The Growth Of Mathematical Knowledge

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Mathematical Knowledge: Its Growth Through Teaching

Alan Bishop,Stieg Mellin-Olsen,Joop van Dormolen
Mathematical Knowledge Its Growth Through Teaching Book PDF

Author : Alan Bishop,Stieg Mellin-Olsen,Joop van Dormolen
Publisher : Springer Science & Business Media
Page : 214 pages
File Size : 48,8 Mb
Release : 2013-11-27
Category : Education
ISBN : 9789401721950

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Mathematical Knowledge: Its Growth Through Teaching by Alan Bishop,Stieg Mellin-Olsen,Joop van Dormolen Pdf

In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A. G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was the fundamental problem area of the relationships between mathematical knowledge and the teaching and learning processes. The subsequent project BACOMET 2, whose outcomes are presented in this book, continued this work, especially by focusing on the genesis of mathematical knowledge in the classroom. The book developed over the period 1985-9 through several meetings, much discussion and considerable writing and redrafting. Our major concern was to try to analyse what we considered to be the most significant aspects of the relationships in order to enable mathematics educators to be better able to handle the kinds of complex issues facing all mathematics educators as we approach the end of the twentieth century. With access to mathematics education widening all the time, with a multi tude of new materials and resources being available each year, with complex cultural and social interactions creating a fluctuating context of education, with all manner of technology becoming more and more significant, and with both informal education (through media of different kinds) and non formal education (courses of training etc. ) growing apace, the nature of formal mathematical education is increasingly needing analysis.

The Growth of Mathematical Knowledge

Emily Grosholz,Herbert Breger
The Growth of Mathematical Knowledge Book PDF

Author : Emily Grosholz,Herbert Breger
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 52,5 Mb
Release : 2013-04-17
Category : Philosophy
ISBN : 9789401595582

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The Growth of Mathematical Knowledge by Emily Grosholz,Herbert Breger Pdf

Mathematics has stood as a bridge between the Humanities and the Sciences since the days of classical antiquity. For Plato, mathematics was evidence of Being in the midst of Becoming, garden variety evidence apparent even to small children and the unphilosophical, and therefore of the highest educational significance. In the great central similes of The Republic it is the touchstone ofintelligibility for discourse, and in the Timaeus it provides in an oddly literal sense the framework of nature, insuring the intelligibility ofthe material world. For Descartes, mathematical ideas had a clarity and distinctness akin to the idea of God, as the fifth of the Meditations makes especially clear. Cartesian mathematicals are constructions as well as objects envisioned by the soul; in the Principles, the work ofthe physicist who provides a quantified account ofthe machines of nature hovers between description and constitution. For Kant, mathematics reveals the possibility of universal and necessary knowledge that is neither the logical unpacking ofconcepts nor the record of perceptual experience. In the Critique ofPure Reason, mathematics is one of the transcendental instruments the human mind uses to apprehend nature, and by apprehending to construct it under the universal and necessary lawsofNewtonian mechanics.

The Nature of Mathematical Knowledge

Philip Kitcher
The Nature of Mathematical Knowledge Book PDF

Author : Philip Kitcher
Publisher : Oxford University Press, USA
Page : 300 pages
File Size : 40,6 Mb
Release : 1984
Category : Electronic books
ISBN : 9780195035414

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The Nature of Mathematical Knowledge by Philip Kitcher Pdf

This book argues against the view that mathematical knowledge is a priori, contending that mathematics is an empirical science and develops historically, just as natural sciences do. Kitcher presents a complete, systematic, and richly detailed account of the nature of mathematical knowledge and its historical development, focusing on such neglected issues as how and why mathematical language changes, why certain questions assume overriding importance, and how standards of proof are modified.

The Origins of Mathematical Knowledge in Childhood

Catherine Sophian
The Origins of Mathematical Knowledge in Childhood Book PDF

Author : Catherine Sophian
Publisher : Routledge
Page : 212 pages
File Size : 54,6 Mb
Release : 2017-09-25
Category : Education
ISBN : 9781351541756

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The Origins of Mathematical Knowledge in Childhood by Catherine Sophian Pdf

This book examines the origins and development of children's mathematical knowledge. It contrasts the widely held view that counting is the starting point for mathematical development with an alternative comparison-of-quantities position. According to the comparison-of-quantities position, the concept of number builds upon more basic concepts of equality, inequality, and less-than and greater-than relations, which derive from comparisons between unenumerated quantities such as lengths. The concept of number combines these basic comparative concepts with the concept of a unit of measure, which allows one quantity to be described as a multiple of another. Sophian examines these alternative accounts of children's developing mathematical knowledge in the light of research: on children's counting; on their reasoning about continuous quantities such as length and area; on the development of the concept of unit; on additive and multiplicative reasoning; and on knowledge about fractions. In the closing chapters, Sophian draws out the developmental and the educational implications of the research and theory presented. Developmentally, the comparison-of-quantities position undermines the idea that numerical knowledge develops through domain-specific learning mechanisms in that it links numerical development both to physical knowledge about objects, which is the starting point for the concept of unit, and to the acquisition of linguistic number terms. Instructionally, the comparison-of-quantities perspective diverges from the counting-first perspective in that it underscores the continuity between whole-number arithmetic and fraction learning that stems from the importance of the concept of unit for both. Building on this idea, Sophian advances three instructional recommendations: First, instruction about numbers should always be grounded in thinking about quantities and how numbers represent the relations between them; second, instruction in the early years should always be guided by a long-term perspective in which current objectives are shaped by an understanding of their role in the overall course of mathematics learning; and third, instruction should be directly toward promoting the acquisition of the most general mathematical knowledge possible. The Origins of Mathematical Knowledge in Childhood is intended for researchers, professionals, and graduate students in developmental psychology, educational psychology, and mathematics education, and as a supplementary text for advanced undergraduate courses in cognitive development, educational psychology, and mathematics education.

Forms of Mathematical Knowledge

Dina Tirosh
Forms of Mathematical Knowledge Book PDF

Author : Dina Tirosh
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 55,6 Mb
Release : 2013-03-14
Category : Education
ISBN : 9789401715843

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Forms of Mathematical Knowledge by Dina Tirosh Pdf

What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing.

The Development of Mathematics

E. T. Bell
The Development of Mathematics Book PDF

Author : E. T. Bell
Publisher : Courier Corporation
Page : 657 pages
File Size : 40,7 Mb
Release : 2012-09-11
Category : Mathematics
ISBN : 9780486152288

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The Development of Mathematics by E. T. Bell Pdf

Time-honored study by a prominent scholar of mathematics traces decisive epochs from the evolution of mathematical ideas in ancient Egypt and Babylonia to major breakthroughs in the 19th and 20th centuries. 1945 edition.

Mathematics Classrooms That Promote Understanding

Elizabeth Fennema,Thomas A. Romberg
Mathematics Classrooms That Promote Understanding Book PDF

Author : Elizabeth Fennema,Thomas A. Romberg
Publisher : Routledge
Page : 217 pages
File Size : 51,7 Mb
Release : 1999-04-01
Category : Education
ISBN : 9781135676490

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Mathematics Classrooms That Promote Understanding by Elizabeth Fennema,Thomas A. Romberg Pdf

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.

Teaching Math to Multilingual Students, Grades K-8

Kathryn B. Chval,Erin Smith,Lina Trigos-Carrillo,Rachel J. Pinnow
Teaching Math to Multilingual Students Grades K 8 Book PDF

Author : Kathryn B. Chval,Erin Smith,Lina Trigos-Carrillo,Rachel J. Pinnow
Publisher : Corwin Press
Page : 317 pages
File Size : 46,6 Mb
Release : 2021-01-07
Category : Education
ISBN : 9781071810835

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Teaching Math to Multilingual Students, Grades K-8 by Kathryn B. Chval,Erin Smith,Lina Trigos-Carrillo,Rachel J. Pinnow Pdf

Using strengths-based approaches to support development in mathematics It’s time to re-imagine what’s possible and celebrate the brilliance multilingual learners bring to today’s classrooms. Innovative teaching strategies can position these learners as leaders in mathematics. Yet, as the number of multilingual learners in North American schools grows, many teachers have not had opportunities to gain the competencies required to teach these learners effectively, especially in disciplines such as mathematics. Multilingual learners—historically called English Language Learners—are expected to interpret the meaning of problems, analyze, make conjectures, evaluate their progress, and discuss and understand their own approaches and the approaches of their peers in mathematics classrooms. Thus, language plays a vital role in mathematics learning, and demonstrating these competencies in a second (or third) language is a challenging endeavor. Based on best practices and the authors’ years of research, this guide offers practical approaches that equip grades K-8 teachers to draw on the strengths of multilingual learners, partner with their families, and position these learners for success. Readers will find: • A focus on multilingual students as leaders • A strength-based approach that draws on students’ life experiences and cultural backgrounds • An emphasis on maintaining high expectations for learners’ capacity for mastering rigorous content • Strategies for representing concepts in different formats • Stop and Think questions throughout and reflection questions at the end of each chapter • Try It! Implementation activities, student work examples, and classroom transcripts With case studies and activities that provide a solid foundation for teachers’ growth and exploration, this groundbreaking book will help teachers and teacher educators engage in meaningful, humanized mathematics instruction.

The Construction of New Mathematical Knowledge in Classroom Interaction

Heinz Steinbring
The Construction of New Mathematical Knowledge in Classroom Interaction Book PDF

Author : Heinz Steinbring
Publisher : Springer Science & Business Media
Page : 242 pages
File Size : 44,7 Mb
Release : 2006-03-30
Category : Education
ISBN : 9780387242538

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The Construction of New Mathematical Knowledge in Classroom Interaction by Heinz Steinbring Pdf

Mathematics is generally considered as the only science where knowledge is uni form, universal, and free from contradictions. „Mathematics is a social product - a 'net of norms', as Wittgenstein writes. In contrast to other institutions - traffic rules, legal systems or table manners -, which are often internally contradictory and are hardly ever unrestrictedly accepted, mathematics is distinguished by coherence and consensus. Although mathematics is presumably the discipline, which is the most differentiated internally, the corpus of mathematical knowledge constitutes a coher ent whole. The consistency of mathematics cannot be proved, yet, so far, no contra dictions were found that would question the uniformity of mathematics" (Heintz, 2000, p. 11). The coherence of mathematical knowledge is closely related to the kind of pro fessional communication that research mathematicians hold about mathematical knowledge. In an extensive study, Bettina Heintz (Heintz 2000) proposed that the historical development of formal mathematical proof was, in fact, a means of estab lishing a communicable „code of conduct" which helped mathematicians make themselves understood in relation to the truth of mathematical statements in a co ordinated and unequivocal way.

Mathematical Mindsets

Jo Boaler
Mathematical Mindsets Book PDF

Author : Jo Boaler
Publisher : John Wiley & Sons
Page : 311 pages
File Size : 44,7 Mb
Release : 2015-10-12
Category : Education
ISBN : 9781118418277

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Mathematical Mindsets by Jo Boaler Pdf

Banish math anxiety and give students of all ages a clear roadmap to success Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler—Stanford researcher, professor of math education, and expert on math learning—has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students. There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets: Explains how the brain processes mathematics learning Reveals how to turn mistakes and struggles into valuable learning experiences Provides examples of rich mathematical activities to replace rote learning Explains ways to give students a positive math mindset Gives examples of how assessment and grading policies need to change to support real understanding Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals—until now. Mathematical Mindsets provides a proven, practical roadmap to mathematics success for any student at any age.

Mathematics Education for Sustainable Economic Growth and Job Creation

David Burghes,Jodie Hunter
Mathematics Education for Sustainable Economic Growth and Job Creation Book PDF

Author : David Burghes,Jodie Hunter
Publisher : Routledge
Page : 218 pages
File Size : 48,9 Mb
Release : 2021-08-24
Category : Education
ISBN : 9781000426212

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Mathematics Education for Sustainable Economic Growth and Job Creation by David Burghes,Jodie Hunter Pdf

Mathematics Education for Sustainable Economic Growth and Job Creation considers the need for young employees to be capable and confident with transferable knowledge and skills in mathematics and statistics in order to support economic growth in developing countries in an increasingly digital age. This book draws on differing international perspectives in relation to mathematics education for sustainable economic growth and job creation. The contributors include education researchers and those involved in policymaking for both developing countries and beyond. Within each chapter, there is a reflection from the authors on their experiences in educational systems and policy development or research studies, which contribute to sustainable economic growth in different countries. As well as considerations of economies and job creation, the scholarship delves further into developing a critically aware citizenship through mathematics education. Extending current thinking about the role of mathematics education and educating students for future needs, this book will be of great interest for academics, researchers and postgraduate students in the field of mathematics education, STEM education and sustainability education.

Constructing Mathematical Knowledge

Paul Ernest
Constructing Mathematical Knowledge Book PDF

Author : Paul Ernest
Publisher : Routledge
Page : 295 pages
File Size : 41,8 Mb
Release : 2012-10-12
Category : Education
ISBN : 9781136364723

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Constructing Mathematical Knowledge by Paul Ernest Pdf

First published in 1994. This book and its companion volume, Mathematics, Education and Philosophy: An International Perspective are edited collections. Instead of the sharply focused concerns of the research monograph, the books offer a panorama of complementary and forward-looking perspectives. They illustrate the breadth of theoretical and philosophical perspectives that can fruitfully be brough to bear on the mathematics and education. The empathise of this book is on epistemological issues, encompassing multiple perspectives on the learning of mathematics, as well as broader philosophical reflections on the genesis of knowledge. It explores constructivist and social theories of learning and discusses the rile of the computer in light of these theories.

Teaching to the Math Common Core State Standards

F. D. Rivera
Teaching to the Math Common Core State Standards Book PDF

Author : F. D. Rivera
Publisher : Springer Science & Business Media
Page : 186 pages
File Size : 45,7 Mb
Release : 2014-02-05
Category : Education
ISBN : 9789462095038

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Teaching to the Math Common Core State Standards by F. D. Rivera Pdf

This is a methods book for elementary majors and preservice/beginning elementary teachers. It takes a very practical approach to learning to teach elementary school mathematics in an emerging Age of the Common Core State Standards. The Common Core State Standards in Mathematics (CCSSM) is not meant to be “the” official mathematics curriculum; it was purposefully developed primarily to provide clear learning expectations of mathematics content that are appropriate at every grade level and to help prepare all students to be ready for college and the workplace. A quick glance at the Table of Contents in this book indicates a serious engagement with the recommended mathematics underlying the kindergarten through grade 5 portions of the CCSSM first, with issues in content-practice assessment, learning, teaching, and classroom management pursued next and in that order. In this book we explore what it means to teach to the CCSSM within an alignment mindset involving content-practice learning, teaching, and assessment. The CCSSM content standards, which pertain to mathematical knowledge, skills, and applications, have been carefully crafted so that they are teachable, learnable, coherent, fewer, clearer, and higher. The practice standards, which refer to institutionally valued mathematical actions, processes, and habits, have been conceptualized in ways that will hopefully encourage all elementary students to engage with the content standards more deeply than merely acquiring mathematical knowledge by rote and imitation. Thus, in the CCSSM, proficiency in content alone is not sufficient, and so does practice without content, which is limited. Content and practice are both equally important and, thus, must come together in teaching, learning, and assessment in order to support authentic mathematical understanding. This blended, multisourced text is a “getting smart” book. It helps elementary majors and preservice/beginning elementary teachers work within the realities of accountable pedagogy and develop a proactive disposition that is capable of supporting all elementary students in order for them to experience growth in mathematical understanding necessary for middle school and beyond, including future careers.

The Mathematical Education of Teachers

Conference Board of the Mathematical Sciences
The Mathematical Education of Teachers Book PDF

Author : Conference Board of the Mathematical Sciences
Publisher : American Mathematical Soc.
Page : 164 pages
File Size : 46,8 Mb
Release : 2001
Category : Education
ISBN : 0821828991

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The Mathematical Education of Teachers by Conference Board of the Mathematical Sciences Pdf

Now is a time of great interest in mathematics education. Student performance, curriculum, and teacher education are the subjects of much scrutiny and debate. Studies on the mathematical knowledge of prospective and practicing U. S. teachers suggest ways to improve their mathematical educations. It is often assumed that because the topics covered in K-12 mathematics are so basic, they should be easy to teach. However, research in mathematics education has shown that to teach well,substantial mathematical understanding is necessary--even to teach whole-number arithmetic. Prospective teachers need a solid understanding of mathematics so that they can teach it as a coherent, reasoned activity and communicate its elegance and power. This volume gathers and reports current thinkingon curriculum and policy issues affecting the mathematical education of teachers. It considers two general themes: (1) the intellectual substance in school mathematics; and (2) the special nature of the mathematical knowledge needed for teaching. The underlying study was funded by a grant from the U.S. Department of Education. The mathematical knowledge needed for teaching is quite different from that required by students pursuing other mathematics-related professions. Material here is gearedtoward stimulating efforts on individual campuses to improve programs for prospective teachers. This report contains general recommendations for all grades and extensive discussions of the specific mathematical knowledge required for teaching elementary, middle, and high-school grades, respectively.It is also designed to marshal efforts in the mathematical sciences community to back important national initiatives to improve mathematics education and to expand professional development opportunities. The book will be an important resource for mathematics faculty and other parties involved in the mathematical education of teachers. Information for our distributors: This series is published in cooperation with the Mathematical Association of America.

Mindset Mathematics

Jo Boaler,Jen Munson,Cathy Williams
Mindset Mathematics Book PDF

Author : Jo Boaler,Jen Munson,Cathy Williams
Publisher : John Wiley & Sons
Page : 288 pages
File Size : 45,6 Mb
Release : 2017-08-28
Category : Education
ISBN : 9781119358800

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Mindset Mathematics by Jo Boaler,Jen Munson,Cathy Williams Pdf

Engage students in mathematics using growth mindset techniques The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the first-grade level through visualization, play, and investigation. During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message—that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that: There is no such thing as a math person - anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.