Download or read book Understanding the Generality of Mathematical Statements written by Milena Damrau and published by Springer Nature. This book was released on with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Oxford Handbook of Generality in Mathematics and the Sciences written by Karine Chemla and published by Oxford University Press. This book was released on 2016-08-25 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how collectives have valued generality and how they have worked with specific types of "general" entities, procedures, and arguments. The books connects history and philosophy of mathematics and the sciences at the intersection of two of the most fruitful contemporary lines of research: historical epistemology, in which values (e.g. "objectivity", "accuracy") are studied from a historical viewpoint; and the philosophy of scientific practice, in which conceptual developments are seen as embedded in networks of social, instrumental, and textual practices. Each chapter provides a self-contained case-study, with a clear exposition of the scientific content at stake. The collection covers a wide range of scientific domains - with an emphasis on mathematics - and historical periods. It thus allows a comparative perspective which suggests a non-linear pattern for a history of generality. The introductory chapter spells out the key issues and points to the connections between the chapters.

Download or read book Undergraduate Students Approaches to Constructing Mathematical Generalities written by Duane Graysay and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation reports results of a study of the ways that mathematics-intending university students construct mathematical generalities, which are general statements associated with domains of mathematical objects. Such generalities are important elements of mathematical knowledge, and there is little research examining how such generalities are constructed outside of attention to the process of generalizing among K- 12 students. The goal of this research was to develop descriptions of the process of generality-constructing among University students in mathematics-intending majors to provide new insights into how individuals approach the process.Data consisted of responses from ten students to tasks of three types: Characterizing tasks, which request constructing a general property statement about a given domain; Populating tasks, which request constructing a domain of objects that satisfy a given general property; and Reconstructing tasks, which request revising a given generality to encompass a broader domain. Findings with respect to Characterizing type tasks are that individuals used superficial features to sort collections, peculiarized examples that did not fit the sorting criteria, and regularized collections by suggesting alternatives to peculiarized elements. Findings from Populating type tasks indicate that participants represented elements of the domain at various levels of generality, which had implications for the approaches through which they constructed domains to satisfy the given property statement.Findings from Reconstructing type tasks suggest some participants attempted to understand relationships between given information in order to scaffold construction of aiiiiv generality by logical deduction. Others reduced such tasks to those of the Characterizingtype or of the Populating type.Implications include potential learning goals for the development of generality-constructing and highlight the importance of attending to the ways in which curriculum provides opportunities for students to engage generality-constructing. Directions for further research include exploring ways of promoting the development of each approach.

Download or read book Functions and Generality of Logic written by Hourya Benis-Sinaceur and published by Springer. This book was released on 2015-06-24 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book examines three connected aspects of Frege’s logicism: the differences between Dedekind’s and Frege’s interpretation of the term ‘logic’ and related terms and reflects on Frege’s notion of function, comparing its understanding and the role it played in Frege’s and Lagrange’s foundational programs. It concludes with an examination of the notion of arbitrary function, taking into account Frege’s, Ramsey’s and Russell’s view on the subject. Composed of three chapters, this book sheds light on important aspects of Dedekind’s and Frege’s logicisms. The first chapter explains how, although he shares Frege’s aim at substituting logical standards of rigor to intuitive imports from spatio-temporal experience into the deductive presentation of arithmetic, Dedekind had a different goal and used or invented different tools. The chapter highlights basic dissimilarities between Dedekind’s and Frege’s actual ways of doing and thinking. The second chapter reflects on Frege’s notion of a function, in comparison with the notions endorsed by Lagrange and the followers of the program of arithmetization of analysis. It remarks that the foundational programs pursued by Lagrange and Frege are crucially different and based on a different idea of what the foundations of mathematics should be like. However, despite this contrast, the notion of function plays similar roles in the two programs, and this chapter emphasizes the similarities. The third chapter traces the development of thinking about Frege’s program in the foundations of mathematics, and includes comparisons of Frege’s, Russell’s and Ramsey’s views. The chapter discusses earlier papers written by Hintikka, Sandu, Demopoulos and Trueman. Although the chapter’s main focus is on the notion of arbitrary correlation, it starts out by discussing some aspects of the connection between this notion and Dedekind Theorem.

Download or read book Constructing and Modeling Algebraic Statements in the Multiplicative Domain written by and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This study of fourth-grade students and teachers explores mathematics teaching and learning that focuses on discovering and modeling algebraic relationships. The study has two parts: an investigation of how students learn to construct algebraic statements and models for comparisons and measurement situations in the multiplicative domain, and an investigation of teacher learning that occurred as their teachers implemented and discussed this mathematics work inside and outside of the classroom. The instructional materials that were the context for this work are unlike those typically used in the United States. They comprise an integrated unit, with tasks and discussions that make quantitative relationships salient for young students and offer rich sites for discovery, modeling, and demonstration. These materials were inspired by V. V. Davydov's work in mathematics education - a Russian innovative approach to teaching elementary school mathematics. The study centered on implementation of this mathematics unit in three classrooms. Participants included 67 fourth-grade students and 3 teachers in the same large, urban, public school. Classroom lessons and teacher interviews and meetings were videotaped, coded, and analyzed. Student work and teachers' notes were also coded and analyzed. Student and teacher learning was the focus of the study. In particular, I examined the nature of students' discourse and participation in activities, and how and what in-service teachers attended to, needed, and learned when teaching this unit. Results of this study show that students could access and engage successfully in comparing quantities and describing relationships between them; and constructing and modeling algebraic statements with words, drawings, and mathematical symbols. Some students also demonstrated a rudimentary understanding of mathematical generality. Noteworthy challenges that arose for students are discussed. Analysis of teacher learning identified three shifts in teachers' participation in activities inside and outside the classroom as their work progressed. These changes showed developed use of the language and questions involved in quantitative reasoning; increased ability to use measurement tasks, models, and hands-on tools to support algebraic thinking; and developing roles and ways of participating in learning about early algebra. Results will be useful to curriculum designers, teacher educators, and teachers working in the early algebra domain.

Download or read book Constructing and Modeling Algebraic Statements in the Multiplicative Domain written by and published by . This book was released on 2013 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: This study of fourth-grade students and teachers explores mathematics teaching and learning that focuses on discovering and modeling algebraic relationships. The study has two parts: an investigation of how students learn to construct algebraic statements and models for comparisons and measurement situations in the multiplicative domain, and an investigation of teacher learning that occurred as their teachers implemented and discussed this mathematics work inside and outside of the classroom. The instructional materials that were the context for this work are unlike those typically used in the United States. They comprise an integrated unit, with tasks and discussions that make quantitative relationships salient for young students and offer rich sites for discovery, modeling, and demonstration. These materials were inspired by V. V. Davydov's work in mathematics education - a Russian innovative approach to teaching elementary school mathematics. The study centered on implementation of this mathematics unit in three classrooms. Participants included 67 fourth-grade students and 3 teachers in the same large, urban, public school. Classroom lessons and teacher interviews and meetings were videotaped, coded, and analyzed. Student work and teachers' notes were also coded and analyzed. Student and teacher learning was the focus of the study. In particular, I examined the nature of students' discourse and participation in activities, and how and what in-service teachers attended to, needed, and learned when teaching this unit. Results of this study show that students could access and engage successfully in comparing quantities and describing relationships between them; and constructing and modeling algebraic statements with words, drawings, and mathematical symbols. Some students also demonstrated a rudimentary understanding of mathematical generality. Noteworthy challenges that arose for students are discussed. Analysis of teacher learning identified three shifts in teachers' participation in activities inside and outside the classroom as their work progressed. These changes showed developed use of the language and questions involved in quantitative reasoning; increased ability to use measurement tasks, models, and hands-on tools to support algebraic thinking; and developing roles and ways of participating in learning about early algebra. Results will be useful to curriculum designers, teacher educators, and teachers working in the early algebra domain.

Download or read book The Oxford Handbook of Generality in Mathematics and the Sciences written by Karine Chemla and published by Oxford University Press. This book was released on 2016 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of original essays aims to inquire into the diversity of Generality. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts.

Download or read book Mathematics in 10 Lessons written by Jerry P. King and published by Prometheus Books. This book was released on 2010-12-29 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Traditional Chinese edition of Mathematics in 10 Lessons: The Grand Tour. This is one of the best books to help lay a solid foundation of math skills for children and for adults who are a little rusty. It goes into details to explain concepts and wordings from the very beginning and build up step-by-step. In Chinese. Distributed by Tsai Fong Books, Inc.

Download or read book Teaching Secondary Mathematics written by Gregory Hine and published by Cambridge University Press. This book was released on 2021-09-24 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: Secondary mathematics teachers working in the Australian education sector are required to plan lessons that engage with students of different genders, cultures and levels of literacy and numeracy. Teaching Secondary Mathematics engages directly with the Australian Curriculum: Mathematics and the Australian Professional Standards for Teachers to help preservice teachers develop lesson plans that resonate with students. This edition has been thoroughly revised and features a new chapter on supporting Aboriginal and Torres Strait Islander students by incorporating Aboriginal and Torres Strait Islander cultures and ways of knowing into lessons. Chapter content is supported by new features including short-answer questions, opportunities for reflection and in-class activities. Further resources, additional activities, and audio and visual recordings of mathematical problems are also available for students on the book's companion website. Teaching Secondary Mathematics is the essential guide for preservice mathematics teachers who want to understand the complex and ever-changing Australian education landscape.

Download or read book Absolute Generality written by Agustín Rayo and published by Oxford University Press. This book was released on 2006-11-23 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: Is it possible to quantify over absolutely all there is? Or must all of our quantifiers range over a less-than-all-inclusive domain? It has commonly been thought that the question of absolute generality is intimately connected with the set-theoretic antinomies. But the topic of absolute generality has enjoyed a surge of interest in recent years. It has become increasingly apparent that its ramifications extend well beyond the foundations of set theory. Connections include semanticindeterminacy, logical consequence, higher-order languages, and metaphysics.Rayo and Uzquiano present for the first time a collection of essays on absolute generality. These newly commissioned articles -- written by an impressive array of international scholars -- draw the reader into the forefront of contemporary research on the subject. The volume represents a variety of approaches to the problem, with some of the contributions arguing for the possibility of all-inclusive quantification and some of them arguing against it. An introduction by the editors draws ahelpful map of the philosophical terrain.

Download or read book Mathematics Classrooms That Promote Understanding written by Elizabeth Fennema and published by Routledge. This book was released on 1999-04-01 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics Classrooms That Promote Understanding synthesizes the implications of research done by the National Center for Research in Mathematical Sciences on integrating two somewhat diverse bodies of scholarly inquiry: the study of teaching and the study of learning mathematics. This research was organized around content domains and/or continuing issues of education, such as equity and assessment of learning, and was guided by two common goals--defining the mathematics content of the K-12 curriculum in light of the changing mathematical needs of citizens for the 21st century, and identifying common components of classrooms that enable students to learn the redefined mathematics with understanding. To accomplish these goals, classrooms in which instruction facilitated the growth of understanding were established and/or studied. This volume reports and discusses the findings which grew out of this research, and subsequent papers and discussions among the scholars engaged in the endeavor. Section I, "Setting the Stage," focuses on three major threads: What mathematics should be taught; how we should define and increase students' understanding of that mathematics; and how learning with understanding can be facilitated for all students. Section II, "Classrooms That Promote Understanding," includes vignettes from diverse classrooms that illustrate classroom discourse, student work, and student engagement in the mathematics described in Chapter 1 as well as the mental activities described in Chapter 2. These chapters also illustrate how teachers deal with the equity concerns described in Chapter 3. Section III addresses "Developing Classrooms That Promote Understanding." The knowledge of the teaching/learning process gained from the research reported in this volume is a necessary prerequisite for implementing the revisions called for in the current reform movement. The classrooms described show that innovative reform in teaching and learning mathematics is possible. Unlike many volumes reporting research, this book is written at a level appropriate for master's degree students. Very few references are included in the chapters themselves; instead, each chapter includes a short annotated list of articles for expanded reading which provides the scholarly basis and research substantiation for this volume.

Download or read book Mathematical Thinking and Problem Solving written by Alan H. Schoenfeld and published by Routledge. This book was released on 2016-05-06 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the early 1980s there was virtually no serious communication among the various groups that contribute to mathematics education -- mathematicians, mathematics educators, classroom teachers, and cognitive scientists. Members of these groups came from different traditions, had different perspectives, and rarely gathered in the same place to discuss issues of common interest. Part of the problem was that there was no common ground for the discussions -- given the disparate traditions and perspectives. As one way of addressing this problem, the Sloan Foundation funded two conferences in the mid-1980s, bringing together members of the different communities in a ground clearing effort, designed to establish a base for communication. In those conferences, interdisciplinary teams reviewed major topic areas and put together distillations of what was known about them.* A more recent conference -- upon which this volume is based -- offered a forum in which various people involved in education reform would present their work, and members of the broad communities gathered would comment on it. The focus was primarily on college mathematics, informed by developments in K-12 mathematics. The main issues of the conference were mathematical thinking and problem solving.

Download or read book Philosophy of Logic written by and published by Elsevier. This book was released on 2006-11-29 with total page 1219 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter

Download or read book New Mathematics Education Research and Practice written by and published by BRILL. This book was released on 2006-01-01 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics education research has blossomed into many different areas which we can see in the programmes of the ICME conferences as well as in the various survey articles in the Handbooks. However, all of these lines of research are trying to grapple with a common problem, the complexity of the process of learning mathematics.

Download or read book Mathematical Thought and its Objects written by Charles Parsons and published by Cambridge University Press. This book was released on 2007-12-24 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite.

Download or read book Interpreting Carnap written by Alan Richardson and published by Cambridge University Press. This book was released on 2024-01-31 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: Rudolf Carnap (1891–1970), one of the most important philosophers of the twentieth century, helped found logical positivism, was one of the originators of the field of philosophy of science, and was a leading contributor to semantics and inductive logic. This volume of new essays, written by leading international experts, places Carnap in his philosophical context and studies his topics, his interests, and the major stages of his thought. The essays reassess Carnap's place in the history of analytic philosophy through his approach to metaphysics, values, politics, epistemology and philosophy of science. They delve into important topics of Carnap's mature thought, namely explication, naturalism, and his defence of analyticity; and they recover the logical and the linguistic components of philosophy and how they unfolded in the syntax-semantics relation, induction, and language-planning. The resulting interpretation of Carnap will be illuminating for both current and future research.

Download or read book Debates in Mathematics Education written by Dawn Leslie and published by Routledge. This book was released on 2013-10-01 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: Debates in Mathematics Education explores the major issues that mathematics teachers encounter in their daily lives. It engages with established and contemporary debates, promotes and supports critical reflection and aims to stimulate both novice and experienced teachers to reach informed judgements and argue their point of view with deeper theoretical knowledge and understanding. Written by experts in the field of mathematics education, it investigates and offers fresh insight into topics of central importance, including: Gender, social inequality and mathematics Mathematics, politics and climate change The history and culture of mathematics Using popular culture in the mathematics classroom The concept of ‘ability’ and its impact on learning What we mean by ‘teaching for understanding’ Choosing and using examples in teaching The fitness of formal examinations. Designed to stimulate discussion and support you in your own research, writing and practice, Debates in Mathematics Education will be a valuable resource for any student or practising teacher engaged in initial teacher training, continuing professional development or Masters level study. It also has much to offer to those leading initial teacher education programmes, and to beginning doctoral students looking for a survey of the field of mathematics education research.