Mathematical Connection Research Articles

Revisiting Lorenz’s Error Growth Models: Insights and Applications

This entry examines Lorenz’s error growth models with quadratic and cubic hypotheses, highlighting their mathematical connections to the non-dissipative Lorenz 1963 model. The quadratic error growth model is the logistic ordinary differential equation (ODE) with a quadratic nonlinear term, while the cubic model is derived by replacing the quadratic term with a cubic one. A variable transformation shows that the cubic model can be converted to the same form as the logistic ODE. The relationship between the continuous logistic ODE and its discrete version, the logistic map, illustrates chaotic behaviors, demonstrating computational chaos with large time steps. A variant of the logistic ODE is proposed to show how finite predictability horizons can be determined, emphasizing the continuous dependence on initial conditions (CDIC) related to stable and unstable asymptotic values. This review also presents the mathematical relationship between the logistic ODE and the non-dissipative Lorenz 1963 model.

МАТЕМАТИКА САБАГЫНДА ИЧКИ ЖАНА СЫРТКЫ ПРЕДМЕТТЕР АРАЛЫК БАЙЛАНЫШТАРДЫ МИСАЛДАРДА КӨРСӨТҮҮ

This article shows that internal and external interdisciplinary connections in mathematics also play an important role in physics, chemistry, economics, finance, philosophy and many other fields. Mathematics is a science that unites different fields of knowledge, giving them structure, accuracy and meaning. Many scientific and technical achievements are impossible without mathematical methods, therefore, understanding, knowledge, and application of mathematics is a prerequisite for success in almost any professional field. The main purpose of teaching the basics of general sciences is to understand these objective relationships in nature and society. Such relationships must be taken into account when determining the content, forms, methods and techniques of teaching students.

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Nonlinear shifts and dislocations in financial market structure and composition.

This paper develops new mathematical techniques to identify temporal shifts among a collection of US equities partitioned into a new and more detailed set of market sectors. Although conceptually related, our three analyses reveal distinct insights about financial markets, with meaningful implications for investment managers. First, we explore a variety of methods to identify nonlinear shifts in a market sector structure and describe the mathematical connection between the measure used and the captured phenomena. Second, we study a network structure with respect to our new market sectors and identify meaningfully connected sector-to-sector mappings. Finally, we conduct a series of sampling experiments over different sample spaces and contrast the distribution of Sharpe ratios produced by long-only, long-short, and short-only investment portfolios. In addition, we examine the sector composition of the top-performing portfolios for each of these portfolio styles. In practice, the methods proposed in this paper could be used to identify regime shifts, optimally structured portfolios, and better communities of equities.

Teaching at the Right Level-based Project-based Learning on Mathematical Connections of Fourth Grade Elementary School Students

Learning in primary schools currently follows the direction of the Merdeka Curriculum which prioritises independent learning according to the learners' own abilities. Mathematical connections are still important things that students must have for their daily lives. This study aims to analyse the effect of implementing the TaRL-based PjBL model on mathematical connections, especially in mathematics subjects. This quasi-experimental study used saturated sampling of 18 grade IV students. A pretest-posttest group design was used in one group. The research used tests. In this study, descriptive analysis technique was used for data analysis, which used sequential sample t test. Before hypothesis testing, prerequisite tests such as normality and homogeneity tests were conducted. When compared to students before treatment, the application of TaRL-based PjBL model was proven to improve their mathematical connection. Furthermore, data processing was carried out with tcount ≥ ttable 5.702 > ttable 2.10982) which indicates that H0 is rejected and H1 is accepted, meaning that there is a positive effect of the TaRL-based PjBL model on students' mathematical connections in class IV elementary schools.

Brain Connectivity Dynamics and Mittag–Leffler Synchronization in Asymmetric Complex Networks for a Class of Coupled Nonlinear Fractional-Order Memristive Neural Network System with Coupling Boundary Conditions

This paper investigates the long-time behavior of fractional-order complex memristive neural networks in order to analyze the synchronization of both anatomical and functional brain networks, for predicting therapy response, and ensuring safe diagnostic and treatments of neurological disorder (such as epilepsy, Alzheimer’s disease, or Parkinson’s disease). A new mathematical brain connectivity model, taking into account the memory characteristics of neurons and their past history, the heterogeneity of brain tissue, and the local anisotropy of cell diffusion, is proposed. This developed model, which depends on topology, interactions, and local dynamics, is a set of coupled nonlinear Caputo fractional reaction–diffusion equations, in the shape of a fractional-order ODE coupled with a set of time fractional-order PDEs, interacting via an asymmetric complex network. In order to introduce into the model the connection structure between neurons (or brain regions), the graph theory, in which the discrete Laplacian matrix of the communication graph plays a fundamental role, is considered. The existence of an absorbing set in state spaces for system is discussed, and then the dissipative dynamics result, with absorbing sets, is proved. Finally, some Mittag–Leffler synchronization results are established for this complex memristive neural network under certain threshold values of coupling forces, memristive weight coefficients, and diffusion coefficients.

Profile of Students' Mathematical Literacy Based on Level of Mathematical Connection Ability

Profile of Students' Mathematical Literacy Based on Level of Mathematical Connection Ability

How are the classification of students’ mathematical connections in solving non-routine problems?

How are the classification of students’ mathematical connections in solving non-routine problems?

Mathematics Learning Innovation for Student Mathematics Process Skills in Higher Grade of Elementary Schools

The process skills approach can improve students' mathematical connection abilities, because it requires students to be active and creative in knowing the relationship between mathematical concepts and everyday life. If students do not have process skills in learning mathematics, they will have difficulty understanding and working on mathematical problems. The purpose of this research is to analyze processing skills differ in grade 4 mathematics learning on multiples and number factors in elementary school. The research method used is a mixed method. This research was conducted in class 4 of elementary school. The sample used in this research was 18 students selected using purposive sampling technique. The instruments used were observation, interviews and questionnaires. Data analysis techniques use descriptive and inferential statistics. Descriptive statistics to determine minimum values, maximum values, mean, median, and mode as well as inferential statistics to test assumptions and test hypotheses. Meanwhile, the process skills carried out in this research can improve students' mathematical connection abilities because students are required to be active and creative so they can understand the relationship between concepts in mathematics and their application in everyday life.

ANALISIS KEMAMPUAN KONEKSI MATEMATIS SISWA DALAM MENYELESAIKAN SOAL-SOAL CERITA DI UPTD SMP NEGERI 5 KUPANG

This study aims to determine the ability of students' mathematical connections in solving story problems. The research used was qualitative research, with the research subject of class VIII B students of UPTD SMP Negeri 5 Kupang. The research instruments used are tests, interviews and documentation. The data that has been obtained is then analyzed using question instrument analysis, data classification, and conclusions. The results showed that 3 students with high mathematical connection skills with a value of 12% were able to master all three indicators, namely linking concepts and procedures in mathematics, linking concepts in mathematics with other sciences, and applying mathematical concepts to everyday life. 11 Students with moderate mathematical connections with a score of 44% only mastered a few indicators, namely relating concepts in mathematics to other sciences, and applying mathematical concepts to everyday life.

Mathematical Connections of Students Viewed From Adversity Quotient in Problem-Based Learning Using Math City Map

Mathematical Connections of Students Viewed From Adversity Quotient in Problem-Based Learning Using Math City Map

The use of soroban as a significant learning tool in basic operations in a mathematical connection

Research into tools and mechanisms that facilitate the process of teaching and learning mathematics is an inexhaustible topic of study, since understanding operations is the basic foundation of students' educational development in the exact sciences. The aim of this study was to investigate the extent to which the use of the Soroban as a teaching mechanism facilitates the method of building students' knowledge within a mathematical connection, with the aim of fostering meaningful learning. The methodology used was qualitative research, due to its concern with analyzing ideosyncratic aspects such as values, attitudes and understandings related to learning practices and how this method can be adapted to different realities and logistical challenges, in order to guarantee significant and functional results for the learning process. The literary research went through the analysis of media documents, articles, theses, dissertations and publications related to the topic, seeking new understandings and knowledge to be added to the methodological tools associated and structured with the purpose of corroborating and increasing the body of existing research. The results show that the success of meaningful learning depends largely on the teacher being able to provide new mechanisms for students to carry out activities that activate cognitive development and favor the acquisition of knowledge in a mathematical connection.

Experimental research on the fracture of Laminated bamboo lumber using compact tension specimens based on the net section method and digital image correlation technique

This investigation evaluates the significance of thickness (B) and initial crack length-to-width ratio (α) on the fracture toughness of Laminated bamboo lumber (LBL) compact tension (CT) specimens under mode Ⅰ fracture conditions. The digital image correlation (DIC) technique is employed to analyse the result, while the net section method is utilised to determine the critical strain energy release rate (GIc). The fracture mode observed in LBL is characterised as brittle fracture, and the critical stress intensity factor (KIc) of CT specimens achieved a maximum value of 25.58 MPa·mm1/2. The fracture energy, calculated by the net section method, was determined by analysing the crack extension length (ae) of the CT specimen obtained by the DIC technique. This study aims to establish the mathematical connection between GIc, ae, and critical load (PQ) in terms of the strain energy release rate context, which offers an innovative approach to analysing GIc data of CT specimens subjected to mode I fracture. The calculated value for GIc was determined to be 12.18 J/mm2.

The role of mathematical connections in object-level and meta-level learning: a potential lens for supporting pre-service teacher learning

ABSTRACT The importance of mathematical connections in teaching mathematics is broadly acknowledged in the literature. Nevertheless, more research is needed to clarify how different types of mathematical connections manifest in classroom interactions and how they can support student learning. This article uses the commognitive framework to analyse 15 pre-service secondary teachers’ (PSTs’) discussions on mathematical connections that hypothetically could be made in a pattern generalisation task. The study seeks to explore the types of connections produced by the PSTs and to examine the potential opportunities for object-level and meta-level learning in algebra discourse. The findings are discussed in light of ongoing efforts to refine teachers’ pedagogical knowledge and practice regarding mathematical connections and to develop better pedagogical tools to communicate about mathematical connections.

Mathematical connections involved in area measurement processes

ABSTRACT The present study seeks to explore the mathematical connections that 13–14-year-old secondary school students establish when solving area tasks. Emphasis is placed on different mathematical objects, and the connections between them, that allow students to successfully solve the tasks. The study follows a mixed methodology using qualitative and quantitative method of analysis. The results show that representations play a key role in solving area tasks, as they condition the use of alternative procedures to the use of formulas. Likewise, the properties involved in area measurement processes may condition the use of geometric procedures, such as surface decomposition and reorganisation. Finally, results show that to accurately carry out area measurement processes, it is necessary to bring different mathematical objects into play simultaneously. If these connections between mathematical objects do not occur, there is a risk of using the formulas in a mechanical way.

Re-thinking mathematical connections with theories of difference

ABSTRACT In this theoretical article, we explore the possibility that instead of a mathematical connection arising “in-between” two (or more) pre-existing ideas or objects, which have presumably been known or understood, that connection is itself the motor of understanding. The standard view of connection, in which two existing ideas or concepts are brought together, is based on a philosophy of identity which seeks to establish objects and their properties and then compare them. In contrast, a philosophy of difference sees objects as “becoming” through difference. We explore this new way of conceptualising connection not only because it reflects our commitment to a relational ontology, but because it enables us to better handle the enmeshment of affect in mathematical learning without subordinating it to the cognitive. We draw on illustrative data to show what analysing connections as acts of differencing might look like and how it involves affective, social and technical dimensions.

Using the TIMSS curriculum model to develop a framework for coherence and its role in developing mathematical connections

ABSTRACT Developing mathematical connections for pupils continues to be at the forefront of educational policies worldwide. One strategy suggested to support the connections pupils make is the idea of developing “coherence”. Research regarding coherence has increased within mathematics education, however, the term is used widely. It therefore becomes necessary for coherence to be further conceptualised. In this systematic scoping review, five databases were searched for literature on the topic of coherence in school-based mathematics education. A total of 11,587 pieces of literature were extracted. Following further screening, 35 papers were reviewed. The findings suggest that there is limited consensus regarding the conceptualisation of coherence, leading to fragmented perceptions across the literature. This paper presents a framework for coherence based on the TIMSS curriculum model to support international comparisons of coherence and provide clarity for mathematics education researchers and school colleagues when considering which aspect of coherence they are addressing.

Onto-semiotic analysis of Colombian engineering students’ mathematical connections to problems-solving on vectors: A contribution to the natural and exact sciences

The mathematical connections Colombian engineering students activated when they solved vector problems were explored. The study was based on the extended theory of connections and the onto-semiotic approach. We followed a qualitative methodology that consisted of three stages: (1) selection of engineering students as participants; (2) application of a questionnaire with 15 tasks on vectors to the participating students; and (3) analysis of these data based on the theoretical articulation. The results show that students perform arithmetic operations with vectors, find the scalar and vector product, the norm of a vector, the angle between vectors, and unit vector based on mathematical connections (<i>procedural, meaning, different representations, </i>and<i> implication</i>), detail from an onto-semiotic point of view. However, some students have difficulty finding the angle between vectors because they misuse the norm. Furthermore, the new metaphorical connection based on mnemonics activated by the “<i>law of the ear</i>” is reported. The connections activated by engineering students to solve problems about vectors may have been influenced by the explanations provided by their calculus teacher, who promotes connections for the teaching and learning mathematical concepts.

Mathematical Connections in Preservice Secondary Mathematics Teachers’ Solution Strategies to Algebra Problems

This study investigated the mathematical connections found in the solutions provided by 22 preservice secondary mathematics teachers to a set of algebra problems. Interest in, and research on, mathematical connections has gained prominence of the past decade. Here, we use the Extended Theory of Mathematical Connections or ETMC to explore the types of connections that this framework does and does not capture in the preservice teachers’ solutions. The ETMC surfaced four types of mathematical connections across four problems: ‘different representations’, ‘procedural’, ‘part-whole’ and ‘meaning’. The other types of connections defined in ETMC such as ‘reversibility’ or ‘feature’ were not found in our data, perhaps because of the specific problems that were used. Some mathematical connections were not highlighted when examining the solutions through the lens of ETMC (‘meaning’, ‘implication or if/then’ and modelling) addressing areas in which ETMC might be limited in its capacity to support researchers in identifying mathematical connections in different contexts.

The Role Played by Extra-mathematical Connections in the Modelling Process

Literature review in Mathematics Education on connections makes evident the necessity of delving into extra-mathematical connections. In this line, we answer the question: What types of mathematical connections are required to develop the modelling process? To this end, we consider three theoretical references: Onto-Semiotic Approach, Extended Theory of Connections, and Mathematical Modelling Cycle from a Cognitive Perspective, following a methodology used in two theoretical articulations previously developed by the authors between these frameworks, based on the use of models for the analysis of mathematical activity proposed by the Onto-Semiotic Approach for connections and the modelling process. The results of this analysis consist of, on one hand, the evidence of the types of intra- and extra-mathematical connections intervening in the different phases of the modelling cycle and, on the other hand, the proposal of a more detailed classification of mathematical connections.