Relevant Mathematical Concepts Research Articles

Utilizing task with scaffolding: growth of students' mathematical understanding from layers of image having to property noticing

Introduction. Learning mathematics accompanied by understanding is an essential component of factual and procedural knowledge skills practical for students in solving real-life problems. Errors in solving systems of linear equations in two variables (SPLDV) among students can be caused by various factors, including students' lack of understanding of this concept, so this problem must be studied in depth by utilizing task with scaffolding. The research aimed to explore the potential of a task to foster mathematical understanding of students from the image-having layer to the property noticing layer in the learning context of a system of linear equations in two variables (SPLDV). Study participants and methods. The study involved three students in grade 8 at a junior high school in Indonesia. They were selected based on their mathematical ability (low, moderate, and high) based on their accumulated score reports in grade 7 and information from the teacher who taught them. This study employed a qualitative research approach, data collection was carried out by giving task to students, observing, and interviews them while doing the task, and data analysis began with transcription and organizing the results of interviews and student task, followed by identification and classification of difficulties faced by students. The results. The study showed that student with low mathematical ability has difficulty recognizing patterns or relationships between coefficients and variables in equations. Even with the help of scaffolding, the student still did not succeed in strengthening her understanding to complete the task given, so her understanding only shifted slightly from the image having layer and could not move completely to the property noticing layer. Next, a student with moderate mathematical ability had a sufficient understanding of the concept of variables and coefficients. However, she still needed additional help to complete task, especially in determining the value of variable. She was observed to be already at the property noticing layer of understanding because she had been able to identify the property of two linear equations in two variables that is equivalence and complete the task given. Lastly, the student with high mathematical ability could understand the relevant mathematical concepts well. She could complete the task assigned well despite folding back and scaffolding. Thus, this student's understanding reached the property noticing layer. Conclusions. The research showed differences in the growth of mathematical understanding from the image having layer to the property noticing layer among junior high school students with low, medium, and high mathematical ability. The student with lower math skills was only slightly shifted from the image having layer and could not be transferred completely to the property noticing layer. The student with moderate mathematical ability had reached the layer of understanding properties because she was able to identify equivalent properties of two linear equations in two variables and complete the given task. Lastly, the student with high mathematical skills has reached the layer of property noticing.

Integrating Sustainable Development Principles in Learning Mathematics to Stimulate Sustainable Skills in Future Generations

This research constitutes a literature review employing a qualitative approach, analyzing scholarly articles, books, and other documents related to sustainable development. This article aims to summarize and analyze previous studies concerning sustainable development in the context of mathematics education, as well as strategies that can be employed to integrate the principles of sustainable education. Integrating the principles of sustainable development into education, including mathematics education, is crucial in fostering a more environmentally responsible society and promoting sustainability across all sectors. However, its implementation remains limited. Educators face various challenges, including a lack of time, resources, and understanding of sustainable education, along with a dearth of supportive teaching materials. The principles of sustainable development can serve as a framework for developing curricula and teaching practices that are more sustainable. Educators can select mathematical problems related to environmental or social issues, discuss relevant mathematical concepts in connection with these problems, and help students comprehend the impact of mathematical decisions on the environment and society. Integrating the principles of sustainable development into mathematics education not only aids in producing a generation with sustainable skills but also motivates students to learn mathematics in more engaging and meaningful ways. A learning approach centered around sustainable development can be an effective way to prepare students for a sustainable future. The article also underscores the necessity for curriculum development, training, and professional advancement for educators.

A novel approach in understanding the basic modes of ventilation in a recent generation mechanical ventilator- A review of the underlying principles.

Understanding the basic concepts and the dynamics of the equation of motion will help the anaesthesiologist and the intensive care physician in gaining an insight into the basics of modern mechanical ventilation. An equation of the type Vt = V0 × (1-e-kt) is often encountered while studying the concepts of mechanical ventilation. This leaves one wondering about the meaning of 'e'. It is explained as the base of a natural logarithm, and the base e as an irrational constant approximately equal to 2. 7182. In medical literature, various physiological mechanisms are explained by using the exponential function e. However, the explanations do not sufficiently help the learner to demystify the enigmatic term e. In this article, an effort is made to explain this function with the help of simple analogies and relevant mathematical concepts. The process of volume build-up in the lungs during mechanical ventilation is used as a model for the explanations.

Öğretimsel Dokümanlardaki Etkinliklerin Amaçlarının Ortaokul Matematik Öğretmenlerinin Perspektifinden İncelenmesi

This study aims to examine how mathematics teachers perceive the purposes of text book activities. The participants include 15 secondary school math teachers enrolled in a Master program at a state university in the south of Turkey. The professional experience of the participants varies from 1 to 6 years. The study was designed as a descriptive research. The data were collected from participants before they attended to a course about design and implementation of activities. As a data collection tool, participants were given a form with four randomly selected text book activities and asked to write the purposes for each of these activities. Among the qualitative analysis methods, the content and descriptive analysis methods were used to analyze the data. The content analysis revealed that teachers had disparate perceptions about the purposes of the same activity. On the other hand, according to the results obtained from the descriptive analyses, it was established that the answers of the participants could be categorized in different levels in terms of classification according to cognitive demand levels. Teachers' varying perceptions of the purposes of a given activity can be attributed to the fact that the activity was designed and implemented for different purposes. In addition, students taught by different teachers may have different learning outcomes. This may cause the students who go through the same educational processes to be at different cognitive levels for the relevant mathematical concepts. On that basis, it has been put forward how important it is to perceive activities as a tool, and to determine and highlight their objectives accordingly. Especially with the activities given in textbooks, the importance of expressing the purpose of the activity in a clearer and more understandable way emerged.

Students’ Mathematical Connection Ability in Solving Real-world Problems

Being able to solve a variety of mathematical problems is one of the main factors and indicators that reflect the success of mathematics learning, including real-world problems. In solving real-world problems, students need a mathematical connection to be able to choose and use a mathematical concept that fits properly with conditions given in the problem. Therefore, it is important to observe students’ mathematical connection in order to improve their mathematical achievement. This study is qualitative descriptive which aimed to describe the ability of eighth-graders’ mathematical connections in solving real-world problems. The subjects of this study were 29 students of a Private Junior High School in Bantul Yogyakarta. Mathematical connection test was employed for collecting the data. Based on the analysis result, students’ mathematical connection falls in the low category. Students’ difficulties in solving real-world problems were restricted to understanding the problem and connecting the problem with relevant mathematical concepts.

Higher-Order Interactions: Understanding the knowledge capacity of social groups using simplicial sets

Abstract A predominant benefit of social living is the ability to share knowledge that cannot be gained through the information an individual accumulates based on its personal experience alone. Traditional computational models have portrayed sharing knowledge through interactions among members of social groups via dyadic networks. Such models aim at understanding the percolation of information among individuals and groups to identify potential limitations to successful knowledge transfer. However, because many real-world interactions are not solely pairwise, i.e., several group members may obtain information from one another simultaneously, it is necessary to understand more than dyadic communication and learning processes to capture their full complexity. We detail a modeling framework based on the simplicial set, a concept from algebraic topology, which allows elegant encapsulation of multi-agent interactions. Such a model system allows us to analyze how individual information within groups accumulates as the group’s collective set of knowledge, which may be different than the simple union of individually contained information. Furthermore, the simplicial modeling approach we propose allows us to investigate how information accumulates via sub-group interactions, offering insight into complex aspects of multi-way communication systems. The fundamental change in modeling strategy we offer here allows us to move from portraying knowledge as a “token”, passed from signaler to receiver, to portraying knowledge as a set of accumulating building blocks from which novel ideas can emerge. We provide an explanation of relevant mathematical concepts in a way that promotes accessibility to a general audience.

Dedekind–MacNeille completion and Cartesian product of multi-adjoint lattices

Fuzzy logic programming tries to introduce fuzzy logic into logic programming in order to provide new generation computer languages which incorporate comfortable programming resources for helping the development of applications where uncertainty could play an important role. In this sense, the mathematical concept of multi-adjoint lattice has been successfully exploited into the so-called Multi-Adjoint Logic Programming approach, MALP in brief, for modelling flexible notions of truth-degrees beyond the simpler case of true and false. In this paper, we focus on two relevant mathematical concepts for this kind of domains useful for evaluating MALP. On the one side, we adapt the classical notion of Dedekind–MacNeille completion in order to relax some usual hypothesis on such kind of ordered sets, and next we study the advantages of generating multi-adjoint lattices as the Cartesian product of previous ones. On the practical side, we show that the formal mechanisms described before have direct correspondences with interesting debugging tasks into the ‘Fuzzy Logic Programming Environment for Research’, FLOPER in brief, developed in our research group.

Mathematical concepts for the micromechanical modelling of dislocation dynamics with a phase-field approach

This contribution reviews the mathematical concepts of micromechanical modelling in the phase-field approach applied to dislocation dynamics. The intention is two-fold. On the one hand, modelling of dislocation dynamics is a very recent field of development in phase-field theory, in comparison to the simulation of diffusional phase transformation and related microstructure evolution problems in materials science. The reason is that modelling dislocation dynamics poses several challenges for phase-field concepts which go beyond purely diffusional problems in materials science such as, e.g. dendritic solidification, as we point out in Section 3. On the other hand, the modelling of dislocations has triggered further wide-ranging developments of phase-field based models for deformation problems. This is an important development, since a comprehensive model for deformation problems should include displacive as well as diffusional degrees of freedom from the atomic scale to the microscale. This is something phase-field theory is capable of, as discussed in this review article. We aim to give an overview of relevant mathematical concepts, and to stimulate further steps in this direction.

Reverse mathematics and the equivalence of definitions for well and better quasi-orders

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Permutation orbifolds

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Λ-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S 3 is described to illustrate the general theory.

From ‘TO POLY’ to generalized poly‐figures and their classification: a learning experience

A heuristic description of a personal learning experience in a LOGO micro‐world is given. The figures produced by the familiar POLY procedure are generalized, and an attempt is made to develop a workable definition and a useful classification. It depicts the gradual evolution of relevant mathematical concepts, and the part intuition played in this process. Moreover, it shows mathematics both as a process of discovery and creative work, illustrating the difference between the final product and the untidy processes used to arrive there.

Workshop on stochastic modeling in biology: relevant mathematical concepts and recent applications : Heidelberg, Federal Republic of Germany 8–12 August 1988

Workshop on stochastic modeling in biology: relevant mathematical concepts and recent applications : Heidelberg, Federal Republic of Germany 8–12 August 1988

Reading up books: Discrete mathematics for computer science: Studying this branch of mathematics can provide you with a solid background for computer theory and applications

The importance of mathematics to the study of both electrical engineering and computer science and engineering is well understood by educators and students alike. The topics learned in mathematics courses are often reviewed in engineering courses that partly focus on relevant mathematical concepts.

Topological Dynamics in Neurobiology

A mathematical model consisting of three interrelated parts is proposed. The first part is the space of representation in terms of physico-chemical elements, the second the static topologies imposed on this space, and the third the dynamics of the representation space endowed with topological properties. In the present study various relevant mathematical concepts are introduced for each of the three parts. The classification of generic elements for the representation spaces, the group representation of the pathways in terms of braid groups and an algebra for the n-dimensional spatial and m-dimensional temporal space of the dynamic system for the neural activities form the basic ideas of the model. Some experimental results are interpreted as illustrative examples of the formal ideas.