Divisibility Of Natural Numbers Research Articles

Divisão de Números Naturais: do saber previsto ao saber efetivamente ensinado em classes do Ensino Fundamental

Esse artigo é recorte de uma pesquisa de mestrado acadêmico que adota a noção de transposição didática de Yves Chevallard e objetiva analisar aspectos relacionados ao processo de transposição didática interna da operação de divisão de números naturais no Ensino Fundamental. Para subsidiar a análise das aulas de Matemática de três professores, foi realizado um estudo dos documentos legais do Brasil e do Estado de Pernambuco, análise de livros didáticos para o 4º, 5º e 6º anos e entrevistas com os professores que atuam nesses anos escolares. Nas aulas, o foco de análise foram os diferentes significados da divisão, os procedimentos de cálculo, o tratamento dado ao resto e à interpretação dos resultados obtidos, abordados ou não nas aulas observadas. Ao comparar o saber previsto para ser ensinado e o saber efetivamente ensinado verificou-se mudanças, deformações e lacunas, que poderiam acarretar dificuldades no processo de aprendizagem da operação. Palavras-chave: Transposição Didática Interna. Divisão de Números Naturais. Prática Docente.

The ability of seventh-grade disabilities students in solving number operation problems

This study aims to determine the ability of seventh-grade disabilities students (mental retardation) in solving number operation problems including addition, subtraction, multiplication, and division of natural numbers at SLB Bhakti Kencana I Berbah. The research method used is descriptive research to see the abilities and activities of students in solving number operation problems given by a researcher. The instruments of this study are a video to see students’ processes, test questions to see students’ abilities, and interviews with a teacher to encourage the student prior knowledge. The results of this study indicate that students still have difficulties in calculating the operation of natural numbers and forget how to calculate number operation according to what the teacher learns. In addition, the teacher mentioned that difficult to teach the concept of number operation.

Um olhar psicopedagógico sobre o ensino das operações de multiplicação e divisão de números naturais proposto do 1º ao 3º ano nos Livros Didáticos dos anos iniciais do ensino fundamental

O objetivo desse artigo, é investigar a proposta de ensino das operações de multiplicação e divisão de números naturais no bloco referente à Alfabetização Matemática em uma coleção de livros didáticos aprovada pelo PNLD/2013, anos iniciais do ensino fundamental.  Para tanto, foi pensado trazer um olhar psicopedagógico voltado para o ensino de Matemática, pautado ainda em algumas teorias de aprendizagem de PIAGET (1998) e VIGOTSKY (1998), acerca das concepções de aprendizagem de ambos. A escolha desse tema foi feita a partir da busca da compreensão de como se dá o ensino dessas operações nesse nível de ensino. Em meio as análises da proposta de ensino por uma coleção de livros didáticos, são trazidas discussões que remetem a reflexões acerca da formação do professor desse nível de ensino. Os resultados das análises mostram que a proposta de ensino das operações de multiplicação e divisão de números naturais no bloco referente à Alfabetização Matemática coaduna com os documentos oficiais em relação às sugestões de ensino nos níveis investigados.

ДЗЕТА-ФУНКЦИЯ МОНОИДОВ НАТУРАЛЬНЫХ ЧИСЕЛ С ОДНОЗНАЧНЫМ РАЗЛОЖЕНИЕМ НА ПРОСТЫЕ МНОЖИТЕЛИ1

In this paper we consider a new class of Dirichlet series, the zeta functions of monoids of natural numbers. The inverse Dirichlet series for the zeta function of monoids of natural numbers are studied. It is shown that the existence of an Euler product for the zeta function of a monoid is related to the uniqueness of the factorization into prime factors in this monoid. The notion of coprime sets of natural numbers is introduced and it is shown that for such sets the multiplicativity of minimal monoids and corresponding zeta-functions of monoids takes place. It is shown that if all prime elements of a monoid are prime numbers, then the characteristic function of the monoid is a multiplicative function and in this case the zeta function of the monoid is a generalized L-function. Various examples of monoids and corresponding zeta functions of monoids are considered. The relation between the inversion of the zeta function of a monoid and the generalized Mo¨bius function on a monoid as a partially ordered set is studied by means of the divisibility of natural numbers. A number of properties of the zeta functions of monoids of natural numbers with a unique decomposition into prime factors are obtained. The paper deals with taking the logarithm of an Eulerian product as a function of a complex argument. It is shown that a continuous function that determines the value of the logarithm of an Euler product runs through all branches of the infinite-valued function of the logarithm near its pole. The corollaries on the value of a complex-valued function of a special form near a singular point are obtained. These properties imply statements about the values of the Riemann zeta function near the boundary of the region of absolute convergence. Using Bertrand’s postulate, infinite exponential sequences of prime numbers are introduced. It is shown that corresponding zeta-functions of monoids of natural numbers converge absolutely in the whole half-plane with a positive real part. Since such zeta-functions of monoids of natural numbers can be decomposed into an Euler product in the whole region of absolute convergence, they do not have zeros in the entire half-plane with a positive real part. In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

초등수학영재의 분수 나눗셈의 이해에 관한 연구

The purpose of this study was to analyze the understanding of the meaning of fraction division and fraction division algorithm of elementary mathematical gifted students through the process of problem posing and solving activities. For this goal, students were asked to pose more than two real-world problems with respect to the fraction division of <TEX>${\frac{3}{4}}{\div}{\frac{2}{3}}$</TEX>, and to explain the validity of the operation <TEX>${\frac{3}{4}}{\div}{\frac{2}{3}}={\frac{3}{4}}{\times}{\frac{3}{2}}$</TEX> in the process of solving the posed problems. As the results, although the gifted students posed more word problems in the 'inverse of multiplication' and 'inverse of a cartesian product' situations compared to the general students and pre-service elementary teachers in the previous researches, most of them also preferred to understanding the meaning of fractional division in the 'measurement division' situation. Handling the fractional division by converting it into the division of natural numbers through reduction to a common denominator in the 'measurement division', they showed the poor understanding of the meaning of multiplication by the reciprocal of divisor in the fraction division algorithm. So we suggest following: First, instruction on fraction division based on various problem situations is necessary. Second, eliciting fractional division algorithm in partitive division situation is strongly recommended for helping students understand the meaning of the reciprocal of divisor. Third, it is necessary to incorporate real-world problem posing tasks into elementary mathematics classroom for fostering mathematical creativity as well as problem solving ability.

Difficulties of 6th Grade Elementary School Students in Solving the Four Basic Fundamental Operations: Addition, Subtraction, Multiplication and Division of Natural Numbers

The results of this study point out the contributions of the theory of Conceptual Fields to identify the nature of the mistakes made by students of the 6th year of elementary school in solving mathematical questions with ten natural numbers. We ask: What are the concepts and theorems put into action to carry out the operations of addition, subtraction, multiplication and division with natural numbers? The concepts and theorems in action related to the understanding of the decimal numbering system structure are responsible for the errors? The mistakes are due to the understanding of the organization of representation registers (word and Arabic notation)? As a result, we find that students have the addition of errors and subtraction errors by misunderstanding of the structure of the Numbering Decimal Positional System, errors due to lack of meaning assignment to the operations of addition, subtraction, multiplication and division, and also the numbers of Arabic script which induce errors in the handling of algorithms.

Reducing Cognitive Load in Learning by Analogy

The ability to draw connections between concepts, problems, or representations is crucial to mathematicalproficiency, though teachers face challenges in capitalizing on classroom opportunities to develop suchconnected thinking. An experiment with fifth-grade students tests theoretically grounded cuing strategies forsupporting learning from instructional analogies. Posttest results indicate that an analogy between division ofnatural numbers and rational numbers improved students’ proficiency in dividing fractions regardless of theaddition of analogical cuing strategies, but that the additional cuing led to more conceptual, flexible problemsolving. These cuing strategies may be effective ways to enhance students’ ability to draw connections acrossmathematics and capitalize on classroom learning opportunities.

O CONHECIMENTO PROFISSIONAL DOCENTE E SUA RELAÇÃO COM A IDEIA DE PROPORCIONALIDADE

Este artigo tem como objetivo analisar o conhecimento profissional docente de professores que ensinam Matemática no 5.º ano do Ensino Fundamental, cujos alunos (média de idade de 9 a 10 anos) se destacaram no Sistema de Avaliação de Rendimento Escolar do Estado de São Paulo (SARESP). Trata-se de pesquisa de campo que envolve um grupo de cinco professores de uma escola da rede pública de São Paulo, Brasil. A coleta de dados incluiu a aplicação de um questionário e entrevistas semiestruturadas. Teoricamente, esta investigação fundamenta-se tanto em teorias que discutem questões relacionadas à formação de professores como em estudos que investigam questões didáticas sobre a multiplicação e divisão de números naturais. Quanto ao primeiro enfoque, apoiou-se especialmente em estudos de Shulman. Em relação às questões didáticas, associadas ao objeto matemático, utilizou-se a Teoria dos Campos Conceituais de Vergnaud. De modo geral, a análise dos dados permitiu identificar que o grupo de professores da escola investigada trabalha de forma coletiva. Entretanto, pudemos verificar que as dificuldades relativas à prática docente podem ser influenciadas pela falta de domínio dos professores a respeito do conteúdo matemático, uma vez que observamos fragilidades em seus argumentos para o uso do procedimento de cálculo. Este estudo indica a necessidade de se rediscutirem as formas de se tratar a multiplicação e a proporcionalidade, em especial, nos cursos de formação inicial e continuada de professores. Palavras-chave: Educação Matemática; Conhecimento Profissional Docente, Ensino e Aprendizagem; Proporcionalidade. 

The effects of computer-assisted instruction on the achievement, attitudes and retention of fourth grade mathematics students in North Cyprus

The purpose of this study was to examine the effects of the educational software Frizbi Mathematics 4 on 4th grade student's mathematics achievement, retention, attitudes toward mathematics and attitude toward computer assisted learning. Two groups (experimental and control) of students from the state primary school in Gazimagusa, North Cyprus were used in this study. The control group was taught using a lecture-based traditional instruction and experimental group was taught using educational software, namely Frizbi Mathematics 4. The control group consisted of 26 students while the experimental group consisted of 29 students. The groups were compared on achievement of mathematics, retention, and attitude toward mathematics and computer assisted learning. The study included three units, Multiplication of Natural Numbers, Division of Natural Numbers, and Fractions. Scores on achievement tests were collected three times; at the beginning of the study, immediately after the intervention, and 4 months later. The mathematics attitude scale and computer assisted learning attitude scale were administrated only two times; at the beginning of the study and immediately after the completion of the study. A series of ANOVAs for repeated measures revealed significant difference between the groups on the post achievement tests and attitude scales in favor of experimental group. However, statistically significant differences in favor of the treatment group, on the retention tests were attained on the multiplication and division units but not on fractions. The evidence indicates that Frizbi Mathematics 4 for learning and teaching mathematics at the primary school level in North Cyprus is an effective tool.