Nature Of Mathematics Research Articles

Research on mathematical beliefs: systematic literature review

This study uses the systematic literature review (SLR) method to identify knowledge trends in beliefs about the nature of mathematics, learning mathematics and identify new development opportunities for further research. The Scopus database was used, and 41 documents were found as the primary source for further analysis. The results show that mathematical beliefs have become a current research trend. The number of publications increased significantly starting in 1989, accompanied by many citations. Based on these primary documents, it is known that the development of mathematical beliefs research topics. The foundation of this topic begins with studying the topic of knowledge, activities, and learning mathematics. Then it is developed into other, more varied studies, including teacher’s beliefs transition; mathematical beliefs comparison; mathematics teacher’s belief, teaching, and learning; mathematical beliefs and problem-solving; inquiry-based learning (IBL); affective factors; technology utilization; common misconceptions; and development of mathematics teacher’s beliefs instrument. This research investigates the core of scientific work to provide information to researchers and institutions as material for consideration of research to be carried out.

Discovering causal models for structural, construction and defense-related engineering phenomena

Causality, the science of cause and effect, has made it possible to create a new family of models. Such models are often referred to as causal models. Unlike those of mathematical, numerical, empirical, or machine learning (ML) nature, causal models hope to tie the cause(s) to the effect(s) pertaining to a phenomenon (i.e., data generating process) through causal principles. This paper presents one of the first works at creating causal models in the area of structural and construction engineering. To this end, this paper starts with a brief review of the principles of causality and then adopts four causal discovery algorithms, namely, PC (Peter-Clark), FCI (fast causal inference), GES (greedy equivalence search), and GRaSP (greedy relaxation of the sparsest permutation), have been used to examine four phenomena, including predicting the load-bearing capacity of axially loaded members, fire resistance of structural members, shear strength of beams, and resistance of walls against impulsive (blast) loading. Findings from this study reveal the possibility and merit of discovering complete and partial causal models. Finally, this study also proposes two simple metrics that can help assess the performance of causal discovery algorithms.

Modified Barnacles Mating Optimizing Algorithm for the Inversion of Self-potential Anomalies Due to Ore Deposits

The self-potential method (SP) has been used extensively to reveal some model parameters of various ore deposits. However, estimating these parameters can be challenging due to the mathematical nature of the inversion process. To address this issue, we propose here a novel global optimizer called the Modified Barnacles Mating Optimizer (MBMO). We improved upon the original approach by incorporating a variable genital length strategy, a novel barnacle offspring evolving method, and an out-of-bounds correction approach. The MBMO has not been previously applied to geophysical anomalies. Prior to inversion of real data sets, modal and sensitivity Analyzes were conducted using a theoretical model with multiple sources. The Analyzes revealed that the problem is modal in nature, model parameters have varying levels of sensitivity, and an algorithm that can well balance global exploration with local exploitation is required to solve this problem. The MBMO was tested on theoretical SP anomalies and four real datasets from Türkiye, Canada, India, and Germany. Its performance was compared to the original version under equal conditions. Uncertainty determination studies were carried out to comprehend the reliability of the solutions obtained via both algorithms. The findings indicated clearly that the MBMO outperformed its original version in estimating the model parameters from SP anomalies. The modifications presented here improved its ability to search for the global minimum effectively. In addition to geophysical datasets, experiments with 11 challenging benchmark functions demonstrated the advantages of MBMO in optimization problems. Theoretical and field data applications showed that the proposed algorithm can be used effectively in model parameter estimations from SP anomalies of ore deposits with the help of total gradient anomalies.

Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review

Hyperbolic systems with stiff relaxation constitute a wide class of evolutionary partial differential equations which describe several physical phenomena, ranging from gas dynamics to kinetic theory, from semiconductor modeling to traffic flow. Peculiar features of such systems is the presence of a small parameter that determines the smallest time scale of the system. As such parameter vanishes, the system relaxes to a different one with a smaller number of equations, and sometime of a different mathematical nature. The numerical solution of such systems may present some challenges, in particular if one is interested in capturing all regimes with the same numerical method, including the one in which the small parameter vanishes (relaxed system). The design, analysis and application of numerical schemes which are robust enough to solve this class of systems for arbitrary value of the small parameter is the subject of the current paper. We start presenting different classes of hyperbolic systems with relaxation, illustrate the properties of implicit–explicit (IMEX) Runge–Kutta schemes which are adopted for the construction of efficient methods for the numerical solution of the systems, and then illustrate how to apply IMEX schemes for the construction of asymptotic preserving schemes, i.e. scheme which correctly capture the behavior of the systems even when the relaxation parameter vanishes.

Empowering Students through a Self-Directed Learning Pedagogy in Mathematics Education

Mathematics is important subject in our daily life. It is taught in school level all over the world. Nowadays students are indifferent towards learning mathematics. Those indifferences are related to learning strategy in mathematics teaching. The main purpose of this study is introduce Self-Directed Learning (SDL) pedagogy in the context of mathematics education. The research employed a qualitative document analysis methodology, synthesizing information from diverse sources to explore a self-directed learning pedagogy through the development of conceptual and theoretical categories. We have prepared different theme related to self-directed learning such as concept of self-directed learning, nature of self-directed learning, mathematics and self-directed learning, role of SDL learners and teacher, philosophical underpinning of SDL, SDL models in practice, self-directed learning and its strategy. From those themes, we concluded that SDL is a teaching and learning pedagogy that encourages students to utilize their imaginative and intuitive abilities, fostering mindfulness, constructiveness, critical thinking, independence, cooperativeness, confidentiality, and autonomy, and promoting self-motivation, self-management, and self-monitoring. Moreover, Self-Directed Learning (SDL) is a self-directed pedagogy that encourages students to take charge of their education, focusing on engaging, creative, and independent learning methods. Under those themes self-directed emphasizes in fostering autonomy, creativity, and active engagement, while acknowledging the need for more specific models and stages for effective implementation in school mathematics education.

Mathematical definition of the fine-structure constant: A clue for fundamental couplings in astrophysics

Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value ∼1/137 of the fine-structure constant α = e2/ℏc vary with astronomical time?) are a very active area of observational research. Using a specific α-free non-relativistic and nonlinear isotropic quantum model compatible with its quantum electrodynamics (QED) counterpart yields the 99% accurate solution α = 7.364 × 10−3 vs its experimental value 7.297 × 10−3. The ∼1% error is due to the deliberate use of mean-field Hartree approximation involving lowest-order QED in the calculations. The present theory has been checked by changing the geometry of the model. Moreover, it fits the mathematical solution of the original nonlinear integro-differential Hartree system by use of a rapidly convergent series of nonlinear eigenstates [G. Reinisch, Phys. Lett. A 498, 129347 (2024)]. These results strongly suggest the mathematical transcendental nature—e.g., like for π or e—of α’s numerical value of ∼1/137 and, hence, its astrophysical as well as its cosmological stability.

Do teachers’ beliefs about the nature and learning of mathematics affect students’ motivation and enjoyment of mathematics? Examining differences between boys and girls across six countries

Mathematics teachers’ beliefs are central to mathematics teaching and student learning. Because different aspects of motivation and affect—particularly enjoyment—primarily develop within the classroom context, examining how different teachers’ beliefs may affect student outcomes in mathematics is imperative. The current study examines teachers’ beliefs about the nature and learning of mathematics in connection to students’ motivation (i.e. intrinsic value, utility value and perceived competence) and enjoyment of mathematics across different settings by considering students’ mathematics achievement, gender and classroom composition (i.e. socioeconomic and behavioural). Data were collected from 3rd- and 4th-grade mathematics teachers (N = 686) and their students (N = 11,782) in six countries (i.e. Norway, Finland, Sweden, Portugal, Estonia and Serbia). A two-level structural equation modelling technique (TSEM) (i.e. student level and classroom level) with random slopes was employed to address our research questions. The results indicate that students’ intrinsic value and perceived competence positively relate to their enjoyment of mathematics in all six countries. Teachers’ beliefs about the nature and learning of mathematics moderate the within-classroom relationship between boys and girls and the motivation and enjoyment of learning mathematics in Portugal and Norway. Unlike boys, girls consistently perceive themselves as less competent in mastering mathematics, even in primary school. Classroom socioeconomic composition had a more pronounced influence on teachers’ beliefs in Sweden, Norway and Serbia. In relation to teachers’ beliefs, classroom behavioural composition was relevant in Estonia and Sweden. In Finland and Norway, classroom composition was essential to boys’ and girls’ differential motivation and enjoyment of mathematics learning.

Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei

There is a surge of research devoted to the formalism and physical manifestations of non-Lorentzian kinematical symmetries, which focuses especially on the ones associated with the Galilei and Carroll relativistic limits (the speed of light taken to infinity or to zero, respectively). The investigations have also been extended to quantum deformations of the Carrollian and Galilean symmetries, in the sense of (quantum) Hopf algebras. The case of 2+1 dimensions is particularly worth to study due to both the mathematical nature of the corresponding (classical) theory of gravity, and the recently finalized classification of all quantum-deformed algebras of spacetime isometries. Consequently, the list of all quantum deformations of (anti-)de Sitter-Carroll algebra is immediately provided by its well-known isomorphism with either Poincaré or Euclidean algebra. Quantum contractions from the (anti-)de Sitter to (anti-)de Sitter-Carroll classification allow to almost completely recover the latter. One may therefore conjecture that the analogous contractions from the (anti-)de Sitter to (anti-)de Sitter-Galilei r-matrices provide (almost) all coboundary deformations of (anti-)de Sitter-Galilei algebra. This scheme is complemented by deriving (Carrollian and Galilean) quantum contractions of deformations of Poincaré algebra, leading to coboundary deformations of Carroll and Galilei algebras.

Assessment of In Situ Compactness and Air Void Content of New Asphalt Layers Using Ground-Penetrating Radar Measurements

This paper deals with the possibilities of ground-penetrating radar (GPR)-based quality control testing, which was demonstrated on an experimental road section of a ~220–240 m long Hungarian residential street. The measurements and their assessment aimed to control the prescribed compactness and air void content of newly built asphalt layers. Research has discussed the relationship between the air void content and the dielectric constant of asphalt layers, and provided empirical results for this relationship. We suggest a new logistic model with lower and upper asymptotes instead of the exponential formula often used in the literature. Contrary to this newly developed robust model, existing models are sensitive to extreme dielectric constant values due to the mathematical nature of their exponential function. The results of the new logistic model are compared to those of the Hoegh–Dai (HD) and Minnesota Department of Transportation (MnDOT) models on the basis of a few calibration data points. Through systematic data collection and analysis, the developed robust empirical model demonstrates a significant correlation between the relative permittivity and air void content in asphalt mixes, enabling accurate estimation of the air void content within a ±0.5% margin of error. The air void content can be applied to estimate the asphalt layer modulus. The developed model can be further exploited by utilizing a combination of GPR and drone technology. The “symbiosis” of these technologies can lead to a totally non-destructive imaging system, which can then be applied to environmental monitoring of roads and their surroundings in terms of quality control of asphalt compaction work and the hot asphalt mix behind the compaction roller during pavement construction.

The myth of being modern: Digital machines and the loss of discovery

Abstract‘Digitization’ is a logical operation that deconstructs information and transforms it into digits, rendering it a logical construct. Digital operations are fast, infinitely replicable, objective, and absolute. But digitization is not without costs: the result of operationalizing information is that it becomes abstract—disconnected from its referent, and subject to processes that alter its representation without detection. Visions of modernity that draw on the potential for digital technology to invariably raise standards of living fail to consider the intrinsic properties of the digital that tend toward replicability, speed, and scalability, which favor globalization. This article argues how, through reductive processes, the discrete, mathematical nature of the digital provides a framework for rationality and order that is fundamentally incompatible with multiple modernities. Moreover, the history of digital machines is intricately linked to education and epistemology across North America and South Africa. The mechanization of learning illustrates the promise, power, and potential consequence of digital operations; by limiting our horizons, the Modern and now the Digital, limits our ability to think, thereby concealing the destruction of society, nature, and us as a species.

Mathematical beliefs: What topics appear, and how has this changed over time?

This research uses the Systematic Mapping Study (SMS) method to identify knowledge trends in beliefs about the nature of mathematics, learning, and learning mathematics and identify new development opportunities for further research. The Systematic Mapping Study (SMS) method is the initial stage for selecting primary sources in a literature study. The Scopus database was used for SMS, then 171 articles were found ready for further analysis. The results show that mathematical belief has become a current research trend. The keywords are "beliefs." The new trending keyword is "pre-service teachers." The thematic map shows thirteen clusters, where the motivation, pre-service mathematics teachers, mathematics education, beliefs, and attitudes clusters are the motor themes currently a global challenge. This means there are still great opportunities to research this theme to complete the knowledge puzzle. In closing, 41 articles are primary articles related to this theme. Three of them, namely articles from Stipek et al. (2001), Beswick (2012), and Felbrich et al. (2012), can be seen as primary references.

Исследование парадоксов в развитии математического мышления: культурно-историческая перспектива

<p>Primary school teachers are the educators responsible for introducing children into the world of mathematics. In the process of learning mathematics, children may experience negative emotions and develop negative attitudes or mathematics anxiety. Some authors approach these issues by focusing on primary teachers’ mathematics anxiety or proficiency in mathematics. In this article, we explore whether and how certain social and educational practices shape or not the development of mathematical thinking in prospective primary teachers, from a cultural-historical perspective. The study’s sample consists of twelve prospective primary teachers in their last year of studies. The participants were interviewed about their experience with mathematics throughout their educational years. By examining and contrasting the different developmental trajectories of the participants, we identified their “imprisonment” in situational thinking and some of the paradoxes that characterise their development. Adopting a historical perspective in analysing participants’ trajectories was crucial in uncovering the paradoxical nature of the development of mathematical thinking. We conclude by highlighting the socially mediated nature of mathematics and the obstacles posed by the competitive and exam-driven nature of educational practices</p>

Aesthetical entanglements in mathematics education

In this study, we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière's approach to aesthetics and politics. We call “Silencing Aesthetics” a dominant aesthetic that Rota has characterized as a “copout (…) intended to keep our formal description of mathematics as close as possible to the description of a mechanism”. The challenge this study attempts to explore is how to question silencing aesthetics to make space for inclusive ones. Our efforts have focused on setting up and studying inclusive and pluralist “Studios”, gathering craftworkers, anthropologists, mathematics educators, and mathematics enthusiasts. We include here a case study based on a conversation amongst basket weavers, anthropologists, and mathematics educators focused on the artisanal and mathematical nature of knots. We discuss the implications of aesthetical entanglements, such as those in our case study, for mathematics learning.

Opportunities to develop student’s math-related agency in primary education: the role of teacher beliefs

Teacher beliefs and practices related to them could provide varying opportunities for students to develop their agency related to learning math. The aim of the current paper is to explore how different math teacher beliefs (concerning the nature of mathematics, constructivist beliefs, and self-efficacy) relate to different dimensions of student agency in primary education. A two-level multilevel model was analyzed to find out to what extent selected teacher beliefs are associated with student math competence and agency dimensions. The study was conducted in Estonia and data from 3rd and 4th grade students (N = 1557) and their teachers (N = 121) was included in the analysis. The results showed that teacher self-efficacy beliefs, constructivist beliefs, and static views on the nature of mathematics were related to students’ agency dimensions. Possible explanations and directions for further studies are discussed.

Modeling the structure and thermodynamics of multicomponent and polydisperse hard-sphere dispersions with continuous potentials.

A model system of identical particles interacting via a hard-sphere potential is essential in condensed matter physics; it helps to understand in and out of equilibrium phenomena in complex fluids, such as colloidal dispersions. Yet, most of the fixed time-step algorithms to study the transport properties of those systems have drawbacks due to the mathematical nature of the interparticle potential. Because of this, mapping a hard-sphere potential onto a soft potential has been recently proposed [Báez et al., J. Chem. Phys. 149, 164907 (2018)]. More specifically, using the second virial coefficient criterion, one can set a route to estimate the parameters of the soft potential that accurately reproduces the thermodynamic properties of a monocomponent hard-sphere system. However, real colloidal dispersions are multicomponent or polydisperse, making it important to find an efficient way to extend the potential model for dealing with such kind of many-body systems. In this paper, we report on the extension and applicability of the second virial coefficient criterion to build a description that correctly captures the phenomenology of both multicomponent and polydisperse hard-sphere dispersions. To assess the accuracy of the continuous potentials, we compare the structure of soft polydisperse systems with their hard-core counterpart. We also contrast the structural and thermodynamic properties of soft binary mixtures with those obtained through mean-field approximations and the Ornstein-Zernike equation for the two-component hard-sphere dispersion.

Student Academic Success in Linear Algebra in an Open Distance Learning Environment

Academic success in first year university mathematics in been problematic for decades and mathematics educators keep looking for the causes. What has been universally agreed, is that the theoretical and abstract nature of mathematics plays a role. A module on Linear Algebra at an Open Distance eLearning (ODeL) institution, was identified for investigation due to very high dropout and failure rates. This article concentrates on identifying the types of knowledge (e.g., procedural, conceptual, strategic, schematic and declarative) necessary for academic success in the subject. Using the literature, a conceptual framework is developed to classify students’ answers into the various types of knowledge. The research question asks what types of knowledge contributes to academic success in Linear Algebra. Script analysis is used to answer the research question. The results showed that lack of the necessary declarative knowledge which forms the basis for the other forms of knowledge as well as procedural knowledge were the main causes of the resulting misconceptions and errors. It was established that students were more engaged in surface learning rather than deep learning that results in conceptual understanding and acquisition of conceptual knowledge.

Addressing discontinuous root-finding for subsequent differentiability in machine learning, inverse problems, and control

There are many physical processes that have inherent discontinuities in their mathematical formulations. This paper is motivated by the specific case of collisions between two rigid or deformable bodies and the intrinsic nature of that discontinuity. The impulse response to a collision is discontinuous with the lack of any response when no collision occurs, which causes difficulties for numerical approaches that require differentiability which are typical in machine learning, inverse problems, and control. We theoretically and numerically demonstrate that the derivative of the collision time with respect to the parameters becomes infinite as one approaches the barrier separating colliding from not colliding, and use lifting to complexify the solution space so that solutions on the other side of the barrier are directly attainable as precise values. Subsequently, we mollify the barrier posed by the unbounded derivatives, so that one can tunnel back and forth in a smooth and reliable fashion facilitating the use of standard numerical approaches. Moreover, we illustrate that standard approaches fail in numerous ways mostly due to a lack of understanding of the mathematical nature of the problem (e.g. typical backpropagation utilizes many rules of differentiation, but ignores L'Hopital's rule).

Exploring the multifaceted roles of mathematics learning in predicting students' computational thinking competency

BackgroundThere exist shared competencies between computational thinking (CT) and mathematics, and these two domains also mutually benefit from various teaching approaches. However, the linkages between mathematics and computational thinking lack robust empirical support, particularly from student-centered learning perspectives. Our study aimed to enhance our understanding of the connections between students' mathematics learning and computational thinking. To assess students' mathematics learning, we measured their beliefs about mathematics learning and their level of mathematical literacy (ML). Our hypothesis posited that students' beliefs concerning mathematics learning, encompassing their views on the nature of mathematics and their attitude towards the subject, can both directly and indirectly influence their CT, with ML serving as a mediating factor. Our data were gathered through surveys and tests administered to eighth- and ninth-grade students. Data were analyzed using partial least squares–structural equation modeling (PLS–SEM).ResultsThe evaluation of the measurement model indicated strong internal consistency for each construct. Both convergent and discriminant validity were also established. Upon assessing the structural model, it was found that beliefs about the nature of mathematics positively predicted attitudes towards mathematics, and this belief also indirectly predicted ML through positive attitudes towards mathematics. In addition, ML directly and positively predicted both CT subscales. Notably, a comprehensive mediating effect of ML on beliefs about mathematics learning and CT was identified in the analysis.ConclusionsThis study advances the understanding of the relationships between mathematics learning and CT. We have further confirmed the importance of mathematical literacy in predicting CT and its mediating role between beliefs about mathematics learning and CT. It is suggested that teachers could promote students’ CT competence by enhancing their mathematical literacy or integrating mathematics and CT into the same learning activities. Finally, we propose that upcoming investigations treat CT assessments as formative constructs, diverging from their reflective counterparts.

Generalized viscoelastic flow with thermal radiations and chemical reactions

Background:A generalized model of mathematical nature is considered to address the viscoelastic flow problem using fractional derivatives. Control/freedom of the flow mechanism is achieved with these derivatives. In simulations of industrial interest, more variations are available with fractional derivatives when compared with ordinary derivatives. Relaxation times are incorporated to handle the abrupt changes in the flow domain. Fluid flow is carried out under the influence of thermal radiations and when a heat source or sink is present. Chemical reactions of the first order are observed in the mathematical modeling of the flow. Methods:Flow is induced with the movement of the lower surface while applying force on the x-plane. Simulations of the governing mathematical problem are computed with the combination of finite element and finite difference algorithms. Significant Findings:It is noted that velocity, temperature, and concentration change with the variation of fractional order derivatives which was not possible with the classical derivatives. Moreover, with greater relaxation times, velocity, temperature, and concentration remained at a lower level. The modeled mechanism can be considered to avoid costly trials in chemical and polymer casting industries.

The Effects of GeoGebra-Assisted Transformation Geometry Instruction on Student Achievement, Attitudes, and Beliefs

This study aims to investigate the effects of using GeoGebra, a popular dynamic geometry software, on students’ mathematics achievement, attitudes toward geometry, and beliefs about mathematics and its teaching in transformation geometry teaching. The study was a quasi-experimental design including experimental and control groups. The sample of the study was 34 7th-grade students from a public middle school in a city in Turkey. The study was conducted in the 2015-2016 academic year, lasting ten lesson hours (three weeks). The data were collected through achievement test and surveys regarding attitudes toward geometry, beliefs about the nature of mathematics, and beliefs about the teaching of mathematics. The quantitative data analyses were carried out by using the Mann-Whitney U test. Results revealed that, although the use of GeoGebra in teaching transformation geometry demonstrated student improvement in achievement, attitudes toward geometry, and beliefs about