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An Approach to Teach Calculus/Mathematical Analysis (For Sandra Isabel Cardoso Gaspar Martins Mestre em Matemática Aplicada An approach to teach Calculus/Mathematical Analysis (for engineering students) using computers and active learning – its conception, development of materials and evaluation Dissertação para obtenção do Grau de Doutor em Ciências da Educação Orientador: Vitor Duarte Teodoro, Professor Auxiliar, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa Co-orientador: Tiago Charters de Azevedo, Professor Adjunto, Instituto Superior de Engenharia de Lisboa do Instituto Politécnico de Lisboa Júri: Presidente: Prof. Doutor Fernando José Pires Santana Arguentes: Prof. Doutor Jorge António de Carvalho Sousa Valadares Prof. Doutor Jaime Maria Monteiro de Carvalho e Silva Vogais: Prof. Doutor Carlos Francisco Mafra Ceia Prof. Doutor Francisco José Brito Peixoto Prof. Doutor António Manuel Dias Domingos Prof. Doutor Vítor Manuel Neves Duarte Teodoro Prof. Doutor Tiago Gorjão Clara Charters de Azevedo Janeiro 2013 ii Título: An approach to teach Calculus/Mathematical Analysis (for engineering students) using computers and active learning – its conception, development of materials and evaluation 2012, Sandra Gaspar Martins (Autora), Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa e Universidade Nova de Lisboa A Faculdade de Ciências e Tecnologia e a Universidade Nova de Lisboa tem o direito, perpétuo e sem limites geográficos, de arquivar e publicar esta dissertação através de exemplares impressos reproduzidos em papel ou de forma digital, ou por qualquer outro meio conhecido ou que venha a ser inventado, e de a divulgar através de repositórios científicos e de admitir a sua cópia e distribuição com objectivos educacionais ou de investigação, não comerciais, desde que seja dado crédito ao autor e editor. iii iv Às minhas doces filhotas Alice, Laura e Marta. E ao meu maridão Rogério. v vi Aknowledgements I want to profoundly acknowledge… …to my advisor, Dr. Vitor Duarte Teodoro, for his constant availability, for his dedication, for his understanding and support. For his intelligent comments based on his scientific vision, aiming to perform honest and excellent quality research. … to my co-advisor, Dr. Tiago Charters de Azevedo, for his interest and support and by his pertinent comments and suggestions. … to my students…those who participated in this research for believing… to all, because they make me love more and more my profession! …to the Instituto Politécnico de Lisboa for the support by a PROTEC scholarship. … to my dear English reviewer, Celina, for the hard work she had. …to Teresa for having suffered with me the difficulties of PhD. … to my parents and sisters for being pillars that are always there! … to Laura for the kisses in eyes that were not given. … to Marta for the close hugs that I did not give her. … to Alice for the nights that I did not snuggle her in bed. … to Rogério, for existing! vii viii Abstract This thesis reports a new approach to the teaching of Mathematical Analysis 1/ Calculus (AM1) to students of engineering, applying results of research on the use of computers and active learning with the aim of enhancing understanding. The main goal of the new approach is to reduce the known problem of failure and superficial understanding in introductory college mathematics in Portugal (and other countries). This researcher created the approach named ActivMathComp where: . Students are active and collaborate with colleagues during classes; . Computer is embedded as a communication, interaction and computational tool; . Students use interactive digital learning documents; . Students explore concepts in order to develop a deep understanding of them; . Students contact with mathematical applications; . Students have frequent short quizzes with immediate feedback on a Learning Management System; . The teacher/student relationship is grounded on trust, on mutual understanding and on students’ involvement on their own learning. The interactive digital documents were created assuming principles such as the zone of proximal development and multiple representations. Towards its comparison with the traditional approach, the ActivMathComp was implemented in a group of 16 AM1 students at the Civil Engineering Undergraduate Program of the Instituto Superior de Engenharia de Lisboa. The participants freely chose to enrol in the group and were required to bring their own laptop to classes. Took place a quasi-experiment where all the other seven classes following AM1 were taken as a comparison group. The participating students got significantly higher grades than the other students and had a higher success rate. Data gathered from questionnaires and tests were screened to identify ix possible bias. The participating students evaluated ActivMathComp as highly positive in nearly all aspects. Keywords: Calculus Mathematical Analysis Computers Technology Active Learning x Resumo Esta tese apresenta uma nova abordagem ao ensino da Análise Matemática 1/ Cálculo (AM1) para estudantes de engenharia, aplicando os resultados da investigação sobre o uso de computadores e da aprendizagem activa com o objectivo de melhorar a sua aprendizagem. O principal objectivo desta nova abordagem é minorar os problemas de insucesso e de compreensão superficial da matemática introdutória de matemática no ensino superior em Portugal (e outros países). Esta investigadora criou esta abordagem chamada ActivMathComp em que: . Nas aulas, os alunos estão activos e colaboram entre si; . O computador é incluído como ferramenta computacional, de interacção e de comunicação. Os alunos utilizam documentos digitais interactivos; . Os alunos exploram os conceitos tendo em vista uma compreensão profunda; . Os alunos contactam com as aplicações da matemática; . Os alunos têm frequentes mini-testes com feedback imediato num Sistema de Gestão da Aprendizagem; . A relação professor/aluno assenta na confiança, compreensão mutua e no envolvimento dos estudantes na sua própria aprendizagem. Os documentos digitais interactivos foram criados usando princípios como a zona de desenvolvimento próximo e as múltiplas representações. Com vista à sua comparação com a abordagem tradicional, o ActivMathComp foi implementado num grupo de 16 estudantes de AM1 da Licenciatura em Engenharia Civil do Instituto Superior de Engenharia de Lisboa. Os alunos decidiram livremente inscrever-se no grupo e era pedido que trouxessem o seu próprio portátil para as aulas. Realizou-se uma quasi- experiência em que as outras sete turmas de AM1 funcionaram como grupo de comparação. xi Os participantes obtiveram notas significativamente superiores às dos outros alunos e apresentam uma taxa mais alta de aprovação entre os avaliados. Os dados do questionário foram estudados de modo a identificar possíveis motivos de enviesamento. Os participantes avaliaram o ActivMathComp muito positivamente em quase todos os aspectos. Palavras Chave: Cálculo Análise Matemática Computadores Tecnologia Aprendizagem Activa xii Résumé Cette thèse présente une nouvelle approche de l'enseignement de Analyse Mathématique 1/Calcul (AM1) pour les étudiants d'ingénierie appliquant les résultats de recherche sur l'utilisation des ordinateurs et sur l'apprentissage actif pour améliorer leur apprentissage. Le principal objectif de cette nouvelle approche est de pallier les problèmes d'échec et de la compréhension superficielle des mathématiques introductives à l'enseignement supérieur au Portugal (et d'autres pays). Cette rechercher a créé l’approche appelée ActivMathComp où: . Les étudiants sont actifs et travaillent ensemble en classe; . L'ordinateur est inclus comme un outil de calcul et de communication. Les étudiants utilisent documents interactifs; . Les étudiants explorent les concepts pour développer une compréhension profonde; . Les étudiants prennent contact avec les applications des mathématiques; . Les étudiants font souvent petites tests avec une retour immédiate dans un système de gestion de l'apprentissage; . La relation professeur / étudiant est basé sur la confiance, la compréhension mutuelle et la participation des étudiants dans leur apprentissage. Les documents interactifs prendre en considération principes comme la zone de développement proximale et de multiples représentations. En vue de leur comparaison avec l’approche traditionnelle, l’ActivMathComp a été testé avec un groupe de 16 étudiants d’AM1 de la Graduation en Ingénierie Civil, à l’Institut Supérieur d’ Ingénierie de Lisbonne. Les étudiants ont librement décidé d'adhérer au groupe où ont été invités à apporter leur propre ordinateur portable pour les classes. C’est fait une quasi- expérience où les sept autres classes d’AM1 ont été le groupe de comparaison. Les participants ont obtenu des notes significativement supérieurs à ceux des autres étudiants, et ont une plus élevé taux d'approbation parmi ceux évalués. Les données du questionnaire ont été xiii étudiées pour identifier les causes possibles de biais. Les participants ont évaluée ActivMathComp très positivement en presque tous les aspects. Mots-clés: Calcul Analyse Mathématique Ordinateurs Technologie Apprentissage actif xiv Table of Contents Aknowledgements vii Abstract ix Resumo xi Résumé xiii Table of Contents 1 List of Figures 9 List of Tables 14 List of Abbreviations 18 Introduction 19 1 Computers and Learning Software on Mathematics 25 1.1 Use of Computers on Mathematics Learning 26 1.1.1 A brief overview from the past to the present 26 1.1.2 Influences of the computers in teaching and learning 27 1.2
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