Mathematical Objects Research Articles

The laboratory as a pedagogical resource for teaching mathematics: the state of knowledge in ENEM communications

The development of mathematical learning begins through the relationship between the subject and the mathematical object with the use of concrete materials. In view of this, it is believed that the Mathematics Teaching Laboratory (LEM) is an environment that can promote mathematical learning through this relationship. Therefore, this qualitative research sought to understand how LEM can collaborate in the teaching and learning process of Basic Education students. To this end, a survey of articles published in editions of the Annals of ENEM’s (1987 – 2022) that had LEM as its theme was carried out; Subsequently, the texts were read in full, in which ten scientific communications were selected and constituted the object of analysis of this investigation. As main results, we highlight that LEM encourages students to develop activities using concrete and manipulable materials that allow them to review, learn about and deepen characteristics of objects of knowledge related to mathematics, developing interest in the study through the relationship that can be established with the social context to which this subject belongs.

An intuitionistic interpretation of Bishop’s philosophy

Abstract The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop’s rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as someone who maintains that mathematics is a mental creation, mathematics is meaningful and eludes formalization, mathematical objects are mind-dependent constructions given in intuition, and mathematical truths are experienceable. This paper develops and defends an intuitionistic interpretation of Bishop’s philosophical views.

Designed activities on developing mental strategies using alms value-based learning trajectory in elementary school

Mental strategies are recognized as an effective approach for solving arithmetic operations such as addition and subtraction. This study focuses on developing and implementing instructional activities aimed at enhancing students' mental strategy skills through the Indonesian Realistic Mathematics Education (IRME) approach. The research employed a design-based methodology involving 2nd-grade students from an Islamic-integrated school in Surabaya. The study followed a research design that emphasizes the development of local instructional theory, encompassing three key phases: the preparatory phase (initial design development), the experimental phase (classroom teaching experiments), and the retrospective analysis phase. Data collection methods included documentation, field notes, and interviews. In the first trial, three instructional activities were conducted. Based on the analysis of the first trial, the number of activities in the second trial was increased to four: "Let's Get Used to Alms Every Day," "Don't Forget Routine Alms This Week," "Set Your Own Pocket Money," and "Let's Give Away Your Alms to Break Mr. Samad’s Fast." These activities were designed to achieve specific mathematical objectives, enabling students to understand and solve addition and subtraction problems using strategies such as skip counting, borrowing, and saving. The retrospective analysis concluded that students successfully learned to perform addition and subtraction operations on both monetary values and ordinary numbers within the designed learning trajectory.

The [formula omitted]-Adic Schrödinger equation and the two-slit experiment in quantum mechanics

p-Adic quantum mechanics is constructed from the Dirac–von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space, QpN. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in QpN. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength p−1. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by ‘quantum waves’. In the p-adic framework, in the double-slit experiment, each particle goes through one slit only. The p-adic quantum mechanics is an analog (or model) of the standard one under the hypothesis of the existence of a Planck length. The precise connection between these two theories is an open problem. Finally, we propose the conjecture that the classical de Broglie wave-particle duality is a manifestation of the discreteness of space–time.

Mini-Workshop: Poisson and Poisson-type algebras

The first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov–Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects.

About mathematical modeling in limnology, oceanology, ecology and economics in the works of V.V. Menchutkin

The article gives a brief review of the works of Doctor of Biological Sciences, Prof. V.V. Menshutkin for the 65-year period of scientific activity, which began in the 60s of the XX century with experimental works on the hydrophisics of Lake Baikal and the first mathematical models of the perch population of the small lake Kherya-Yarvi (Karelian Isthmus) and fish community in Lake Dalneye (Kamchatka). The latter was of great practical importance when concluding an agreement with Japan on catch quotas for sockeye salmon.V.V. Menshutkin’s range of interests was very wide and included mathematical modeling in biology, physiology, limnology, oceanology, demography, economics, ecology and rational nature management. The works on research of water objects and processes in them include modeling of dynamics of fish and aquatic invertebrate populations, Antarctic krill. He created mathematical models of ecosystems of Ladoga and Onega lakes, and during oceanic expeditions of the Academy of Sciences — of the Sea of Japan and the Peruvian upwelling area.His research in the field of physiology of blood circulation and water-salt metabolism, works related to space subjects, modeling of diagnostics and treatment of mental diseases are interesting. V.V. Menshutkin paid special attention to the issues of biological evolution. V.V. Menshutkin paid special attention to the issues of biological evolution and modeling of evolutionary process. The mathematical apparatus used by V.V. Menshutkin is very broad. The mathematical apparatus used by V.V. Menshutkin is very wide: from systems of differential equations to finite automata, neural networks, fuzzy logic and cognitive modeling (a new direction in artificial intelligence), with which the studies of ecological-socio-economic systems of water bodies catchments were connected.O.P. Savchuk and V.V. Menshutkin united research methods — mathematical modeling and objects under study: oceans, seas and lakes. However, their only joint work was the collective monograph on the results of “Nevskaya Guba” project, which conclusions became the basis for the important decision to complete the construction of the dam in St. Petersburg.

A novel application of artificial intelligence technology for outdoor high-voltage composite insulator

The distribution of the electric field plays a crucial role in evaluating the effectiveness of composite insulators. This research presents an innovative approach for optimizing the design of corona rings, aiming to decrease the electric field intensity in critical areas to satisfactory levels. The investigation examined the correlation among corona ring design variables and the intensity of the electric field close to the energized-end fitting in a 220 kV composite insulator equipped with a corona ring. Using design of experiment methods, the impact of three corona ring design parameters – corona ring diameter (R), ring tube diameter (Dr), and corona ring height (H) – was assessed. A novel nonlinear mathematical objective function was formulated, connecting the electric field magnitude to the structural parameters of the corona ring. Subsequently, the Bat algorithm, a bio-inspired optimization technique, was employed to enhance the initial corona ring design and mitigate the electric field intensity. Finally, the Finite Element Method (FEM) was utilized to simulate and evaluate the voltage distribution and electric field stress. The optimization process resulted in a substantial decrease in the highest electric field magnitude, by 58.6 % compared to the insulator with the threshold value, and 75 % when devoid of a corona ring. Additionally, the research highlighted the significant influence of ring tube thickness on the distribution of the electric field. The combination of experimental design and the Bat algorithm proved to be a powerful tool for optimizing the design of corona rings on transmission line composite insulators, providing precise solutions for optimizing corona discharge problems and enhancing the reliability of power transmission systems.

Ray Tracing Harmonic Functions

Sphere tracing is a fast and high-quality method for visualizing surfaces encoded by signed distance functions (SDFs). We introduce a similar method for a completely different class of surfaces encoded by harmonic functions , opening up rich new possibilities for visual computing. Our starting point is similar in spirit to sphere tracing: using conservative Harnack bounds on the growth of harmonic functions, we develop a Harnack tracing algorithm for visualizing level sets of harmonic functions, including those that are angle-valued and exhibit singularities. The method takes much larger steps than naïve ray marching, avoids numerical issues common to generic root finding methods and, like sphere tracing, needs only perform pointwise evaluation of the function at each step. For many use cases, the method is fast enough to run real time in a shader program. We use it to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction. We also use it to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, mesh "exoskeletons", and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces. Finally we show that, at least in theory, Harnack tracing provides an alternative mechanism for visualizing arbitrary implicit surfaces.

GAME EDUKASI AUGMENTED REALITY BERBASIS RME (REALISTIC MATHEMATICS EDUCATION) UNTUK MENINGKATKAN COMPUTATIONAL THINKING SISWA

This research addresses the common problem in mathematics education in Indonesia, which is the students' difficulties, as reflected in Indonesia's declining ranking in the Program for International Student Assessment (PISA). The research aims to create an innovative Augmented Reality (AR) Educational Game based on the Realistic Mathematics Education (RME) approach to improve students' understanding of the spatial geometry mathematics subject at the elementary school level. AR technology allows students to interact with mathematical objects in real-world contexts using Android devices. This game is designed to make mathematics more engaging and relevant to elementary school students. The research was conducted at Purwantoro 1 Public Elementary School in Malang City over four months, following the Game Development Lifecycle (GDLC). The test results showed an n-gain improvement of 0.71, indicating a positive impact of the game's development on students' learning outcomes.

The MTSK model as a tool for designing tasks for teacher education

This paper studies the knowledge developed by a group of sixty-two pre-service primary teachers during a training session on the definition of a polygon and the mathematical practice of defining. We explore the knowledge developed by pre-service teachers when they carry out a series of training tasks oriented around the practice of defining with primary pupils, within the theoretical framework of the Mathematics Teachers’ Specialised Knowledge model, and using videos of an authentic lesson as prompt. Data were collected by means of video recordings of the implementation of the training tasks, in conjunction with the pre-service teachers’ own written observations as they watched the recorded lesson. A content analysis was carried out on the collected data using the Mathematics Teachers’ Specialised Knowledge model. The results highlight the way in which knowledge of the mathematical practice of defining is constructed, along with the mathematical objects involved in this practice, and pedagogical aspects of defining with young pupils. They also illustrate how a sequence of tasks oriented around a model of teachers’ knowledge can contribute to the development of this knowledge in initial training programmes, and more especially provide orientation about the training of pre-service teachers in the mathematical practice of defining.

New kinds of embodied interactions that arise in augmented reality dynamic geometry software

Dynamic geometry software (DGS) has long been studied in mathematics education as a way for students to explore and interact with geometric objects and figures. Recent advances in Augmented Reality (AR) technologies that allow dynamic three-dimensional mathematical objects to appear in students’ environment as holograms have changed the nature of what is possible for a DGS, particularly with respect to embodiment. New forms of embodied interactions may arise in AR-based DGS, as students gesture and move their bodies through their environment, taking different perspectives to interact with these immersive shapes projected in three dimensions. In the present study, we examine videos of 28 high school students interacting with an AR-based version of the DGS GeoGebra, while wearing the Microsoft HoloLens 2 headsets. We document the novel kinds of embodied interactions that the AR environment affords, relating to (1) perspective and orientation, (2) scale, (3) three dimensions. Based on our analysis, we give important directions for future research on DGS and implications for the design of the next generation of holographic DGS.

Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as the uniform ball property, which ensures \mathscr{C}^{1,1} regularity of the hypersurface. In this paper, we revisit and generalize the results of Dalphin (2018 and 2020) and Guo and Yang (2013). We provide a simpler framework and more concise proofs of some of the results contained in these references and extend them to a new class of problems involving PDEs. Indeed, by using the signed distance, we avoid the intensive and technical use of local maps, as was the case in the above references. Our approach, originally developed to solve an existence problem in Privat, Robin, and Sigalotti’s 2022 paper, can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.

Semiotic transformations to analyze interpretation and representation problem solving strategie

Introduction: The theory of registration of semiotic representations establishes that the use of sign systems, which allow the cognitive activities of formation, conversion and treatment, is essential to have access to mathematical objects or processes. Objective: This research aims to analyze the strategies used by eleventh-grade students to solve problems of interpretation and representation through treatment and conversion activities. Methodology: A questionnaire was selected with questions of the type Saber tests, in which the competence of interpretation and representation is evaluated. Also, to show what each student does and says when answering the questions, video recordings and transcriptions are made. Results: An a priori analysis is proposed per question, to analyze and interpret the cognitive treatment and conversion activities used by thestudents, as well as an a posteriori analysis where the results obtained are analyzed and the hypotheses are verified. Conclusions: It shows how students associate the significant units of a problem posed in a representation register to another register, to find either a contrast of the information or a relationship in the information, , taking into account the semantic correspondence and semantic univocity.

Studi Literatur Review : Geogebra Sebagai Media Belajar Untuk Mengetahui Kemampuan Komputasi Peserta Didik

Geogebra is a computer software that is useful for visualizing objects in mathematics effectively and efficiently. This research aims to look at Geogebra as a learning medium to determine students' computing abilities. This research takes the type of literature study research by identifying and evaluating the results of previous research. In this research, 3 previous journals were identified. From the results of this research, it can be seen that Geogebra can improve students' computational thinking abilities

Weight Vector Definition for MOEA/D-Based Algorithms Using Augmented Covering Arrays for Many-Objective Optimization

Many-objective optimization problems are today ever more common. The decomposition-based approach stands out among the evolutionary algorithms used for their solution, with MOEA/D and its variations playing significant roles. MOEA/D variations seek to improve weight vector definition, improve the dynamic adjustment of weight vectors during the evolution process, improve the evolutionary operators, use alternative decomposition methods, and hybridize with other metaheuristics, among others. Although an essential topic for the success of MOEA/D depends on how well the weight vectors are defined when decomposing the problem, not as much research has been performed on this topic as on the others. This paper proposes using a new mathematical object called augmented covering arrays (ACAs) that enable a better sampling of interactions of M objectives using the least number of weight vectors based on an interaction level (strength), defined a priori by the user. The proposed method obtains better results, measured in inverted generational distance, using small to medium populations (up to 850 solutions) of 30 to 100 objectives over DTLZ and WFG problems against the traditional weight vector definition used by MOEA/D-DE and results obtained by NSGA-III. Other MOEA/D variations can include the proposed approach and thus improve their results.

Towards a Dynabook for verified VM construction

We present Smalltalk-25, a scientific Programm of research towards the synthesis of Smalltalk VMs by formal methods. We approach this through a Dynabook-style implementation of Hilbert’s “manipulating proofs as mathematical objects”. We begin by implementing an automated-reasoning engine based on refinement types. Having that foundation laid, we build a program logics for a simple “model” language; a framework for reasoning about a real intermediate language by translating into the model language; an ahead-of-time bootstrapper; and a framework for debugging thus-constructed VMs.Our Programm is an ongoing research effort; this paper provides a report of its current status. Every part of our system is implemented in Smalltalk and available from GitHub under the MIT license.

Layers of Mathematical Understanding and Folding Back of Prospective Teachers on Arithmetic Sequence and Series

Mathematical understanding is one of the objectives of learning mathematics, so students need to master it. The success of the process of growing students' mathematical understanding is inseparable from the role of the teacher. Therefore, student teacher candidates must have a complete mathematical understanding structure. This is what prompted him to carry out research to describe the layers of mathematical understanding and folding back prospective teacher students on the material of arithmetic sequences and series. The researcher took 18 sixth-semester students of S1 Mathematics Education at the State University of Malang. A purposive sampling technique was used to determine the participants. So, one subject was chosen in each category of mathematical understanding. Researchers used a descriptive approach with three stages of research procedures including planning, implementing, and drawing conclusions. The instruments used consisted of test questions and interview guidelines. Based on analysis carried out using indicators for each layer of mathematical understanding, information was obtained that subjects with high abilities were able to achieve layering arrangement and perform effective folding back independently. Subjects with moderate abilities were able to reach the observation layer and fold back effectively but needed help. Meanwhile, subjects with low abilities were able to reach the formalization layer but had not been able to fold back effectively. It can be concluded that subjects with better mathematical understanding have a more complete layered structure of mathematical understanding and can fold back independently and effectively.

MATEMATIKA DAN PEPUTIQ CINA SEBAGAI ADAT PERNIKAHAN SUKU MANDAR

Culture can determine how students view things, including understanding mathematical material. Therefore, there is a need for a learning approach that is able to connect mathematics with culture, called Ethnomathematics. The concept of ethnomathematics makes a major contribution to improving mathematics learning. Ethnomathematics objects are cultural objects that contain mathematical concepts in a particular society, one of which is Peputiq China. One of the mathematical concepts contained in Peputiq Cina is the concept of geometry. This research aims to determine the various geometric concepts found in Chinese Peputiq. The method in this research uses qualitative research with an ethnographic research design. The data sources in this research use data from observation, documentation and interviews. This research aims to identify and describe the mathematical objects contained in the Chinese peputiq as one of the Mandar tribe's wedding customs. This exploratory research using an ethnographic approach is to explore the concept of geometric shapes in the creation of the Chinese peputiq frame. Through exploration, documentation, observation and interviews, geometric shapes can be found, both flat shapes and spatial shapes, including squares, rectangles, triangles, while geometric shapes include upright prisms and blocks.

A Spearman dependence matrix for multivariate functional data

We propose a nonparametric inferential framework for quantifying dependence among two families of multivariate functional data. We generalise the notion of Spearman correlation coefficient to situations where the observations are curves generated by a stochastic processes. In particular, several properties of the Spearman index are illustrated emphasising the importance of having a consistent estimator of the index. We use the notion of Spearman index to define the Spearman matrix, a mathematical object expressing the pattern of dependence among the components of a multivariate functional dataset. Finally, the notion of Spearman matrix is exploited to analyse two different populations of multivariate curves (specifically, Electrocardiographic signals of healthy and unhealthy people), in order to test if the pattern of dependence between the components is statistically different in the two cases. Abbreviations: ANA: anti-nuclear antibodies; APC: antigen-presenting cells; IRF:interferon regulatory factor

AlephZero and mathematical experience

This essay explores the impact of automated proof construction on three key areas of mathematical cognition: on how we judge the role one piece of mathematics plays in another, on how we make mistakes in reasoning about mathematical objects, and on how we understand what our theorems are truly about. It concludes by speculating on a new form of mathematical experience that these methods could make possible: “glitching”, a game-like search for uncanny consequences of our definitions.