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Title: Working with numerical and algebraic expressions in the compulsory education Author: Bc. Lenka Pantůčková Department: Department of Mathametics and Mathematical Education Supervisor: Prof. RNDr. Jarmila Novotná, CSc. This Diploma Thesis deals with numerical and algebraic expressions in elementary school. The theoretical part deals with the framework educational programme for elementary education and grammar schools. It further mentions the propaedeutic of algebraic expressions at the first and second levels. It also introduces the most frequent mistakes done in calculating algebraic expressions. Meanings for the variable are stated as well. The last pre-chapter of the theoretical part is focused on statistics, which is further used for the experiment result evaluation. The experimental part first states the aims and methods of the research. Selected elementary schools taking part in the experiment are further introduced. An analysis of mathematics text books for the eighth grade (mainly chapters concerning the algebraic expressions), the selected schools use in their lessons, is also carried out. The experimental part also introduces the test presented to 128 pupils. Further pre-chapters are devoted to hypotheses and expected mistakes in single subtasks. In the conclusion of the experimental...

The diploma with the help of algebraic tiles thesis focuses on teaching expressions with variables in the eighth grade in middle school. It is divided into theoretical and experimental parts. A particular chapter is devoted to an analysis of selected textbooks. First of all, the aim of this thesis was to analyse mathematics textbooks from the perspective the algebraic expressions. Some knowledge acquired thanks to the textbook analysis was used for my own educational experiment, whose preparation, realisation and assessment were the second objective of this thesis. The theoretical part provides a look at expressions with variables in curricular documents. Thereafter the pillars of teaching algebraic expressions are presented. This part is completed by the classification of errors made by students in relation to the topic. An analysis of selected textbooks follows the theoretical part. The core of this thesis is the experimental part, where teaching is based on constructivist principles. The preparation of the course explains the manipulation of the algebra tiles used during the lessons. After the teaching experiments, we may conclude that the application of algebra tiles helps students get into the environment of algebraic expressions, improves the understanding of the rules of manipulations with...

This thesis focuses on errors and difficulties that students face when solving problems with algebraic expressions in mathematics at secondary school. Its aim was to describe the factors that affect pupils' achievement while dealing with algebraic expressions, classify them on the basis of a classification of pupils' errors used in mathematics and identify the biggest pupils' difficulties. The thesis consists of theoretical and experimental parts. The theoretical part focuses on the factors that influence the success of pupils in their learning process. I present their summary based on information gained from literature and I complete them with my own teaching experience of mathematics at secondary school. Next I deal with the concept of error, error classification and one of the most important phases of learning process, which is a description of teacher's work with pupil's error (again on the basis of information gained from the literature). The theoretical part ends with definitions of basic concepts from specialized literature on algebraic expressions at the end of the theoretical part. In the experimental part I deal with my own experiment during teaching of mathematics at secondary school, which is based on individual written work of students in the first year of their study and on the...

Made available in DSpace on 2016-04-27T16:57:24Z (GMT). No. of bitstreams: 1 TATIANA ALBIERI BARBOSA.pdf: 3147539 bytes, checksum: 085577e83e34af251e8ba627515ea0e1 (MD5) Previous issue date: 2013-05-14 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / This paper presents a Masters qualitative study which aimed to investigate if and how High School students observe regularities and generalize them, building expressions of Geometric Progressions with the use of simple calculators or with printer. To collect data for the research three sessions were held, with volunteer students from the first grade of High School from the same State School in the city of São Paulo, which according to André (2005) is a case study. The analyzes made were based mainly on the ideas of Devlin (2002), Mason (1996a and 1996b), Orton and Orton (1999), Silva, Loureiro and Veloso (1989) and Abelló (1997). It was concluded that the calculator was used as a tool that allowed students to focus on solving the activities, and not on the calculations to be performed. It was used as "leverage", enabling students to manipulate the statements data, perceive the presented patterns, establish relations and give sense to them in a way to generalize patterns, and express them as algebraic expression / O presente trabalho apresenta uma pesquisa qualitativa de Mestrado que teve como objetivo investigar se e como estudantes do Ensino Médio observam regularidades e as generalizam, construindo expressões relativas às Progressões Geométricas com o recurso das calculadoras simples ou com impressora. Para a coleta de dados da pesquisa foram realizadas três sessões, com alunos voluntários da 1ª série do Ensino Médio de uma mesma Escola Estadual, da cidade de São Paulo, o que, de acordo com André (2005) caracteriza um estudo de caso. As análises realizadas foram baseadas, sobretudo, nas ideias de Devlin (2002), Mason (1996a e 1996b), Orton e Orton (1999), Silva, Loureiro e Veloso (1989) e Abelló (1997). Concluiu-se que a calculadora serviu como um instrumento que possibilitou aos alunos focar na resolução das atividades, e não nos cálculos a serem realizados. Ela serviu como alavanca , possibilitando aos alunos manipular os dados dos enunciados, perceber os padrões apresentados, estabelecer relações e dar sentido às mesmas, de forma generalizar os padrões, e expressá-los por meio de expressões algébricas

O presente trabalho trata basicamente do conteúdo expressões algébricas no ensino fundamental. Motivados pela crença de que a ausência do método de argumentação matemática e do pensamento genérico no atual ensino de Matemática têm se apresentado muito nociva ao estudo de expressões algébricas, desenvolvemos, em uma turma de sétima série, uma proposta didática onde apresentamos uma sequência de atividades que enfatizam estes dois aspectos. Tais atividades servem de pré-requisitos para o estudo de expressões algébricas. Inclui-se neste trabalho uma análise dos PCN para o ensino fundamental e uma análise crítica de livros didáticos de sétima série/oitavo ano sobre expressões algébricas. A partir destas análises, percebemos, nos livros didáticos, o inadequado emprego (no nosso ponto de vista) de polinômios neste nível de ensino, o que resultou na elaboração de um capítulo de conteúdo matemático versando sobre polinômios, dedicada a professores de Matemática da Educação Básica. / This text is closely related to the subject algebraic expressions in Fundamental School. Believing that the absence of the mathematical method and of what we call generical thought in the treatment of mathematics in the first school years causes much harm in the student’s mathematical education, mainly when one deals with algebraic expressions, we applied in a 7th grade class and present here a sequence of activities which give emphasis to those two aspects mentioned above and which prepare the students for the study of algebraic expressions. Additionally we include here an analysis of part of the PCN (Brazilian Curriculum Recommendations) and of nine 7th grade-school books, namely, the part related to algebraic expressions. Finally, we include a chapter about polynomials which was written for school teachers, motivated by the inadequate use (for this level of mathematical education, at least in our point of view) of polynomials in the school books which we have analyzed.

O presente trabalho trata basicamente do conteúdo expressões algébricas no ensino fundamental. Motivados pela crença de que a ausência do método de argumentação matemática e do pensamento genérico no atual ensino de Matemática têm se apresentado muito nociva ao estudo de expressões algébricas, desenvolvemos, em uma turma de sétima série, uma proposta didática onde apresentamos uma sequência de atividades que enfatizam estes dois aspectos. Tais atividades servem de pré-requisitos para o estudo de expressões algébricas. Inclui-se neste trabalho uma análise dos PCN para o ensino fundamental e uma análise crítica de livros didáticos de sétima série/oitavo ano sobre expressões algébricas. A partir destas análises, percebemos, nos livros didáticos, o inadequado emprego (no nosso ponto de vista) de polinômios neste nível de ensino, o que resultou na elaboração de um capítulo de conteúdo matemático versando sobre polinômios, dedicada a professores de Matemática da Educação Básica. / This text is closely related to the subject algebraic expressions in Fundamental School. Believing that the absence of the mathematical method and of what we call generical thought in the treatment of mathematics in the first school years causes much harm in the student’s mathematical education, mainly when one deals with algebraic expressions, we applied in a 7th grade class and present here a sequence of activities which give emphasis to those two aspects mentioned above and which prepare the students for the study of algebraic expressions. Additionally we include here an analysis of part of the PCN (Brazilian Curriculum Recommendations) and of nine 7th grade-school books, namely, the part related to algebraic expressions. Finally, we include a chapter about polynomials which was written for school teachers, motivated by the inadequate use (for this level of mathematical education, at least in our point of view) of polynomials in the school books which we have analyzed.

O presente trabalho trata basicamente do conteúdo expressões algébricas no ensino fundamental. Motivados pela crença de que a ausência do método de argumentação matemática e do pensamento genérico no atual ensino de Matemática têm se apresentado muito nociva ao estudo de expressões algébricas, desenvolvemos, em uma turma de sétima série, uma proposta didática onde apresentamos uma sequência de atividades que enfatizam estes dois aspectos. Tais atividades servem de pré-requisitos para o estudo de expressões algébricas. Inclui-se neste trabalho uma análise dos PCN para o ensino fundamental e uma análise crítica de livros didáticos de sétima série/oitavo ano sobre expressões algébricas. A partir destas análises, percebemos, nos livros didáticos, o inadequado emprego (no nosso ponto de vista) de polinômios neste nível de ensino, o que resultou na elaboração de um capítulo de conteúdo matemático versando sobre polinômios, dedicada a professores de Matemática da Educação Básica. / This text is closely related to the subject algebraic expressions in Fundamental School. Believing that the absence of the mathematical method and of what we call generical thought in the treatment of mathematics in the first school years causes much harm in the student’s mathematical education, mainly when one deals with algebraic expressions, we applied in a 7th grade class and present here a sequence of activities which give emphasis to those two aspects mentioned above and which prepare the students for the study of algebraic expressions. Additionally we include here an analysis of part of the PCN (Brazilian Curriculum Recommendations) and of nine 7th grade-school books, namely, the part related to algebraic expressions. Finally, we include a chapter about polynomials which was written for school teachers, motivated by the inadequate use (for this level of mathematical education, at least in our point of view) of polynomials in the school books which we have analyzed.

Modelagem e controle de execução são duas abordagens do gerenciamento de processos de negócio que, embora complementares, têm se desenvolvido independentemente. Por um lado, a modelagem é normalmente conduzida por especialistas de negócio e explora aspectos semânticos do processo. Por outro lado, o controle de execução estuda mecanismos consistentes e eficientes de implementação. Este trabalho apresenta um método algorítmico que relaciona modelagem e controle de execução, por meio da geração de expressões algébricas a partir de digrafos acíclicos. Por hipótese, assumimos que modelos de processos de negócio são formados por estruturas baseadas em grafos, e mecanismos de controle de execução são baseados na interpretação de expressões de álgebra de processos. Para a geração de expressões algébricas, esta tese apresenta as propriedades topológicas de digrafos série-paralelo e define um sistema de transformação baseado em redução de digrafos. Além disso, um algoritmo de identificação de digrafos série-paralelo e geração de expressões algébricas é apresentado. O texto também discute o tratamento de digrafos que não são série-paralelo e apresenta, para alguns desses casos, soluções baseadas em mudanças topológicas. Finalmente, o algoritmo é ilustrado com o estudo de caso de uma aplicação real. / Modeling and execution control are complementary approaches of business process management that have been developed independently. On one hand, modeling is usually performed by business specialists and explores semantical aspects of the business process. On other hand, execution control studies consistent and efficient mechanisms for implementation. This work presents an algorithmic method which joins modeling and execution control through algebraic expression generation from acyclic digraphs. By hypothesis, we assume that business process models are defined by graph structures, and execution control mechanisms are based on interpretation of process algebra expressions. For algebraic expression generation, this thesis presents the topological properties of series-parallel digraphs and defines a transformation system based on digraph reduction. Therefore, we present an algorithm for identification of series-parallel digraphs and generation of algebraic expressions. This work also discusses the treatment of non-series-parallel digraphs and presents solutions based on topological changing for some cases. Finally, the algorithm is illustrated with a case study based on a real system.

Made available in DSpace on 2016-04-27T16:57:49Z (GMT). No. of bitstreams: 1 dissertacao karina andrezzo.pdf: 1135963 bytes, checksum: 3dd93f8487f3f29db82548fd1e1745ac (MD5) Previous issue date: 2005-04-08 / ABSTRACT This study aimed to identify factors that contribute to the understandings of algebraic expressions constructed by blind students. To this end, it involved the design of situations that would enable the participation of blind high-school students (Brazilian Ensino Médio) in generalizing activities. The research was inspired by Vygotsky s visions of the social integration and intellectual development of learners with special needs. The use of an approach to algebra emphasizing generalizations based on the patterns underlying sequences presented as spatial arrangements has been supported by various researchers in Mathematics Education, this study was informed particularly by the work of Küchemann (1981), Booth (1988) and Mason (1996). In the first phase of the research, materials based on the use of magnets to represent dynamic elements of sequences were designed and tested. The second phase involved five blind students in completing a preliminary activity, elaborated to obtain data about their performance on traditional algebra exercises followed by a series of interviews, during which each students worked through a set of seven generalizing activities. The analysis of the students responses to the preliminary activities indicated that the students had difficulties in dealing with three of six categories for interpreting letters in algebraic expressions described by Küchemann (1981), letter as specific unknown , letter as generalized number and letter as variable ; and identified errors committed during the simplification of expressions. Analysis of the interview data was based on the strategies the students used to construct mathematical generalizations. These analyses highlighted a process by which the external marks (the magnets and their organizations) were gradually internalized through the manipulations and articulations by which the learners strived to get a sense of algebraic expressions. / Nesta pesquisa, buscamos identificar fatores que contribuíram na apreensão de expressões algébricas por alunos sem acuidade visual. Para isto elaboramos atividades que facilitassem sua participação em atividades de generalização. O estudo se desenvolveu com alunos do Ensino Médio e tem como ponto de partida a posição de Vygotsky referente à integração social do aluno portador de alguma deficiência e seu potencial para um desenvolvimento normal. A sugestão de abordagem algébrica através de generalizações de padrões figurativos teve suporte de vários pesquisadores em educação Matemática, em particular, os estudos e considerações de Küchemann (1981), Booth (1988) e Mason (1996). A primeira fase da pesquisa envolveu a elaboração das atividades e dos materiais manipulativos, através dos quais os alunos s.a.v. poderiam ter acesso às seqüências de padrões figurativos. A segunda fase constituiu-se de uma Atividade de Sondagem composta por itens que envolviam noções algébricas e cinco entrevistas onde foram trabalhadas sete seqüências cujos padrões eram representados por ímãs de formas geométricas. A análise da Atividade de Sondagem possibilitou que identificássemos as dificuldades dos alunos em lidar com três das seis categorias descritas por Küchemann (1981), letra como uma incógnita específica , letra como número generalizado e letra como variável ; e alguns erros cometidos pelos alunos durante as simplificações. Quanto à análise das tarefas, esta, procedeu das estratégias usadas na evolução das generalizações matemáticas. Nossas análises indicaram um processo gradual de internalização das marcas externas (os imãs e suas organizações) através das manipulações , articulações e na tentativa de atribuir significado algébrico às letras em expressões algébricas ( constituindo um sentido para ).