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

Orientador: Marcelo de Carvalho Borba / Banca: Márcia Maria Fusaro Pinto / Banca: Maria Helena Sebastiana Sahão Bizelli / Banca: Edna Maura Zuffi / Banca: Henrique Lazari / Resumo: Baseando-me na noção de coletivo pensante seres-humanos-com-mídias, o objetivo desta pesquisa foi responder à pergunta diretriz Como o coletivo, formado por alunos-comtecnologias, produz o conhecimento acerca de função composta e regra da cadeia, a partir de uma abordagem gráfica? O processo de visualização implícito nessa pergunta é potencializado pelas Tecnologias da Informação e Comunicação (TIC), que transformam o modo como o conhecimento é produzido, reorganizando a forma de interagir e pensar. Os dados foram coletados com alguns alunos ingressantes no Curso de Matemática da UNESP - Rio Claro durante os "Experimentos de Ensino". Foram elaborados cinco episódios que apresentaram subsídios para responder à pergunta diretriz desta pesquisa. Tais episódios indicam que a produção do conhecimento dos alunos envolvidos, acerca de função composta e regra da cadeia, ocorreu por meio de elaborações de conjecturas, formuladas durante o processo de visualização potencializado pelas TIC. Tais conjecturas foram confirmadas ou refutadas levando-se em conta o entrelaçamento das representações múltiplas, que permearam todas as atividades, e um coletivo pensante seres-humanos-com-mídias, no qual o ser humano transforma e é transformado pelas mídias em um processo interativo. A partir desses resultados, outras indagações surgiram sobre o papel do professor-pesquisador e sua prática na sala de aula. / Abstract: Based on the notion of thinking collectives of humans-with-media, the objective of this research was to respond to the research question How does a collective composed of studentswith- technologies produce knowledge about the Composition of Functions and the Chain Rule using a graphic approach? The visualization process implicit in this question is potentiated by Information and Communication Technologies (ICT), which transform the way knowledge is produced, reorganizing interaction and thinking. Data was collected with some university students enrolled within the undergraduate Mathematics Program at UNESP - Rio Claro during "Teaching Experiments". Five episodes were selected that were particularly informative with respect to the research question. The episodes indicate that students' knowledge production regarding composition of functions and the chain rule occurred through the elaborations of conjectures formulated during the process of visualization potentiated by the ICT. These conjectures were confirmed or rejected based on the interweaving of multiple representations that permeated all the activities, and a humans-with-media thinking collective, in which the human transforms and is transformed by the media in an interactive process. Based on these findings, new questions emerged regarding the role of the researcher-professor and teaching practice in the classroom. / Doutor

Made available in DSpace on 2016-04-27T16:57:48Z (GMT). No. of bitstreams: 1 EDM - Mauricio B da Silva.pdf: 2913764 bytes, checksum: 2d022c37920400ed194e2e0e363c0cdd (MD5) Previous issue date: 2006-10-19 / Secretaria da Educação do Estado de São Paulo / This present work of research has as its aim to present my experience in building and applying a WebQuest activity by analyzing the difficulties and the concrete possibilities of such a teaching way. This activity has been applied to a group of 10 students in the third year High School in a private school in Sao Paulo City. While analyzing the WebQuest since its construction and application up to the fi nal results obtained, I have looked for bringing evidences of if such kind of activity, which basically utilizes internet resources, is able to bring any kind of benefit in relation to the traditional classes that make use of books and books of key facts. Some learning aims have been defined starting from the Bloom Taxonomy (1972) and, when reached, have raised a thought of higher level, according to Dodge (1995). I verified that the students have constructed their mathematical knowledge as I noticed that they have reached the level of visualization geometrical comprehension, as defined by Van Hiele (1986) in his classification. I have used, as a way of validating the results obtained during the construction and application phases, a confrontation between the evidences observed through the registrations of such phases and the theoretic-methodological recommendations that have been utilized, which were mainly based in the own principles of the WebQuest activity and in some principles of the teaching experiment methodology, according to Cobb, P. et all (2003). It was possible to conclude that the students` mathematical knowledge of the Spatial Geometry basic notions was, in great part, consolidated throughout the use of the WebQuest activity. A new purpose of a WebQuest activity has been elaborated for application in an opportune moment / Este trabalho de pesquisa tem como objetivo apresentar minha experiência ao construir e aplicar uma atividade WebQuest analisando as dificuldades e possibilidades desta forma de ensinar. Esta atividade foi aplicada em um colégio particular de São Paulo, a 10 alunos do 3º ano do Ensino Médio. Ao analisar a WebQuest desde a sua construção e aplicação até a obtenção dos resultados, procurei evidenciar se esta atividade, que utiliza basicamente recursos da Internet, apresenta algum benefício em relação às aulas tradicionais com o uso de livros e apostilas. Alguns objetivos de aprendizagem foram definidos a partir da Taxonomia de Bloom (1972), que ao serem atingidos proporcionaram um pensamento de nível elevado, conforme Dodge (1995). Constatei que os alunos construíram seus conhecimentos matemáticos ao verificar que os mesmos atingiram o nível de compreensão geométrica de visualização, definido por Van Hiele (1986) em sua classificação. Utilizei, como forma de validar os resultados obtidos nas fases de construção e aplicação, um confronto entre as evidências observadas por meio dos registros destas fases e as recomendações teórico-metodológicas utilizadas, com base principalmente nos próprios princípios da atividade WebQuest e em alguns princípios da metodologia Experimento de Ensino, conforme Cobb et al (2003). Podese concluir que o conhecimento matemático dos alunos das noções básicas de Geometria Espacial foi, em grande parte, consolidado por meio da utilização da atividade WebQuest. Uma nova proposta da atividade Webquest foi elaborada para aplicação em momento oportuno

Made available in DSpace on 2016-04-27T16:58:23Z (GMT). No. of bitstreams: 1 edith desp.pdf: 9443168 bytes, checksum: 373751b634881d1df27b702d1636cc5a (MD5) Previous issue date: 2007-05-25 / The aim of this work was to design and evaluate a musical microworld, created to enable users to explore concepts related to multiplication. Inspired by the constructionist perspective of Papert, the study evolved from the conjecture that by engaging in the construction of a musical rhythm, learners could interact with various aspects of the multiplicative field. To investigate this conjecture, a teaching experiment of two phases was conceived. The analyses in both these phases were based on the distinction made in the research of Confrey between the world of counting and the world of splitting. In the design phase, attention was given to the aspects of the microworld that might permit learners to construct conceptions that go beyond a vision of multiplication as repeated addition. The experimentation phase involved two groups of six students of the 5th Grade of a school within the public education system in the city of São Paulo. Using as their bases the interactions of students during the construction of their own rhythmic compositions, the analyses explored the different ways through which the learners made use of tools of the microworld and, in particular, the ways they chose to represent and express ideas related to notions such as ratio and proportion. The results suggest that, with the support of these tools, the learners gradually came to associate their musical activities with mathematical properties and concepts. They also showed how, during the experiment, the learners were enabled to explore the relations half and double within a perspective coherent with the world of splitting, as well as to use a diverse set of representations of these two relations / Este trabalho tem como objetivo o design e a avaliação de um micromundo musical para a exploração de conceitos multiplicativos. Dentro de uma perspectiva construcionista, partiu-se da hipótese de que, no contexto da construção de um ritmo musical, os aprendizes possam interagir com diferentes aspectos do campo multiplicativo. Foi elaborado um experimento de ensino composto por duas fases: a fase de design do micromundo e a fase de experimentação. As análises em ambas as fases foram baseadas na distinção feita nas pesquisas de Confrey entre o mundo da contagem e o mundo de splitting. Na fase de design, a atenção foi dada aos aspetos do micromundo que permitissem os aprendizes ampliar as concepções do campo multiplicativo, indo além de uma visão de multiplicação como adição repetida. A fase de experimentação contou com dois grupos de seis alunos da 5.ª série do Ensino Fundamental de uma escola pública localizada na cidade de São Paulo. Por meio das interações dos estudantes durante a construção de suas próprias composições rítmicas, as análises exploraram as diferentes maneiras pelas quais os aprendizes interagiram com as ferramentas do micromundo e em particular as formas utilizadas para expressar concepções relativas a noções, tais como razão e proporção. Como resultado, foi observado que, com o apoio destas ferramentas, os alunos gradualmente associaram suas atividades musicais com propriedades e conceitos matemáticos. Em particular, todos os alunos conseguiram, ao longo do experimento, explorar as relações de dobro e metade em uma perspectiva coerente com o mundo de splitting e utilizaram uma variedade de formas para representar estas relações

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

In his keynote address to the National Council of Teachers of Mathematics research pre-session, Sloane (2006b) challenged mathematics education researchers to ‘quantify qualitative insights’. This quasi-experimental study used blended methods to investigate the development of two-digit addition and subtraction strategies. Concurrent classroom teaching experiments were conducted in two intact first grade classrooms (n = 41) in a mid-Atlantic American public school. From a pragmatic emergent perspective, design research (Gravemeijer & Cobb, 2006) was used to develop local instructional theory. An amplified theoretical framework for early base-ten strategies is explicated. Multilevel modelling for repeated measures was used to evaluate the differences in strategy usage between classes across occasions and the association of particular pedagogical practices with the emergence of incrementing and decrementing by ten (N10) or decomposition (1010) strategies (Beishuizen, Felix, & Beishuizen, 1990).The two matched classes were not different in terms of gender, poverty, race, pre-assessment performance, and special education services. After the first unit of instruction with differentiated pedagogical tools, the collection class was significantly (p = .001) more likely to use 1010 than the linear class. No difference was demonstrated during the post-assessment. Students in both classes were more likely to use N10 during the last structured interview than in the first (p < .0001). Furthermore, there was no difference between the two classes in using any advanced strategy; however, students in both classes were more likely to use an advanced strategy at the conclusion of the study than they were initially (p = .033). The order of emergence of 1010 and N10 was not associated with the ability to develop both strategies, but there was an association (p < .001) between use of an advanced strategy and success on a district-mandated written assessment of two-digit addition and subtraction.Two original instructional sequences of contextually-based investigations are presented. Protocols transcribed from videotaped lessons and dynamic assessment interviews are presented to illuminate specific constructs detected and to illustrate the pedagogical techniques. An amplified framework for early place value constructs is proposed. Recommendations for future studies, curricular changes, and the need of early intervention are discussed.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-03-04T11:06:59Z No. of bitstreams: 1 leozartdasilvamatos.pdf: 3656628 bytes, checksum: 2566a15b7d9fd0f2945b842289a7bd23 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-04-24T01:50:11Z (GMT) No. of bitstreams: 1 leozartdasilvamatos.pdf: 3656628 bytes, checksum: 2566a15b7d9fd0f2945b842289a7bd23 (MD5) / Made available in DSpace on 2016-04-24T01:50:11Z (GMT). No. of bitstreams: 1 leozartdasilvamatos.pdf: 3656628 bytes, checksum: 2566a15b7d9fd0f2945b842289a7bd23 (MD5) Previous issue date: 2013-12-12 / O presente estudo teve como objetivo investigar compreensões dos conceitos de Derivadas e Integral construído por um grupo de alunos de terceiro ano do Ensino Médio de uma escola particular brasileira em um ambiente informatizado com o CAS on-line WolframAlpha e applets gerados pelo Geogebra. Embora a literatura sobre o ensino e aprendizagem de Cálculo seja bastante extensa, por outro lado, particularmente no Brasil, ela é quase sempre restrita a cenários do Ensino Superior. O estudo propôs uma nova abordagem para a compreensão dos conceitos de Derivada e de Integral, enfraquecendo manipulações algébricas procedurais, típicas das aulas de Cálculo em nível de graduação. Para este fim, nós exploramos o conceito de derivada como taxa instantânea de variação e a integral como função acumulação e área líquida sob uma curva. Os participantes trabalharam em pares com o CAS on-line WolframAlpha como uma ferramenta para toda a manipulação algébrica e algorítmica considerado necessário durante as atividades propostas. Metodologicamente, o estudo foi realizado como um experimento de ensino com o quadro teórico baseado nas ideias de Imagem de Conceito e Definição de Conceito. Os dados sugeriram que os participantes foram capazes de lidar com a CAS WolframAlpha com relativa facilidade, em parte devido à sua sintaxe bastante flexível. Também tiveram uma dificuldade inicial em termos de pensamento comparativo, o que levou a outra dificuldade inicial com a ideia de taxa instantânea de variação. Os participantes tiveram uma boa compreensão da integral como a área líquida sob uma curva, mas detectamos algumas dificuldades com a função de acumulação. Apesar destes problemas iniciais, os dados sugerem que a taxa de variação funcionou como raiz cognitiva para o conceito de derivada. Finalmente, a pesquisa sugere que ambos os conceitos de derivada e integral, bem como a sua conexão via Teorema Fundamental do Cálculo podem ser explorados conceitualmente de forma significativa para estudantes do ensino médio em contextos similares sob a abordagem proposta. / The present study aimed to investigate understandings of the concepts of Derivative and Integral constructed by a group of third-year students of a Brazilian private high school in digital environment with the CAS online WolframAlpha and applets generated by the Geogebra. Although the literature on teaching and learning Calculus is quite extensive, on the other hand, particularly in Brazil, it is almost always restricted to Higher Education settings. The study proposed a new approach to the concepts of Derivative and Integral with focus on conceptual understanding as well as weakening procedural algebraic manipulations typical in Calculus classes in undergraduate level. For this reason, we explored the concept of derivative as instantaneous rate of change and the integral as accumulation function and net area under a curve. Participants worked in pairs with the CAS online WolframAlpha as a tool for all algebraic and algorithmic manipulations deemed as necessary during the proposed activities. Methodologically, the study was conducted as a teaching experiment with the theoretical framework based on the ideas of Concept Image and Concept Definition. Data have suggested that the participants were able to deal the CAS WolframAlpha to some extent with ease, partially due to its quite flexible syntax. They also had an initial difficulty in terms of comparative thinking, which led to an initial difficulty with the idea of instantaneous rate of change. The participants had a good understanding of the integral as the net area under a curve, but we detected some difficulties with the accumulation function. Despite these initial problems, data suggest that the rate of change had a role as a cognitive root for the concept of derivative. Finally, the research suggests that both concepts of derivative and integral, as well as their connection via Fundamental Theorem of Calculus, can be explored conceptually in a significant way for high school students in similar settings under the proposed approach.

Central to this article is networking in math lessons, whereby concentration is placed on the construction of a student-focused teaching method for the networking of mathematical knowledge in the lower secondary. Firstly, normative standards and descriptive results will be compared. Secondly, several already existing teaching methods for networking in math lessons will be added to the method of „chapter-spanning task variation“. Using this method, attention is be placed on the integration of mathematical content and specific social netowrk-form (e.g. teacher led classes, group-work etc.). This paper will be concluded with the presentation of the testing of the method in the school context).

Made available in DSpace on 2014-06-11T19:31:43Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-17Bitstream added on 2014-06-13T19:21:03Z : No. of bitstreams: 1 barbosa_sm_dr_rcla.pdf: 1736492 bytes, checksum: 916a8396ad40af059b64229204ec6e8b (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Baseando-me na noção de coletivo pensante seres-humanos-com-mídias, o objetivo desta pesquisa foi responder à pergunta diretriz Como o coletivo, formado por alunos-comtecnologias, produz o conhecimento acerca de função composta e regra da cadeia, a partir de uma abordagem gráfica? O processo de visualização implícito nessa pergunta é potencializado pelas Tecnologias da Informação e Comunicação (TIC), que transformam o modo como o conhecimento é produzido, reorganizando a forma de interagir e pensar. Os dados foram coletados com alguns alunos ingressantes no Curso de Matemática da UNESP - Rio Claro durante os “Experimentos de Ensino”. Foram elaborados cinco episódios que apresentaram subsídios para responder à pergunta diretriz desta pesquisa. Tais episódios indicam que a produção do conhecimento dos alunos envolvidos, acerca de função composta e regra da cadeia, ocorreu por meio de elaborações de conjecturas, formuladas durante o processo de visualização potencializado pelas TIC. Tais conjecturas foram confirmadas ou refutadas levando-se em conta o entrelaçamento das representações múltiplas, que permearam todas as atividades, e um coletivo pensante seres-humanos-com-mídias, no qual o ser humano transforma e é transformado pelas mídias em um processo interativo. A partir desses resultados, outras indagações surgiram sobre o papel do professor-pesquisador e sua prática na sala de aula. / Based on the notion of thinking collectives of humans-with-media, the objective of this research was to respond to the research question How does a collective composed of studentswith- technologies produce knowledge about the Composition of Functions and the Chain Rule using a graphic approach? The visualization process implicit in this question is potentiated by Information and Communication Technologies (ICT), which transform the way knowledge is produced, reorganizing interaction and thinking. Data was collected with some university students enrolled within the undergraduate Mathematics Program at UNESP – Rio Claro during “Teaching Experiments”. Five episodes were selected that were particularly informative with respect to the research question. The episodes indicate that students’ knowledge production regarding composition of functions and the chain rule occurred through the elaborations of conjectures formulated during the process of visualization potentiated by the ICT. These conjectures were confirmed or rejected based on the interweaving of multiple representations that permeated all the activities, and a humans-with-media thinking collective, in which the human transforms and is transformed by the media in an interactive process. Based on these findings, new questions emerged regarding the role of the researcher-professor and teaching practice in the classroom.