Mathematical Reasoning: Writing and Proof, Version 2.1

Reviewed by Thao Tran, Assistant Teaching Professor, Massachusetts Department of Higher Education on 6/30/21

Comprehensiveness rating: 5 see less

This textbook is an excellent introduction to mathematical proofs. It helps students learn how to read, understand, and write proofs. Various proof methods are covered, including direct proof, proof by contradiction, and mathematical induction. Along the way, some essential mathematical concepts are introduced and discussed, including logic, sets, functions, and equivalence relations. Many step-by-step examples are provided, and plenty of good problems are included at the end of each section.

Content Accuracy rating: 5

The mathematical content seems to be all correct. I didn't notice any errors aside from some minor typos.

Relevance/Longevity rating: 5

Math majors must learn to read and write proofs, so this textbook is very relevant to them. More generally, the art of writing logical arguments and clear explanations of abstract concepts is relevant to any student in a STEM field.

Clarity rating: 5

The explanations are very clear and appropriate for a student who has completed the typical calculus sequence.

There is an emphasis on active learning in this textbook. In the course of reading the textbook, the reader is expected to do exercises which guide them along. These exercises occur in two forms:
- Preview Activities help students develop an intuition about material that's about to be introduced. There are generally one or two preview activities in each section.
- Progress Checks help students check their reading comprehension. These brief sets of exercises are interspersed through each section.
These exercises are clearly written, simple enough for many students to complete on their own, and very helpful for learning the material.

Consistency rating: 5

I didn't notice any inconsistencies.

Modularity rating: 5

Chapters 1-6 cover essential material on proof techniques and basic mathematical objects. As is the case with many math textbooks, it's necessary to go through this material in the order presented as each new section generally depends on knowing the concepts developed in previous sections.

Chapters 7-9 contains less essential material that can be omitted depending on the instructor's preference.

Organization/Structure/Flow rating: 5

The material is well-organized and is presented in a natural, logical way.

Interface rating: 5

The interface is good. The PDF version of this textbook mostly looks like a standard printed book.

Grammatical Errors rating: 5

I didn't notice any grammatical issues.

Cultural Relevance rating: 5

This textbook doesn't deal with cultural issues.

Reviewed by David Steinberg, Senior Instructor, University of Oregon on 1/15/21

Comprehensiveness rating: 5 see less

The textbook does a good job of not promising more than it can deliver, nor biting off more than it can chew. The material that it presents is thoroughly and carefully articulated, and easy to navigate with and the table of contents and index.

Content Accuracy rating: 5

I do not recall finding any errors or typos.

Relevance/Longevity rating: 5

The material of the textbook is relevant to anyone wishing to understand mathematical proof, and will not made obsolete.

Clarity rating: 5

The textbook is exceptionally well written with the student in mind. There are lots of examples, and problems to attempt before reading the chapter to increase reader engagement. One small issue near the beginning is a lack of communication about what properties of N we are going to assume and what we want to prove later. This would be a relatively small addition to make that would improve clarity.

Consistency rating: 5

The textbook is internally consistent.

Modularity rating: 5

I have taught from the book many times, and have had no problem reorganizing sections to accommodate material from elsewhere

Organization/Structure/Flow rating: 5

The book is very well organized, with each chapter and section leading naturally into the next.

Interface rating: 5

The textbook is easy to navigate.

Grammatical Errors rating: 5

I do not recall any grammatical issues.

Cultural Relevance rating: 5

I do not think that the textbook makes any insensitive references to culture.

Reviewed by Jason Hardin, Assistant Professor, Worcester State University on 6/2/20

Comprehensiveness rating: 5 see less

The textbook thoroughly covers all material typically included in a one-semester introduction to mathematical proof course for mathematics majors. One of the main objectives of the text is to help students learn to think and reason mathematically, and to communicate clear and rigorous written arguments. The textbook does this wonderfully and provides "Progress Checks" throughout to keep the reader engaged in the development of the material.

Content Accuracy rating: 5

The content in the text is correct. There are a couple typographical errors but none which impact understanding.

Relevance/Longevity rating: 5

The primary content of this text pertains to proof writing techniques, which will remain relevant to mathematics majors for the foreseeable future. The mathematical topics included (e.g., divisibility, congruence, sets, functions) are foundational tools for those studying college-level mathematics, and will also remain relevant for a long time. Updating the text, if necessary, can be done relatively easily and would be straightforward to implement.

Clarity rating: 5

The textbook excels in this area. The language throughout is conversational, straightforward, and readable by the target audience. The text is able to clearly communicate complicated and abstract ideas with clear, precise language.

Consistency rating: 5

The text is internally consistent and has issues in this regard.

Modularity rating: 5

The textbook, like most undergraduate mathematics textbooks, is partitioned into chapters (organized by major topic, e.g., functions) and sections (smaller topics falling under the main topic's umbrella, e.g., bijections, composition, inverse functions). Each section is an appropriate length to be covered in one standard 75 minute class meeting. The chapters and sections are designed to be covered in the order presented, but this is a feature of mathematics as a whole and not a product of this particular textbook. There are some instances in which the topics could be presented in a different order with minimal effort from the instructor.

Organization/Structure/Flow rating: 5

The textbook is excellently organized and the topics in the textbook are presented in a clear, logical fashion.

Interface rating: 5

There are no issues with the interface for this textbook.

Grammatical Errors rating: 5

The textbook is well written with only a handful of grammatical errors.

Cultural Relevance rating: 5

The textbook is not culturally insensitive or offensive in any way. As a textbook in pure mathematics, it does not contain any content dealing with race, ethnicity, or background of individuals.

Comments

I have used this textbook the past two times I have taught an introductory proof course (I used a different textbook previously). There is a strong focus on active learning throughout the textbook, and this allows students to become engaged in the course material in a more active way than simply reading examples of good proofs. The clear writing style and organization make this textbook easy to use in a classroom setting, and provides a large variety of problems for students to practice the skills they learn in class.

Reviewed by Matt DeLong, Professor of Mathematics, Marian University on 4/11/19

Comprehensiveness rating: 5 see less

An introduction to proofs, or "bridge" course, provides a chicken-and-egg dilemma. Do you first lay the groundwork for logic, proofs, and writing and then subsequently use these for writing proofs in different mathematical contexts? Or, do you explore different mathematical contexts (e.g. number theory, combinatorics, geometry, calculus/analysis, algebra) as a means to motivate the foundations of mathematical logic, proofs, and writing? Neither can be done completely divorced from the other. Sundstrom's book takes the former approach with chapters on proofs, logic, writing, induction, set theory, functions, and relations that focus heavily on elementary number theory and particularly the notion of congruence. In these chapters, his text is appropriately comprehensive. These foundational chapters are followed by two further chapters on number theory and cardinality, providing some application of the preceding foundations. Other mathematical contexts are found in the exercises throughout. Users who desire the latter approach (using different mathematical areas to motivate the foundations) might find the text to be limited in its scope. However, users who prefer the approach taken here will find a comprehensive treatment appropriate for a one-semester bridge course.

Content Accuracy rating: 5

This open textbook was formerly available as a traditional textbook from a traditional publisher (Pearson), and has been in use for nearly two decades. In addition, the author has numerous textbook credits with both traditional publishers and in open formats. Therefore, it is unsurprising that the accuracy is high.

Relevance/Longevity rating: 5

The basic writing conventions and logical tools of theoretical mathematics change slowly over time, if at all. Thus, like in most mathematics texts, the contents of this book will remain relevant for beginning mathematics students for some time. Moreover, this textbook is structured with an active-learning pedagogy in mind. Given the growing preponderance of evidence for the efficacy of active learning in college mathematics instruction (e.g. https://www.pnas.org/content/111/23/8410), the pedagogical framework of this book should also remain relevant for years to come.

Clarity rating: 5

Bridge courses in mathematics exist for a reason. Many students find the transition from computationally-based mathematics (e.g. the calculus sequence) to theoretically-based mathematics (many upper-level electives) confusing to navigate. Clarity of thought and expression are essential to good mathematics, yet challenging to acquire for many. Moreover, improvement of thought and expression are two of the formational outcomes from studying mathematics. Thus, it is essential that a textbook resource for such a bridge course both be clear in its development and exposition, as well as teach these to students by more than merely example. Sundstrom's book is strong in this area. Not only is it lucid and readable, even for beginning undergraduates, it also tries to intentionally teach its readers the principles of the development of correct mathematical reasoning and the articulation of well-crafted arguments.

Consistency rating: 5

Similar to the question of accuracy, this book demonstrates the quality of its writing and editing through its consistency.

Modularity rating: 5

The book is divided into nine chapters, each of which is divided into 2-6 sections (excluding the chapter summaries). Each section is appropriate for 1-2 standard 50-minute class periods, depending upon the topic and the strength of the class. Thus, the text is readily divisible in a way that fits a standard semester schedule. Since mathematics, in general, builds upon itself in a more linear way than most disciplines, the book really isn't intended to be reorganized and realigned. It is unlikely that adopters will find this to be a deficiency given its context.

Organization/Structure/Flow rating: 5

The textbook is well organized, and provides a clear and logical flow for a semester-long bridge course.

Interface rating: 5

The text seems to be free of any interface issues. The bookmarks and hyperlinks in the pdf function correctly. There are very few figures, other than the occasional Venn diagram, graph, or table, and these all display nicely.

Grammatical Errors rating: 5

Again, this is a well-written and well-edited text without noticeable grammatical errors.

Cultural Relevance rating: 5

Unlike many texts in calculus, linear algebra, statistics, etc., a bridge course textbook provides little in the way of application, except to other areas of mathematics. In addition, most theoretical mathematics textbooks (although not all) are written in an academic tone, with very limited use of humor, colloquialisms, or cultural references. Given these things, the question of cultural sensitivity is essentially moot in this context. The prose is primarily in the first-person plural, using "we" throughout. There are a couple of examples that reference a named (hypothetical) person, although the names seem limited in their cultural diversity (e.g. "Ed" and "Laura").

Comments

The open version of this textbook was the inaugural winner of the Mathematical Association of America's Solow Award, the primary criterion for which is "... the material's impact on undergraduate education in mathematics and/or the mathematical sciences (operations research, statistics, computer science, applied mathematics)." Thus, it is clearly viewed within the mathematical community as being of high quality and having strong impact on student learning.

This is no doubt due to the quality and clarity of the exposition, as well as its presumed active-learning approach. Almost every section begins with two Preview Activities, which are designed to be done by the students before they read the section (and before they have a classroom lesson on the topic). There are also occasional Progress Check activities within the sections that ask the reader to stop and determine if they are understanding the material. Each section includes a healthy, although not overwhelming, number of exercises. Finally, each chapter ends with a summary consisting of hyperlinked compendia of important definitions, results, and proof techniques.

Although not officially part of the textbook itself, there are ancillary materials that help support teaching and learning with this textbook. The author has a web site devoted to this book that contains freely available resources (https://www.tedsundstrom.com/mathematical-reasoning-writing-and-proof). The author is also willing to send solutions to the Preview Activities and Exercises to adopters who request them via email. Finally, one of the author's colleagues at Grand Valley State University has put 107 screencasts on Youtube that are designed to support learning with this textbook (https://www.youtube.com/playlist?list=PL2419488168AE7001).

Reviewed by Kenneth Ward, Professorial Lecturer, Mathematics & Statistics, American University on 2/1/18

Comprehensiveness rating: 4 see less

This text addresses most major components of proof methods, with suitable exploration exercises for the mathematical novice/undergraduate major which do not overly burden the reader with prerequisite knowledge.

Content Accuracy rating: 5

The book has accurate and only contains minimal typographical errors. The mathematics in the book is correct.

Relevance/Longevity rating: 5

This text will be relevant for a long while. Students who become math or statistics majors need to understand proof, and the basic methods used in proof and mathematical logic have not significantly changed (and will not) over time.

Clarity rating: 5

This is a strong point of the text. The writing is extremely clear and simple, making it easy for the undergraduate reader to follow where ,any other books fail. The examples lead the reader gently towards an understanding of logic and proof. Especially good are the sections where the author clarifies how to write a proof for your audience.

Consistency rating: 5

Excellent. There are simply no problems with the consistency of the mathematical work or exposition.

Modularity rating: 4

This can generally be done, although it takes a bit of work. Mathematics courses are often linear in this way. Despite this, with a little extra effort by an instructor, most sections can be separated. For example, it is difficult to speak of correspondences without the notion of a function, but an instructor can simply introduce the function definition to address correspondences without covering the entire chapter on functions. (By the way, one of the topics covered is “modular” arithmetic, so I am inclined to say that those parts are quite modular!)

Organization/Structure/Flow rating: 5

The presentation is clear and allows the instructor to develop a natural flow to a course.

Interface rating: 5

There are no issues with the interface. It is an easy-to-read pdf, of small size.

Grammatical Errors rating: 5

The grammar is mostly good, with only a minor error or two: so minor that it is easy to not notice them.

Cultural Relevance rating: 3

This book is not culturally insensitive. It is simply mathematics, and doesn’t include any offensive content in the main chapters or exercises. I am rating this a 3 as it is therefore neutral in this regard.

Comments

This book is very useful either as a primary course text for an Introduction to Proof course, or as a supplementary text in a course in philosophical logic or mathematics content course at roughly the level of Calculus I or beyond.