Advanced Mathematical Knowledge Research Articles

Exploring the Cultivation Path of Algebraic Thinking Ability for Junior High School Students

With the deepening development of mathematics education, the cultivation of algebraic thinking has gradually become one of the core tasks of junior high school mathematics teaching. As a fundamental ability in mathematics, it not only has a positive promoting effect on students logical thinking and abstract ability, but also provides important tools and methods for students to further learn advanced mathematical knowledge and solve practical problems. Based on the understanding of mathematical algebraic thinking, this article mainly discusses the importance of students algebraic thinking and how to cultivate students algebraic thinking ability in general operations, logical reasoning, mathematical modeling, etc. and explores new methods and approaches to cultivate students algebraic thinking in junior high school teaching.

Evaluating the deep learning software tools for large-scale enterprises using a novel TODIFFA-MCDM framework

Deep learning (DL) is one of the most promising technological developments emerging in the fourth industrial revolution era for businesses to improve processes, increase efficiency, and reduce errors. Accordingly, hierarchical learning software selection is one of the most critical decision-making problems in integrating neural network applications into business models. However, selecting appropriate reinforcement learning software for integrating deep learning applications into enterprises’ business models takes much work for decision-makers. There are several reasons for this: first, practitioners’ limited knowledge and experience of DL makes it difficult for decision-makers to adapt this technology into their enterprises’ business model and significantly increases complex uncertainties. Secondly, according to the authors’ knowledge, no study in the literature addresses deep structured learning solutions with the help of MCDM approaches. Consequently, making inferences concerning criteria that should be considered in an evaluation process is impossible by considering the studies in the relevant literature. Considering these gaps, this study presents a novel decision-making approach developed by the authors. It involves the combination of two new decision-making approaches, MAXC (MAXimum of Criterion) and TODIFFA (the total differential of alternative), which were developed to solve current decision-making problems. When the most important advantages of this model are considered, it associates objective and subjective approaches and eliminates some critical limitations of these methodologies. Besides, it has an easily followable algorithm without the need for advanced mathematical knowledge for practitioners and provides highly stable and reliable results in solving complex decision-making problems. Another novelty of the study is that the criteria are determined with a long-term negotiation process that is part of comprehensive fieldwork with specialists. When the conclusions obtained using this model are briefly reviewed, the C2 “Data Availability and Quality” criterion is the most influential in selecting deep learning software. The C7 “Time Constraints” criterion follows the most influential factor. Remarkably, prior research has overlooked the correlation between the performance of Deep Learning (DL) platforms and the quality and accessibility of data. The findings of this study underscore the necessity for DL platform developers to devise solutions to enable DL platforms to operate effectively, notwithstanding the availability of clean, high-quality, and adequate data. Finally, the robustness check carried out to test the validity of the proposed model confirms the accuracy and robustness of the results obtained by implementing the suggested model.

Preservice teachers' mathematical knowledge for teaching: Focus on lesson planning and reflection

AbstractMathematics teacher educators have suggested preservice mathematics teachers' (PMTs') practices provide evidence of their mathematical knowledge for teaching (MKT). PMTs engage in approximations of practice to rehearse teaching. To explore connections between MKT and PMTs’ practices, we operationalized six MKT subdomains in terms of approximations of practice. We used that operationalization to investigate, which subdomains were evidenced in PMTs’ lesson plans and their lesson plan reflections. Eleven secondary PMTs participated in our study. Using a qualitative content analysis method, we found Knowledge of Content and Teaching most evidenced, suggesting that the PMTs demonstrated MKT when they described mathematical tasks and associated instructional activities. Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks. However, the PMTs struggled to plan for activities by building on students’ potential mathematical thinking and reasoning. Only limited instances of Horizon Content Knowledge suggest that the PMTs had limited opportunities to demonstrate how their advanced mathematical knowledge connected with their mathematics teaching in the approximations of practice provided. We propose alternative forms of approximations of practice to optimize PMTs’ opportunities to demonstrate and develop MKT.

Exploring the Relevance of Advanced Mathematical Knowledge for Secondary-Level Instruction

The usefulness of advanced mathematics in teaching is referred to as its relevance for secondary-level mathematics instruction. This paper is based on research that explores teachers' views on the relevancy of advanced mathematics and their expectations from advanced mathematics courses for teacher preparation programs. Ten secondary-level mathematics teachers participated in this study. semi-structured interviews with open-ended questions and focus group discussions were utilised for data collection and thematic for data analysis. The finding indicates that advanced mathematics is relevant for developing mathematical proficiency, mathematical, and logical thinking, knowledge of mathematical context mathematical activities and increased problem-solving skills and professional competencies. Along with advanced mathematical knowledge, technological and research knowledge, and skills are expected from undergraduate teacher preparation courses. Therefore, it is suggested to incorporate the expected form of knowledge in the teacher preparation program.

Artificial Intelligence (AI) Revolution in Research: Transforming Data into Discovery

Artificial Intelligence (AI) has emerged as a Atransformative force in the dominion of research, reshaping the landscape across diverse fields. AI refers to the simulation of human intelligence by a system or a machine. It implies a vast term that encompasses machine learning and deep learning. AI is based firstly on 'intelligence' which refers to cognitive capacity of an individual to plan learn and interpret. Secondly 'techne', a more wideranging ability to solve problems using technological objects. Although the intelligence remains the same measured by intelligence quotient (IQ) for many decades now, it is the change in techne that leads to amplification of intelligence. Use of AI in healthcare and biomedical sciences is based on development of algorithms. This requires a sound knowledge of programming languages, advanced mathematics and statistics. In healthcare AI cannot be used fairly if these skills are lacking. However visual and no code programming tools available now with free data science tools make it easy for clinicians, researchers and health journalists.

ІМІТАЦІЙНА МОДЕЛЬ РУХУ ГРУПИ АВТОНОМНИХ ОБʼЄКТІВ-ДРОНІВ В ТРИВИМІРНОМУ ПРОСТОРІ

Purpose. To develop a simulation model of the motion of a group of objects in three-dimensional space to generate datasets and individual samples for further use in training neural networks and developing a mutual positioning system. Methodology. Analytical review, the automated control theory, imitation modelling. Results. Using Python, a 3D model has been developed. Digital and graphical representations of the linear trajectory of movement for a group of 4 drones based on input data, including coordinates of initial and final points and an error range, have been received. Trajectories with compensated and uncompensated deviations have been compared. Approximation and simplifications have been justified by the goal of enabling the use of the model by those without fundamental knowledge of control theory, system modelling or advanced mathematics. Code testing and benchmarking have also been conducted, proving the financial feasibility of the solution. The model will be used by the authors for further research, including assessing the likelihood of obtaining biassed data from sensors of some drones in a swarm and finding ways to correct it. The source code of the developed model is provided for open access. Scientific novelty. An imitation model of interaction and movement in three-dimensional space for a group of autonomous objects has been proposed. These objects, capable of collective decision-making based on swarm intelligence, aim to compensate for errors or interference. Practical significance. Testing in the AWS cloud environment has demonstrated that even with minimal computational power the proposed imitation model yields adequate results, suitable for neural network training.

Foreign market selection of suppliers through a novel REF-Sort technique

PurposeThe present study aims to provide a practical and robust assessment technique for assessing countries' investability in global supply chains to practitioners. Thus, the proposed approach can help decision-makers evaluate and select appropriate countries in the expansion process of the global supply chains and reduce risks concerning country (market) selection.Design/methodology/approachThe present study proposes a novel decision-making approach, namely the REF-Sort technique. The proposed approach has many valuable contributions to the literature. First, it has an efficient basic algorithm and can be applied to solve highly complicated decision-making problems without requiring advanced mathematical knowledge. Besides, some characteristics differentiate REF-Sort apart from other techniques. REF-Sort employs the value or value range that reflects the most typical characteristic of the relevant class in assignment processes. The reference values in REF-Sort and center profiles are similar in this regard. On the other hand, class references can be defined as ranges in REF-Sort. Secondary values, called successors, can also be employed to assign a value to the appropriate class. REF-Sort can also determine the reference and successor values/ranges independently of the decision matrix. In addition, the proposed model is a maximally stable and consistent decision-making tool, as it is resistant to the rank reversal problem.FindingsThe current papers' findings indicate that countries have different features concerning investment. Hence, the current paper pointed out that only 22% of the 95 countries are investable, whereas 19% are risky. Thus, decision-makers should make detailed evaluations using robust, powerful, and practical decision-making tools to make more reasonable and logical decisions concerning country selection.Originality/valueThe current paper proposes a novel decision-making approach to evaluate. According to the authors' information, the proposed model has been applied to evaluate investable countries for the global supply chains for the first time.

Some Theoretical References Related to Advanced Mathematical Thinking

Background: In Mathematics Education, the theoretical reference of Advanced Mathematical Thinking has contributed to some research. However, others opt for a different but somehow related theoretical reference. Identifying the similarities and differences between those references can elucidate the researcher’s motivations when adopting one or another reference. Objective: to synthesise three studies involving Mathematical Creativity, Advancing Mathematical Activity and Advanced Mathematical Knowledge, respectively, and to illustrate possible contributions of those theoretical references in the analysis of a written production. Design: the research is qualitative and theoretical and speculative so that theoretical relationships were carried out and used later in the analysis of the written production. Setting and Participants: the research involves a participant with a degree in mathematics with whom the researchers had a virtual interaction to present the questions. Data collection and analysis: the procedures performed in the resolution of questions prepared based on the OBMEP question bank are analysed and, with the support of this example of analysis, possible contributions of each reference in the analysis of a written production are discussed. Results: none of the references covers all the thinking mobilised in the activity, but each one favours the focus on specific aspects that were verified in the written production of the research participant, in line with the similarities and divergences emphasised in the theoretical appreciation of each reference. Conclusions: the knowledge of different theoretical frameworks provides the teacher with foundations to exercise their teaching practice and the researcher with options for an adequate choice for their research.

Integration of recent developments in transient heat conduction and mass transfer for solid sphere into teaching

AbstractTo bring research and teaching together in engineering education, the term scholarship can be categorized into four types: research scholarship, applied research scholarship, integrative research scholarship, and teaching scholarship. Linking traditional research and classroom‐oriented teaching scholarship is a common and recurring challenge for many academics and active involvement in the latest research developments would directly improve the quality of teaching. In the case of research‐teaching of a heat transfer course, the teaching methodologies integrating academic research offer a great experience for students to learn various heat transfer concepts in a limited time frame. From the view of the research‐teaching nexus, analytical solving methods introduced in textbooks are reasonable ways to apply the integrated teaching strategies, which is highlighted by their fast results and versatility. Furthermore, a complete analytical solution to the subject matter could provide the fundamental knowledge necessary to comprehend the behavior of the associated thermal systems. This article demonstrates an analytical solution to the problem of heat and mass transfer for spherical particles in an airflow field. This investigation elucidates the basic principles in combination with advanced mathematical knowledge learned and its application to frontier research to improve the teaching of heat conduction and enhance the understanding of students at an undergraduate or graduate level. It greatly improves the engagement of students, promoting the theoretical concepts in engineering applications. The introduction of the latest developments in analytical solutions to teaching activities in the classroom will undoubtedly enhance the motivation and the horizon of students.

Correlation analysis of platform intelligent electric screen doors and accidents in urban rail transit

In order to determine the main factors affecting the accident of passenger falling off the track, 223 cases of subway station passengers falling off the track in mainland China from 2000 to 2020 (up to September) were collected as the research object. Among them, 13% of the samples had platform screen doors, and 87% had no platform screen doors. Considering the limitations of the statistical sample data, in order to accurately and scientifically study and judge the effectiveness of PSD in reducing the accident of passengers falling off the track, this paper uses the knowledge of advanced mathematics and statistics, combined with the passenger volume information of the station, to analyse and verify the relationship between the existence of PSD and the occurrence of the accident. The statistical results of the case show that the installation of platform screen doors in the station can reduce the occurrence of passenger falling off the rail to a certain extent. The mathematical analysis shows that the correlation between the installation of platform screen doors in the station and the passenger falling off the rail in the operation process of Urban Rail Transit reaches 0.99, which is consistent with the statistical results of this case.

Understanding primary school children’s learning on addition of fractions

Fractions are universally known to be difficult to learn. Learning basic fractions may not be sustainable if one is to apply it to more advanced mathematical content knowledge as it is interlinked with other mathematics topics. In the effort to understand how children grasp the learning of fractions additions, a study was carried out with a random sampling of Years 4, 5 and 6 children in Brunei Darussalam. The research participants consisted of 33 children who are currently studying at government and private primary schools in Brunei. A pen-and-paper self-designed test was used as an instrument in collecting the data. The quantitative data was analysed by using a descriptive analysis method. The overall reliability of the self-designed tests indicated that the items have relatively high internal consistency with a Cronbach alpha of .859. The findings revealed that the children’s understanding is stronger on questions with answers regarding proper fractions rather than improper fractions. This could be further explored through future studies employing a qualitative approach.

학교 수학과 대학 수학 사이의 연계성에 대한 중등교사의 전문성 분석

The purpose of this study is to analyze perceptions and advanced mathematical knowledge of secondary teachers’ connectivity between secondary and undergraduate algebra. Survey, created based on prior research on teacher perception and advanced knowledge teaching algebra, was administered to 49 in-service secondary mathematics teachers. Analysis of the survey suggests that teachers do not well perceive how contents of undergraduate algebra can be applied to teaching secondary one. Moreover, despite that the participant teachers were experienced in teaching upper secondary mathematics, most of them felt difficulties linking undergraduate algebra to their school curriculums, and their strength was more on procedural calculation than on conceptual understanding of algebra. The study has an implication towards future research that more research is needed to investigate and to enhance secondary teachers’ capacity to connect what they learn in undergraduate with what they teach in school.

Educational Stakeholders’ Independent Evaluation of an Artificial Intelligence-Enabled Adaptive Learning System Using Bayesian Network Predictive Simulations

Artificial intelligence-enabled adaptive learning systems (AI-ALS) are increasingly being deployed in education to enhance the learning needs of students. However, educational stakeholders are required by policy-makers to conduct an independent evaluation of the AI-ALS using a small sample size in a pilot study, before that AI-ALS can be approved for large-scale deployment. Beyond simply believing in the information provided by the AI-ALS supplier, there arises a need for educational stakeholders to independently understand the motif of the pedagogical characteristics that underlie the AI-ALS. Laudable efforts were made by researchers to engender frameworks for the evaluation of AI-ALS. Nevertheless, those highly technical techniques often require advanced mathematical knowledge or computer programming skills. There remains a dearth in the extant literature for a more intuitive way for educational stakeholders—rather than computer scientists—to carry out the independent evaluation of an AI-ALS to understand how it could provide opportunities to educe the problem-solving abilities of the students so that they can successfully learn the subject matter. This paper proffers an approach for educational stakeholders to employ Bayesian networks to simulate predictive hypothetical scenarios with controllable parameters to better inform them about the suitability of the AI-ALS for the students.

Educational Challenges in Computer-based Finite Element Analysis and Design of Structures

Computer simulations and computational methods, such as the Finite Element Analysis (FEA) have become essential methodologies in science and engineering during the last decades, in a wide variety of academic fields. Six decades after the invention of the digital computer, advanced FE simulations are used to enhance and leapfrog theoretical and experimental progress, at different levels of complexity. Particularly in Civil and Structural Engineering, significant research work has been made lately on the development of FE simulation codes, methodologies and validation techniques for understanding the behavior of large and complex structures such as buildings, bridges, dams, offshore structures and others. These efforts are aimed at designing structures that are resilient to natural excitations (wind loads, earthquakes, floods) as well as human-made threats (impact, fire, explosion and others). The skill set required to master advanced FEA is inherently interdisciplinary, requiring in-depth knowledge of advanced mathematics, numerical methods and their computational implementation, as well as engineering sciences. In this paper, we focus on the importance of sound and profound engineering education and knowledge about the theory behind the Finite Element Method to obtain correct and reliable analysis results for designing real-world structures. We highlight common mistakes made by structural engineers while simulating complex structures and the risk of structural damage because of human-made mistakes or errors in the model assumptions. The event of the collapse and eventual sinking of a concrete offshore platform in the North Sea is presented as a case study where a serious error in the finite element analysis played a crucial role leading to structural failure and collapse.

Teaching Quantum Mechanics with Blended Learning

Quantum mechanics is very difficult for the beginners to learn, because of its high degree of abstraction and the high request to the advanced mathematical knowledge. For undergraduate students majoring in materials physical at the University of Jinan, the quantum theory is particularly difficult. The main reason is their fragile mathematical basis and students' indifference. The aim of our developmental work is to supply the mathematical knowledge reasonable and inspire the students learning interest in the limited class hour, based on the blended learning. We share here the concrete examples of how blended learning has been used into quantum mechanics teaching. Our blended learning practice provides ideas for developing new ways of quantum mechanics teaching. Whether camera-ready or not, please follow the guidelines.

Socio-epistemological research on mathematical modelling: an empirical approach to teaching and learning

This work deals with the role of social practice in the construction of models related to advanced mathematical knowledge; specifically, with the way an individual uses the notion of variation in different phenomena while predictive thinking is the underlying principle in the mathematical process. The mathematical modelling of successive variation in mathematics and science is addressed through the interpretation of graphs in two cases of different nature: the diagnosis of a cardiologist on the health of a patient (a non-deterministic phenomenon) and the arguments of high school students facing the task of filling containers (a deterministic phenomenon). An emerging theory in the field of mathematics education—the socio-epistemological theory—explains the problem of construction of mathematical knowledge. The constructs of this theory are elaborated with an empirical basis; and the legitimacy of all forms of knowledge is assumed, be this knowledge popular, technical or scientific, since all of them constitute human wisdom. (In contrast, other contemporary approaches examine only one of the forms of knowledge).

Complementing Mathematics Teachers’ Horizon Content Knowledge with an Elementary-on-Advanced Aspect

Although Ball and her colleagues provided empirical evidence to support the existence of the six sub-domains in mathematical knowledge for teaching (MKT) and further explained or defined the majority of these sub-domains, there were few explanations of what horizon content knowledge (HCK) embedded in MKT meant and they merely provided ideas about HCK. Many researchers attempted to provide some teaching incidents and exemplification to interpret the construct of HCK. Moreover, they thought teachers’ studies of tertiary mathematics are useful for classroom teaching practice. Their discourse and instantiation of HCK was correspondent with a higher perspective on elementary mathematics mentioned by Felix Klein (1924), but was not entirely coincide with a kind of elementary perspective on advanced knowledge introduced by Ball and Bass (2009). This study lasted 1 years, and data collection included in-depth interviews, classroom observation and video analysis. We provide a shared classroom teaching incidence and illustrations to explain and to describe the construct of HCK. HCK not only is a kind of elementary perspective on advanced mathematical knowledge, but also complements to a higher perspective on elementary mathematics. Furthermore, HCK could be seen as a reciprocal pathway between the elementary and advanced mathematical knowledge.

Predicting the direction of stock markets using optimized neural networks with Google Trends

Abstract The stock market is affected by many factors, such as political events, general economic conditions, and traders’ expectations. Predicting the direction of stock markets movement has been one of the most widely investigated and challenging problems for investors and researchers as well. Many researchers focus on stock market analysis using advanced knowledge of mathematics, computer sciences, economics and many other disciplines. In this paper, we present an improved sine cosine algorithm (ISCA), which introduces an additional parameter into the sine cosine algorithm (SCA), to optimize the weights and basis of back propagation neural networks (BPNN). Thus, ISCA and BPNN are combined to create a new network, ISCA-BPNN, for predicting the directions of the opening stock prices for the S&P 500 and Dow Jones Industrial Average Indices, respectively. In addition, Google Trends data are taken into consideration for improving stock prediction. We analyze two types of prediction: Type I is the prediction without Google Trends and Type II is the prediction with Google Trends. The predictability of stock price direction is verified by using the hybrid ISCA-BPNN model. The experimental results indicate that ISCA–BPNN outperforms BPNN, GWO-BPNN, PSO-BPNN, WOA-BPNN and SCA-BPNN in terms of predicting the direction of the opening price for both types and significantly for Type II. The hit ratios for ISCA-BPNN with Google Trends reach 86.81% for the S&P 500 Index, and 88.98% for the Dow Jones Industrial Average Index. Our results show that Google Trends can help in predicting the direction of the stock market index.

Forms of Knowledge of Advanced Mathematics for Teaching

1.IntroductionTeachers, particularly at the middle and secondary level, are frequently required to take advanced mathematics courses (e.g., real analysis, abstract algebra, etc.) as part of their teacher preparation program. This is based on a general sense of the importance of strong content knowledge for teachers. However, many teachers at some point question the relevance of these advanced courses for their future teaching careers (e.g., Zazkis & Leikin, 2010). Indeed, as Monk (1994) noted, the quantity of advanced mathematics preparation does not guarantee teaching quality; that is, one's own mathematical understanding does not necessarily translate into an ability to enhance the understanding of others. With the recent focus on practice-based approaches to conceptualizing teacher's mathematical knowledge (e.g., Ball, Thames, & Phelps, 2008; Petrou & Goulding, 2011) - which has led to documenting specific ways teachers use their knowledge of mathematics in their professional work - much more needs to be done to identify, define, and document how advanced mathematics may inform the actual work of teaching. Although this problem has been observed for a long time - Felix Klein (1932) made comments about this in the early 20th century - little progress has been made in this regard.In contrast to defining specific content areas in advanced mathematics or undergraduate course recommendations (e.g., CBMS 2001; 2012), this paper focuses instead on more general forms of knowledge of advanced mathematics that may be productive for the teaching of school mathematics. Following a grounded theory approach (e.g., Strauss & Corbin, 1990), we developed a framework of five forms of knowledge of advanced mathematics for teaching that, while not necessarily an exhaustive or exclusive list, moves the discussion of mathematical knowledge for teaching - particularly of advanced mathematics - beyond just listing what content teachers need to know, and toward a more general conception of how knowing advanced mathematics could positively interact with the work of teaching. We begin by reviewing related literature and situating this work within it.2.Literature2.1.Teachers' Knowledge of advanced MathematicsRecent efforts to conceptualize the mathematical knowledge required for teaching have incorporated practice-based approaches to teacher knowledge - that is, the content knowledge teachers need should be relevant for the practices and work of teaching. In particular, researchers have used this perspective to conceptualize different "domains" of knowledge, at both the elementary and secondary levels (e.g., Ball, Thames, & Phelps, 2008; McCrory, et al., 2012; Heid, Wilson, & Blume, 2015). Notably, many of these - and others' (e.g., Zazkis & Leikin, 2010)- allude to the importance of knowing advanced mathematics. For example, the Mathematical Knowledge for Teaching (MKT) framework (Ball, Thames, & Phelps, 2008) included the domain Horizon Content Knowledge (HCK), which is related the conversation about advanced mathematics (e.g., Wasserman & Stockton, 2013; Jakobsen, Thames, & Ribeiro, 2013; Zazkis & Mamolo, 2011); and in translating to secondary mathematics teaching, McCrory, et al. (2012) included knowledge of advanced mathematics as a domain of their Knowledge of Algebra for Teaching (KAT) framework. Heid, Wilson, and Blume (2015) described Mathematical Understandings for Secondary Teaching (MUST), which, although not connected explicitly to advanced courses, have connections to larger mathematical practices in the discipline which are often honed in courses such as abstract algebra, real analysis, or an introduction to proofs course. Thus, many regard advanced mathematics as important for teaching.However, finding explicit connections to practice has been more difficult. This fact is perhaps captured best by the "provisional" nature of the HCK domain - it was left lessdeveloped and less-defined than other parts of the MKT framework. …

A curious case of superscript (−1): Prospective secondary mathematics teachers explain

In mathematics the same symbol – superscript (−1) – is used to indicate an inverse of a function and a reciprocal of a rational number. Is there a reason for using the same symbol in both cases? We analyze the responses of prospective secondary school teachers to this question. The responses are presented in a form of a dialogue between a teacher and a student and are accompanied with participants’ commentary on their choices of instructional approaches. The data show that the majority of participants treat the symbol ☐−1 as a homonym, that is, the symbol is assigned different and unrelated meanings depending on a context. We discuss how knowledge of advanced mathematics (or lack of it) can guide instructional interaction.