Propaedeutics of Differential and Integral Calculi Author: Ing. Martin Malachov Department: Department of Mathematics Education Supervisor: Zdeněk Halas, DiS., Ph.D. Department of Mathematics Education Keywords: propaedeutics, derivative, Riemann integral, applications Differential and integral calculi both are interesting and beautiful branches of mathematics with many interdisciplinary overlaps and significant and practi- cal applications. Learning as well as teaching of these topics is very difficult and demanding. In this thesis we show that the derivatives and integrals have much to offer in the high school education while the schooling can be eased and made attractive with intentional propaedeutics and knowledge of rich background of applications. First part of the thesis presents short contem- plation on the current state of teaching and literature, we focus on the urge for the propaedeutics of the differential and integral calculi revealed by us. We identify key terms that can be used to build useful preconcepts during the whole high school education, even in the elementary education. In the latter part of the thesis we offer teacher innovative texts and a rich set of origi- nal exercises that can be used for motivation, application and propaedeutics of the differential and integral calculi. We also present...

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-08-24T17:24:33Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Felipe Arruda Sodré - 2010 - Doutorado em Filosofia UFPE-UFPB-UFRN.pdf: 1039429 bytes, checksum: e9651daa9c7bf789b76aabcbeffd3de4 (MD5) / Made available in DSpace on 2016-08-24T17:24:33Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Felipe Arruda Sodré - 2010 - Doutorado em Filosofia UFPE-UFPB-UFRN.pdf: 1039429 bytes, checksum: e9651daa9c7bf789b76aabcbeffd3de4 (MD5) Previous issue date: 2010-05-07 / Capes / A pergunta central dessa tese é: qual o sentido da Matemática na Crítica da Razão Pura? Assim, antes de tudo, é preciso deixar claro que na própria pergunta está implícita a informação de que esse é um problema de Metafísica que vai ser tratado ao modo kantiano. Nesse contexto, apresentar o sentido propedêutico da Matemática representa o resgate de um tema metafísico que, apesar de presente, parece que permaneceu latente e inexplorado, pelo menos na forma como o apresentamos nessa tese. Assim, no primeiro capítulo, o nosso tema é definido e resgatado a partir de uma breve história das interpretações da filosofia de Kant. Isso, por sua vez, projeta a nossa investigação no horizonte da tradição filosófica, conduzindo-nos até Platão, através do neokantismo de Cohen. Contudo, a leitura cientificista de Cohen joga sombra sobre o nosso tema. Por isso, o nosso distanciamento de Cohen permite, ao mesmo tempo, que a nossa investigação incida diretamente sobre os Diálogos de Platão, especificamente a República. O recurso a Platão revela a estratégia argumentativa capaz de subordinar o sentido epistemológico da Matemática ao seu sentido metafísico. Isso determina um modelo interpretativo que  sem comprometer-se com os princípios filosóficos de Platão  serve para mostrar, no segundo capítulo, como a Matemática, ao ser submetida ao crivo da filosofia crítica, impulsiona a razão na sua totalidade a se reposicionar voltando-se sobre si mesma. O idealismo crítico da Matemática descreve o primeiro impulso propedêutico do pensamento em direção ao interior da razão pura. Portanto, no último capítulo, mostramos como o sentido metafísico da Matemática para Kant não pode servir como método para a Metafísica, mas, ao contrário, determina uma legislação negativa para a razão pura. Essa é, precisamente, a centralidade da Matemática no Idealismo Transcendental de Kant: a crítica da Matemática desencadeia o processo de educação da razão, regulando o seu uso teórico e esclarecendo o caminho metodológico da própria Filosofia. / The central question of this thesis is: what is the meaning of Mathematics in the Critique of Pure Reason? So, first of all, it must make it clear that is implicit the information in this question that this is a problem of metaphysics that will be treated to the Kantian way. In this context, to show the propaedeutic sense of the Mathematics represents to regain a metaphysical theme that, despite to be present, it appears that remained latent and unexplored, at least in the way we presented in this thesis. Thus, in the first chapter, our theme is defined and regained from a brief history of the interpretations of Kant's philosophy. This, at its turn, projects our investigation into the horizon of the philosophical tradition, leading us back to Plato, through Cohen's neo-Kantianism. However, the Cohen’s scientistic reading throws shadow over our theme. Therefore, at the same time, our detachment of Cohen allows our research focuses directly on Plato´s Dialogues, particularly the Republic. The use of Plato reveals the argumentative strategy capable of subordinate the epistemological sense of mathematics to its metaphysical sense. This determines an interpretive model that - without compromising it with the philosophical principles of Plato - goes to show, in the second chapter, how the mathematics, when submitted to the appraisement of critical philosophy, gives impulse to the reason in its totality to reposition itself coming back to itself. The critical idealism of Mathematics describes the first propaedeutic impulse of thought into the direction of the pure reason interior. So in the last chapter, we show how the metaphysical sense of Mathematics for Kant can not serve as method for Metaphysics, but rather, determines a negative legislation to the pure reason. This is precisely the centrality of Mathematics in Kant's Transcendental Idealism: a critique of mathematics triggers the education process of the reason, that regulate its theoretical use and that clarify the methodological way of the philosophy itself.

This thesis deals with an introduction to variables. The aim of the thesis was to compare different approaches to teaching the introduction to variables in teachers who teach a method based on building schemes (known as Hejny method). The theoretical part of the thesis is a search of literature on the topic of introduction to variables and the theoretical background of teaching methods based on building schemes. Further, the thesis analyses textbooks designed for this method with a focus on tasks important for the propaedeutics of the variables. The research itself consisted of interviews with four teachers and observations of the lessons they taught in the 7th and 8th grades of secondary school, the topic of which was an introduction to the "language of letters". In total, there were fifteen lessons taught by these teachers in six classes (2-3 lessons in each class). Data were analyzed by qualitative methods. Although the observed teachers taught using the same method, I identified differences in the approach to the introduction of the variables and in the style of their teaching. There were differences in the organization of the lesson, in the selection of tasks and their style of assignments to pupils, teaching methods and ways in which teachers communicate with pupils. At the moment of entering the...