Algebraic Arithmaric of The Remainders

The new Algebraic language has the purpose of describing the demonstrations of modular arithmatic in a diferente language, providing its theorems and axioms in totally new and diferente way from what has been done until now. And in it has the proof and demonstration of Catalan Conjecture.

Quasi-Nelson algebras and fragments

The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒ ew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Jogos e álgebra voltados para formação de professores de matemática da EJA: o que revelam as pesquisas?

ResumoEsse trabalho é de caráter bibliográfico e traz a revisão de literatura realizada para a construção da tese de doutorado, da primeira autora e orientada pela segunda, cujo tema é: Os jogos e a linguagem algébrica para a formação de professores de Matemática da EJA do Ensino Fundamental II. A revisão de literatura foi embasada no site de Catálogo de teses e dissertações da CAPES e no site da Biblioteca da PUC-SP de 2015 à 2018. Foram selecionadas cinco dissertações de mestrado e duas teses de doutorado nas universidades brasileiras nos campos da Educação, Matemática e da Educação Matemática, pois tinham maior ligação com o nosso objeto de investigação que foram divididas em três categorias: formação de professores da EJA, os jogos no ensino da Matemática e gamificação e a Matemática. Os trabalhos foram lidos em seus resumos e a partir dos dados foi possível concluir que há falta de formação para professores de Matemática da modalidade EJA e que os jogos são recursos que possibilitam a interação, a motivação e promovem aprendizagem Matemática. Palavras-chave: Jogos; Álgebra; EJA; Professores; Matemática.AbstractThis work is bibliographical and brings the literature review performed for the construction of the doctoral thesis, from the first author and guided by the second one, its theme is: Games and algebraic language for the formation of mathematics teachers of the Education for Youth and Adults students (EJA, in Brazil) of Elementary School II (from 6th to 9th grade, in Brazil). The literature review was based on the CAPES (Coordination for Improvement of Graduated Students, in Brazil) Thesis and Dissertations Catalogue website and on the PUC-SP (Pontifical Catholic University, in São Paulo) Library website from 2015 to 2018. Five master's dissertations and two doctoral thesis were selected in Brazilian universities in the fields of Education, Mathematics and Mathematics Education, as they have a greater connection with our research object they were divided into three categories: teacher training of the EJA, the games in the teaching of Mathematics and Gamification and Mathematics. The papers were read in their abstracts and from the data it was possible to conclude that there is a lack of training for mathematics teachers of the EJA modality and games are resources that enable interaction, motivation and promote mathematical learning.Keywords: Games; Algebra; EJA; Teachers; Mathematics.

Prototipo de juegos serios para el aprendizaje del lenguaje algebraico

En el presente trabajo se describe un prototipo de aplicación móvil para el aprendizaje del álgebra, en específico, el tema de Traducción del lenguaje común al lenguaje algebraico y viceversa. El tema fue seleccionado como resultado de un diagnóstico realizado a 126 estudiantes de segundo semestre del ciclo escolar 2017-2018 del Centro de Bachillerato Tecnológico Agropecuario No. 14. Se diseñó el contenido educativo de la aplicación, así como su secuencia y organización. Posterior a esto, se utilizó la técnica de Storyboarding para representar la interfaz y la herramienta AppInventor de Google para desarrollar tres juegos didácticos. Se observó que más de un 80% de estudiantes dispone un dispositivo móvil y lo emplean para actividades escolares, por lo que se espera que con la implementación del prototipo mejoren las calificaciones y se capte el interés por las matemáticas. In the present work a mobile application prototype is described for the learning of algebra, specifically, the topic of Translation from the common language to the algebraic language and vice versa. The topic was selected as a result of a diagnosis made to 126 students of the second semester of the 2017-2018 school year of Centro de Bachillerato Tecnológico Agropecuario No. 14. The educational content of the application was designed, as well as its sequence and organization. After this, the Storyboarding technique was used to represent the interface and the Google AppInventor tool to develop three didactic games. It was observed that more than 80% of students have a mobile device and use it for school activities, so it is expected that with the implementation of the prototype the grades will improve and the interest in mathematics will be captured.

Addressing Misconceptions in Algebra 1

An algebra 1 teacher collaborated with two university researchers to develop vocabulary minilessons and peer discussions to support students in understanding and using algebraic language.

Método de Pólya aplicado al lenguaje algebraico en primer año de licenciatura / Pólya Method Applied to the Algebraic Language in First Year of Degree

El bajo rendimiento presente en los estudiantes de diversos niveles en el ámbito matemático es un tema preocupante en la actualidad para los docentes. Mediante este estudio se efectuó un análisis documental a través del cual se pudo evidenciar que el método de Pólya, como estrategia didáctica aplicada a las matemáticas, incrementa las habilidades en los alumnos como resolutores. El estudio de alcance correlacional se trabajó con dos grupos, control y experimental, formados a partir de la población de 68 estudiantes de primer año de la Universidad Autónoma del Carmen. El desarrollo de la investigación consistió en una evaluación previa para verificar la homogeneidad de los grupos; posteriormente se implementó la estrategia didáctica al grupo experimental, y finalmente se evaluaron los conocimientos adquiridos. Dentro de los resultados arrojados por las pruebas estadísticas se pudo encontrar evidencia significativa, y se concluyó que el método de Pólya incrementó el rendimiento de los estudiantes con los que se trabajó la estrategia en comparación con aquellos a los que no se les aplicó.

A importância da função discursiva de designações de relações algébricas para o desenvolvimento do pensamento algébricoThe importance of the discursive function of algebraic relations designations for the development of algebraic thought

A investigação contou com um instrumento de coleta de dados subsidiadas pelas ideias de Raymond Duval relacionadas à aprendizagem da álgebra, aplicado a 115 alunos do 7º e 8º anos de escolas do estado do Paraná no Brasil. Foi analisada uma das questões do instrumento relacionada à ideia de que não são as letras que são importantes, mas as operações discursivas de designação dos objetos feitas por meio da língua natural ou formal. Os resultados encontrados revelaram formas de designação e redesignação diretas e verbais em linguagem natural, numérica ou algébrica ou indiretas e descritivas que implicaram na utilização de letras com utilização de léxicos associativos e, dessa forma, a identificação da atribuição de significação à essas letras pelos alunos. The present research used a data collection instrument subsidized by some of Raymond Duval's ideas related to learning algebra. The instrument was applied to 115 students from the 7th and 8th years in the state of Paraná, in Brazil. One of the questions of the instrument was analyzed in relation to the idea that it is not the letters that are important, but the discursive operations of designating objects, which are made through natural or formal languages. The results revealed specific forms of direct and verbal designation and reassignment in natural, numerical or algebraic language or, indirect and descriptive that implied the use of letters through a process of associative lexicons and, therefore, the assignment of meaning to such letters by students.

Linear diquasigroups

Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words.