Proof in Mathematics Education: An Endangered Species? A Review of Teaching and Learning Proof Across the Grades: A K–16 Perspective

Abstract

Every rational discipline has its own epistemology: What is the nature of knowledge? From where does it come? How is it generated? How is it warranted? How is it certified? These questions represent knowledge as the product of actions by individuals and by a disciplinary community, subject to established norms and to social negotiation. Deborah Ball and I (2003) have argued that the teaching and learning of mathematics must attend seriously to the preceding questions, but we are merely picking up an argument made by several major educational thinkers (Bruner, 1960; Dewey, 1902; Schwab, 1961). Still, mathematics has a unique response to these questions, founded, since ancient Greece, on deductive reasoning and leading to the currently refined, formal, and in some ways austere notion of mathematical proof.1 Moreover, because of the largely self-referential nature of mathematics, education in mathematics, more than in most disciplines, has the potential to offer students the opportunity to engage authentically in its epistemological practices. This remains, sadly, a potential rarely fulfilled. On and off, over more than a century, people have argued for a prominent place for proof in the school mathematics curriculum, because the “art of proving,” or deductive reasoning, is at the heart of what it means to do mathematics. Moreover, high school geometry was identified early as the ideal arena in which to cultivate such reasoning skills. The curricular embodiment of this goal is scaffolded by the familiar two-column proof format. But, in many cases, the things that students are asked to prove are intuitively obvious, without surprise or sense of discovery, so that ritualistic execution of the two-column proof has become more an academic etiquette than a purposeful mathematical practice. Herbst (2002) has provided an insightful narrative of this history. Supporters of the New Math reforms of the 1960s exerted pressure so that proof would appear in school subjects beyond geometry. But as these reforms failed to fully permeate the school curriculum, the press for proof also wavered. A curricular