Mathematical Objects Research Articles

Partial Meanings that a Teacher Mobilizes for Learning Limits at the School Level

Objective: The objective of this study is to analyze the partial meanings of the function limit object in a variable that a teacher mobilizes when teaching the Calculus unit at the school level. Theoretical Framework: For the analysis of the partial meanings of the mathematical object, the epistemic configurations proposed by the OntoSemiotic Approach to Mathematical Cognition and Instruction (OSA) were used. Method: The study is qualitative in nature and its methodological design is an instrumental-descriptive case study.. To carry out the analysis of the meanings mobilized by the teacher, a semi-structured interview, teaching material, field notes from class observation and the notes of two students of the course were used. Results and Discussion: The results obtained revealed that the teacher preferentially mobilizes the last two epistemic configurations of the mathematical object and to a limited extent some primary objects of other configurations. Research Implications: The research provides a reflection for the pedagogical practices of teachers regarding the epistemic richness of the mathematical object, mainly to Chilean teachers, who must teach said object promoting all its partial meanings according to the requirements of the Chilean school curriculum. Originality/Value: The research lies in the importance of the pedagogical practices that teachers should consider when teaching said object, in particular, the mathematical richness of the partial meanings of the object.

The objectivity of mathematics in Cassirer’s symbol-theoretical approach

AbstractCassirer’s concept of symbolic forms represents a cultural-philosophical approach in which specific modes of objectification, the symbolic forms themselves, establish an independent truth. This truth does not consist in a purely analytical truth, but in the fact that we live along these symbolic forms in a specific stubbornness. When we use language, we live and experience language in its specific autonomy. The same applies to the other symbolic forms developed by Cassirer. Both before the cultural-philosophical phase of creation and within the philosophy of symbolic forms, Cassirer refers to mathematics several times. An interesting tension arises as to whether Cassirer could have understood mathematics as a symbolic form in its own right. On the one hand, cultural-historical considerations in Cassirer’s work speak in favour of this; on the other hand, Cassirer embeds mathematics as a mere formal science in physics and thus in the philosophical form of science as a whole. The article explores this tension—mathematics as a symbolic form—and shows that only indications within the framework of the historical-hermeneutical treatment would speak in favour of such an interpretation. Furthermore, the article argues that a specific, cultural-historical semiotics can be considered for mathematics in Cassirer’s work.

Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp+logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + \\log c^t$$\\end{document} Function

Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical zp+c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + c$$\\end{document} function used in the definition of Mandelbrot set with logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log c^t$$\\end{document}, where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and t≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 1$$\\end{document}. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.

On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication.

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.

Teaching practice aimed at promoting student engagement with metarules of defining

By viewing students’ learning to define as learning about meta-discursive rules about defining and how to engage with them in their activity of defining, we investigated teaching practices that aim to promote such learning by analyzing a teaching experiment in which Calculus II students reinvented a formal definition of a limit. The teaching practices we identified addressed how students view defining tasks by providing a method to check whether their narratives satisfied the metarules that they aimed to follow and also provided guidance about how to revise their definitions to satisfy those metarules. Our results provide an example for the teaching practice that promotes student learning about and engagement with new meta-discursive rules that existing literature called for in general, especially in their reinvention of a formal definition of a mathematical object.

Are Mathematical Objects ‘sui generis Fictions’? Some Remarks on Aquinas’s Philosophy of Mathematics

Abstract This contribution proposes an interpretation of Thomas Aquinas’s philosophy of mathematics. It is argued that Aquinas’s philosophy of mathematics is a coherent view whose main features enable us to understand it as a moderate realism according to which mathematical objects have an esse intentionale. This esse intentionale involves both mathematicians’ intellectual activity and natural things being knowable mathematically. It is shown that, in Aquinas’s view, mathematics’ constructive part does not conflict with mathematical realism. It is also held that mathematics’ imaginative reasoning is coherent with Aquinas’s doctrine of formal abstraction and his realistism. It focuses on some of Aquinas’s texts, which it places within their textual and doctrinal context and interprets them in the light of some historical elements.

ANALYSIS OF ECOTOURISM AND DISASTER MITIGATION DATA AT THE KUALA LANGSA MANGROVE FOREST BY USING THE ANALYTIC HIERARCHY PROCESS

The problem in the management of a public area is to determine aspects that contribute most as its best practices. This paper explores the contributions of the Kuala Langsa Mangrove Forest located in Langsa City, Aceh Province, for ecotourism and disaster mitigation. This research was intended to analyze the aspects of forest management that has successfully reformed dry coast into an ecotourism site. The Analytic Hierarchy Process (AHP) as a multicriteria decision making method was used to analyze the best indicators data. This method was selected as the suitable method since it utilizes the objectivity of mathematics in comparing some subjective criteria to obtain the best indicators. The implementation of the AHP method consists of three stages: the development of the criteria’s hierarchy table; the development of pairwise matrix related to the intention of giving weights for the criterion’s elements; the logical concistency verification of the matrix. Indicators were written on a five point scale related to a value judgement. The results of the research show that as an ecotourism site, the activities of the mangrove forest are motivated and based on the existing rules which attract people to become interested. The role of the local government has resulted: developed the programs according to the existing mangrove management law that reached 67% among other law related indicators; budget allocation priority by self-generating which get priority of 59% among budget indicators; oriented on the increasing of fishermen, obtained 49% of economical function elements; encouraged the involvement and partnership of the local resources around the mangrove forest which reached 31% among other programs offered and sustainability approach by considering ecological functions aspect. For physical contribution for disaster mitigation, the largest dissipation is given by Rhizophora mucronata due to its height compared to the other types.

Logistics Performance Index-driven in operational planning for logistics companies: A smart transportation approach

The responsibility of Logistics Companies (LCs) extends beyond simply handling logistical processes and choosing transportation routes; they focus on being competitive and efficient in their operations. In multiple origins and destinations networks, LCs attempt to select the most optimal routes from each point. To address this challenge and reduce the time needed to identify potential routes, a novel method has been developed in this research for LCs. In operational planning, additional factors need to be considered to increase efficiency, customer satisfaction, and competitiveness. To confront this issue, the study proposes a novel model and contributions, inspired by the LPI, to apply the transformative impact of smart ports, the integration of smart containers, the choices of clearing and forwarding agents and port operators, and time uncertainty in the multi-objective framework. Specifically, the mathematical model's objectives include cost, time, customer satisfaction, and environmental impact in the periodic multimodal network. An innovative customer satisfaction function is introduced by integrating the selection of smart containers and smart ports, emphasizing their impact on enhancing customer satisfaction, while addressing time uncertainty through a chance-constrained approach and coefficients of variation. The model is solved using goal programming, and the results show that smart ports and smart containers can majorly affect transportation time and customer satisfaction. Furthermore, comparing the results to other studies demonstrates its superiority in decision-making for LCs, particularly by including time uncertainty and the role of clearing and forwarding agents and port operators. Therefore, the model holds practical significance in lowering costs, enhancing customer satisfaction, and facilitating smart international logistics. This research offers insights that are not only useful for the LCs but also for other stakeholders in the transport industry.

Умови екстремальності допустимого елемента для задачі відшу-кання узагальненого чебишовського центра кількох точок деякого полінормованого простору відносно множини цього простору

The problems related to the need to approximate complex mathematical objects in the best possible way with simpler and more convenient ones arise in various sections of mathematical science. An important class of approximation theory problems is the best simultaneous approximation of several elements. The problem of finding the Chebyshev center of several points of a linear normalized space relative to the set of this space can be attributed to the problems of best simultaneous approximation of several elements. This task consists in finding in a given set of linear normed space such a point (the relative Chebyshev center) the maximum distance to which from several fixed points of space would be the smallest, in other words not exceeding the maximum distance from the given points to any other point of this set. The problems of the best simultaneous approximation of several elements of a linear normed space by convex sets of this space from single positions were considered, in particular, in works [1, 2]. In practice, one has to deal with such problems, in which, when finding the Chebyshev's center of several given points of a linear normed space relative to the set of this space, appear weighted distances. The task of finding the weighted distances of the Chebyshev’s center was considered, in particular, in the paper [3]. In this work, the criteria of generalized Chebyshov’s center in the sense of the weighted distances of the of several points of a linear normed space relative to the convex set of this space, based on the duality ratio for the corresponding extremal problem, are established. If in the problem of the Chebyshev's center of several points of a linear normed space, in which the distances between points are determined by weighted norms, the weighted norms are replaced, generally speaking, by different norms given on the corresponding linear space, then we obtain the problem of the Chebyshev's center of several points of some polynormed space, which is considered in this work. It is clear that the problems about the Chebyshev's center of several points of a linear normed space, which were discussed above, are partial cases of the problem about the Chebyshev center of several points of some polynormed space.

Mathematical discussion in classrooms as a technologically-supported activity fostering participation and inclusion

AbstractWhole-class mathematical discussion in a problem-solving activity is recognized as a powerful pedagogical activity but also a challenge for teachers who must consider several difficulties that learners might face, particularly in terms of an overload of Working Memory and Executive Functions. This study investigates how the use of a digital platform (Padlet) can support participatory and inclusive mathematical classroom discussion. We proposed a teaching experiment based on graphical tasks anticipating integral calculus to grade 13 students, and we examined how the use of the digital platform plays a role in the construction and interpretation of new mathematical objects emerging from the activity. The use of Instrumental Genesis and Double Instrumental Genesis frameworks allowed us to make the affordances of the tool emerge. As a result, we got evidence of how mathematical discussion may develop as a network of interactions, feedback, and connection of input and discuss examples of how active participation and inclusion are enhanced by the tool affordances. Indeed, the digital platform allowed easy interaction, with many ways to represent and express the ongoing evolution of personal and shared meanings and the possibility to manage the time of the activity. This fostered students’ participation and students which did not participate in previous discussions were actively engaged in it.

A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}=4$$\\end{document} super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for Gr4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ Gr}_{4,n}$$\\end{document}. Secondly, we exhibit a tiling of the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for Gr4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{ Gr}_{4,n}$$\\end{document}. This paper is a companion to our previous paper “Cluster algebras and tilings for the m=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=4$$\\end{document} amplituhedron.”

Investigating the use of ChatGPT to solve a GeoGebra based mathematics+computational thinking task in a geometry topic

ChatGPT is a chatbot with potential educational benefits, particularly in enhancing computational thinking (CT) proficiencies such as programming, debugging, and algorithmic thinking for students. Despite its promise, there is limited research on how ChatGPT can specifically support the integration of CT into mathematics education using tools like GeoGebra. The researchers implemented plugged-computational thinking in mathematics (Math+CT) lessons by means of the utilization of GeoGebra, an application that requires students to input commands in order to generate mathematical objects. The present investigation employed an educational design research (EDR) methodology in which the researchers incorporate ChatGPT into our Math+CT lessons to assist students in accomplishing the task. We purposely selected the participants who are mainly postgraduate students and collected data from the participants’ conversation with ChatGPT and recorded their screens while interacting with ChatGPT and our Math+CT task. We analyzed the data through descriptive qualitative method on the participants’ prompts, the final codes and the number of iterations. The researchers examined how ChatGPT could be utilized to assist the participants in writing GeoGebra commands in terms of its benefits and limitations. ChatGPT assisted most participants in completing the task successfully, with only a basic need for proficiency in GeoGebra commands, mathematics, and critical thinking. However, it revealed that participants did not yet utilize an affective prompt to ChatGPT. Furthermore, ChatGPT has the potential to be utilized for differentiated instruction due to the fact that its responses to individual users vary significantly based on the input prompts. Limited understanding of basic GeoGebra commands, and mathematical concepts could hinder the participants from training ChatGPT or prevent them from arguing with ChatGPT. This study enhances the existing literature by illustrating that ChatGPT can facilitate critical CT aspects, including programming and debugging, in a mathematics education context. This suggests that AI tools such as ChatGPT can contribute to the development of independent problem-solving skills, provide tailored support based on the needs of individual students, and enhance personalized learning experiences. Additional research involving students in school is required in order to gain a deeper understanding of the integration of ChatGPT into Math+CT lessons.

Character-Level Text Generation for Shakespearean Style with LSTMs

This paper presents a pioneering approach to text generation employing Recurrent Neural Networks (RNN) with Long Short-Term Memory (LSTM) architecture, inspired by the rich and timeless prose of William Shakespeare. The motivation stems from the enduring allure of Shakespearean language, which has captivated audiences across centuries, and the challenge of replicating itsintricate style using modern computational techniques. Our research contributes a novel methodology that leverages the capabilities of RNN LSTM networks to emulate the linguistic nuances of Shakespeare with remarkable fidelity. The paper begins by providing a comprehensive overview of RNN LSTM networks, highlighting their suitability for sequential data processing tasks and their ability to capture long-rangedependencies. A review of related work in the field sets the stage for our proposed approach, shedding light on recent advancements and methodologies employed in text generation using similar techniques. We formulate the problem by defining the mathematical framework, optimization objectives, and evaluation metrics for our proposed model. The architecture consists of three layers: the data layer for preprocessing input text data, the intelligence layer comprising multiple LSTM units for capturing different aspects of Shakespearean language, and the application layer for generating output text based on learned representations. Experimental results demonstrate the effectiveness of our approach, with evaluations conducted on a corpus of Shakespearean texts.In conclusion, our research presents a significant advancement in the field of natural language generation, opening new avenues for exploring the intersection of literature and artificial intelligence.

On Recursive Tiling of some Mathematical Objects.

This essay, in a novel way, uses various tiles, such as dominoes, square-, interlocking-, and flow-tiles, to visualize approximations to recursively defined, potentially infinite mathematical objects, such as binary code tables, and variants of the space-filling Hilbert curve. The power of L-systems is shown by defining these tiled objects as L-systems.

Enhancing Measurement and Calculation Education in High School through 3D Printing Technology

Mathematics education often grapples with the challenge of teaching abstract mathematical concepts, particularly those existing in 3D space. Visualizing, manipulating, and comprehending these abstract objects can be a formidable task for learners. While 3D printing technology has found applications in various fields, its utilization in mathematics education remains limited. This research explores the potential and effectiveness of 3D printing technology in enhancing the teaching of measurement and calculation topics in high school mathematics. We designed 3D printed models with a specific focus on shapes, polyhedra, pyramids, and cylinders. These models were created to facilitate hands-on learning experiences for students, enabling them to directly measure, calculate, and experiment with these tangible mathematical objects. The experimental results reveal that students not only learned how to employ measuring tools to gauge the dimensions of tangible mathematical objects but also conducted mathematical experiments related to the 3D printed models. This practical engagement allowed students to gain valuable experiences and foster a deeper understanding of abstract mathematical concepts.

Implementation of the practice-oriented mathematics teaching methodology to future rescue engineers

Introduction. In modern conditions, the practice-oriented training of rescue engineers, whose professional activities are associated with high-risk situations, is being updated. The purpose of the study is to describe the implementation of the author's methodology of practical-oriented training in mathematics of future specialists in fire and technosphere safety and experimental verification of efficiency. Materials and methods. The experimental training was attended by 685 students of the Donetsk Institute of the State Fire Service (Russian Federation), studying in the specialty "Fire safety" and the areas of training "Technosphere safety" (321 students in experimental groups, 347 in control groups). Methods of mathematical statistics: Wilcoxon signed-rank test, Fisher's exact test, Chi square test, Student's t-test. Results. The methodology of practice-oriented teaching mathematics to future rescue engineers has been developed. Mathematical training of students based on a practice-oriented approach ensures the formation of students' practice-oriented skills and mathematical modeling skills in the field of civil protection, as well as the development of personal qualities of a rescuer. Practice-oriented learning technologies are used: integration of educational and future professional activities of students, educational and practice-oriented research activities. Practice-oriented teaching methods are used (imitation of the practical activities of rescue engineers, "rapid response" and practice-oriented visualization of mathematical objects). The implementation of the developed teaching methodology contributes to the effective formation of practice-oriented mathematical competence among students (T = 0.32, α = 0.05). Conclusions. The implementation of practical-oriented training in mathematics of future specialists methodology in fire and technosphere safety contributes to increasing the level of development of practical-oriented mathematical competence due to the formation of personal qualities of the rescue engineer and his value orientations, knowledge of mathematical and practical-oriented knowledge, skills and methods of activity in mathematical and computer modeling.

TÍCH HỢP WEB SKETCHPAD VÀO KHÓA HỌC TRỰC TUYẾN TRONG DẠY HỌC CHUYÊN ĐỀ PHÉP QUAY

Web Sketchpad is a new technology based on The Geometer’s Sketchpad software, featuring a highly simplified user interface and customizable tools, allowing learners to focus on mathematical exploration experiences to construct their own knowledge. This study integrated two dynamic mathematical representations designed on Web Sketchpad into an online course to support students in exploring the topic of Rotations in the plane. The results show that Web Sketchpad on the Moodle platform supports students well in conducting mathematical explorations related to rotations. It can serve as a cognitive tool and support different stages of mathematical problem-solving. The students in the experiment demonstrated problem-solving and creative competency, independently constructing new mathematical objects, searching for and discovering mathematical relationships. Additionally, they had the opportunity to practice self-directed learning and self-study through heuristic exploratory activities with dynamic mathematical representations. Keywords: Web Sketchpad; rotation; online-learning; moodle platform

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.

Useful properties of phenomenological-based models

This work compiles the conceptual and practical proposals of various research groups regarding interesting properties of mathematical models. Although different types of mathematical models are mentioned, phenomenological-based semi-physical models (PBSM) are the main focus of this paper. The partitioning of the process to be modeled, the scalability of the model, and the interpretability of its parameters are presented as important properties of the model that help to understand and use the mathematical model. The use of these properties is illustrated with examples, providing the respective bibliographical reference where an expanded presentation of each example can be reviewed. The importance of each model property presented is explained from the inherent condition of the model as a mathematical object and from the expectation of the final users of the model. The confidence in the model is increased when the user knows how the model is obtained. These properties provide valuable information to the end use of the model. Therefore, the modeler must understand and apply this knowledge during model deduction and write, in the model report, how the mentioned properties were tested.

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.