Book Review: Problem Solving and Reasoning with Discrete Mathematics

by Cathy Liebars, The College of New Jersey

Problem Solving and Reasoning with Discrete Mathematics (2014), by Rutgers mathematics professor Joseph Rosenstein, is a mathematics textbook focused on the Mathematical Practices of problem solving, reasoning, and mathematical modeling, situated in the content of discrete mathematics. It is intended for a course taken by high school students or by college students preparing to become elementary or middle school mathematics teachers. Although the mathematical topics in this book are not explicitly addressed in the Common Core State Standards for Mathematics, the issues related to problem solving, reasoning, and modeling that are addressed in this book play a major role in the Standards for Mathematical Practice.

Problem Solving and Reasoning with Discrete Mathematics consists of a textbook and an “Activity Book.” The activities from the “Activity Book” are also embedded in the text and meant to be completed at the appropriate point in the reading. Problem Solving and Reasoning with Discrete Mathematics is intended to be used interactively with the reader. Each page in the textbook is divided into two columns.  The column at the right is the main text; the column at the left contains notes that expand on topics mentioned briefly in the text, provides reflections on the problem solving, modeling, and reasoning that appears in the text, and provides space for the many diagrams that appear in the book.

The discrete mathematics topics that appear in Problem Solving and Reasoning with Discrete Mathematics include graph coloring, systematic listing and counting, paths and circuits, spanning trees, weighted graphs, and apportionment, just to name a few. The Mathematical Practices and specific types of mathematical proofs, such as proof by contradiction, are integrated into these topics.

I have used an earlier publication co-authored by Rosenstein and Valerie DeBellis in courses I have taught for both preservice and inservice middle school teachers.The book worked well for both audiences. The parts of the book that were at too high of a level for the inservice teachers who had little mathematical background could easily be skipped, and there is more than enough material for a one-semester course so one has to make decisions about what to skip in any case. The activities and approaches used in Problem Solving and Reasoning with Discrete Mathematics helped to make concepts that are typically difficult for this audience much easier to understand. For example, the approach of the “slot method” and the “choose numbers” and the activities that are provided made permutations and combinations easily understood in a way that I have never witnessed before. There are also many connections in the book to other areas of mathematics with topics such as factor graphs and planar graphs, which have connections to number theory and geometry, respectively.

Although parts of Problem Solving and Reasoning with Discrete Mathematics are similar to the earlier publication, this book has a substantial amount of both new and revised material. I am excited to teach my course, Discrete Mathematics for Middle School Teachers, again to use this new book! The book can be found at


Rosenstein, J. G. (2014). Problem Solving and Reasoning with Discrete Mathematics. New Jersey: Shiviti Publications.