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.