Mathematical Connection Research Articles

Analogy between Boltzmann Machines and Feynman Path Integrals.

Machine learning has had a significant impact on multiple areas of science, technology, health, and computer and information sciences. Through the advent of quantum computing, quantum machine learning has developed as a new and important avenue for the study of complex learning problems. Yet there is substantial debate and uncertainty in regard to the foundations of machine learning. Here, we provide a detailed exposition of the mathematical connections between a general machine learning approach called Boltzmann machines and Feynman's description of quantum and statistical mechanics. In Feynman's description, quantum phenomena arise from an elegant, weighted sum over (or superposition of) paths. Our analysis shows that Boltzmann machines and neural networks have a similar mathematical structure. This allows the interpretation that the hidden layers in Boltzmann machines and neural networks are discrete versions of path elements and allows a path integral interpretation of machine learning similar to that in quantum and statistical mechanics. Since Feynman paths are a natural and elegant depiction of interference phenomena and the superposition principle germane to quantum mechanics, this analysis allows us to interpret the goal in machine learning as finding an appropriate combination of paths, and accumulated path-weights, through a network, that cumulatively captures the correct properties of an x-to-y map for a given mathematical problem. We are forced to conclude that neural networks are naturally related to Feynman path-integrals and hence may present one avenue to be considered as quantum problems. Consequently, we provide general quantum circuit models applicable to both Boltzmann machines and Feynman path integrals.

Energy-level shift of quantum systems via the scalar electric Aharonov-Bohm effect

A novel version of the electric Aharonov-Bohm effect is proposed where the quantum system which picks up the Aharonov-Bohm phase is confined to a Faraday cage with a time varying, spatially uniform scalar potential. The electric and magnetic fields in this region are effectively zero for the entire period of the experiment. The observable consequence of this version of the electric Aharonov-Bohmn effect is to shift the energy levels of the quantum system rather than shift the fringes of the 2-slit interference pattern. We show a strong mathematical connection between this version of the scalar electric AB effect and the AC Stark effect.

Investigation of the mathematical connection’s ability of 9th grade students

This study aims to investigate students' mathematical connection ability in solving the problem of mathematical and correlation between indicators of mathematical connection ability. This research is correlation research. The sample of the study is 35 girls’ students in the 9th grade in a middle school in Kerman province (south-east of Iran). Data collection is done by giving a test on mathematical connection skills. The variables in this study are indicators of mathematical connection ability, namely the connection between mathematical concepts, connections between mathematics and other sciences, and connections between mathematics and everyday life. Spearman's correlation coefficient was used to analyze the data after calculating the score and percentage of test answers. The findings showed that qualification of students in the indicator of the connection between mathematical concepts was good enough (69%), in the indicator of the connection between mathematics and other sciences was also good enough (66%), and the indicator of the connection between mathematics and everyday life was not enough (42%). Also, the results of Spearman's correlation coefficient showed that there is a significant relationship between both indicators of mathematical connection ability. Therefore, the connection between mathematics and everyday life of Students must be improved.

Exploring grade 11 learners’ mathematical connections when solving two-dimensional trigonometric problems in an activity-based learning environment

In this paper, we explored the type of mathematical connections Grade 11 learners make when solving two-dimensional (2D) trigonometric problems in an Activity-Based Learning (ABL) environment. We followed a qualitative case study design within an interpretive paradigm. Convenience sampling was used to select a whole class of 45 Grade 11 learners from one of the public non-fee-paying secondary schools in Capricorn District, Limpopo Province of South Africa. Group work presentations and classroom interactions were used to collect data. Data were analyzed using deductive thematic analysis guided by the mathematical connections’ framework. The findings indicated that learners managed to make procedural, meaning, reversibility, different representations, feature, and inclusion part whole as well as integrated connections as they worked on 2D trigonometric problems in an ABL environment. We established that learners did not make generalization part-whole connections. In addition, we found that some learners lacked mathematical connections skills and failed to solve the problems. Engaging learners in an ABL environment provided a fine-grained approach that allowed them to make mathematical connections. We, therefore, recommend that teachers should create an ABL environment to enable learners to make different types of mathematics connections during the teaching and learning of trigonometric concepts.

PENERAPAN MODEL PEMBELAJARAN CONTEXTUAL TEACHING AND LEARNING (CTL) UNTUK MENINGKATKAN KEMAMPUAN KONEKSI DAN PEMECAHAN MASALAH MATEMATIS SERTA SELF REGULATED LEARNING SISWA

The study is intended to examine the issue of increasing the capability of connection and mathematical problem solving and self regulated learning (SRL) students through the learning of contextual teaching and learning (CTL), in terms of overall student and Early Math Ability category (KAM) students (high, medium, low). The research method is the mix method range of embedded. The population in this study are all the students of SMK Nur Pandeglang and Human Samples randomly selected two classes, namely class X 2 TKJ as class experiments, and class X 1 of the control class as TKJ. Instruments in this research in the form of a test and non-test. 6 the question of mathematical ability test connection, and 4 the question of mathematical problem solving ability tests, while tests of non instrument in the form of the now self regulated learning, observation sheets, and interviews. The results of this research were: 1) increased ability of Mathematical connections students who learn learning Contextual Teaching and Learning (CTL) is better compared to conventional learning when are reviewed from tegori Early Mathematical Ability (KAM) students; 2) increased Mathematical problem solving abilities students who learn learning Contextual Teaching and Learning (CTL) is better compared to conventional learning when are reviewed from tegori Early Mathematical Ability (KAM) students; 3) self regulated learning students who learn learning Contextual Teaching and Learning (CTL) is better compared to conventional learning: 4) there is a relationship between the ability of problem-solving and mathematical connections assembled and self regulated learning.

Physics-Informed Supervised Residual Learning for Electromagnetic Modeling

In this study, physics-informed supervised residual learning (PhiSRL) is proposed to enable an effective, robust, and general deep learning framework for 2-D electromagnetic (EM) modeling. Based on the mathematical connection between the fixed-point iteration method and the residual neural network (ResNet), PhiSRL aims to solve a system of linear matrix equations. It applies convolutional neural networks (CNNs) to learn updates of the solution with respect to the residuals. Inspired by the stationary and nonstationary iterative scheme of the fixed-point iteration method, stationary and nonstationary iterative physics-informed ResNets (SiPhiResNet and NiPhiResNet) are designed to solve the volume integral equation (VIE) of EM scattering. The effectiveness and universality of PhiSRL are validated by solving VIE of lossless and lossy scatterers with the mean squared errors (MSEs) converging to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10^{-4}$ </tex-math></inline-formula> (SiPhiResNet) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10^{-7}$ </tex-math></inline-formula> (NiPhiResNet). Numerical results further verify the generalization ability of PhiSRL.

Exploring the pedagogical practices of seasoned elementary mathematics teachers and connections with their content knowledge and beliefs

AbstractThis inquiry examined the pedagogical practices in mathematics of elementary teachers (N = 27) who had been identified as experienced and successful and were working in an urban school district with underserved student populations. Also investigated were relationships between their instructional practices and other elements of proficient teaching of mathematics, including specialized content knowledge and beliefs. Quantitative and qualitative data were collected via a knowledge assessment, belief surveys, classroom observations, and individual interviews. The findings related to learner‐centered, equitable mathematics instruction reveal a mixed picture of understanding and enactment by the participants, illuminating variability and complexity, especially within the context of a standardized model for instructional delivery. Participants expressed constraints in implementing learner‐centered, equitable mathematics instruction, particularly: prescribed, scripted lesson plans; teaching roles that involved instruction of many students thus contributing to lack of familiarity; and a mix of learners who were in‐class and remote due to the COVID‐19 pandemic. The results also demonstrate that pedagogical practices were shaped by participants’ specialized content knowledge and to a more limited extent their beliefs. These data further reveal mixed endorsements of the different belief constructs. Considerations for teacher development are discussed.

Pengaruh Pendekatan RME (Realistic Mathematics Education) Terhadap Kemampuan Koneksi Matematis Siswa Kelas VIII MTs Al-Aziziyah Putri Kapek Gunungsari

The research aims to describe the effect of the RME (Realistic Mathematics Education) approach on students' mathematical connection abilities of class VIII MTs. Al-Aziziyah Putri Kapek Gunungsari for the 2022/2023 school year. This research is experimental research using a posttest-only control group design. The population in this study was all class VIII with a total of 7 classes. While the sample in this study there were 2 classes, namely class VIII-7 as the experimental class and VIII-6 as the control class. The sampling technique used is cluster random sampling. Data collection was carried out using a mathematical connection ability test. The data obtained were processed with SPSS. The results of the statistical analysis showed that the test results of the students' mathematical connection abilities with the RME approach are higher than conventional learning. This can be seen from the average value of the experimental class 65 and the control class 56.57. The results of the data analysis shows. The value means that there is an influence of the RME approach on the students' mathematical connection abilities. The magnitude of the influence given is 5.75%, which means that the effect of the RME approach on the ability of mathematical connections is not large.

Applying RME in the CoViD-19 pandemic: An effort to improve sixth graders’ mathematical connection ability

AbstrakKoneksi matematis merupakan sebuah keterampilan yang penting bagi siswa. Sementara itu, penelitian terkait implementasi pendekatan pembelajaran sebagai upaya peningkatan keterampilan ini masih terbatas. Oleh karena itu, penelitian ini bertujuan untuk meningkatkan kemampuan koneksi matematis siswa kelas VI melalui pendekatan RME dengan setting PTM terbatas di era pandemi CoViD-19. Indikator koneksi matematis yang digunakan: (1) memahami masalah kehidupan sehari-hari dalam bentuk model matematika, (2) menuliskan hubungan antar objek dan konsep matematika, dan (3) menuliskan konsep matematika yang mendasari jawaban. Penelitian ini menggunakan sequential explanatory mixed-methods. Sampel sekaligus populasi dalam penelitian ini adalah 15 siswa kelas VI di salah satu SD Negeri di Sragen, Jawa Tengah. Teknik pengumpulan data yang digunakan yaitu tes, dokumentasi, dan wawancara. Pada bagian kuantitatif, data dianalisis dengan sign test sedangkan pada bagian kualitatif, data di analisis dalam tiga tahap: reduksi data, penyajian data, dan penarikan kesimpulan. Penelitian ini menggunakan triangulasi teknik dengan wawancara dan dokumentasi. Berdasarkan uji tanda, terdapat peningkatan secara signifikan pada kemampuan koneksi matematis siswa. Selanjutnya, berdasarkan analisis data kualitatif: (1) siswa dengan konektivitas matematis yang baik hanya menyelesaikan satu indikator pada pretest dan tidak indikator pada posttest, (2) siswa dengan konektivitas matematis sedang atau rendah hanya menyelesaikan satu indikator pada pretest dan posttest. Hal ini menunjukkan bahwa peningkatan indikator kinerja hanya terjadi pada siswa dengan koneksi matematis tinggi.Kata kunci: Koneksi Matematis, Pandemi CoViD-19, Realistic Mathematic Education. AbstractMathematical connection ability is an essential skill for students. There are still limited studies focus on developing the skills through applying specific learning approaches. Therefore, this study aims to improve students’ mathematical connection ability through the Realistic Mathematics Education (RME) approach with a limited face-to-face learning setting in the CoViD pandemic era. This study used the sequential explanatory mixed-methods. The sample and the population in this study were 15 sixth-grade students at one of the state elementary schools in Sragen, Central Java, Indonesia. The data collection techniques used in this study were tests, documentation, and interviews. In the quantitative part, the data analysis technique used is a sign test. In contrast, the qualitative aspect of the data analysis technique involves three stages: data reduction, data presentation, and conclusion drawing. This study used triangulation with interviews and documentation. Based on the sign test, there was a significant improvement in students’ mathematical connection ability. Furthermore, based on qualitative data analysis: (1) In the pretest, students with high mathematical connection ability, only one indicator was met. In contrast, in the post-test, three indicators were met, (2) in both pretest, and post-test, students with medium and low mathematical connection ability can only be met one indicator. These findings show that the achievement of indicators increases only in students with high mathematical connection ability.Keywords: CoViD-19 pandemic; Mathematical connection ability; Realistic Mathematics Education.

MATHEMATICAL CONNECTION OF GEOMETRY PROBLEMS BASED ON NUMERACY LITERACY THROUGH THE POLYA FRAMEWORK

The results of the 2018 Program for International Student Assessment (PISA) study released by The Organization for Economic Cooperation and Development (OECD) show that the numeracy literacy skills of Indonesian students are at a low level, namely getting a score of 379 out of the OECD average score of 487 (OECD, 2019). As a result, students tend not to be able to solve problems correctly, and the cause is a need for mathematical connection skills. This study aims to describe the mathematical connections based on numeracy literacy in geometry. This research is a qualitative descriptive study. The participants in this study were 38 grade 7 students from 3 different Madrasas in Maluku. Two students will take the research subjects from saturated data after being given a numeracy literacy-based test. In this study, students were asked to state the results of their thoughts on solving geometry concept problems given orally. Analyzing the research data was carried out through three stages: data reduction, data presentation, and drawing conclusions. The findings in this study of mathematical connections within the Polya framework can be divided into two: structured analytics and unstructured analytics.

DEVELOPMENT OF STEAM-BASED E-HANDOUT TO IMPROVE MATHEMATICAL CONNECTION SKILLS AT UPT SMP NEGERI 41 MEDAN

This paper discusses an e-handout of STEAM-based circle material to improve mathematical connection skills at UPT SMP Negeri 41 Medan. The research and development design (R&amp;D) uses the Thiagarajan model. The research results imply that the STEAM-based e-handout met the criteria of valid, practical, and improved mathematical connection skills. Material experts provided an average score of 81% (very valid), linguists provided an average of 80% (fairly valid), and media experts provided an average score of 100% (very valid). The average score of the teacher's response to the practicality of the e-handout was 75% (practical) and the student's percentage was 67% (practical), which means that teachers and students have a positive response. Mathematical connection skills at UPT SMP Negeri 41 Medan met the criteria of effectiveness. It is shown in the second trial in which the learning objectives that were achieved from the three mathematical connection indicators exceeded 75% and students’ learning completeness was 96%. The improvement in students' mathematical connection skills with an average N-gain score in trial I and trial II indicate an improvement score of 0.134. It can be concluded that the developed e-handout is valid, practical, and effective, as well as it can improve mathematical connection skills.

PENGARUH PENDEKATAN OPEN-ENDED DENGAN PEMBELAJARAN KOOPERATIF TIPE STAD TERHADAP KEMAMPUAN KONEKSI MATEMATIS SISWA DI SMPIT BINA SALAM BEKASI

This study aims to determine the effect of an open-ended approach with cooperative learning type Student Team Achievement Division (STAD) on students' mathematical connection abilities. The research was conducted at SMPIT Bina Salam, Bekasi City in the framework of the independent learning program in the form of teaching assistance learning activities in educational units. This research method is an experiment. The number of students as many as 54 students in this study. The results showed that there was no difference in the mathematical connection abilities of students who used an open-ended approach with STAD type cooperative learning and expository learning.&#x0D; Keywords: Open-Ended Approach, STAD type Cooperative, Mathematical Connection capabilities.&#x0D;

PHILOSOPHY PERSPECTIVE OF MATHEMATICS EDUCATION AS A FIELD OF KNOWLEDGE

Philosophy of education is essentially the application of philosophical ways of thinking into the realm of education. Philosophy helps the process of implementing mathematics education. Philosophy directs and guides all processes that take place in mathematics education, so that education is carried out according to the right path and direction. The purpose of this study is to examine in depth how the philosophy of mathematics education perspective as a field of knowledge. The research method in this study used the literature study method. Data collection was carried out by collecting data from the library, from research journals and books and then reading, recording and processing these sources as research data. Philosophy of mathematics education as a field of knowledge provides a clear basis and direction for mathematics to become a means of thinking in discovering and developing various new knowledge and will then communicate the results of this knowledge so that it becomes useful and usable. Based on the study of ontology and epistemology, in carrying out mathematics education activities as a field of knowledge it is necessary to apply a philosophy of mathematics which involves the use of logic and the involvement of students in constructing their knowledge in solving existing phenomena or problems. Based on the axiological study, mathematics education as a field of knowledge will have value if students in the implementation of mathematics education are organized by involving the connections of mathematics with other sciences and connections with previously acquired knowledge.

PENGGUNAAN MODEL DISCOVERY LEARNING TERHADAP PENINGKATKAN KEMAMPUAN KONEKSI DAN KOMUNIKASI MATEMATIK

This study aims to determine the better improvement of students' mathematical connection and communication abilities between those participating in learning with the Discovery Learning model and those participating in direct learning, as well as to find out the differences in increasing the mathematical connection and communication abilities of students in high, medium and low groups who take learning with Discovery Learning models. Data analysis used a two-mean difference test to determine better improvement between the experimental class and the control class and one-way ANOVA with the Scheffe test to determine better improvement between the high, medium and low groups. Based on the results of the study, it can be concluded that the increase in the mathematical connection and communication skills of students who take part in learning with the Discovery Learning model is better than students who take part in direct learning. The improvement of students' mathematical connection and communication skills in the middle group was better than the high and low groups who took part in learning using the Discovery Learning model

연결성 관점에서 본 수학과 교육과정 국제 비교·분석 연구: 한국, 싱가포르, 홍콩을 중심으로

The purpose of this study This study aimed to analyze the curricula of Korea, Singapore, and Hong Kong from the perspective of connection — one of the competences of the 2022 national revised mathematics curriculum — and to suggest theoretical ideas for the Korean curriculum. The curricula were divided into three categories, and the criteria for the classification of connection were set into four categories: horizontal and vertical connections in terms of mathematical internal connection, connection through other subjects, and real life in terms of mathematical external connection. The analysis showed that none of the three countries revealed horizontal connections in goals and characteristics. Regarding achievement standards, horizontal connections did not appear in Korea; moreover, this was the case for vertical connections in Singapore as well as vertical connections and connections through other subjects in Hong Kong. All types of connections were reflected in teaching, learning, and assessment, with the exception of Hong Kong’s horizontal connection. Therefore, Korea needs to stipulate internal connections in goals and characteristics, reflect horizontal connections in achievement standards, and specify vertical connections in teaching, learning, and assessment. Moreover, specific methods were discussed for Singapore and Hong Kong.

제수가 자연수인 소수의 나눗셈 계산 방법의 교육에 대한 논의

In this study, I analyzed and structured the calculation methods for (decimal numbers)÷(whole numbers). Using the structure as a framework, I examined the current education regarding calculation methods for (decimal numbers)÷(whole numbers) in Korea. There are three calculation methods for (decimal numbers)÷(whole numbers): [Method 1] Extending the method of long division of whole numbers, [Method 2] Considering the number of minimum units, and [Method 3] Converting to division of fractions. [Method 1], which is based on the inherent conceptual aspect of decimal numbers, has been treated insufficiently. Furthermore, [Method 2] seems to be implemented by unit conversions, which do not highlight the essential idea of the method. By contrast, excessive emphasis is placed on [Method 3], which is based on the relationship between decimal numbers and fractions. It is more suitable to focus on [Method 1] and [Method 2] until students understand the essence of each method. [Method 1] and [Method 2] may be introduced in real-life situations. Subsequently, [Method 3] can be used as an opportunity to examine the division of decimals, fractions, and whole numbers together, focusing on mathematical internal connections.

Interdisciplinary integration in teaching Mathematics as a means of forming pupils' mathematical literacy

This article discusses the problem of implementing interdisciplinary connections between a school course in Mathematics and other natural science disciplines at a secondary school as topical means of developing functional literacy among pupils. The mechanisms of interdisciplinary connection of Mathematics with other disciplines are listed. As a means of interdisciplinary communication of Mathematics with other disciplines, there can be the interaction of pupils in the information educational environment to search for new knowledge, to identify properties of a generalised character, to solve practice-oriented problems, to combine knowledge from various general education subjects, to teach Computer Science based on the transfer and generalisation of knowledge from related subjects. The article provides examples of practice-oriented tasks that illustrate the relationship of Mathematics with such secondary school disciplines as Computer Science, Biology, Geography, and Physics. It is noted that when solving such problems, schoolchildren learn various aspects of multi-level practice-oriented activities, including the use of computer capabilities and the global Internet. It is shown that mathematical constructions and their analogues are found even in areas of knowledge quite distant from exact sciences, such as literature, for example. It is substantiated that it is the solution of interdisciplinary problems that forms the meta-skills necessary for everyday life in pupils.

ANALYSIS OF MATHEMATICAL CONNECTION ABILITY IN LINEAR EQUALIES AND INEQUALITY ONE VARIABLE OF CLASS VII STUDENTS OF SMP DOMICILI VILLAGE NANOWA LEARNING YEAR

This study aims to describe students' mathematical connection abilities, especially in the matter of one-variable linear equations and inequalities. This research uses a type of qualitative research with a descriptive research approach. The informants of this study were 20 students of grade VII junior high school who live in Nanowa Village. The data were analyzed with the steps of data collection, data reduction, data presentation and drawing conclusions. The research instrument used was a test of students' mathematical connection abilities and interviews. The research findings show that: students do not understand the material concept of one-variable linear equations and inequalities, students do not understand the relationship between the concepts of one-variable linear equations and inequalities with other mathematical material topics such as exponents and squares, and in everyday life, students are unable to find the relationship between the procedure for solving linear equations and inequalities of one variable and the procedure for solving other mathematical material. This is because students do not understand the questions, students do not master the previous prerequisite material associated with the questions, students are confused in choosing the right completion procedure and students lack accuracy in carrying out arithmetic operations so they often make calculation errors. From these results and findings it can be concluded that students' mathematical connection abilities in the matter of linear equations and inequalities of one variable are still low. Through this research, it is suggested that students' mathematical connection abilities be further enhanced by working on more questions related to mathematical connections.

The implementation of RME approach using concrete objects media to improve the mathematical connection skill of fifth-grade elementary school students

Mathematical connection skill needs to be mastered. This study aims to determine: 1) implementing a realistic approach to enhance mathematical connection skills using concrete objects as media and 2) improving the mathematical connection skill by applying the Realistic Mathematics Education approach using concrete objects as media in fifth-grade elementary school students. The research subjects consisted of 17 students. The method of this study is Classroom Action Research. The data collection method is by tests, using mathematical connection questions as the instrument. The research results show that in Cycle I, eleven students (64.7%) passed the minimum score, while six students (35.3%) did not, with a classical average of 73.83. In Cycle II, 14 students (82.35%) passed, and three (17.65%) did not, with a classical average of 82.65. These findings indicate that applying the Realistic Mathematics Education (RME) approach with concrete objects as media can improve the mathematical.

STUDI LITERATUR MEDIA PEMBELAJARAN YANG DIGUNAKAN PADA MODEL PEMBELAJARAN PROBLEM BASED LEARNING

This research aims to describe the analysis result of the use of learning media in the Problem Based Learning model. The methods used in this research is literature study of 6 sources consists of 3 thesis and 3 journals discuss the use of learning media in the Problem Based Learning model in Indonesia published in the span of 2016-2022. From the calculation of the effect size, the average value is 0.79 which is categorized in the medium category. This shows that the use of learning media with Problem Based Learning in mathematics learning had a positive effect on improving the mathematical ability of students and the effect obtained is in the medium category. The mathematical ability could be improved based on the literature study such as student’s mathematical connection ability, student’s mathematical problem soving ability dan student’s mathematical numeracy literacy ability. The type of learning media according this research are animation, geogebra, microsoft power point, fun math book, cabri 3D v2 dan youtube. Hope for further research can be analyzed more deeply by using more sources so the result will be more maximal and varied.