This book is a thorough critique of geometry instruction prevalent in American schools during the early 1900s. Smith argues that the rigid, axiomatic approach – starting with definitions and postulates and building up to theorems – stifled students’ understanding and appreciation for the subject. He proposes a shift towards a more 'discovery' based learning, emphasizing the historical development of geometric principles, using concrete examples, and connecting geometry to real-world applications. The book isn't simply about changing *what* is taught, but fundamentally *how* it is taught, with a focus on stimulating thought and fostering genuine mathematical insight.