planning (re)organization. About the teachers, it is important emphasizing that the main challenge is to approach them of the academic researches and keep them in continuous learning process / O presente trabalho tem como objetivos: investigar a construção, discussão e avaliação de um planejamento de ensino para a aprendizagem de noções introdutórias à ideia de função, dentro de uma perspectiva construtivista de aprendizagem; investigar como as pesquisas, na área de Educação Matemática, contribuem para a organização do ensino deste tema e analisar a atuação de professores de Matemática, diante de uma proposta de ensino com esta característica. A fundamentação teórica está baseada nas formulações propostas por Simon (1995), sobre Trajetórias Hipotéticas de Aprendizagem (THA). O estudo realizado é de natureza qualitativa envolvendo dois professores de Matemática de uma escola da rede pública do Estado de São Paulo, e suas atuações junto a 77 alunos distribuídos em duas turmas que frequentam o primeiro ano do ensino médio. Os dados foram coletados por entrevistas semiestruturadas, questionário e observações. Embora a THA, tendo sido elaborada com tarefas que envolvam resolução de problemas, uso de tecnologia, abordagens interdisciplinares, aplicações em situações do cotidiano e em outras áreas do conhecimento e sejam potencialmente ricas, no sentido de produzir situações de aprendizagem, sem a participação constante do professor na (re)organização do planejamento, a THA por si só não garante uma aprendizagem sob perspectivas construtivistas. Ainda em relação ao professor, cabe ressaltar que o principal desafio é aproximá-lo das pesquisas acadêmicas e que continuem sempre em processo de formação

Made available in DSpace on 2016-04-27T16:57:08Z (GMT). No. of bitstreams: 1 Denilson Goncalves Pereira.pdf: 7924777 bytes, checksum: 477a51645572c92238dd48b6a98b1d8e (MD5) Previous issue date: 2011-05-26 / Secretaria da Educação do Estado de São Paulo / The present work is part of a project that aims to examine how teaching proposal in the classroom can be organized and developed, exploring the everyday contexts in other areas of knowledge and of mathematics itself, with a view to building some expectations learning. Our research involves three mathematics teachers from public schools in São Paulo, each one with a class of Grade 3 High School, during the night, totaling 122 students. The guest teachers work in school EE Padre Anchieta located in Diadema (SP). The project aims to contribute to the professional development of teachers in a consistent manner which is intended to be his practice in the classroom, through a strategy of closely linking theory, teaching practice and research. Through the collaborative work and the hypothetical learning trajectory (HLT), we believe that teachers can reflect about their teaching practice, focusing on a teaching strategy in which the student is motivated to investigate and assign meaning to their learning autonomously. After the teachers know the whole project and the primary HLT by the teacher researcher, with assistance from teachers applicators, that develop a course in class. During this stage, the researcher observed and recorded with audio recordings and notes, teachers reported applicators, along with the protocols of the students, which occurred in each proposed activity, pointing out the mistakes, difficulties, interactions, assumptions that were questioned in elaboration of the trajectory and other observations / O presente trabalho faz parte de um projeto que tem por objetivo analisar o modo como podem ser organizadas e desenvolvidas propostas didáticas em sala de aula, explorando contextos do cotidiano, de diversas áreas de conhecimento e da Matemática, com vistas a contemplar algumas expectativas de aprendizagem. Nossa pesquisa envolve três professores de Matemática da rede pública estadual paulista, cada qual com uma turma da terceira série do Ensino Médio, período noturno, totalizando 122 alunos. Os professores convidados atuam na E.E. Padre Anchieta, localizada em Diadema. O projeto visa a contribuir para o desenvolvimento profissional dos professores de forma coerente com o que se pretende que seja sua prática em sala de aula, mediante uma estratégia de estreita articulação entre teoria, prática docente e pesquisa. Por meio de uma trajetória hipotética de aprendizagem (THA) e do trabalho colaborativo, acreditamos que o professor pode refletir sobre sua prática docente, privilegiando uma estratégia de ensino na qual o aluno seja motivado a investigar e atribuir significado a sua aprendizagem de forma autônoma. A THA preliminar foi elaborada pelo professor pesquisador, com a contribuição dos professores aplicadores, e cada um destes a desenvolveu em uma turma. Nessa etapa, o pesquisador fez observações e registro, por meio de gravações de áudio e anotações, enquanto os professores aplicadores relataram, junto com os protocolos dos alunos, o que ocorreu em cada atividade proposta, apontando os erros, dificuldades, interações, hipóteses que foram questionadas na elaboração da trajetória, além de outras observações

Conceptions of area and volume of pupils at the elementary school Veronika Tůmová ABSTRACT: The aim of my thesis is to investigate how the conceptions of area and volume are built, what the major pitfalls and problems are, what skills and strategies are helpful for solving problems and what are the frequent unsuccessful strategies and pupils' misconceptions. I used the concept of the hypothetical learning trajectory as a tool to describe this process. Based on existing research review, I compiled two hypothetical learning trajectories - one for area and one for volume. The crucial building blocks that were identified based on these trajectories are: space abilities, structuration of space into arrays of units and multiplicative thinking. A test was designed to measure these factors and the correlation of these factors with success in volume and area problems ranged from week (multiplicative thinking) to very strong (spatial abilities). These findings confirm that these factors constitute an important part of the hypothetical learning trajectory for both concepts. Several structuration tasks were selected to investigate pupils' structuration skills and mistakes in more detail. Three main categories of problems were identified in the pupils' solutions: incorrect space structuration, disconnection between...

Thesis (MEd)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Research has confirmed the educational value of mathematical modelling for learners of all abilities. The development of modelling competencies is essential in the modelling approach. Little research has been done to identify and develop the mathematising modelling competency for specific sections of the mathematics curriculum. The study investigates the development of mathematising competencies during the modelling of number pattern problems. The RME theory has been selected as the theoretical framework for the study because of its focus on mathematisation. Mathematising competencies are identified from current literature and developed into models for horizontal and vertical (complete) mathematisation. The complete mathematising competencies were developed for number patterns and mapped on a continuum. They are internalising, interpreting, structuring, symbolising, adjusting, organising and generalising. The study investigates the formulation of a hypothetical trajectory for algebra and its associated local instruction theory to describe how effectively learning occurs when the mathematising competencies are applied in the learning process. Guided reinvention, didactical phenomenology and emergent modelling are the three RME design heuristics to form an instructional theory and were integrated throughout the study to comply with the design-based research’s outcome: to develop a learning trajectory and the means to support the learning thereof. The results support research findings, that modelling competencies develop when learners partake in mathematical modelling and that a heterogeneous group of learners develop complete mathematising competencies through the learning of the modelling process. Recommendations for additional studies include investigations to measure the influence of mathematical modelling on individualised learning in secondary school mathematics. / AFRIKAANSE OPSOMMING: Navorsing steun die opvoedkundige waarde van modellering vir leerders met verskillende wiskundige vermoëns. Die ontwikkeling van modelleringsbevoegdhede is noodsaaklik in 'n modelleringsraamwerk. Daar is min navorsing wat die identifikasie en ontwikkeling van die bevoegdhede vir matematisering vir spesifieke afdelings van die wiskundekurrikulum beskryf. Die studie ondersoek die ontwikkeling van matematiseringsbevoegdhede tydens modellering van getalpatrone. Die Realistiese Wiskundeonderwysteorie is gekies as die teoretiese raamwerk vir die studie, omdat hierdie teorie die matematiseringsproses sentraal plaas. Matematiseringsbevoegdhede vanuit die bestaande literatuur is geïdentifiseer en ontwikkel tot modelle wat horisontale en vertikale (volledige) matematisering aandui. Hierdie matematiseringsbevoegdhede is spesifiek vir getalpatrone ontwikkel en op ‘n kontinuum geplaas. Hulle is internalisering, interpretasie, strukturering, simbolisering, aanpassing, organisering en veralgemening. Die studie lewer die formulering van ‘n hipotetiese leertrajek vir algebra, die gepaardgaande lokale onderrigteorie en beskryf hoe effektiewe leer plaasvind wanneer die ontwikkelde matematiseringsbevoegdhede volledig in die leerproses toegepas word. Die RME ontwikkellingsheuristieke, begeleidende herontdekking, didaktiese fenomenologie en ontluikende modellering, is geïntegreer in die studie sodat dit aan die uitkoms van ‘n ontwikkelingsondersoek voldoen. Die uitkoms is ‘n leertrajek en ‘n beskrywing hoe die leerproses ondersteun kan word. Die analise het tot die formulering van ‘n lokale-onderrig-teorie vir getalpatrone gelei. Die resultate van die studie kom ooreen met navorsingsbevindings dat modelleringsbevoegdhede ontwikkel wanneer leerders deelneem aan modelleringsaktiwiteite, en bewys dat ‘n groep leerders met gemengde vermoëns volledige matematiseringsbevoegdhede ontwikkel wanneer hulle deur die modelleringsproses werk. 'n Aanbeveling vir verdere navorsing is om die uitwerking van die modelleringsperspektief op individuele leer in hoërskool klaskamers te ondersoek.

Made available in DSpace on 2016-04-27T16:57:00Z (GMT). No. of bitstreams: 1 Luciane Santos Rosenbaum.pdf: 6907932 bytes, checksum: 73fa79dd753c44164be51ff0aab23315 (MD5) Previous issue date: 2010-10-04 / Secretaria da Educação do Estado de São Paulo / This present work aims to verify: as compatible constructivist perspectives of learning with the planning of Trigonometric Functions; teaching the as researches in mathematics education field , which brings important results on the learning process may contribute to the organization of the Trigonometric Functions teaching that leverage best learning situations for students, as the performance of teachers of mathematics is revealed, with regard to planning activities in the teaching of Trigonometric Functions, consistent with a constructivist view of learning. We developed a qualitative study with two teachers and 70 students from the 2nd Grade of high school in a public school in the State of São Paulo. Its theoretical work of Simon (1995) on the use of HLT in teaching mathematics to formulate models of teaching based on constructivism. As a component of the Mathematics Teaching Cycle developed by Simon, the elaborate HLT was made use of the research findings for the development of Trigonometric Functions through activities and solve problems involving: constructions with ruler and compass, manipulative material, scientific calculator, construct graphs using software GeoGebra and paper and pencil. The results led us to conclude that the use of research contributes to the education organization of Trigonometric Functions; however you must provide access to such teachers to such research. Although the HLT are potentially rich, complex is the task of developing activities to accomplish a constructivist learning perspective. We note that participation in tasks involving the use of technology and material handling enhances the learning of Trigonometric Functions. However, the HLT is not prepared enough for learning to occur, because the teacher performance has a decisive role in mediating the construction of knowledge of the students. In the same way we experience the interaction and participation between students and teacher which is essential for learning / O presente trabalho tem como objetivo verificar: como compatibilizar perspectivas construtivistas de aprendizagem com o planejamento do ensino de Funções Trigonométricas; como as pesquisas na área de Educação Matemática, que trazem resultados importantes sobre a aprendizagem, podem contribuir para a organização do ensino de Funções Trigonométricas que potencialize boas situações de aprendizagem aos alunos; como a atuação do professor de Matemática se revela, no que se refere às atividades de planejamento do ensino de Funções Trigonométricas, de forma compatível com uma perspectiva construtivista de aprendizagem. Desenvolvemos um estudo de natureza qualitativa com 2 professores e 70 alunos da 2.ª série do Ensino Médio de uma escola da rede pública do Estado de São Paulo. Este trabalho, tem como fundamentação teórica os trabalhos de Simon (1995) sobre o uso de THA no ensino de Matemática para formular modelos de ensino baseados no construtivismo. Como componente do Ciclo de Ensino de Matemática desenvolvido por Simon, a THA elaborada fez uso de resultados de pesquisas para o desenvolvimento de Funções Trigonométricas por meio de atividades e resolução de problemas que envolveram: construções com régua e compasso, material manipulativo, calculadora científica, construção de gráficos usando o software Geogebra e papel e lápis. Os resultados obtidos nos levaram a concluir que o uso de pesquisas contribui para a organização do ensino de Funções Trigonométricas, no entanto é necessário possibilitar o acesso dos professores a tais pesquisas. Verificou-se que embora as THAs sejam potencialmente ricas, é complexa a tarefa de elaboração de atividades para que se efetive uma aprendizagem numa perspectiva construtivista. Constatamos que a participação em tarefas que envolvem o uso de tecnologia e manipulação de materiais potencializa o aprendizado de Funções Trigonométricas. Porém, a THA elaborada não é suficiente para que a aprendizagem ocorra, pois a atuação do professor tem papel decisivo na mediação da construção do conhecimento dos seus alunos. Da mesma forma vimos que a interação entre alunos, e estes com o professor são essenciais para uma aprendizagem significativa