Calculus at the university level is taken by thousands of undergraduate students each year. However, a significant number of students struggle with the subject, resulting in poor problem solving, low achievement, and high failure rates in the calculus courses overall (e.g., Kaput, 1994; Szydlik, 2000; Tall, 1985; Tall & Ramos, 2004; White & Mitchelmore, 1996). This is cause for concern as the lack of success in university calculus creates further barriers for students who require the course for their programs of study. This study examines this issue from the perspective of Tall’s Three Worlds of Mathematics (Tall, 2004a, 2004b, 2008), a theory of mathematics and mathematical cognitive development. A fundamental argument of Tall’s theory suggests that connecting between the different mathematical worlds, named the Embodied-Conceptual, Symbolic-Proceptual, and Formal-Axiomatic worlds, is essential for full cognitive development and understanding of mathematical concepts. Working from this perspective, this research examined, through the use of calculus task questions and semi-structured interviews, how fifteen undergraduate calculus students made connections between the different mathematical worlds for the calculus topics of limits and derivatives. The analysis of the findings suggests that how the students make connections can be described by eight different Response Categories. The study also found that how the participants made connections between mathematical worlds might be influenced by the type of questions that are asked and their experience in calculus courses. I infer that these Response Categories have significance for this study and offer potential for further study and educational practice. I conclude by identifying areas of further research in regards to calculus achievement, the Response Categories, and other findings such as a more detailed study of the influence of experience.

Calculus at the university level is taken by thousands of undergraduate students each year. However, a significant number of students struggle with the subject, resulting in poor problem solving, low achievement, and high failure rates in the calculus courses overall (e.g., Kaput, 1994; Szydlik, 2000; Tall, 1985; Tall & Ramos, 2004; White & Mitchelmore, 1996). This is cause for concern as the lack of success in university calculus creates further barriers for students who require the course for their programs of study. This study examines this issue from the perspective of Tall’s Three Worlds of Mathematics (Tall, 2004a, 2004b, 2008), a theory of mathematics and mathematical cognitive development. A fundamental argument of Tall’s theory suggests that connecting between the different mathematical worlds, named the Embodied-Conceptual, Symbolic-Proceptual, and Formal-Axiomatic worlds, is essential for full cognitive development and understanding of mathematical concepts. Working from this perspective, this research examined, through the use of calculus task questions and semi-structured interviews, how fifteen undergraduate calculus students made connections between the different mathematical worlds for the calculus topics of limits and derivatives. The analysis of the findings suggests that how the students make connections can be described by eight different Response Categories. The study also found that how the participants made connections between mathematical worlds might be influenced by the type of questions that are asked and their experience in calculus courses. I infer that these Response Categories have significance for this study and offer potential for further study and educational practice. I conclude by identifying areas of further research in regards to calculus achievement, the Response Categories, and other findings such as a more detailed study of the influence of experience.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2016-10-04T17:17:29Z No. of bitstreams: 1 Jayme do Carmo Macedo Leme.pdf: 2250748 bytes, checksum: 2ae7e03aa0d8c8057f0dcb3624011c98 (MD5) / Made available in DSpace on 2016-10-04T17:17:29Z (GMT). No. of bitstreams: 1 Jayme do Carmo Macedo Leme.pdf: 2250748 bytes, checksum: 2ae7e03aa0d8c8057f0dcb3624011c98 (MD5) Previous issue date: 2016-06-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This thesis has as main theme the learning of the derivative. The research was motivated by the questioning: What are the different thoughts related to the learning process of a concept? Thus, the theory of the Three Worlds of Mathematics proposed by David Tall was used as basis, what brought elements that allow us to understand how humans learn to think mathematically. From this framework, was developed nine thought flows involved in learning a mathematical concept. A study of the derivative is also held to compose the material analyzed in this research and, from the perspective of thought flows, specific flows that allow us to observe learning elements of derivative were highlighted, as in the embodied world, which presents sensitive and noticeable aspects, when it is treated as the tangent line slope, place of righteousness instant variation curve or rate. In the symbolic world, it’s possible to observe routines and tacit processes to this concept, which are run in order to create their own symbols; and in the formal world, it was observed the axiomatic structure that a derivative can be designed. This thesis follows the exploratory theoretical proposal, because it is a research that relies only on documents to build its own concepts and arguments, using the accuracy and logical consistency to produce a base material for further research. Another result observed, which resulted in a breakthrough among the theories discussed, refers to noticeable feature of the derivative, belonging to the embodied world as this unlinked is procedural characteristics, providing thus provide a reference basis for further investigations in the process teaching and learning derivative / A presente tese tem como temática a aprendizagem da derivada. A pesquisa foi motivada pelo problema: Quais são os diferentes pensamentos relacionados ao processo de aprendizagem de um conceito? Assim, buscou-se na teoria dos Três Mundos da Matemática, proposta por David Tall, elementos que permitem compreender, como os humanos aprendem a pensar matematicamente. A partir desse referencial, foram desenvolvidos nove fluxos de pensamento envolvidos na aprendizagem de um conceito matemático. Também foi realizado um estudo da derivada para compor o material de análise desta pesquisa e, na perspectiva dos fluxos de pensamento, destacou-se os fluxos que permitem observar elementos de aprendizagem da derivada, como no mundo corporificado, que apresenta aspectos sensíveis e perceptíveis aos sentidos, ocasião em que é tratado como a inclinação da reta tangente, a retidão local de uma curva ou a taxa de variação instantânea; no mundo simbólico, são observadas rotinas e processos tácitos a esse conceito, que são executados a fim de se criar simbologias próprias; e no mundo formal, observou-se a estrutura axiomática que a derivada pode ser concebida. Esta tese apresenta uma proposta teórica exploratória, pois se trata de uma pesquisa que utiliza apenas documentos para construir os próprios conceitos e argumentos, utilizando o rigor e a coerência lógica, para produzir um material de base para novos estudos. Outro resultado observado, que resultou em um avanço dentre as teorias discutidas, refere-se a característica perceptível da derivada, pertencente ao mundo corporificado, pois esta desvincula-se de características processuais, propiciando dessa forma, oferecer um referencial base para novas investigações no processo de ensino e aprendizagem da derivada

Submitted by MARCIA ROVADOSCHI (marciar@unifra.br) on 2018-08-20T16:54:25Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_GabrielDeOliveiraSoares.pdf: 3563656 bytes, checksum: 8232661e148b570fa94209b37941261c (MD5) / Made available in DSpace on 2018-08-20T16:54:25Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_GabrielDeOliveiraSoares.pdf: 3563656 bytes, checksum: 8232661e148b570fa94209b37941261c (MD5) Previous issue date: 2018-01-03 / This qualitative research has as general objective to analyze the limit concept of a function in a point, presented by students of two undergraduate mathematics teaching courses, as well as its strategies of solving questions, in the light of the Theory of the Three Worlds of Mathematics. Firstly, was made an analysis of the introduction of the concept of limit in Calculus textbooks, based on the theoretical framework. Then, a mapping of articles on the ideas of David Tall was done, as well as a search of dissertations and theses that were based on these ideas. In the last phase, a test on limits of functions was applied to students of mathematics teaching courses, who had already studied the discipline of Calculus I and were interviewed professors of this discipline in both courses. The analysis of the textbooks made it possible to verify that, in the introductory chapters, there is a work that advocates features of the symbolic and embodied worlds, especially the first, with some aspects of the formal world. In relation to the interviews with the teachers, they affirmed that begin with graphical examples and tables of functions values to introduce the concept and only then try to reach the formal definition, crossing the Three Worlds of Mathematics. In the test applied to the students, it was concluded that most of them use the natural language to conceptualize limit, presenting characteristics of the embodied world, with some symbolic elements, but without achieving a development compatible with the formal axiomatic world. / Esta pesquisa, de caráter qualitativo, tem como objetivo geral analisar o conceito de limite de uma função em um ponto, apresentado por estudantes de dois cursos de Licenciatura em Matemática, bem como suas estratégias de resolução de questões, à luz da Teoria dos Três Mundos da Matemática. Primeiramente, foi realizada uma análise da introdução do conceito de limite em livros didáticos de Cálculo, com base no quadro teórico. Em seguida, foi feito um mapeamento de artigos sobre as ideias de David Tall, bem como uma busca de dissertações e teses que se basearam nessas ideias. Na última etapa, foi aplicado um teste sobre limites de funções a alunos que já haviam cursado a disciplina de Cálculo I e foram entrevistados docentes dessa disciplina nos dois cursos. A análise dos livros didáticos possibilitou verificar que, nos capítulos introdutórios, há um trabalho que preconiza características dos Mundos Simbólico e Corporificado, especialmente do primeiro, com alguns aspectos do Mundo Formal. Em relação às entrevistas com os professores, estes declararam partir de exemplos gráficos e tabelas de valores de função para introduzir o conceito e só depois tentam chegar à definição formal, perpassando os Três Mundos da Matemática. Em se tratando do teste aplicado aos estudantes, concluiu-se que a maior parte dos alunos utiliza a linguagem natural para conceituar limite, apresentando características do Mundo Corporificado, com alguns elementos simbólicos, mas sem atingir um desenvolvimento compatível com o Mundo Axiomático Formal.

Made available in DSpace on 2016-04-27T16:58:26Z (GMT). No. of bitstreams: 1 Rosana Nogueira de Lima.pdf: 65809143 bytes, checksum: 5ac1a0f9257d6350b3604e303ed3c9d3 (MD5) Previous issue date: 2007-09-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This thesis presents a study on the conceptions of equations held by students from first and second years of High School. Five mathematics teachers collaborated in the design of the instruments for data collection: a concept map, a questionnaire, a equation solving task and interviews. Two of these teachers were also responsible for the application of the instruments with their classes: one of first year students and one of second year students from a public school, and one of second year students from a private school, both from the Greater São Paulo area. The data collected was analysed in the light of the theoretical framework of the Three Worlds of Mathematics (Tall, 2004a, 2004b). This analysis is mainly focused on the embodied and symbolic worlds, and the met-befores and met-afters that interfere in the students work with equations. Results indicate that the most evident conception of equation among these students is equation as a calculation. The unknown and the equals sign do not seem to be important characteristics of an equation, and the main met-befores used by the students are from Arithmetic with integer numbers and from Algebra. The quadratic formula is the only solving method for quadratic equations that is used successfully, and it acts as a met-after in the work of some students with linear equations. The analysis shows that the students use techniques to solve equations which are disconnected from the mathematical principle of performing the same operation in both sides. The students create their own ways of working and end up using procedural embodiments, treating the symbols as physical entities that can be moved from one side to the other of the equation / Apresentamos, neste trabalho, um estudo sobre as concepções de equações apresentadas por alunos de primeira e segunda séries do Ensino Médio. Trabalhamos com cinco professores de Matemática, que colaboraram na confecção dos instrumentos de coleta de dados: um mapa conceitual, um questionário, uma atividade de resolução de equações e entrevistas. Dois desses professores, ainda, foram responsáveis pela aplicação dos instrumentos às turmas de alunos para as quais lecionavam: uma turma de primeira e uma de segunda séries do Ensino Médio, de uma escola pública, e uma turma de segunda série do Ensino Médio de uma escola particular, ambas as escolas localizadas na Grande São Paulo. Os dados coletados foram analisados à luz do quadro teórico dos Três Mundos da Matemática (Tall, 2004a, 2004b). Esta análise teve como enfoque, principalmente, os mundos corporificado e simbólico, e os já-encontrados e os a-encontrar que interferem no trabalho, com equações, feito pelos alunos. Os resultados obtidos indicam que a concepção de equação como conta é a mais evidente entre os sujeitos desta pesquisa. A incógnita e o sinal de igual não parecem ser considerados como características importantes de uma equação, e os principais já-encontrados usados são provenientes da Aritmética com números inteiros e da Álgebra. A fórmula de Bhaskara é o único método de resolução de equações quadráticas usado com sucesso, e age como a-encontrar no trabalho de alguns alunos com equações lineares. Evidências mostram que a resolução de equações é feita com o uso de técnicas desconectadas do princípio matemático de efetuar a mesma operação em ambos os membros. Os alunos criam seus próprios meios de trabalho, derivados dessas técnicas, e acabam por usar corporificações procedimentais, tratando os símbolos como entidades físicas que são movimentadas de um lado a outro da equação