Combinatorics Through Guided Discovery

Reviewed by Kristen Barnard, Associate Professor, Berea College on 12/14/23

Comprehensiveness rating: 5 see less

This text incorporates all areas that one would typically cover in an upper-level undergraduate course in Combinatorics.

Content Accuracy rating: 5

There may still be some typographical errors, but none that would impact understanding the material.

Relevance/Longevity rating: 5

The mathematical content is current.

Clarity rating: 5

The whole point of this text is to learn through solving problems. As such, it has to do a good job of introducing concepts clearly so that the reader can make their own examples.

Consistency rating: 5

This text is focused on learning through guided inquiry, and it remains consistent throughout. It delivers a minimal amount of information to set up the problem and then lets the reader discover the content.

Modularity rating: 4

This text is generally linearly ordered, but it really needs to be. It starts with no basic assumptions of the student as a combinatorist, and it gradually builds up the content. There is little modularity in the content, but the chapters are well broken into subsections, easy to focus on for assignments and discussion.

Organization/Structure/Flow rating: 5

The text starts with no assumptions of the reader other than basic arithmetic skills, and gradually builds up to very sophisticated counting techniques. Nothing seems disjoint or out of order.

Interface rating: 5

All formats of this book (print, pdf, and interactive text) are well designed for the student.

Grammatical Errors rating: 5

Any grammar or punctuation issues are small and do not alter the understanding of the material.

Cultural Relevance rating: 4

This text was originally written in the early 2000s and, like most math texts of that time and before, has issues on some problems in treating gender as a binary. Otherwise it does not address cultural issues.

Comments

I have used this text for quite a few years with students in an upper-level combinatorics class. I believe strongly in inquiry-based education and good texts are hard to find. I find that this one causes students to think, which is a good thing, but the level is not prohibitively hard for students that have not had previous experience with counting.

Reviewed by Heath Hart, Instructor, Virginia Tech on 6/10/15

Comprehensiveness rating: 4 see less

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).

Content Accuracy rating: 4

Apart from the significant errors in the book's index, there are few errors in the body of the text.

Relevance/Longevity rating: 2

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.

Clarity rating: 5

No issues; the tone is conversational, but can be precise when called-for.

Consistency rating: 5

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.

Modularity rating: 3

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.

Organization/Structure/Flow rating: 5

This is one of the book's strong points.

Interface rating: 2

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.

Grammatical Errors rating: 5

Although I haven't scrutinized every page, I have not noticed anything objectionable in this area.

Cultural Relevance rating: 3

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.

Comments

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.