Construction Of Mathematical Knowledge Research Articles

Making Mathematics Accessible to Visually Impaired High School Learners in South Africa: Teachers’ Instructional Approaches, Assistive Devices and Challenges

ABSTRACT This case study explored the instructional approaches and assistive devices employed by high school mathematics teachers in teaching visually impaired (VI) learners and the challenges encountered by the teachers in facilitating the learners’ construction of mathematical knowledge. The participants were two high school mathematics teachers from a special school for the visually impaired in South Africa. Data was collected via semi-structured interviews and lesson observation. Inadequate skills in using Braille codes, time constraints, lack of learners’ home support structure, and technology failure are some of the challenges encountered by the teachers in teaching the VI learners. The approaches used by the teachers to teach mathematics to the VI learners in this study are direct instruction, contextual linking, scaffolding, peer tutoring, and problem-solving. The technologies the teachers used included text enlargers, text translators, text embossers, structure embossers, and scientific talking calculators. The study recommends in-service training for teachers teaching mathematics to

THE CONSTRUCTION OF MATHEMATICAL KNOWLEDGE AND ENVIRONMENTAL KNOWLEDGE IN CHILDHOOD EDUCATION

Several studies and statistical data point to Mathematics as the subject that most fails Elementary School students. In addition, children reach the fi nal grades seeing this subject as the bogeyman of schools. Given this context, this study presents refl ections and analyzes about the construction of mathematical knowledge in Early Childhood Education, whose purpose is to understand the implications of teaching practice in teaching Mathematics, understanding that Early Childhood Education is the primordial stage of Basic Education, in which the The child is in a phase of development of natural curiosity experiencing the discovery of the environment that surrounds him. In this context, Mathematics is a discipline that, if well worked, can contribute to the development of signifi cant skills that will help the child to better understand this world. However, the teacher’s intervention in working on mathematical knowledge linked to environmental issues in classroom activities will enable the student to develop such skills. The work is a part of the theoretical contribution that underlies the Doctoral Thesis that investigates the teaching practice in the construction of mathematical knowledge in preschool children. The study shows the importance of teaching Mathematics in Early Childhood Education, relating the construction of knowledge to children’s knowledge for meaningful learning in Preschool.

Mathematics teacher's continuing education in Professional and Technological Education: concepts and questions

Professional and Technological Education (PTE) has as its basic assumption, to qualify the subject for a universal training based on social parameters predisposed to the work and the full exercise of citizenship. Based on a continuous formation, this study aimed to present suggestions that can refine the teaching praxis of the PTE teacher, minimizing the discrepancies between the dichotomy of the propedeutic education and the technical education, effective in the integrated high school technical courses (EMI), particularly in the Federal Institutes (IFs). This is a bibliographical research that selects works that discuss the continuing education of the PTE mathematics teacher, who works in the EMI. It is believed that this article can help in the reflection about this theme, making possible for the student a new perspective in the construction of mathematical knowledge, with the purpose of entering the labor market.

The construction of algebraic thinking by deaf students from a systematic review of the literature

This article aims to analyze the specificities of the construction of algebraic thinking by deaf students in the early and final years of elementary school through a Systematic Review. The specific objectives are: to identify the curricular recommendations - National Common Curricular Base (2017) and Pernambuco Curriculum (2019) related to the construction of algebraic thinking by elementary school students; to retrieve from the CAPES Periodicals Portal, Brazilian Digital Library of Theses (BDTD) and Google Scholar research that deals with the construction of algebraic thinking or the learning of mathematical knowledge by deaf students; to point out the recommendations of the literature on the development of algebraic thinking by deaf elementary school students. The study was guided by a systematic literature review methodology based on the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA), which started with searches for articles and dissertations in the CAPES Periodicals Portal, BDTD and Google Scholar databases. A total of 2 727 studies were identified. After applying the inclusion or exclusion criteria, nine studies were selected for review. The results indicated that deaf students, like hearing students, present particularities in the construction of mathematical knowledge, especially in the Algebraic Unit Theme.

DANDO VEZ E VOZ AOS ALUNOS DA EJA:

Este artigo é fruto da escuta realizada no início do ano letivo de 2019 com os alunos da Educação de Jovens e Adultos - EJA, Ensino Médio, do Colégio Polivalente de Caravelas. A fim de tornar mais atraente e dinâmica a Matemática desenvolvida nesta modalidade de educação, o presente trabalho tem como objetivo mostrar o uso do material impresso e do jogo, como possibilidades didático-pedagógicas de construção do conhecimento matemático. São muitos os alunos desta turma que desenvolveram reações repulsivas em relação a este campo do saber. Nesta interação, contatou-se de que a forma que esta área do conhecimento foi apresentada para estes discentes contribuiu negativamente, tornando-a desinteressante. Por esta razão, optou-se no desenvolvimento de uma prática de ensino e aprendizagem baseado em elementos que estimulassem, viabilizassem e favorecessem o relacionamento destes alunos com os conceitos e ideias matemáticas. Para isso, apoia-se tanto em Freire (1996), que defende uma educação baseada no principio da dialogicidade, na prática da escuta, no querer bem ao educando, quanto em Muniz (2001), que propõe uma educação matemática a partir de situações ligadas aos jogos, ao uso de materiais impressos, dentre outros, como contextos favoráveis ao conhecimento matemático. Nesta fundamentação, também foram enfatizadas orientações que refletem a EJA na Bahia. Os resultados das atividades aplicadas comprovam que quando aprendemos assim, através do jogo e do uso do material impresso, os conteúdos matemáticos tem grandes chances de serem incorporados mais naturalmente e facilmente, o que de certa forma favorece o seu entendimento, a sua compreensão. 
 
 Palavras-chave: EJA; Matemática; Material impresso; Jogo.

DESAIN PENYAJIAN MATERI PERSAMAAN GARIS LURUS DI SMP BERORIENTASI KETERAMPILAN BERPIKIR TINGKAT TINGGI

The purpose of this literature studies is to gain knowledge in designing straight line equation material in middle school oriented to higher order thinking skills. The theory used in this study is constructivism in mathematics learning and the concept of higher order thinking skills. The presentation design of straight line equation material is designed through a presentation in the form of four problem situations are gradient, straight line equations, characteristics of straight line equations and applications of straight line equations. The four problem situations, based on theoretical studies, can be said to have fulfilled the three basic principles in constructing mathematical knowledge and three aspects of higher order thinking skills. The three basic principles in constructing mathematical knowledge are active student construction of mathematical knowledge, students construct new mathematical knowledge from the results of a physical and mental reflection and mathematical knowledge is formed by students through discussion and dialogue in their social learning environment. The three aspects of higher order thinking skills are deep understanding, reasoning and problem solving. The results of this literature studies can be concluded that the presentation design of straight line equations material in middle school in this study can be said to be oriented to higher order thinking skills. This is because theoretically able to generate activities that are oriented towards higher-order thinking skills, so that there is a great opportunity to equip students with the thoughts needed to form quality human resources in the 21st century.

Tarefa investigativa no ensino de derivadas em uma turma de licenciatura em matemática

O presente trabalho teve como objetivo analisar estratégias e conjecturas que os alunos da disciplina de Cálculo I de uma turma de Licenciatura em Matemática elaboram ao utilizarem tarefas investigativas envolvendo derivadas. Como base teórica, este estudo esteve pautado em Ponte, Brocardo e Oliveira (2016), Magalhães e Varizo (2016) em relação aos fundamentos da Investigação Matemática. A metodologia da pesquisa foi qualitativa, assistida pela observação participante e pela análise descritiva. Em especial, foi enfocado taxa de variação, máximos e mínimos de funções. Os resultados enfatizam que o desenvolvimento de tarefas investigativas em sala de aula, contribui para que o aluno seja ativo na busca da construção de conhecimento matemático. Para a formulação de conjecturas utilizaram diversas estratégias: simulações por meio da construção de quadros, desenhos, modelagem de funções e uso do Excel. Pode-se inferir que esta metodologia tem potencial para auxiliar o professor a desenvolver aulas mais dinâmicas e oferecer possibilidades de o aluno assumir o protagonismo na sua formação educacional.

A argumentação e a experimentação investigativa no ensino de matemática

O presente estudo teve início a partir da reflexão de nossa prática, em que buscamos analisar as contribuições das intervenções da professora-monitora para o surgimento e desenvolvimento da argumentação entre discentes participantes de um clube de ciências, durante uma atividade experimental investigativa sobre os conceitos introdutórios de área e perímetro. A pesquisa se caracteriza como qualitativa, sendo utilizada a Análise de Conteúdo para interpretação das informações levantadas a partir de videogravações. Participaram como sujeitos sete discentes. O espaço investigado é considerado um ambiente alternativo destinado ao ensino, pesquisa e extensão de ações didáticas voltadas às Ciências e Matemáticas. Durante as análises realizadas percebemos que nossas intervenções como professora-monitora auxiliaram no surgimento da argumentação e na construção do conhecimento matemático. Verificamos que surgiram organismos argumentativos completos de acordo com o padrão de Toulmin, assim como diversas formas de ação e pensamento.

Investigating Fifth-Grade Students’ Construction of Mathematical Knowledge through Classroom Discussion

This study is descriptive-qualitative in nature, aims to examine, (1) How do the students construct the mathematical knowledge (i.e. statistics for elementary school) through classroom discussion?, (2) What role are the students and the teacher playing in the discussion?, (3) What impact do their contributions have in the construction of new mathematical knowledge? As subjects of this study are 21 fifth-grade students of SD Labschool Unesa. Data were collected by using observation sheets, and by videotaping the class lessons with three cameras. The results suggest that both the teacher and the students participated actively in small group discussion, and played the majority of roles offered in an effective discussion. At the end of the lesson, there was a whole class discussion that functioned as meaning negotiation – to facilitate the students to share solutions and strategies with the whole class, and promote student reflection on the different strategies. This suggests that even ‘difficult materials’ can be successfully constructed by fifth-grade students using classroom discussion. Therefore, it is suggested that classroom discussion can be chosen as one of learning methods in a mathematics classroom in which the teacher provides appropriate mathematics contents and uses productive strategies to facilitate the learning processes.

Teachers’ Instrumental Genesis and Their Geometrical Understanding in a Dynamic Geometry Environment

Investigating how learners appropriate technological tools while engaging in mathematical tasks provides an insight into how learners’ tool-mediated activities shape their mathematical knowledge. The theory of instrumental genesis (Lonchamp in International Journal of Computer-Supported Collaborative Learning, 7(2), 211–237, 2012; Rabardel and Beguin in Theoretical Issues in Ergonomics Science, 6(5), 429–461, 2005) explains how learners appropriate technological tools and accomplish tasks while interacting with these tools. In this study, we investigate how teachers appropriate a collaborative, dynamic geometry environment, the Virtual Math Teams with GeoGebra (VMTwG), and how their appropriation shapes their geometrical knowledge. The environment contains a chat panel and multiuser version of GeoGebra (a dynamic geometry software) that teachers share. The participants are seven middle and high school mathematics teachers who participated in a 15-week professional development course in which they collaborated synchronously in VMTwG to solve geometrical tasks. We analyzed the work of one team of two high school teachers. Our analysis shows that the teachers appropriated collaboratively technological and mathematical tools, which supported their mathematical problem solving. These tools enabled them to manipulate and explore geometric objects and to notice and discuss relations among them. Our study contributes an analysis of longitudinal data to understand the co-emergence of learners’ collaborative appropriation of digital technologies and their construction of mathematical knowledge. This study suggests further research to understand how to orchestrate learners’ appropriation of collaborative technological tools in which instructors are not contemporaneously present during learners’ problem-solving activity.

مدى التوافق في محتوى كتب الرياضيات المطورة ومعيار حل المسألة الرياضية والإستراتيجيات المستخدمة من قبل المعلمين في تدريسها

هدفت هذه الدراسة إلى تحليل کتب الرياضيات المطورة للمرحلة الأساسية العليا في المملکة العربية السعودية في ضوء معايير(NCTM;2000) للتعرف على درجة توافر معيار المسألة الرياضية فيها والاستراتيجيات التي عرضت من خلالها. کما تهدف أيضا التعرف على استراتيجيات المسألة الرياضية التي يستخدمها معلمو الرياضيات في المرحلة الأساسية العليا. وقد استخدمت منهجية التحليل النوعي من خلال بناء أداة للتحليل مشتقة من معيار المسألة الرياضية المتضمن في وثيقة (NCTM;2000)، استخدمت في تحليل کتب الرياضيات المطورة للصفوف الرابع والخامس والسادس الابتدائي. کما تم بناء استبيان خاص باستراتيجيات المسألة الرياضية المستخدمة في تدريس المحتوى الرياضي تم توزيعه على عينة الدراسة المکونة من (26) مدرسا من مدرسي الرياضيات في المدارس الحکومية التابعة لوزارة التربية والتعليم في محافظة الرس. وأظهرت نتائج تحليل کتب عينة الدراسة أن المحتوى الرياضي لکتب الرياضيات المطورة للمرحلة الأساسية العليا قد وفر فرص کبيرة للنجاح في بناء المعرفة الرياضية حيث عرض 88% منها من خلال المسألة. وقد أظهر التحليل أن متوسط نسبة المسائل التي ظهرت في سياقات غير رياضية 64% مقابل 36% للمسائل التي ظهرت في سياقات رياضية. کما أظهر التحليل أن متوسط نسب استخدام الاستراتيجيات قد تفاوت بشکل ملحوظ. حيث أن استراتيجية انشاء رسومات للمساعدة في تصور المسألة جاءت في المرتبة الأولى في الاستخدام بنسبة 60%. کما تبين أيضا أن متوسط نسب الآليات التي تم استخدامها في مراقبة وملاحظة عمليات المشکلات والتأمل بها تراوحت بين 8% إلى 40.7%. وقد أظهر التحليل أن آلية التقدير ومعقولية الحل جاءت في المرتبة الأولى في الاستخدام بنسبة 40.7%. ومن خلال تحليل الاستبيان الذي وزع على مدرسي عينة الدراسة تبين أن استراتيجية حل مسألة أبسط هي أکثر الاستراتيجيات المستخدمة فعليا من قبل المعلمين في المسألة الرياضية وبدرجة تحقق عالية جدا. وحصلت جميع الاستراتيجيات الأخرى على درجة تحقق عالية. وقد أوصى الباحث بضرورة استفادة مدرسي الرياضيات من المسائل الموجودة في کتب الرياضيات المطورة وتوظيف استراتيجيات المسألة في أثناء تدريسها، وأخيرا أوصى بتضمين کتب الرياضيات کافة استراتيجيات المسألة. This study aimed to analyze developed Mathematics textbooks for higher primary stage in Kingdom of Saudi Arabia in light of standards (NCTM ، 2000) to identify degree of availability of solving standard and strategies presented through them. It also aimed to identify solving strategies used by teachers in higher primary stage. Qualitative analysis methodology has been used by building a tool for analysis is derived from solving standard included in document (NCTM; 2000), were used in analysis of developed mathematics textbooks for primary grades fourth، fifth and sixth. A special questionnaire was built for strategies for mathematical solving used in teaching of mathematical content was distributed to study sample consisting of (26) mathematics teachers for primary grades fourth، fifth and sixth in government schools under Ministry of Education in ArRass. The results of textbooks of study sample analysis showed that mathematical content had provided great opportunities for success in construction of mathematical knowledge where it offered 88% of them by solving. Analysis has shown that average rate of problems that have emerged in non-mathematical contexts is 64% compared to 36% of problems that have emerged in mathematical contexts. The analysis also showed that average rates of using strategies varied significantly. Where that strategy create drawings to help imagine problem came first by 60%. It also showed that average mechanisms that used in monitoring and observing operations of solving and contemplate it, ranged from 8% to 40.7%. Analysis has shown that mechanism the appreciation and reasonableness of solution came first in use by 40.7%. Through analyzing questionnaire, which distributed to teachers in study sample shows that solve a simpler problem is most strategy actually used by teachers in mathematical and a very high degree of verification, and all other strategies achieved a high degree of verification Depending on these results، researcher recommended need for mathematics teachers to benefit from problems existing in developed mathematics textbooks and employ solving strategies in teaching it, finally he recommended to include all problem-solving strategies in mathematics textbooks.

Zig-zagging in geometrical reasoning in technological collaborative environments: a Mathematical Working Space-framed study concerning cognition and affect

This study highlights the importance of cognition-affect interaction pathways in the construction of mathematical knowledge. Scientific output demands further research on the conceptual structure underlying such interaction aimed at coping with the high complexity of its interpretation. The paper discusses the effectiveness of using a dynamic model such as that outlined in the Mathematical Working Spaces (MWS) framework, in order to describe the interplay between cognition and affect in the transitions from instrumental to discursive geneses in geometrical reasoning. The results based on empirical data from a teaching experiment at a middle school show that the use of dynamic geometry software favours students’ attitudinal and volitional dimensions and helps them to maintain productive affective pathways, affording greater intellectual independence in mathematical work and interaction with the context that impact learning opportunities in geometric proofs. The reflective and heuristic dimensions of teacher mediation in students’ learning is crucial in the transition from instrumental to discursive genesis and working stability in the Instrumental-Discursive plane of MWS.

A METAFÍSICA FREUDIANA: UM ESTUDO PARA A DEMARCAÇÃO METODOLÓGICA DA FILOSOFIA FREUDIANA A PARTIR DE KANT

Da exposição oferecida por Kant em seu tratado Investigação sobre a evidência dos princípios da teologia natural e da moral (1764), aproveitamos as observações acerca da forma metodológica adequada às investigações em filosofia, distinguindo-a daquela que é própria da construção do conhecimento matemático. Essas observações servirão de base para uma tentativa de aproximação entre a forma metodológica oferecida pela metafísica, entendida como ontologia, e a metapsicologia freudiana que poderá, então, ser concebida como uma ontologia funcionando no interior de sua psicologia. Nosso intuito é entender a metapsicologia freudiana como obediente à forma metodológica que atribuímos à ontologia. Nesse sentido, a metapsicologia freudiana funcionaria como um tipo de gerador/processador de conceitos psicológicos muito fundamentais. Tomaremos o exemplo da concepção da série qualitativa prazer-desprazer, como apresentada no Projeto de uma psicologia, para descortinar o surgimento de um princípio geral para o funcionamento do aparelho psíquico: o princípio de constância. Em seguida, nos encarregaremos de mostrar que o esforço de fundamentação conceitual de Freud o conduz, quase vinte e cinco anos depois, a uma profunda reconsideração do lugar desse princípio no seio de sua teoria, principalmente com a análise da hipótese de que todos os organismos partilhariam de uma tendência ainda mais fundamental que a busca pelo prazer: a pulsão de morte.

Construction of mathematical knowledge and learning in and through individual education plans

This study, which is grounded in Bernstein’s theories of symbolic power and control in the context of pedagogy, explores how pupils’ learning in mathematics is constructed by a Swedish assessment practice called Individual Education Plans (IEPs). A systematic selection of 233 IEPs from five municipalities constitutes the empirical basis for the study. The results show that the content of IEPs primarily stresses the mastery of elementary skills, with an emphasis on automatization and routinization skills. In addition, regulative discourses about how to perform while learning mathematics are emphasized, especially for boys. The study also shows that IEPs, as they appear in the study, do not reflect the aims expressed in the current curriculum; on the contrary, the practice of using IEPs tends to reproduce a traditional discourse on the content of mathematics and how mathematics should be taught.

Teacher Belief: An Influencing Factor in Mathematics Teaching-Learning

This article explores the concept of belief and the belief system of mathematics teachers. The methodology used was mainly document analysis. About forty publications were reviewed which included journal articles, books and reports. It was found that beliefs develop through personal experience and culture. They might not always be reflected in the actions of a person or readily observable, but they play an important role in the decisions that people make. Beliefs of mathematics teachers are expressed through a system that has three major components: belief about nature of mathematics, about mathematics teaching, and about mathematics learning. Three philosophies are recognized to express the nature of mathematics: i) Instrumentalist, ii) Platonist, and iii) Constructivist. Beliefs about mathematics teaching and learning are classified as traditional and contemporary. Traditional beliefs perceive teachers as the transmitter of knowledge and students as submissive and compliant receivers. In contemporary constructivist beliefs teachers are viewed as guides or leaders and facilitate construction of mathematical knowledge and ability among students. Learning is an active construction, and autonomy and development of child interest are encouraged.

The use of graphs in specific situations of the initial conditions of linear differential equations

In this article, we present a discussion on the role of graphs and its significance in the relation between the number of initial conditions and the order of a linear differential equation, which is known as the initial value problem. We propose to make a functional framework for the use of graphs that intends to broaden the explanations of the traditional analytical frames that are widely favoured in school mathematics. As a result, the different forms and functions of graphs support their redefinition in a specific situation of the initial conditions. To that end, graphs are presented as a means to explore the nature of differential equations; thus, considering an epistemology based on the uses of graphs to explain the construction of mathematical knowledge.

O ler e o escrever na construção do conhecimento matemático no Ensino Médio

Esta pesquisa teve como objetivo investigar a utilização de diferentes estratégias de Leitura e Escrita no ensino de Matemática do Ensino Médio, e de instrumentos nos quais os alunos externaram as suas percepções durante o processo de ensino e aprendizagem. Para tanto, realizou-se uma pesquisa qualitativa com análise interpretativa sobre as atividades e os instrumentos propostos, os quais foram organizados em um Portfólio. A formulação e a aplicação da pesquisa, bem como a análise dos resultados e as conclusões foram baseadas no enfoque histórico-cultural de Vigotski e estudos sobre a linguagem matemática e a utilização da Leitura e Escrita nas aulas de Matemática, de diversos pesquisadores. As categorias emergentes da análise dos dados foram: a construção do conhecimento matemático, a semântica dos termos matemáticos, os valores demonstrados pelos alunos e a potencialidade das atividades. Os resultados, considerando-se todas as categorias, indicam que o processo pode tornar a relação professor-aluno e aluno-aluno mais interativa e mais efetiva para a construção do conhecimento matemático.

Construcción social del conocimiento matemático durante la obtención de genes en una práctica toxicológica

Esta investigación centra su interés en el estudio de las prácticas sociales en la construcción del conocimiento matemático. Abandona la tradicional centración en el estudio de los conceptos matemáticos explícitos para encontrarlos en el ambiente científico de un toxicólogo que los pone en funcionamiento implícitamente. Se utiliza el constructo que temporalmente llamamos juego de prácticas, con la intención de analizar cómo construyen conocimiento innovador mediante procesos de institucionalización de las prácticas; tomando a la práctica social como generadora del conocimiento y reconociendo la importancia de los saberes en uso. Se propone un modelo en el que creencias, conocimientos, experiencia, expectativas, Representaciones Sociales y concepciones, juegan un papel en dicha construcción. Los resultados obtenidos son relativos al carácter funcional de la matemática en tanto herramienta para la construcción de conocimiento, el mecanismo localizado de optimización - estandarización tuvo la función normativa de la práctica estudiada.

A modelagem como um ambiente de aprendizagem para a conversão do conhecimento matemático

A investigação da modelagem como um ambiente de aprendizagem para a construção do conhecimento matemático está se tornando predominante nos meios acadêmicos. Esse ambiente facilita uma estrutura pedagógica que promove a identificação e a disseminação de ambos os conhecimentos, explícito e tácito, dos alunos, através do estudo e da codificação das ideias matemáticas presentes nos sistemas retirados da realidade. Neste artigo, discutimos as principais características da conversão dos conhecimentos matemáticos, explícito e tácito, através da utilização do ambiente de aprendizagem da modelagem.

Equação e conhecimento matemático para o ensino: relações e potencialidades para a Educação Matemática

A noção de conhecimento matemático para o ensino, desenvolvida por Deborah Ball e seus colaboradores, surgiu a partir dos trabalhos de Shulman acerca do conhecimento pedagógico do conteúdo. O conhecimento matemático para o ensino refere-se a um tipo de conhecimento necessário para o professor poder desenvolver a sua "tarefa" de ensinar matemática. No presente ensaio teórico, pretende-se apontar relações e potencialidades entre diferentes significados de equação - concebidos a partir dos trabalhos de Alessandro Ribeiro - e o conhecimento matemático para o ensino, no caso do conceito de equação. Nesse sentido, desenvolveram-se análises dos resultados de pesquisas que contemplam tais significados à luz do modelo teórico proposto por Ball. Dentre os principais resultados, observaram-se as convergências entre tais estudos, reconhecendo potencialidades que a abordagem de diferentes significados de equação pode propiciar para a constituição do conhecimento matemático para o ensino de equações, por exemplo, a partir da epistemologia do conceito de equação.