Twenty Key Ideas in Beginning Calculus


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Twenty Key Ideas in Beginning Calculus
A book by Dan Umbarger

Twenty Key Ideas in Beginning Calculus, a book by Dan UmbargerThis calculus book for beginners was conceived when the author noticed that many high school AP programs, especially English, often required summer reading for their students. Some math programs have been known to experiment with requiring the reading of historical books about mathematics or famous mathematicians, but they do not get much curricular payback for their students’ time. Meaningful, accessible, materials that could be assigned to students for summer reading and that would support the calculus curriculum simply do not exist. Such materials sound like an oxymoron. Common wisdom has it that calculus materials are inherently too difficult for students to read and study on their own. Any attempt to create such materials would fail.

Twenty Key Ideas in Beginning Calculus does not claim or intend to be a calculus text. It is a creative sequencing and presentation of a subset of topics in the standard calculus curriculum. The author makes heavy use of “anticipatory sets,” “schedules of reinforcement,” “connection of mathematical ideas,” “connection to real-world applications,” pacing, patterns, visuals, examples and counterexamples, evolution and organization of ideas, repeated threading and spiraling of concepts, and especially repetition, repetition, and repetition. Four of the fourteen chapters (1, 2, 3, & 11) are written using only introductory algebra skills to introduce both calculus vocabulary and concepts. Limits and other major calculus concepts are taught intuitively using tables and visuals. All major proofs are relegated to the appendices, allowing students to customize their learning experience according to their ability and interest for rigor. 

Click to Download Table of Contents & Sample Text (Pdf)


In the book you'll meet Limitman in the Limitman-Piggy challenge games!