Concepts of Mathematics (21-127 at CMU) is a course designed to introduce students to the world of abstract mathematics, guiding them from more calculation-based math (that one learns in high school) to higher mathematics, which focuses more on abstract thinking, problem solving, and writing "proofs." This transition tends to be a shock: new notation, terminology, and expectations (particularly the requirement of "writing" mathematical thoughts formally) give students much trouble. Many standard texts for this course are found to be too dense, dry and formal by the students. Ultimately, this means these texts are unhelpful for student learning, despite their intentions. For this project, I have written a new textbook designed specifically for this course and the way it has been taught. My emphases--writing more informal prose, properly motivating all new material, and including a wider variety of exercises and questions for the reader--all reflect the goal of increasing the reader's interest and potential for learning. The selection of material, examples, explanations, and problems has been supported by in-classroom experience, and their presentation has been influenced greatly by educational research. I have also included accompanying class notes, meant to supplement the teaching of this course, as well as assignments and exams, and their corresponding solutions and grading rubrics. Ultimately, this project constitutes a standalone packet for future instructors of this course and similar ones. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]