Twenty Key Ideas in Beginning Calculus

Reviews for Twenty Key Ideas in Beginning Calculus

Interesting...impressive visual effects!       Dr. Bruce Edwards, co-author Calculus by Larson
A review by Dr. Tom Butts, UT Dallas

Twenty Key Ideas in Beginning Calculus is a labor of love by a passionate teacher/author with a sincere interest in wanting his students to achieve an understanding of a difficult subject. The book is intended to serve as a bridge for beginning calculus student to study independently in preparation for a traditional calculus curriculum or a supplemental material for students who are currently in a calculus class. It aims to prepare a student to appreciate concepts involving the idea of a limit – a sophisticated concept sometimes called the “third plateau of mathematics” following “number” and “variable”.

Chapters 1-14 teach “intuitive calculus” while the appendices contain “traditional calculus” proofs allowing the reader to customize their learning experience according to their ability and interest for rigor. There is deserved emphasis on reviewing, re-teaching, and rephrasing throughout the book.

The author repeatedly uses three proven techniques of effective teachers:
1.) go from the “specific” to the “general” – abstract general principles from consideration of several well-chosen (specific) examples,
2.) use visual representations – in this case over 100 detailed graphs and several interactive applets that demonstrate various concepts,
3.) connect new ideas to previously learned knowledge or historical precedent to help the student see the “big picture”.
While some teachers might have some different favorite examples to illustrate a given idea, all teachers and students will benefit from studying calculus using this approach.
A guide that introduces students to the notoriously tough subject’s fundamental blueprint.
One can easily imagine Umbarger (Explaining Logarithms, 2006), with his 30-plus years of teaching various math courses to various grade levels, has frequently encountered the question “Why do I need to know this?” His recognition of this popular question’s legitimacy lays the groundwork for his uncommon teaching here. Practical application and big-picture context—two ideas normally reserved as cake toppers in mathematics—are integrated at each stage of Umbarger’s guide, allowing readers to continually examine their current level of knowledge in terms of the “whole story” of calculus. Proceeding on the belief that having a reason to care about calculus is a prerequisite for grasping the subject, Umbarger moves from one point to the next by virtue of practical application scenarios. Students edging their way into calculus for the first time will find that, by calling on their cumulative body of learned methods up to a given point, they can solve a given, real-life problem in the text—to a point. The author utilizes students’ organic frustration with the limits of their ability as a hinge leading into his next topic, encouraging a self-propelled way of learning. In mimicking the original, need-based discovery of the founding ideas of calculus, this manual steps outside the linear organization of traditional calculus texts; at the same time, it makes itself accessible to a wide array of math-shy students who have never had their basic questions (such as “How will I ever use this?”) answered. Populated by the expected charts, graphs and tables, this manual also includes well-placed explanatory photos and the occasional cartoon for comic relief. While the book’s layout seems most conducive to students preparing to enter calculus, it also includes helpful hints for instructors who may have become so familiar with the ins, outs and overview of calculus that they have difficulty seeing it from the perspective of a student new to the subject.

Well-organized and student-friendly, this book thoroughly covers the mainstays of calculus.

Kirkus Indie, Kirkus Media LLC