Combinatorics Through Guided Discovery
Kenneth Bogart, Dartmouth College
Pub Date: 2004
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Conditions of Use
In order to comment on the comprehensiveness of the book, I first have to describe the book's unusual pedagogical structure: the author poses a list read more
In order to comment on the comprehensiveness of the book, I first have to describe the book's unusual pedagogical structure: the author poses a list of questions for each major topic beginning with questions that are simple and concrete and gently moving to questions that are more difficult and abstract. Even when answers are given, they are often given in a sketch form for the student to complete; the "Socratic" approach indicates a potential dialogue between students and their mentor. So the book is "comprehensive" in that it asks questions about a lot of topics appropriate for an undergraduate combinatorics class, but you should not expect to find many worked-out examples or completed proofs in the book. There is an index, but it is of limited use (see question 8).
Apart from the significant errors in the book's index, there are few errors in the body of the text.
Longevity of the book is an issue, due to the unfortunate circumstances of its creation. The author of the text was killed in a vehicle accident during a sabbatical taken for the purpose of updating the text, and his family was not able to locate source files for the last printed version. There are source files available, but they represent an unfinished state of an intended next edition of the text; as the book's website explains, the PDF "does not correspond to the current state of the source TeX files". This complicates the ability of an adopter to update or adapt the text.
No issues; the tone is conversational, but can be precise when called-for.
The "framework" of using long, deepening lists of questions is Bogart's choice of pedagogy, and he uses it consistently, possibly to the point of overwhelming a potential adopter who is inexperienced in teaching from this approach.
Reordering would be difficult, because the nature of the material is developmental and new sections build on old. Some reordering is possible, but the book offers no hints on how to do so effectively.
This is one of the book's strong points.
The index lists page numbers that are off by a few pages for many topics; I surmise that there were some last-minute changes to the print edition that were not reflected in its index. For example, the index says the Pólya-Redfield Theorem can be found on page 269; it's actually on page 265. Reading online and using the search function of your PDF reader is more reliable. Other, minor comments: The author uses a nonstandard notation for the quotient n!/(k-1)!. Color would have made some of the graphs easier to follow.
Although I haven't scrutinized every page, I have not noticed anything objectionable in this area.
Most of the examples involve vertices, functions, maps and similar mathematical objects; there are entire chapters that mention no people. The author does not go out of his way to highlight contributions to the field by women or mathematicians of color.
I have adopted this textbook for the junior-level combinatorics course that I teach, because the pedagogy is strong enough to overcome the mechanical defects in the index and the divergent state of the source TeX files. My students respond positively to the book; they appreciate the cost, but they also find the book to be engaging. I've found it difficult to cover as much breadth and content with this book as I have with a more traditional book, but conversely, I believe the students emerge from the course with a deeper understanding of the content that we do cover.
Table of Contents
Chapter 1: What is Combinatorics?
Chapter 2: Applications of Induction and Recursion in Combinatorics and Graph Theory
Chapter 3: Distribution Problems
Chapter 4: Generating Functions
Chapter 5: The Principle of Inclusion and Exclusion
Chapter 6: Groups Acting on Sets
About the Book
This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not exclusively on the part of combinatorics that mathematicians refer to as “counting.” The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. There are problems that some people will solve quickly, and there are problems that will take days of thought for everyone. Probably the best way to use this book is to work on a problem until you feel you are not making progress and then go on to the next one. Think about the problem you couldn’t get as you do other things. The next chance you get, discuss the problem you are stymied on with other members of the class. Often you will all feel you’ve hit dead ends, but when you begin comparing notes and listening carefully to each other, you will see more than one approach to the problem and be able to make some progress. In fact, after comparing notes you may realize that there is more than one way to interpret the problem. In this case your first step should be to think together about what the problem is actually asking you to do. You may have learned in school that for every problem you are given, there is a method that has already been taught to you, and you are supposed to figure out which method applies and apply it. That is not the case here. Based on some simplified examples, you will discover the method for yourself. Later on, you may recognize a pattern that suggests you should try to use this method again.
About the Contributors
Kenneth P. Bogart arrived at Dartmouth in 1968 after receiving his Ph.D. at the California Institute of Technology in that year. At the time of his death in 2005, Ken was in California on a sabbatical and working to complete revisions on his books, Introductory Combinatorics and Discrete Mathematics in Computer Science, while continuing his research on graph theory and partially ordered sets. During his career, Ken published nine books and over 60 articles. His many years of service to Dartmouth were marked by a dedication to teaching, which included participation in Math Across the Curriculum and his own grant for Teaching Introductory Combinatorics by Guided Discovery.