Working paper Open Access
Michael Range (2016) "What is Calculus?" (WIC) and its chapter "Prelude" and Thomas Colignatus (2011) "Conquest of the Plane" (COTP) are proofs of concept, that show how one might implement a course in calculus starting with an algebraic approach and avoiding limits as long as possible. A proof of concept comes with notes for instructors and discussion of didactics, but not all is explained, since the idea is to show how it works. Thus, evaluations by others are useful to highlight not only the explicit explanations but also the actual (implicit) implementations that only transpire from following the method step by step. In the present discussion, teachers and other readers will find information about WIC Prelude that cannot be found in the WIC "Notes for instructors".
This comparison concerns the algebraic approach to calculus and thus concerns the WIC Prelude and not the main body of WIC. WIC shows an approach without awareness or reference to COTP. As author of COTP I may have a bias but I will try to evaluate WIC Prelude as unbiasedly as possible. This discussion should highlight aspects of COTP as well. The reader is invited not to mistake this highlighting as sign of bias.
WIC claims this readership: "Undergraduates, high school students, instructors and teachers, and scientifically literate readers with special interest in calculus and analysis." This would be too ambitious. The WIC Prelude relies on (group theory) notions of "rational function" and "polynomial theory" that will only fit the matricola of science students and up. On the other hand, COTP is a primer and thus targets teachers and researchers of didactics. It only relies on notions for non-mathematics majors in matricola and highschool and thus can support such students as well.