An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number theory, and more. The exposition guides and mentors the reader through an adventure in mathematical discovery, requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Ultimately, this is really a book about productive struggle and learning how to learn.
This book is intended to be used for a one-semester/quarter introduction to proof course (sometimes referred to as a transition to proof course). The purpose of this book is to introduce the reader to the process of constructing and writing formal and rigorous mathematical proofs. The intended audience is mathematics majors and minors. However, this book is also appropriate for anyone curious about mathematics and writing proofs. Most users of this book will have taken at least one semester of calculus, although other than some familiarity with a few standard functions in Chapter 8, content knowledge of calculus is not required. The book includes more content than one can expect to cover in a single semester/quarter. This allows the instructor/reader to pick and choose the sections that suit their needs and desires. Each chapter takes a focused approach to the included topics, but also includes many gentle exercises aimed at developing intuition.
This book is unique. The book has been an open-source project since day one. The source and PDF versions of the book will always be available for FREE (see links below). In addition, the book has undergone the traditional review and editorial process with the American Mathematical Society/MAA Press and is available for purchase as a low-cost paperback. I am extremely grateful to the AMS/MAA Press for being willing to publish the book while maintaining the open-source license. I hope more textbooks can be published using the same model. Each year, I will donate any proceeds from the print version of the book to one or both of the Association for Women in Mathematics or the National Association of Mathematicians.
The first draft of the book was written in 2009. At that time, several of the sections were adaptations of course materials written by Matthew Jones (CSU Dominguez Hills) and Stan Yoshinobu (University of Toronto). The current version of the book is the result of many iterations that involved the addition of new material, retooling of existing sections, and feedback from instructors that have used the book. The current version of the book is a far cry from what it looked like in 2009.
If you’ve found an error or have suggestions for improvements, please let me know by sending me an email or submitting an issue via GitHub. You can find the most up-to-date version of this textbook on GitHub. I would be thrilled if you used this textbook and improved it. If you make any modifications, you can either make a pull request on GitHub or submit the improvements via email. You are also welcome to fork the source and modify the text for your purposes as long as you maintain the Creative Commons Attribution-Share Alike 4.0 International License.
Mathematics is not about calculations, but ideas. My goal as a teacher is to provide students with the opportunity to grapple with these ideas and to be immersed in the process of mathematical discovery. Repeatedly engaging in this process hones the mind and develops mental maturity marked by clear and rigorous thinking. Like music and art, mathematics provides an opportunity for enrichment, experiencing beauty, elegance, and aesthetic value. The medium of a painter is color and shape, whereas the medium of a mathematician is abstract thought. The creative aspect of mathematics is what captivates me and fuels my motivation to keep learning and exploring.
While the content we teach our students is important, it is not enough. An education must prepare individuals to ask and explore questions in contexts that do not yet exist and to be able to tackle problems they have never encountered. It is important that we put these issues front and center and place an explicit focus on students producing, rather than consuming, knowledge. If we truly want our students to be independent, inquisitive, and persistent, then we need to provide them with the means to acquire these skills. Their viability as a professional in the modern workforce depends on their ability to embrace this mindset.
When I started teaching, I mimicked the experiences I had as a student. Because it was all I knew, I lectured. By standard metrics, this seemed to work out just fine. Glowing student and peer evaluations, as well as reoccurring teaching awards, indicated that I was effectively doing my job. People consistently told me that I was an excellent teacher. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to depend on me to be successful, and second, they retained only some of what I had taught them. In the words of Dylan Retsek:
“Things my students claim that I taught them masterfully, they don’t know.”
Inspired by a desire to address these concerns, I began transitioning away from direct instruction towards a more student-centered approach. The goals and philosophy behind inquiry-based learning (IBL) resonate deeply with my ideals, which is why I have embraced this paradigm. According to the Academy of Inquiry-Based Learning, IBL is a method of teaching that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate—all those wonderful skills and habits of mind that mathematicians engage in regularly. This book has IBL baked into its core.
This book is intended to be a task sequence for an introduction to proof course that utilizes an IBL approach. The primary objectives of this book are to:
Ultimately, this is really a book about productive struggle and learning how to learn. Mathematics is simply the vehicle.
The table of contents is listed below. The book includes more content than one can expect to cover in a single semester/quarter. This allows the instructor/reader to pick and choose the sections that suit their needs and desires.
The following sections form the core of the book and are likely the sections that an instructor would focus on in a one-semester introduction to proof course.
Time permitting, instructors can pick and choose topics from the remaining sections. I typically cover the core sections listed above together with Chapter 6: Three Famous Theorems during a single semester. The Instructor Guide contains examples of a few possible paths through the material, as well as information about which sections and theorems depend on material earlier in the book.
Several instructors and students have provided extremely useful feedback, which has improved the book at each iteration. Moreover, due to the open-source nature of the book, I have been able to incorporate content written by others. Below is a partial list of people (alphabetical by last name) who have contributed content, advice, or feedback.
Mathematics & Teaching
Northern Arizona University
MAT 123: First Year Seminar
MAT 136: Calculus I
MAT 526: Combinatorics
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