Establishing a custom of proving in american school geometry: evolution of the two-column proof in the early twentieth century

Abstract

Having high school students prove geometrical propositions became the norm in the United States with the reforms of the 1890's — when geometry was designated as the place for students to learn the ‘art of demonstration.’ A custom of asking students to produce and write proofs in a ‘two-column format’ of statements and reasons developed as the teaching profession responded to the demands of reform. I provide a historical account for how proving evolved as a task for students in school geometry, starting from the time when geometry became a high school subject and continuing to the time when proof became the centerpiece of the geometry curriculum. I use the historical account to explain how the two-column proof format brought stability to the course of studies in geometry by making it possible for teachers to claim that they were teaching students how to prove and for students to demonstrate that their work involved proving. I also uncover what the nature of school geometry came to be as a result of the emphasis in students' learning to prove by showing that students' acquisition of a generic notion of proof was made possible at the expense of reducing students' participation in the development of new ideas. I draw connections between that century-old reform and current reform emphases on reasoning and proof. I use observations from history to suggest that as we carve a place for proof in present-day school mathematics we must be leery of isolating issues of proving from issues of knowing.