Combinatorial Reasoning
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| Author | : Duane DeTemple |
| Publisher | : John Wiley & Sons |
| Total Pages | : 478 |
| Release | : 2014-04-08 |
| Genre | : Mathematics |
| ISBN | : 1118652134 |
Written by two well-known scholars in the field, Combinatorial Reasoning: An Introduction to the Art of Counting presents a clear and comprehensive introduction to the concepts and methodology of beginning combinatorics. Focusing on modern techniques and applications, the book develops a variety of effective approaches to solving counting problems. Balancing abstract ideas with specific topical coverage, the book utilizes real world examples with problems ranging from basic calculations that are designed to develop fundamental concepts to more challenging exercises that allow for a deeper exploration of complex combinatorial situations. Simple cases are treated first before moving on to general and more advanced cases. Additional features of the book include: • Approximately 700 carefully structured problems designed for readers at multiple levels, many with hints and/or short answers • Numerous examples that illustrate problem solving using both combinatorial reasoning and sophisticated algorithmic methods • A novel approach to the study of recurrence sequences, which simplifies many proofs and calculations • Concrete examples and diagrams interspersed throughout to further aid comprehension of abstract concepts • A chapter-by-chapter review to clarify the most crucial concepts covered Combinatorial Reasoning: An Introduction to the Art of Counting is an excellent textbook for upper-undergraduate and beginning graduate-level courses on introductory combinatorics and discrete mathematics.
| Author | : Duane DeTemple |
| Publisher | : John Wiley & Sons |
| Total Pages | : 214 |
| Release | : 2014-09-29 |
| Genre | : Mathematics |
| ISBN | : 1118830784 |
COMBINATORIAL REASONING Showcases the interdisciplinary aspects of combinatorics and illustrates how to problem solve with a multitude of exercises Written by two well-known scholars in the field, Combinatorial Reasoning: An Introduction to the Art of Counting presents a clear and comprehensive introduction to the concepts and methodology of beginning combinatorics. Focusing on modern techniques and applications, the book develops a variety of effective approaches to solving counting problems. Balancing abstract ideas with specific topical coverage, the book utilizes real-world examples with problems ranging from basic calculations that are designed to develop fundamental concepts to more challenging exercises that allow for a deeper exploration of complex combinatorial situations. Simple cases are treated first before moving on to general and more advanced cases. Additional features of the book include: Approximately 700 carefully structured problems designed for readers at multiple levels, many with hints and/or short answers Numerous examples that illustrate problem solving using both combinatorial reasoning and sophisticated algorithmic methods A novel approach to the study of recurrence sequences, which simplifies many proofs and calculations Concrete examples and diagrams interspersed throughout to further aid comprehension of abstract concepts A chapter-by-chapter review to clarify the most crucial concepts covered Combinatorial Reasoning: An Introduction to the Art of Counting is an excellent textbook for upper-undergraduate and beginning graduate-level courses on introductory combinatorics and discrete mathematics.
| Author | : Iddo Gal |
| Publisher | : |
| Total Pages | : 300 |
| Release | : 1997 |
| Genre | : Mathematics |
| ISBN | : 9784274901584 |
This book discusses conceptual and pragmatic issues in the assessment of statistical knowledge and reasoning skills among students at the college and precollege levels, and the use of assessments to improve instruction. It is designed primarily for academic audiences involved in teaching statistics and mathematics, and in teacher education and training. The book is divided in four sections: (I) Assessment goals and frameworks, (2) Assessing conceptual understanding of statistical ideas, (3) Innovative models for classroom assessments, and (4) Assessing understanding of probability.
| Author | : Carolyn A. Maher |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 226 |
| Release | : 2010-11-25 |
| Genre | : Education |
| ISBN | : 9400706154 |
Combinatorics and Reasoning: Representing, Justifying and Building Isomorphisms is based on the accomplishments of a cohort group of learners from first grade through high school and beyond, concentrating on their work on a set of combinatorics tasks. By studying these students, the editors gain insight into the foundations of proof building, the tools and environments necessary to make connections, activities to extend and generalize combinatoric learning, and even explore implications of this learning on the undergraduate level. This volume underscores the power of attending to basic ideas in building arguments; it shows the importance of providing opportunities for the co-construction of knowledge by groups of learners; and it demonstrates the value of careful construction of appropriate tasks. Moreover, it documents how reasoning that takes the form of proof evolves with young children and discusses the conditions for supporting student reasoning.
| Author | : Alina Galvão Spinillo |
| Publisher | : Springer Nature |
| Total Pages | : 324 |
| Release | : 2021-05-24 |
| Genre | : Education |
| ISBN | : 303069657X |
This book adopts an interdisciplinary approach to investigate the development of mathematical reasoning in both children and adults and to show how understanding the learner’s cognitive processes can help teachers develop better strategies to teach mathematics. This contributed volume departs from the interdisciplinary field of psychology of mathematics education and brings together contributions by researchers from different fields and disciplines, such as cognitive psychology, neuroscience and mathematics education. The chapters are presented in the light of the three instances that permeate the entire book: the learner, the teacher, and the teaching and learning process. Some of the chapters analyse the didactic challenges that teachers face in the classroom, such as how to interpret students' reasoning, the use of digital technologies, and their knowledge about mathematics. Other chapters examine students' opinions about mathematics, and others analyse the ways in which students solve situations that involve basic and complex mathematical concepts. The approaches adopted in the description and interpretation of the data obtained in the studies documented in this book point out the limits, the development, and the possibilities of students' thinking, and present didactic and cognitive perspectives to the learning scenarios in different school settings. Mathematical Reasoning of Children and Adults: Teaching and Learning from an Interdisciplinary Perspective will be a valuable resource for both mathematics teachers and researchers studying the development of mathematical reasoning in different fields, such as mathematics education, educational psychology, cognitive psychology, and developmental psychology.
| Author | : Bruce E. Sagan |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 304 |
| Release | : 2020-10-16 |
| Genre | : Education |
| ISBN | : 1470460327 |
This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance. The book assumes minimal background, and a first course in abstract algebra should suffice. The exposition is very reader friendly: keeping a moderate pace, using lots of examples, emphasizing recurring themes, and frankly expressing the delight the author takes in mathematics in general and combinatorics in particular.
| Author | : Istvan Mezo |
| Publisher | : CRC Press |
| Total Pages | : 438 |
| Release | : 2019-08-19 |
| Genre | : Computers |
| ISBN | : 1351346377 |
Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations. The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics. In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too. Features The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems. An extensive bibliography and tables at the end make the book usable as a standard reference. Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.
| Author | : Stasys Jukna |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 389 |
| Release | : 2013-03-09 |
| Genre | : Computers |
| ISBN | : 3662046504 |
This is a concise, up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods.
| Author | : Richard P. Stanley |
| Publisher | : Springer |
| Total Pages | : 268 |
| Release | : 2018-06-06 |
| Genre | : Mathematics |
| ISBN | : 3319771736 |
Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author’s extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound understanding to mathematical, engineering, and business models. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and rudiments of group theory. The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix-Tree Theorem, de Bruijn sequences, the Erdős–Moser conjecture, electrical networks, the Sperner property, shellability of simplicial complexes and face rings. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labeled trees. The new edition contains a bit more content than intended for a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Instructors may pick and choose chapters/sections for course inclusion and students can immerse themselves in exploring additional gems once the course has ended. A chapter on combinatorial commutative algebra (Chapter 12) is the heart of added material in this new edition. The author gives substantial application without requisites needed for algebraic topology and homological algebra. A sprinkling of additional exercises and a new section (13.8) involving commutative algebra, have been added. From reviews of the first edition: “This gentle book provides the perfect stepping-stone up. The various chapters treat diverse topics ... . Stanley’s emphasis on ‘gems’ unites all this —he chooses his material to excite students and draw them into further study. ... Summing Up: Highly recommended. Upper-division undergraduates and above.” —D. V. Feldman, Choice, Vol. 51(8), April, 2014
| Author | : Robin Wilson |
| Publisher | : |
| Total Pages | : 392 |
| Release | : 2013-06-27 |
| Genre | : Mathematics |
| ISBN | : 0199656592 |
Combinatorics is the branch of discrete mathematics that studies (and counts) permutations, combinations, and arrangements of sets of elements. This book constitutes the first book-length survey of the history of combinatorics and uniquely assembles research in the area that would otherwise be inaccessible to the general reader.
