Contributions of Neuroscience to Develop Teaching Strategies and Learning of Mathematics

The goal of the present work is to develop some strategies based on research in neurosciences that contribute to the teaching and learning of mathematics. The interrelationship of education with the brain, as well as the relationship of cerebral structures with mathematical thinking was discussed. Strategies were developed taking into consideration levels that include cognitive, semiotic, language, affect and the overcoming of phobias to the subject. The fundamental conclusion was the imperative educational requirement in the near future of a new teacher, whose pedagogic formation must include the knowledge on the cerebral function, its structures and its implications to education, as well as a change in pedagogy and curricular structure in the teaching of mathematics.

Strategies to Promote Research and Logical Reasoning in the Teaching of Mathematics in the Schools

The purpose of this paper is to raise a debate on the urgent need for teachers to generate innovative situations in the teaching-learning process, in the field of Mathematics, as a way for students to develop logical reasoning and research skills applicable to everyday situations. It includes some statistical data and possible reasons for the poor performance and dissatisfaction of students towards Mathematics. Since teachers are called to offer meaningful and functional learning experiences to students, in order to promote the pleasure of learning, teacher training should include experiences that can be put into practice by teachers in the education centers. This paper includes a work proposal for Mathematics Teaching to generate discussion, curiosity and logical reasoning in students, together with the Mathematical problem solving study.

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.

La logique ordinale de Turing

Le sujet visé par cette dissertation est la logique ordinale de Turing. Nous nous référons au texte original de Turing «Systems of logic based on ordinals» (Turing [1939]), la thèse que Turing rédigea à Princeton sous la direction du professeur Alonzo Church. Le principe d’une logique ordinale consiste à surmonter localement l’incomplétude gödelienne pour l’arithmétique par le biais de progressions d’axiomes récursivement consistantes. Étant donné son importance considérable pour la théorie de la calculabilité et les fondements des mathématiques, cette recherche méconnue de Turing mérite une attention particulière. Nous retraçons ici le projet d’une logique ordinale, de ses origines dans le théorème d’incomplétude de Gödel jusqu'à ses avancées dans les développements de la théorie de la calculabilité. Nous concluons par une discussion philosophique sur les fondements des mathématiques en fonction d’un point de vue finitiste.

A investigação do teorema fundamental do cálculo com calculadoras gráficas

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

O aprendizado de regras matemáticas: uma pesquisa de inspiração wittgensteiniana com crianças da 4ª série no estudo da divisão

Neste trabalho, investigamos o aprendizado de regras matemáticas no contexto da sala de aula, com ênfase, principalmente, nas discussões sobre a linguagem. Nosso objetivo principal foi pesquisar as dificuldades de ordem lingüística, enfrentadas pelos alunos no decurso do aprendizado das regras matemáticas, em especial, o conceito/algoritmo da divisão. Para tanto, discutimos, entre outras coisas, o tema “seguir regras”, proposto pelo filósofo austríaco Ludwig Wittgenstein em sua obra Investigações Filosóficas. Nosso trabalho e nossas análises foram fundamentadas, principalmente, na filosofia deste autor, que discute, entre outros temas, a linguagem e sua significação e os fundamentos da matemática, bem como nas reflexões do filósofo Gilles-Gaston Granger que analisa as linguagens formais. Realizamos uma pesquisa de campo que foi desenvolvida na Escola de Aplicação da Universidade Federal do Pará, em uma turma da quarta série do ensino fundamental. As aulas ministradas pela professora da turma foram observadas e, posteriormente, foi solicitado aos alunos que resolvessem problemas de divisão verbais e não-verbais, seguido de uma breve entrevista, na qual indagamos, entre outras questões, como os alunos resolveram os problemas envolvendo a divisão. Em nossas análises destacamos algumas dificuldades dos alunos, percebidas nas observações e em seus registros escritos ou orais: alguns alunos, em suas estratégias de resolução, inventam novas “regras matemáticas”. Há ainda aqueles que “confundem” os contextos na resolução de problemas matemáticos verbais, bem como a dificuldade de compreensão de problemas que trazem informações implícitas.

As concepções de função de Frege e Russell: um estudo de caso em filosofia e história da matemática

Pós-graduação em Educação Matemática - IGCE

O Raciocínio lógico-matemático: classificação e uso em concursos públicos

This study offers an analysis of classification of the main issues of logic and logical thinking found in competitive tendering and math tests, according to their concepts and characteristics, whether involving mathematics, or not. Moreover, a research on the evolutionary historic processes of logic according to three major crises of the foundations of mathematics was conducted. This research helped to define Logic as a science that is quite distinctive from Mathematics. In order to relate the logical and the mathematical thinking, three types of knowledge, according to Piaget, were presented, with the logical-mathematical one being among them. The study also includes an insight on the basic concepts of propositional and predicative logic, which aids in the classification of issues of logical thinking, formal logic or related to algebraic, and geometric or arithmetic knowledge, according to the Venn diagrams. Furthermore, the key problems - that are most frequently found in tests are resolved and classified, as it was previously described. As a result, the classification in question was created and exemplified with eighteen logic problems, duly solved and explained

O Raciocínio lógico-matemático: classificação e uso em concursos públicos

This study offers an analysis of classification of the main issues of logic and logical thinking found in competitive tendering and math tests, according to their concepts and characteristics, whether involving mathematics, or not. Moreover, a research on the evolutionary historic processes of logic according to three major crises of the foundations of mathematics was conducted. This research helped to define Logic as a science that is quite distinctive from Mathematics. In order to relate the logical and the mathematical thinking, three types of knowledge, according to Piaget, were presented, with the logical-mathematical one being among them. The study also includes an insight on the basic concepts of propositional and predicative logic, which aids in the classification of issues of logical thinking, formal logic or related to algebraic, and geometric or arithmetic knowledge, according to the Venn diagrams. Furthermore, the key problems - that are most frequently found in tests are resolved and classified, as it was previously described. As a result, the classification in question was created and exemplified with eighteen logic problems, duly solved and explained

Unpicking Priest's Bootstraps

Date of Acceptance: 13/07/2015

Unpicking Priest's Bootstraps

Date of Acceptance: 13/07/2015

Students as decoders of graphics in mathematics

This paper reports on students’ ability to decode mathematical graphics. The findings were: (a) some items showed an insignificant improvement over time; (b) success involves identifying critical perceptual elements in the graphic and incorporating these elements into a solution strategy; and (c) the optimal strategy capitalises on how information is encoded in the graphic. Implications include a need for teachers to be proactive in supporting students’ to develop their graphical knowledge and an awareness that knowledge varies substantially across students.

An evaluation of the pattern and structure mathematics awareness program in the early school years

This paper reports a 2-year longitudinal study on the effectiveness of the Pattern and Structure Mathematical Awareness Program (PASMAP) on students’ mathematical development. The study involved 316 Kindergarten students in 17 classes from four schools in Sydney and Brisbane. The development of the PASA assessment interview and scale are presented. The intervention program provided explicit instruction in mathematical pattern and structure that enhanced the development of students’ spatial structuring, multiplicative reasoning, and emergent generalisations. This paper presents the initial findings of the impact of the PASMAP and illustrates students’ structural development.

Understanding graphicacy : students’ making sense of graphics in mathematics assessment tasks

The ability to decode graphics is an increasingly important component of mathematics assessment and curricula. This study examined 50, 9- to 10-year-old students (23 male, 27 female), as they solved items from six distinct graphical languages (e.g., maps) that are commonly used to convey mathematical information. The results of the study revealed: 1) factors which contribute to success or hinder performance on tasks with various graphical representations; and 2) how the literacy and graphical demands of tasks influence the mathematical sense making of students. The outcomes of this study highlight the changing nature of assessment in school mathematics and identify the function and influence of graphics in the design of assessment tasks.