| concrete to abstract math: Concrete Approach to Abstract Algebra W. W. Sawyer, 2018-08-15 Brief, clear, and well written, this introductory treatment bridges the gap between traditional and modern algebra. Includes exercises with complete solutions. The only prerequisite is high school-level algebra. 1959 edition. |
| concrete to abstract math: Concrete Mathematics Ronald L. Graham, Donald E. Knuth, Oren Patashnik, 1994-02-28 This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. More concretely, the authors explain, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. |
| concrete to abstract math: Concrete Abstract Algebra Niels Lauritzen, 2003-10-16 This book presents abstract algebra based on concrete examples and applications. All the traditional material with exciting directions. |
| concrete to abstract math: Guided Math Workshop Laney Sammons, Donna Boucher, 2017-03-01 This must-have resource helps teachers successfully plan, organize, implement, and manage Guided Math Workshop. It provides practical strategies for structure and implementation to allow time for teachers to conduct small-group lessons and math conferences to target student needs. The tested resources and strategies for organization and management help to promote student independence and provide opportunities for ongoing practice of previously mastered concepts and skills. With sample workstations and mathematical tasks and problems for a variety of grade levels, this guide is sure to provide the information that teachers need to minimize preparation time and meet the needs of all students. |
| concrete to abstract math: Abstract and Concrete Categories Jiri Adamek, Horst Herrlich, George E. Strecker, 2009 This up-to-date introductory treatment employs category theory to explore the theory of structures. Its unique approach stresses concrete categories and presents a systematic view of factorization structures, offering a unifying perspective on earlier work and summarizing recent developments. Numerous examples, ranging from general to specific, illuminate the text. 1990 edition, updated 2004. |
| concrete to abstract math: Symbolizing, Modeling and Tool Use in Mathematics Education K.P Gravemeijer, R. Lehrer, H.J. van Oers, Lieven Verschaffel, 2013-03-09 This book explores the option of building on symbolizing, modeling and tool use as personally meaningful activities of students. It discusses the dimension of setting: varying from the study of informal, spontaneous activity of students, to an explicit focus on instructional design, and goals and effects of instruction; and the dimension of the theoretical framework of the researcher: varying from constructivism, to activity theory, cognitive psychology and instructional-design theory. |
| concrete to abstract math: Abstract Algebra John A. Beachy, William D. Blair, 1996 |
| concrete to abstract math: An Introduction to Abstract Mathematics Robert J. Bond, William J. Keane, 2007-08-24 Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant disciplineits long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher- level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments. |
| concrete to abstract math: A Concrete Approach to Abstract Algebra Jeffrey Bergen, 2009-12-28 A Concrete Approach to Abstract Algebra presents a solid and highly accessible introduction to abstract algebra by providing details on the building blocks of abstract algebra.It begins with a concrete and thorough examination of familiar objects such as integers, rational numbers, real numbers, complex numbers, complex conjugation, and polynomials. The author then builds upon these familiar objects and uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices. The final four chapters present the more theoretical material needed for graduate study.This text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics which arise in courses in algebra, geometry, trigonometry, precalculus, and calculus. Presents a more natural 'rings first' approach to effectively leading the student into the the abstract material of the course by the use of motivating concepts from previous math courses to guide the discussion of abstract algebra Bridges the gap for students by showing how most of the concepts within an abstract algebra course are actually tools used to solve difficult, but well-known problems Builds on relatively familiar material (Integers, polynomials) and moves onto more abstract topics, while providing a historical approach of introducing groups first as automorphisms Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices |
| concrete to abstract math: Making Mathematics Accessible for Elementary Students who Struggle Margaret Flores, Megan Burton, Vanessa Hinton, 2018 Making Mathematics Accessible for Elementary Students Who Struggle: Using CRA/CSA for Interventions provides educators with focused methods for students who struggle in elementary mathematics. The methods and procedures revolve around the CRA/CSA (concrete-representational/semi-concrete-abstract) instructional sequence. These strategies are intended for small group intensive instruction, one that addresses students' need for increased repetition and explicitness that cannot be provided within a large group of students with diverse learning needs. Current research on the CRA/CSA instructional sequence is scattered across multiple resources. This book provides comprehensive coverage of the most up-to-date research in one user-friendly resource. The sequence is included in every chapter and addresses instruction related to number sense, counting, basic operations, complex operations, basic fraction concepts, and operations with fractions. This resource is written by experienced professors spanning the fields of special education and curriculum and teaching. Its professional insight, aligned with current mathematical teaching standards and CRA/CSA research, makes this text invaluable to upcoming or current teachers in elementary mathematics. Features: Explicit and hands-on examples of CRA/CSA's use aligned with current mathematics standards and practices, Suggestions and tips for various classroom situations, Application questions for every chapter, Drawings and diagrams associated with steps of the teaching process, Classroom-tested strategies Book jacket. |
| concrete to abstract math: Why Don't Students Like School? Daniel T. Willingham, 2009-06-10 Easy-to-apply, scientifically-based approaches for engaging students in the classroom Cognitive scientist Dan Willingham focuses his acclaimed research on the biological and cognitive basis of learning. His book will help teachers improve their practice by explaining how they and their students think and learn. It reveals-the importance of story, emotion, memory, context, and routine in building knowledge and creating lasting learning experiences. Nine, easy-to-understand principles with clear applications for the classroom Includes surprising findings, such as that intelligence is malleable, and that you cannot develop thinking skills without facts How an understanding of the brain's workings can help teachers hone their teaching skills Mr. Willingham's answers apply just as well outside the classroom. Corporate trainers, marketers and, not least, parents -anyone who cares about how we learn-should find his book valuable reading. —Wall Street Journal |
| concrete to abstract math: A Concrete Introduction to Higher Algebra Lindsay N. Childs, 2012-12-04 An informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials, with much emphasis placed on congruence classes leading the way to finite groups and finite fields. New examples and theory are integrated in a well-motivated fashion and made relevant by many applications -- to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises, ranging from routine examples to extensions of theory, are scattered throughout the book, with hints and answers for many of them included in an appendix. |
| concrete to abstract math: Learning Begins Andrew C. Watson, 2017-03-08 Learning Begins, written by a teacher for teachers, translates current brain research into practical classroom strategies. Because students learn with their brains, it simply makes sense for teachers to explore educational psychology and neuroscience. And yet, information in these fields can be daunting and contradictory. Worse still, few researchers can clearly explain the specific classroom uses of their remarkable discoveries. Learning Begins both explains this research and makes it useful for teachers and administrators. Part I investigates the science of working memory: a cognitive capacity essential to all school work. When teachers recognize the many classroom perils that can overwhelm working memory, they can use research-aligned strategies to protect it, and thereby promote student learning. Part II reveals the complexities of student attention. By understanding the three neural sub-processes that create attention, teachers can structure their classrooms and their lessons to help students focus on and understand new material. Written in a lively and approachable voice, based on years of classroom experience and a decade of scientific study, Learning Begins makes educational psychology and neuroscience clear and useful in schools and classrooms. |
| concrete to abstract math: An Invitation to Abstract Mathematics Béla Bajnok, 2020-10-27 This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic.... The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH |
| concrete to abstract math: Contexts for Learning Mathematics Catherine Twomey Fosnot, Pearson Education, Fosnot, 2007-05 Contexts for Learning consists of: Investigations and Resource Guides - workshop structure involves students in inquiring, investigating, discussing, and constructing mathematical solutions and strategies - investigations encourage emergent learning and highlight the developmental landmarks in mathematical thinking - strings of related problems develop students' deep number sense and expand their strategies for mental arithmetic Read-Aloud Books and Posters - create rich, imaginable contexts--realistic and fictional--for mathematics investigations - are carefully crafted to support the development of the big ideas, strategies, and models - encourage children to explore and generate patterns, generalize, and develop the ability to mathematize their worlds Resources for Contexts for Learning CD-ROM - author videos describe the series' philosophy and organization - video overviews show classroom footage of a math workshop, including minilessons, investigations, and a math congress - print resources include research base, posters, and templates |
| concrete to abstract math: A Book of Abstract Algebra Charles C Pinter, 2010-01-14 Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition. |
| concrete to abstract math: Algebra: Abstract and Concrete, edition 2.6 Frederick Goodman, 2014-01-10 This text provides a thorough introduction to “modern” or “abstract” algebra at a level suitable for upper-level undergraduates and beginning graduate students. The book addresses the conventional topics: groups, rings, fields, and linear algebra, with symmetry as a unifying theme. This subject matter is central and ubiquitous in modern mathematics and in applications ranging from quantum physics to digital communications. The most important goal of this book is to engage students in the ac- tive practice of mathematics. |
| concrete to abstract math: Companion to Concrete Mathematics Z. A. Melzak, 1973 |
| concrete to abstract math: Mathematical Modelling and Applications Gloria Ann Stillman, Werner Blum, Gabriele Kaiser, 2017-11-05 This volume documents on-going research and theorising in the sub-field of mathematics education devoted to the teaching and learning of mathematical modelling and applications. Mathematical modelling provides a way of conceiving and resolving problems in the life world of people whether these range from the everyday individual numeracy level to sophisticated new problems for society at large. Mathematical modelling and real world applications are considered as having potential for multi-disciplinary work that involves knowledge from a variety of communities of practice such as those in different workplaces (e.g., those of educators, designers, construction engineers, museum curators) and in different fields of academic endeavour (e.g., history, archaeology, mathematics, economics). From an educational perspective, researching the development of competency in real world modelling involves research situated in crossing the boundaries between being a student engaged in modelling or mathematical application to real word tasks in the classroom, being a teacher of mathematical modelling (in or outside the classroom or bridging both), and being a modeller of the world outside the classroom. This is the focus of many of the authors of the chapters in this book. All authors of this volume are members of the International Community of Teachers of Mathematical Modelling (ICTMA), the peak research body into researching the teaching and learning of mathematical modelling at all levels of education from the early years to tertiary education as well as in the workplace. |
| concrete to abstract math: Clojure for the Brave and True Daniel Higginbotham, 2015-10-15 For weeks, months—nay!—from the very moment you were born, you’ve felt it calling to you. At long last you’ll be united with the programming language you’ve been longing for: Clojure! As a Lisp-style functional programming language, Clojure lets you write robust and elegant code, and because it runs on the Java Virtual Machine, you can take advantage of the vast Java ecosystem. Clojure for the Brave and True offers a dessert-first approach: you’ll start playing with real programs immediately, as you steadily acclimate to the abstract but powerful features of Lisp and functional programming. Inside you’ll find an offbeat, practical guide to Clojure, filled with quirky sample programs that catch cheese thieves and track glittery vampires. Learn how to: –Wield Clojure’s core functions –Use Emacs for Clojure development –Write macros to modify Clojure itself –Use Clojure’s tools to simplify concurrency and parallel programming Clojure for the Brave and True assumes no prior experience with Clojure, the Java Virtual Machine, or functional programming. Are you ready, brave reader, to meet your true destiny? Grab your best pair of parentheses—you’re about to embark on an epic journey into the world of Clojure! |
| concrete to abstract math: Number Talks Sherry Parrish, 2010 A multimedia professional learning resource--Cover. |
| concrete to abstract math: A Primer of Abstract Mathematics Robert B. Ash, 2020-03-02 The purpose of this book is to prepare the reader for coping with abstract mathematics. The intended audience is both students taking a first course in abstract algebra who feel the need to strengthen their background and those from a more applied background who need some experience in dealing with abstract ideas. Learning any area of abstract mathematics requires not only ability to write formally but also to think intuitively about what is going on and to describe that process clearly and cogently in ordinary English. Ash tries to aid intuition by keeping proofs short and as informal as possible and using concrete examples as illustration. Thus, it is an ideal textbook for an audience with limited experience in formalism and abstraction. A number of expository innovations are included, for example, an informal development of set theory which teaches students all the basic results for algebra in one chapter. |
| concrete to abstract math: The Teaching of Fractions Edward Wildeman, 1923 |
| concrete to abstract math: Visual Group Theory Nathan Carter, 2021-06-08 Recipient of the Mathematical Association of America's Beckenbach Book Prize in 2012! Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts, but its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective. The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. |
| concrete to abstract math: Abstract Algebra Dan Saracino, 2008-09-02 The Second Edition of this classic text maintains the clear exposition, logical organization, and accessible breadth of coverage that have been its hallmarks. It plunges directly into algebraic structures and incorporates an unusually large number of examples to clarify abstract concepts as they arise. Proofs of theorems do more than just prove the stated results; Saracino examines them so readers gain a better impression of where the proofs come from and why they proceed as they do. Most of the exercises range from easy to moderately difficult and ask for understanding of ideas rather than flashes of insight. The new edition introduces five new sections on field extensions and Galois theory, increasing its versatility by making it appropriate for a two-semester as well as a one-semester course. |
| concrete to abstract math: An Elementary Transition to Abstract Mathematics Gove Effinger, Gary L. Mullen, 2019-11-05 An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role. The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part introduces key objects from abstract algebra; and the final part focuses on polynomials. Features: The material is presented in many short chapters, so that one concept at a time can be absorbed by the student. Two looking back chapters at the outset (pre-calculus and calculus) are designed to start the student’s transition by working with familiar concepts. Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns. A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student. |
| concrete to abstract math: Abstract Mathematical Cognition Wolfgang Grodd, Philippe Chassy, 2016-08-09 Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell recording experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts. |
| concrete to abstract math: Advanced Mathematical Thinking David Tall, 2006-04-11 This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. It is highly appropriate for the college professor in mathematics or the general mathematics educator. |
| concrete to abstract math: Undergraduate Algebra Serge Lang, 2013-06-29 The companion title, Linear Algebra, has sold over 8,000 copies The writing style is very accessible The material can be covered easily in a one-year or one-term course Includes Noah Snyder's proof of the Mason-Stothers polynomial abc theorem New material included on product structure for matrices including descriptions of the conjugation representation of the diagonal group |
| concrete to abstract math: Young Mathematicians at Work Catherine Twomey Fosnot, Maarten Ludovicus Antonius Marie Dolk, 2001 Explains how children between the ages of four and eight construct a deep understanding of numbers and the operations of addition and subtraction. |
| concrete to abstract math: Learning and Teaching Early Math Douglas H. Clements, Julie Sarama, 2009-04-01 In this important new book for pre- and in-service teachers, early math experts Douglas Clements and Julie Sarama show how learning trajectories help teachers become more effective professionals. By opening up new windows to seeing young children and the inherent delight and curiosity behind their mathematical reasoning, learning trajectories ultimately make teaching more joyous. They help teachers understand the varying level of knowledge and thinking of their classes and the individuals within them as key in serving the needs of all children. In straightforward, no-nonsense language, this book summarizes what is known about how children learn mathematics, and how to build on what they know to realize more effective teaching practice. It will help teachers understand the learning trajectories of early mathematics and become quintessential professionals. |
| concrete to abstract math: Powerful Problem Solving Max Ray, 2013 How can we break the cycle of frustrated students who drop out of math because the procedures just don't make sense to them? Or who memorize the procedures for the test but don't really understand the mathematics? Max Ray-Riek and his colleagues at the Math Forum @ Drexel University say problem solved, by offering their collective wisdom about how students become proficient problem solvers, through the lens of the CCSS for Mathematical Practices. They unpack the process of problem solving in fresh new ways and turn the Practices into activities that teachers can use to foster habits of mind required by the Common Core: communicating ideas and listening to the reflections of others estimating and reasoning to see the big picture of a problem organizing information to promote problem solving using modeling and representations to visualize abstract concepts reflecting on, revising, justifying, and extending the work. Powerful Problem Solving shows what's possible when students become active doers rather than passive consumers of mathematics. Max argues that the process of sense-making truly begins when we create questioning, curious classrooms full of students' own thoughts and ideas. By asking What do you notice? What do you wonder? we give students opportunities to see problems in big-picture ways, and discover multiple strategies for tackling a problem. Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be over and done, but to realize the many different ways to approach problems. Read a sample chapter. |
| concrete to abstract math: Teaching Elementary Mathematics to Struggling Learners Bradley S. Witzel, Mary E. Little, 2016-01-25 Packed with effective instructional strategies, this book explores why certain K-5 students struggle with math and provides a framework for helping these learners succeed. The authors present empirically validated practices for supporting students with disabilities and others experiencing difficulties in specific areas of math, including problem solving, early numeracy, whole-number operations, fractions, geometry, and algebra. Concrete examples, easy-to-implement lesson-planning ideas, and connections to state standards, in particular the Common Core standards, enhance the book's utility. Also provided is invaluable guidance on planning and delivering multi-tiered instruction and intervention. |
| concrete to abstract math: Computation of Fractions Bradley S. Witzel, Paul J. Riccomini, 2009 In response to the lack of intervention resources available to middle school teachers to help their struggling students, the authors present this collection of easy-to-implement, research-based lesson plans to help those who struggle with mathematical fractions. |
| concrete to abstract math: Algebraic Geometry and Commutative Algebra Siegfried Bosch, 2022-04-22 Algebraic Geometry is a fascinating branch of Mathematics that combines methods from both Algebra and Geometry. It transcends the limited scope of pure Algebra by means of geometric construction principles. Putting forward this idea, Grothendieck revolutionized Algebraic Geometry in the late 1950s by inventing schemes. Schemes now also play an important role in Algebraic Number Theory, a field that used to be far away from Geometry. The new point of view paved the way for spectacular progress, such as the proof of Fermat's Last Theorem by Wiles and Taylor. This book explains the scheme-theoretic approach to Algebraic Geometry for non-experts, while more advanced readers can use it to broaden their view on the subject. A separate part presents the necessary prerequisites from Commutative Algebra, thereby providing an accessible and self-contained introduction to advanced Algebraic Geometry. Every chapter of the book is preceded by a motivating introduction with an informal discussion of its contents and background. Typical examples, and an abundance of exercises illustrate each section. Therefore the book is an excellent companion for self-studying or for complementing skills that have already been acquired. It can just as well serve as a convenient source for (reading) course material and, in any case, as supplementary literature. The present edition is a critical revision of the earlier text. |
| concrete to abstract math: An Introduction to Ontology Nikk Effingham, 2013-08-26 In this engaging and wide-ranging new book, Nikk Effingham provides an introduction to contemporary ontology - the study of what exists - and its importance for philosophy today. He covers the key topics in the field, from the ontology of holes, numbers and possible worlds, to space, time and the ontology of material objects - for instance, whether there are composite objects such as tables, chairs or even you and me. While starting from the basics, every chapter is up-to-date with the most recent developments in the field, introducing both longstanding theories and cutting-edge advances. As well as discussing the latest issues in ontology, Effingham also helpfully deals in-depth with different methodological principles (including theory choice, Quinean ontological commitment and Meinongianism) and introduces them alongside an example ontological theory that puts them into practice. This accessible and comprehensive introduction will be essential reading for upper-level undergraduate and post-graduate students, as well as any reader interested in the present state of the subject. |
| concrete to abstract math: How to Think about Abstract Algebra Lara Alcock, 2021 How to Think about Abstract Algebra provides an engaging and readable introduction to its subject, which encompasses group theory and ring theory. Abstract Algebra is central in most undergraduate mathematics degrees, and it captures regularities that appear across diverse mathematical structures - many people find it beautiful for this reason. But its abstraction can make its central ideas hard to grasp, and even the best students might find that they can follow some of the reasoning without really understanding what it is all about. This book aims to solve that problem. It is not like other Abstract Algebra texts and is not a textbook containing standard content. Rather, it is designed to be read before starting an Abstract Algebra course, or as a companion text once a course has begun. It builds up key information on five topics: binary operations, groups, quotient groups, isomorphisms and homomorphisms, and rings. It provides numerous examples, tables and diagrams, and its explanations are informed by research in mathematics education. The book also provides study advice focused on the skills that students need in order to learn successfully in their own Abstract Algebra courses. It explains how to interact productively with axioms, definitions, theorems and proofs, and how research in psychology should inform our beliefs about effective learning. |
| concrete to abstract math: A Certain Ambiguity Gaurav Suri, Hartosh Singh Bal, 2010-07-01 While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier--and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge--including religion. Together, the two men discover the power--and the fallibility--of what has long been considered the pinnacle of human certainty, Euclidean geometry. As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity--and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex. Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent--and the limits--of human knowledge. |
| concrete to abstract math: Abel’s Theorem in Problems and Solutions V.B. Alekseev, 2007-05-08 Do formulas exist for the solution to algebraical equations in one variable of any degree like the formulas for quadratic equations? The main aim of this book is to give new geometrical proof of Abel's theorem, as proposed by Professor V.I. Arnold. The theorem states that for general algebraical equations of a degree higher than 4, there are no formulas representing roots of these equations in terms of coefficients with only arithmetic operations and radicals. A secondary, and more important aim of this book, is to acquaint the reader with two very important branches of modern mathematics: group theory and theory of functions of a complex variable. This book also has the added bonus of an extensive appendix devoted to the differential Galois theory, written by Professor A.G. Khovanskii. As this text has been written assuming no specialist prior knowledge and is composed of definitions, examples, problems and solutions, it is suitable for self-study or teaching students of mathematics, from high school to graduate. |
| concrete to abstract math: Educational Assessment of Students Anthony J. Nitko, Susan M. Brookhart, 2013-11-01 For a wide variety of courses in classroom assessment. This highly respected text offers the most comprehensive discussion of traditional and alternative assessments of any classroom assessment text-explaining, giving examples, discussing pros and cons, and showing how to construct virtually all of the traditional and alternative assessments teachers use in the classroom. The author explores assessment theories and research findings as they affect teaching and learning, and examines why, when, and how teachers should use assessment in the classroom. To the text's hundreds of practical examples are added checklists to aid in evaluating assessment vehicles and scores of strategies for assessing higher-order thinking, critical-thinking, and problem-solving skills. |
Concrete-Representational-Abstract Instructional Approach
What Is the Concrete-Representational-Abstract (CRA) Instructional Approach? CRA is an intervention for mathematics instruction that research suggests can enhance the mathematics …
Concrete-Representational- Abstract: Instructional Sequence …
Using the Concrete –Representational– Abstract Sequence With Integrated Strategy Instruction to Teach Subtraction With Regrouping to Students With Learning Disabilities
CRA Approach - Accelerate Learning
Specifically, research-based studies show that students who use concrete materials to learn math develop more precise and comprehensive mental representations, often show more motivation …
Using Concrete Manipulatives in Mathematical Instruction - ed
rocess for teaching and learning mathemati-cal concepts. Starting with manipulation of concrete materials (counters, beans, Unifix cubes), the process moves students to the representational …
MATHEMATICS TEACHING PRACTICE 6 - Kentucky
In the concrete stage, students use physical manipulatives to explore, build and experience math. In the early years, this includes students understanding quantities and foundational numeracy …
Applied Problems Intervention Strategy Concrete, …
When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math …
A Simple Strategy to Teach Almost Any Math Skill - Positively …
Can be used to teach various skills from basic math (such as adding and subtracting) to more advanced concepts such as algebra.
Research-Based Practice: Concrete-Representational-Abstract …
The abstract notation is presented along with the concrete models to support the ultimate goal of students demonstrating mathematical concepts at the abstract level.
Teaching Algebra with the Concrete-Representational …
Mastery of fractions and early division is a predictor of students' later success with algebra and other higher-level mathematics. Need to focus on whole number division and fractions and …
From Concrete to Abstract: Teaching for Transfer of Learning …
We provide an example and a model, with specific steps of how teacher educators can effectively demonstrate connections between concrete objects and abstract math concepts.
Handout 3 C-R-A peFINAL - CEEDAR
Abstract – The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols). Students are provided many opportunities to practice and demonstrate …
Developing a consistent concrete, pictorial, abstract approach …
It is widely recognised and understood that a concrete, pictorial, abstract, (CPA) approach to teaching mathematics supports children to develop a deep and secure understanding. A …
Concrete-Representational-Abstract (CRA) Instructional
CRA supports understanding underlying mathematical concepts before learning “rules,” that is, moving from a concrete model of chips or blocks for multiplication to an abstract representation …
Teaching strategy: Concrete, representational, abstract (CRA)
The Concrete, Representational (Pictorial), Abstract model (CRA) is based on Jerome Brunner’s theory of cognitive development: enactive (action-based), iconic (image-based) and symbolic …
Mathematics Learning from Concrete to Abstract (1968-2021): …
Specifically, this study sought to profile the landscape of research on mathematics learning from concrete to abstract published in 1968 to 2021 by conducting a bibliometric analysis. The …
Concrete Pictorial Abstract (CPA) Model as a Teaching …
Jun 5, 2024 · Abstract — The Concrete Pictorial Abstract (CPA) model as a teaching strategy in General Mathematics simply helps students visualize abstract problems and make them more …
CONCRETE-REPRESENTATIONAL- ABSTRACT SEQUENCE OF …
The Concrete-Representational-Abstract sequence was also used to enhance the mathematics performance of the students. It help students involves the use of three distinct teaching phases …
Experiencing Mathematics From Concrete to Abstract
These papers are rooted in the GECDSB core beliefs and the Ontario Mathematics Curriculum, 2005. They are meant to elevate, enrich and extend the discourse of mathematics education …
The concrete-representational-abstract sequence of …
Teaching learners mathematics, using concrete objects, pictorial representations, followed by abstract symbols and numerals, is called the Concrete to Representational to Abstract (CRA) …
Re-thinking ‘Concrete to Abstract’ in Mathematics ... - Springer
In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract.
Concrete-Representational-Abstract Instructional Approach
What Is the Concrete-Representational-Abstract (CRA) Instructional Approach? CRA is an intervention for mathematics instruction that research suggests can enhance the mathematics …
Concrete-Representational- Abstract: Instructional Sequence …
Using the Concrete –Representational– Abstract Sequence With Integrated Strategy Instruction to Teach Subtraction With Regrouping to Students With Learning Disabilities
CRA Approach - Accelerate Learning
Specifically, research-based studies show that students who use concrete materials to learn math develop more precise and comprehensive mental representations, often show more motivation …
Using Concrete Manipulatives in Mathematical Instruction …
rocess for teaching and learning mathemati-cal concepts. Starting with manipulation of concrete materials (counters, beans, Unifix cubes), the process moves students to the representational …
MATHEMATICS TEACHING PRACTICE 6 - Kentucky
In the concrete stage, students use physical manipulatives to explore, build and experience math. In the early years, this includes students understanding quantities and foundational numeracy …
Applied Problems Intervention Strategy Concrete, …
When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math …
A Simple Strategy to Teach Almost Any Math Skill - Positively …
Can be used to teach various skills from basic math (such as adding and subtracting) to more advanced concepts such as algebra.
Research-Based Practice: Concrete-Representational …
The abstract notation is presented along with the concrete models to support the ultimate goal of students demonstrating mathematical concepts at the abstract level.
Teaching Algebra with the Concrete-Representational …
Mastery of fractions and early division is a predictor of students' later success with algebra and other higher-level mathematics. Need to focus on whole number division and fractions and …
From Concrete to Abstract: Teaching for Transfer of Learning …
We provide an example and a model, with specific steps of how teacher educators can effectively demonstrate connections between concrete objects and abstract math concepts.
Handout 3 C-R-A peFINAL - CEEDAR
Abstract – The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols). Students are provided many opportunities to practice and …
Developing a consistent concrete, pictorial, abstract …
It is widely recognised and understood that a concrete, pictorial, abstract, (CPA) approach to teaching mathematics supports children to develop a deep and secure understanding. A …
Concrete-Representational-Abstract (CRA) Instructional
CRA supports understanding underlying mathematical concepts before learning “rules,” that is, moving from a concrete model of chips or blocks for multiplication to an abstract …
Teaching strategy: Concrete, representational, abstract (CRA)
The Concrete, Representational (Pictorial), Abstract model (CRA) is based on Jerome Brunner’s theory of cognitive development: enactive (action-based), iconic (image-based) and symbolic …
Mathematics Learning from Concrete to Abstract (1968 …
Specifically, this study sought to profile the landscape of research on mathematics learning from concrete to abstract published in 1968 to 2021 by conducting a bibliometric analysis. The …
Concrete Pictorial Abstract (CPA) Model as a Teaching …
Jun 5, 2024 · Abstract — The Concrete Pictorial Abstract (CPA) model as a teaching strategy in General Mathematics simply helps students visualize abstract problems and make them more …
CONCRETE-REPRESENTATIONAL- ABSTRACT SEQUENCE OF …
The Concrete-Representational-Abstract sequence was also used to enhance the mathematics performance of the students. It help students involves the use of three distinct teaching phases …
Experiencing Mathematics From Concrete to Abstract
These papers are rooted in the GECDSB core beliefs and the Ontario Mathematics Curriculum, 2005. They are meant to elevate, enrich and extend the discourse of mathematics education …
The concrete-representational-abstract sequence of …
Teaching learners mathematics, using concrete objects, pictorial representations, followed by abstract symbols and numerals, is called the Concrete to Representational to Abstract (CRA) …
Re-thinking ‘Concrete to Abstract’ in Mathematics ... - Springer
In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract.
