Every Tree is a Subtree of a Graceful Unicyclic Graph
A decomposition of a graph K into a set of graphs {G1, G2, ..., Gt} is a partition (E1, E2, ..., Et) of E(K) such that ∼= Gi, for i,1
Reciprocity and contrapositive in the Pythagorean theorem
The reciprocal of the Pythagorean theorem is the same as explaining that a triangle is right-angled. For example, consider a triangle whose lengths are 3, 4 and 5. To explain that this triangle is a right-angled triangle, we need to use the reciprocal of the Pythagorean theorem. On the one hand, we have 32+42=9+16=25. On the other hand, it's true that 52=25. So 32+42=52 and, by the reciprocal of the Pythagorean theorem, it's a right-angled triangle. If we want to explain that a triangle is a right-angled triangle, we need to sum the squares of the two shortest lengths and check whether this sum is equal to the square of the longest side.
A Treatise on the Theory of Invariants
A Treatise on the Theory of Invariants, a classical book, has been considered important throughout the human history, and so that this work is never forgotten we at Alpha Editions have made efforts in its preservation by republishing this book in a modern format for present and future generations. This whole book has been reformatted, retyped and designed. These books are not made of scanned copies of their original work and hence the text is clear and readable.
Preparing for Calculus
The book will enable the reader to assess their readiness for calculus. It provides a review and practice with the math needed for early success in a calculus course.High school, home-schooled and college students who face calculus as their next math course will benefit from this book.
Maths in Construction
This work is the result of Marcelo Azevedo de Souza's dissertation in natural sciences and maths at the University of Blumenau (FURB) - Blumenau/Santa Catarina - Brazil. The main purpose of this book is to show readers, teachers and students of youth and adult education and ethnomathematics researchers, that the everyday mathematics used in construction can be related to the mathematics taught in schools and that both can go hand in hand in an organised way, with the aim of developing excellence in the teaching and learning of this subject. The subjects chosen in this book are varied and come from the student's everyday life, so they can be applied in any classroom in basic and higher education. The book features a brief introduction, explaining the choice of topic, general and specific objectives, the structure of each chapter, followed by creative and challenging activities, a timetable for application, research methods, data collection, resources to be used, mathematical games for the construction industry, and finally, tips for future application.
Formal Methods Teaching
This book constitutes the proceedings of the 6th International Workshop on Formal Methods Teaching, FMTea 2024, which was held in Milan, Italy, on September 10, 2024. The 7 full papers included in these proceedings were carefully reviewed and selected from 9 submissions. The book also contains one invited talk in full paper length. The papers focus on learning formal methods for the purpose of teaching and self-learning.
Information Theoretic Measures
Communication system is vital for message transmission, affected by noise. Researchers contribute to improving communication. Information theory development is significant, enhanced by C. E. Shannon's work. Shannon model encodes and decodes messages. Researchers generalize information measures for various applications. This book focuses on information theoretic measures, their analysis and applications. Information measures are applied in data mining, including weighted residual information measure. Classification methods and results are compared for improvement. Information theoretic measures in ID3 algorithm find information gain as splitting criteria for data. ID3 algorithm applied to census 2011 data of India to classify attributes into States and Union Territories. Decision tree made using Renyi entropy, rules developed to decide project implementation region. Varma entropy based ID3 algorithm used to analyze health services in India up to 2015. C4.5 algorithm applied on same data with discrete attribute values to develop decision tree. Weighted generalized residual information measure of order α and type β developed as extension of information theoretic measures.
Enhancing Security in Fuzzy Networks Through Efficient Domination
Let G(V, E) be a finite simple connected graph of order m with vertex set V and edge set E. A dominating set S ⊆ V(G) is called an efficiently dominating set if, for every vertex u ∈ V(G), N[u] ∩ S = 1, where N[u] denotes the closed neighborhood of the vertex. Using efficient domination techniques and labeling, we constructed the different types of fuzzy networks. An algorithm has been framed to encrypt and decrypt the secret information present in the network. The mathematical modeling of a strong type of fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of efficient domination on types of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. The main purpose of types of fuzzy network are encryption and decryption, our contributions to this research is to build a novel combinatorial technique to encrypt and decrypt the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided, along with the encryption and decryption algorithms.
Teaching Knowledge and Inclusive Mathematics Education
"Even with the advances in Brazilian legislation related to Inclusive Education, the daily practice of schools and teacher training courses is still a long way from what is desirable. In the specific case of teaching mathematics to deaf students, despite the obligation to have interpreters in schools and the subject of Libras in degree courses, the scenario is not very different. Training teachers to work in this scenario is an urgent task, both in degree courses and in spaces dedicated to continuing education. The purpose of this book, which is the result of qualitative research, is to analyse the possible contributions of an extension course to the mobilization of teaching knowledge related to the inclusion of deaf students in Mathematics classes in regular classes".(...) Viviane C.Costa*
Transition to Advanced Mathematics
This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics.
Introduction to Calculus
Introduction to Calculus provides a comprehensive introduction to fundamental concepts in calculus and their applications, covering all of Calculus 1 and some of Calculus 2. The book starts with functions and limits, followed by differential calculus, and then moves on to integral calculus and a brief discussion of differential equations. It ends with the complete solution for a sample exam based on the AP Calculus AB course. Students can access a free, interactive ebook version of the text using a URL provided inside the print version. Further supplemental materials, including videos, practice problems, exercises and quizzes, are available online in the companion Wolfram U course.
Boundary Value Problems on Time Scales, Volume I
This book is devoted to the qualitative theory of boundary value problems on time scales. It summarizes the most recent contributions in this area.
Statistics and reality in students' daily lives
Statistics is a part of applied mathematics that studies methods for collecting, organising, analysing and interpreting data, since its study is aimed, among other things, at decision-making. So we have to experience it so that we can understand it better without skipping any steps. In this way we will better interpret and consequently improve the skills and competences that, although important, should not be measured only through a written test, but through the application of a concept constructed and experienced by the student in everyday life. Statistics therefore seeks situations in which problem-solving is meaningful for the student and mobilises their cognitive resources. According to the PCN's (2002), the teaching of mathematics can help students develop skills related to representation, comprehension and investigation, as well as socio-cultural contextualisation. This means placing students in a teaching-learning process that values mathematical (statistical) reasoning and also enables the teacher to work on content in an interdisciplinary way and broaden the understanding of its meanings, thus contributing to the formation of knowledge.
Mathematics Success Mantra
"I aspire to be a good human teacher who possesses wisdom to teach my students, and show them that I genuinely care, and am concerned with their learning, a good human teacher who cannot be replaced by technology. As the old saying goes... "Give a man a fish, and you feed him for a day." I myself identify the despite of efforts, some students struggle to secure good results. I believe that the key to success lies in applying hard work correctly. The plethora of books which available today can often confuse students, . They putting them in a tough spot. Are these books truly beneficial? It's doubtful.After much consideration, I've decided to publish a book that caters to all types of students. I have intentionally focused on presenting fundamental ideas and equations. Mastering these basic concepts and principles will help students achieve top grades. However, in this competitive environment, securing good grades is crucial.Here are my recommendations for problem-solving: *Read the problem thoroughly;*Understand what is being asked of you;*Gather relevant concepts and formulae;*Apply them to the problem;*Draw diagrams when necessary.I believe that the fundamental ideas, rules, and equations are essential for solving most problems. This is why I introduced the book 'Mathematics Success Mantra.' I hope 'The Success Mantra' will be well-received and beneficial to all students.I dedicate this book to Sri J.R.L.V. Sharma, my math teacher, who has been a significant influence on my life. He transformed me from an average student into a successful math teacher, and Ravi Kumar kothapalli, the author of maths books.I have worked diligently to ensure this book is error-free. However, some printing errors may still be present. I would greatly appreciate it if you could inform me of any errors you find. We welcome constructive feedback and suggestions for improvement, which will be included in the next edition.-Ravi Kumar kothapalli
Mining Complex Networks
This book concentrates on mining networks, a subfield within data science. Mining complex networks to understand the principles governing the organization and the behaviour of such networks is crucial for a broad range of fields of study.
Artificial Intelligence and Deep Learning in Sensors and Applications
The aim of this reprint is to address increasingly complex human problems by utilizing various sensors to collect data, enabling the formulation of solutions through deep learning and artificial intelligence (AI). This trend creates a high demand for sensors while presenting new challenges in developing sensor devices and applications across various fields, such as healthcare, manufacturing, agriculture, transportation, construction, and environmental monitoring. For instance, in environmental monitoring, AI-integrated sensors rapidly analyze large datasets to identify real-time patterns and trends, enhancing weather forecasting accuracy by gathering data from multiple sources. In industrial settings, AI-enhanced sensors optimize manufacturing by monitoring equipment health, predicting failures, and proactively scheduling maintenance. This reprint compiles contributions on AI and sensor technology, sharing ideas, designs, applications, and deployment experiences across various fields, including smart manufacturing, construction, autonomous vehicles, traffic monitoring, object recognition, image classification, speech processing, and human behavior analysis.
Mathematics for Business Decisions
Mathematics for Business Decisions has been prepared for the students who want to learn mathematics and mathematical calculations from elementary level to advance level successively. The more specialty of this book is that, all the definitions are described with easy way and relevant examples to develop the theoretical knowledge of the students and the mathematical calculations have been presented meticulously with step by step of every part so that any student can figure out independently at home effortlessly. Finally, by studying this book, students will be able to decide how to pricing the products and services after analyzing all the costs and then calculating the gross or net profit. Also they will able to calculate all the business taxes and the actual business profit after deduction of all the taxes and other deductions. In addition, the knowledge gathering from this book helps improving decision making about different aspects of business such as costs, raw materials, marketing, advertising, strategies for the short and long term etc. and analyzing unforeseen situations and making decisions in favour of the company.
Expertise Scilab
Embark on a comprehensive journey into the world of Scilab, a powerful open-source software for scientific computing and data visualization. This meticulously crafted guide will equip you with the essential knowledge and skills to harness Scilab's capabilities for a wide range of computational tasks. This book is primarily intended for graduate, and postgraduate students especially scholars working in any domain of computation techniques. In the author's opinion, the finest software package for solving engineering problems is MATLAB, which is capable of conducting advanced mathematical and engineering computations whose versatility has transformed the personal computer industry. This book is intended to teach the student/reader how to use MATLAB through the use of practical examples. MATLAB is a software package that needs to be purchased for use. This is one of the reasons why Scilab exists. Scilab is currently being developed for programmers who wish to totally migrate their work to an open-source platform but are willing to amend their MATLAB files if necessary. This document contains Scilab commands, functions, and statements that are executable. The author makes no claim to content or style originality but has taken care to present the material clearly and sequentially with detailed discussion.
The Little Book of Math Into English
This colorful and concise little book is uniquely tailored for those who write mathematical texts at any level and are eager to improve their English writing skills. The easy-to-read guide focuses on helping the writer avoid common English mistakes in mathematical writing. With just a few minutes of engaging, light reading each day, the reader will learn to create clearer, more readable, math texts. The book covers 23 crucial topics, ranging from correct article and preposition usage to proper usage of dashes, conjunctions, and prepositions. It also addresses the construction of direct sentences, effective introductory phrases for formulas, and more. As a bonus to the reader, 'Practice makes perfect' exercises relating to each topic are freely accessible on this book's Springer website. Appendix A gives a quick tutorial on grammatical terms and constructs. Appendix B looks at ChatGPT and the positive aspects of its powerful capabilities. Additionally, Paul Halmos's article on 'How to write mathematics' is included in Appendix C. It deals with the mathematical aspects of writing.
The Elements of Advanced Mathematics
This edition offers a more instructive Preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
Boundary Value Problems on Time Scales, Volume II
This book is devoted to the qualitative theory of boundary value problems on time scales. It summarizes the most recent contributions in this area.
Modeling Change and Uncertainty
This book offers a problem-solving approach. The authors introduce a problem to help motivate the learning of a particular mathematical modeling topic. The problem provides the issue or what is needed to solve using an appropriate modeling technique.
Ergodic Theory and Related Topics
No detailed description available for "Ergodic Theory and Related Topics".
Numerical Analysis of Selected Semilinear Differential Equations
No detailed description available for "Numerical Analysis of Selected Semilinear Differential Equations".
A New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients
No detailed description available for "A New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients".
Examining the Challenges and New Methods
The importance and necessity of evaluation in the process of educational activities can be examined from two perspectives. The first point of view is related to the planner or teacher. In fact, in planning teaching and making decisions during educational activities, in order to obtain reliable information on the preparation and academic progress of learners and the effectiveness of their classroom activities through evaluation, the teacher finds information and awareness about three categories. These categories include: A) Problems and deficiencies in the elements of the educational plan (goals, content, methods, materials and educational tools); B) The level of academic progress of learners and their preparation for further education and C) The efficiency of teaching and the adequacy of the tools used in measuring the academic progress of learners. The second point of view is related to learners. In such a way that through evaluation, learners are encouraged to develop their positive aspects and solve their educational deficiencies with the help and guidance of the teacher, while being aware of the positive and negative aspects of their performance in learning.
An Introduction to the Distance Laplacian Spectra of Graphs
In this book, we determine the distance Laplacian spectra of graphs obtained by various graph operations.We first obtain the distance Laplacian spectrum of the join of two graphs in terms of adjacency spectra. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is a union of two graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs. We then find the universal adjacency spectra of the join of two regular graphs with distinct degrees and the join of two regular graphs in which of the graphs is a union of two regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. We obtain the distance Laplacian spectrum of Indu-Bala product of graphs in terms of their Laplacian spectra. We then obtain the distance signless Laplacian spectrum of Indu-Bala product of graphs in terms of their signless Laplacian spectra. Finally, we prove that the distance energy of the complete split graph always increases when an edge is deleted from it.
Scalar Conservation Laws
This book are notes prepared for the PhD courses that the author has been teaching during the last 10 years. The material available in the already existing literature (papers and essays) has been collected in this unique text, presenting the results with all the details for the reader's convenience, fixing a unified notation, and providing a consistent framework for the subject. These notes cover many of the arguments that usually can be found in high level essays, where the proofs are simply sketched, and in papers, which are not easily available and not always self-contained.This book is intended for1. PhD students in Mathematics, Physics and Mechanical Engineering in order to learn the basic features of nonlinear scalar equations, 2. researchers interested in nonlinear hyperbolic PDEs in order to learn the details behind some known and deep results on nonlinear scalar equations, 3. teachers of courses on nonlinear PDEs.The readers are expected to know the basic measure theory and Sobolev spaces.
Fixed Point Results on Various Metric Spaces with Applications
This book "Fixed Point Results on Various Metric Spaces with Applications" aims to present advanced results in fixed point theory that expand upon, enhance, and generalize classical and recent findings in this field. It also explores various research avenues within the metric fixed-point theory. It offers opportunities for more investigation and serves as a readily available reservoir of information. This book is suitable for young researchers interested in pursuing research in fixed-point theory. It provides the latest advancements in the field, offering new techniques for proving the existence of various types of fixed points, such as fixed points, fixed interval circles, fixed discs, coincidence points, common fixed points, coupled common fixed points, quadruple fixed points, quadruple coincidence points and common quadruple fixed points. These tech In addition to conventional applications, it provides innovative applications that can be used to solve integral equations and Homotopy Theory. The book is both self-contained and unified, making it an invaluable resource for researchers seeking innovative ideas. It will be a valuable addition to the library.
Fixed Point Theorem in Different Spaces
The development of computers has led to the emergence of numerous scientific and technological fields of study. At the same time as it has replaced and replicated the uses of mathematics in science and technology, it has also highlighted a lot of issues with mathematical theories. Functional analysis and topology are closely linked topics that were discussed before. The fixed point theorem has emerged as a key instrument for research in both mathematics and other branches of management science and technology. The purpose of our research is to illuminate the novel features of fixed point theorems in various spaces with regard to applications. Our study aims to uncover innovative aspects of Banach fixed point theorems or Banach contraction theorem in different metric spaces such as TVS-Cone metric space, c distance in cone metric space, c -complete complex valued metric space, complex valued b-metric space and Modular metric space . Henceforth, it has become a very important study for correlating and coordinating various aspects of fixed point theorems.
New Directions in Term Logic
The systematic account of deductive reasoning and the development of a formal logic to reveal the principles of such reasoning began with Aristotle's syllogistic. It was a term logic, a logic that dominated the field until the rise of modern predicate logic at the end of the Nineteenth century. That system quickly supplanted the old logic of terms. However, in the middle of the Twentieth century Fred Sommers took up the challenge to build a revised and strengthened term logic, one fit to challenge the hegemony of predicate logic. The aim was to devise a formal logic that could better serve as the logic of natural language.In recent years a new group of logicians have taken this version of term logic into many new directions. This book presents here for the first time some of the best examples of their work with new essays by philosophers, logicians, mathematicians, computational theorists, and historians of logic. The topics addressed include relative terms, logical copulation, Aristotelian diagrams, modality, truth, semantics, epistemic term logic, non-classical quantifiers, and more.
Mathematics and problem solving
Problem-solving in early childhood education is both possible and fantastic. This paper aims to provide motivation for educators in the field and others interested in the subject. The approach is based on my experience as an educator in the field of mathematics, which every day results in various learning experiences and arouses interest and the need for research into the teaching of mathematics in early childhood education and the search for effective methodologies.
Debt Dance
Debt Dance: A Global Journey Through Fiscal Tightropes" takes you on a whirlwind tour of government borrowing around the world. It explores the reasons why countries, from emerging economies to established powers, choose to take on debt. You'll discover how these borrowed funds can be used to invest in infrastructure, social programs, or stimulate economic growth. But this book isn't all about the tango. It also delves into the delicate balancing act - the very real risks of excessive debt. We'll explore concepts like inflation, how debt can crowd out private investment, and the burden placed on future generations. "Debt Dance" uses engaging explanations and global examples to shed light on the complex world of sovereign debt. It's a must-read for anyone curious about how nations manage their finances in an interconnected world.
Foundations and requirements of teacher preparation
At the international level, Angola's results in the Mathematics Olympiads are very poor, and at the national level, in Huambo province, these competitions are not developed with the required systematicity. This is largely due to the existence of theoretical and methodological gaps in the teachers' training process.This book provides the fundamentals and requirements for the preparation of teachers in order to improve their performance in the development of the mathematics Olympiads.
Introduction to the Structure of Groups
One of the major and vital concerns that occupies the central positions in Mathematic is the theory of groups.The origin of group theory can be traced to the ardent desire and attempt to determine solutions to some trending problems.Knowledge of groups now plays indispensable roles in such areas like coding theory, counting and the study of symmetries.It is notable to know that biology, chemistry and physics are mutual areas which have greatly benefited from the study and theory of groups.
Mathematical Biology
Rich in applications, this text can be used for a semester-long course in differential equations or mathematical biology or serve as a companion text for a 2-semester sequence beginning with discrete time systems, through a wide array of topics in differential equations, culminating in systems, SIR models, & other applications.
Decomposing G = ( V, E ) into Planar Graphs
This book is our attempt to answer the question, "Is it possible to decompose any graph G with n vertices, lacking any specific symmetry of the underlying structures?" This book presents a solution to this question by developing iterative techniques in an intuitive, simple, and visual way without losing the essence of the underlying mathematics. We trust that this method of decomposing any graph into known subgraphs will open new directions in graph decomposition. This book will be of interest to researchers in graph theory and discrete mathematics.
Fundamentals and Principles of Differential Calculus
Differential calculus is a pivotal concept in mathematics and computer science, focusing on techniques to analyze how variations in input parameters influence mathematical quantities or computational results. This book presents a differential calculus theoretical framework and practical applications, emphasizing its significance across various disciplines. This work delves into the study of derivatives differentials. It provides a formalized approach to quantifying rates of change and conducting sensitivity analysis, underscoring the relevance and utility of differential calculus in diverse fields.
The mathematical knowledge of miners in the Matup獺 region
This paper describes an investigation based on Ethnomathematics. This approach, based on its remarkable and comprehensive approach, seeks to demonstrate the ways in which a more productive and lasting teaching practice can be implemented. The aim is to reveal the mathematical knowledge preserved by the miners who live on the outskirts of the municipality of Matup獺-MT, particularly those who did not attend school until the end of secondary school.
2021-2022 High School Contest Materials
This book contains one year of mathleague.org high school tests and answer keys: 7 qualifying-level test sets, one state-level test set, and one national-level test set. Each test set contains a Sprint, Target, Team, and Relay test. Three of the test sets also contain a Power Question.
Studies on Some Aspects of Difference Sequence Spaces
In the book we have introduced various new sequence spaces in different chapters which generalize many known spaces introduced by several mathematicians. First chapter of this book is the introduction, where as Chapter-II to VI provide certain new classes of difference sequence spaces via Orlicz function. The last chapter is devoted to calculate the spectral sub-divisions of forward difference operator of order "r".
Incidence of Technology as a Pedagogical Tool
Technologies have revolutionized education, transforming the way we learn and teach, allowing us to explore their impact on the different levels of curricular concretion, at the macro curricular level: the general objectives, the competencies to be developed and the evaluation criteria are established at a global level, by integrating technologies the teaching strategies are enhanced and fostering the acquisition of digital competencies necessary in today's society; at the meso-curricular level, the specific contents, teaching, learning and evaluation strategies for each particular area or subject are detailed, having as resources the educational platforms such as videos, animations and presentations enrich the educational content and facilitate the understanding of the concept; at the micro-curricular level, the daily activities are planned, defining the didactic resources to be used, the evaluation criteria are established at the level of the didactic units, the technology will improve the interaction in the classroom and personalize the learning process.
Difficulties in understanding and using rational numbers
This work is concerned with the development of science and technology. It aims to investigate how we can contribute to minimizing the countless situations that interfere with the development of fractional activities in EJA - Youth and Adult Education, and to research the development of strategies for working with tools that can act by enabling and making EJAI students feel more comfortable with rational numbers.
Changing methodologies for better maths learning
This research aims to analyse the potential of the connection between Mathematical Modelling and constructivism and its contribution as a methodology in pedagogical practices in the classroom in the teaching-learning process in General Secondary Education (1st Cycle) at Matola Secondary School. This study came about as a result of the realisation that the pedagogical practices in place in the majority of schools do not respond to current social demands, as students are limited to receiving the knowledge transmitted by the teacher without criticising it, which compromises the quality of education and their skills. In view of this, the use of mathematical modelling combined with the constructivist approach contributes to an effective teaching practice, making students active, reflective, participative and capable of producing their own knowledge.
Mathematical modelling Use of public transparency
Teaching maths isn't just about asking students to solve complicated problems, it's about giving them the thoughts, methods and logical reasoning, as well as providing them with the ability to deal with the exact science, giving some the opportunity to choose to follow this marvellous universal path. This book, which is innovative in its teaching ideas due to the encouragement of new technologies, recalls the history of maths teaching and enables the reader to understand why it is necessary to move forward with the use of the most up-to-date resources. It shows the step-by-step process of setting up games or working with mathematical modelling and provides examples that will make it easier for the teacher to understand what is proposed and then work not just with small inspectors, but with curious people with unlimited interests. Learners, who have already been synchronised for a long time, will certainly have the intention of continuing the human legacy on planet Earth. The use of Transparency made available by public bodies during the Modelling games brings the lessons into the reality where the future citizen must act wisely and confidently in the proper understanding of Mathematics. It makes the lesson fun and interesting, and helps solve problems.
The Challenge of Computing Homotopy Groups
In the grand tapestry of mathematics, topology stands out for its unique approach to geometric shapes. It doesn't care about size or angles, but rather how spaces can be continuously deformed into one another, like stretching a clay model without tearing or gluing separate pieces. This seemingly abstract concept holds immense power, unlocking hidden properties of shapes and revealing a universe beyond our everyday perception of space. One of the central ideas in topology is the notion of a homotopy group. Imagine a loop drawn on a surface, like a circle on a sphere or a square on a plane. Homotopy groups classify these loops based on how they can be continuously deformed into one another without breaking the loop or letting it touch the edge of the surface. Intuitively, a loop that can be shrunk to a point without leaving the surface represents a "trivial" element in the group. More intricate loops that cannot be shrunk but can be continuously deformed into each other belong to the same class within the group.
Teacher Noticing of Pre-Service and In-Service Secondary Mathematics Teachers
In light of increasing demands on teachers and the need to develop teaching-related competences, this book examines the situation-specific skill of teacher noticing in pre-service and in-service secondary mathematics teachers. A video-based test instrument is used to measure teachers' noticing skills in perception, interpretation, and decision-making from both general and mathematics pedagogical perspectives. The aim is to understand the structure and characteristics of teacher noticing across different groups, as well as the influences of teaching experience and opportunities to learn. Three quantitative studies are conducted: two cross-sectional studies with 457 participants, including master's students, early career teachers, and experienced teachers, and one longitudinal study with 175 master's students. The results support the conceptualization of teacher noticing as comprising three facets. They also reveal positive influences of teaching experience on the development of teacher noticing, with in-service teachers outperforming master's students. However, experienced teachers perform similarly to early career teachers in general and worse in certain areas, suggesting saturation or forgetting effects. The longitudinal study finds that interpretation skills facilitate the development of perception and decision-making, emphasizing the knowledge-based nature of teacher noticing.
