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.

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.

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.

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.

"T. 1. Graph Theory. 1. Ch. 1. Elements of Graph Theory. 3. Ch. 2. Covering Circuits and Graph Coloring. 53. Ch. 3. Trees and Searching. 95. Ch. 4. Network Algorithms. 129. Pt. 2. Enumeration. 167. Ch. 5. General Counting Methods for Arrangements and Selections. 169. Ch. 6. Generating Functions. 241. Ch. 7. Recurrence Relations. 273. Ch. 8. Inclusion-Exclusion. 309. Pt. 3. Additional Topics. 341. Ch. 9. Polya's Enumeration Formula. 343. Ch. 10. Games with Graphs. 371. . Appendix. 387. . Glossary of Counting and Graph Theory Terms. 403. . Bibliography. 407. . Solutions to Odd-Numbered Problems. 409. . Index. 441.

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

Among other subjects explored are the Clements-Lindström extension of the Kruskal-Katona theorem to multisets and the Greene-Kleitmen result concerning k-saturated chain partitions of general partially ordered sets. Includes exercises and solutions.

Exploring Probability in School provides a new perspective into research on the teaching and learning of probability. It creates this perspective by recognizing and analysing the special challenges faced by teachers and learners in contemporary classrooms where probability has recently become a mainstream part of the curriculum from early childhood through high school. The authors of the book discuss the nature of probability, look at the meaning of probabilistic literacy, and examine student access to powerful ideas in probability during the elementary, middle, and high school years. Moreover, they assemble and analyse research-based pedagogical knowledge for teachers that can enhance the learning of probability throughout these school years. With the book’s rich application of probability research to classroom practice, it will not only be essential reading for researchers and graduate students involved in probability education; it will also capture the interest of educational policy makers, curriculum personnel, teacher educators, and teachers.

Emphasizes a Problem Solving Approach A first course in combinatorics Completely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics. New to the Second Edition This second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises. Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya’s counting theorem.

These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline.