Teaching and Learning Authentic Mathematics: The Case of Proving

As the goals of mathematics instruction have broadened over the past few decades, there has been a growing appreciation of the idea that there is value in students’ classroom mathematical activity, even in the elementary school, being a representation of some core aspects that are characteristic of mathematicians’ activity in the discipline of mathematics. In this chapter, we use the notion of authentic mathematics to describe the productive intersection between classroom mathematical activity and disciplinary mathematical activity. Deriving from established knowledge in the field of mathematics education, we propose four criteria to define authentic classroom mathematical activity. We illustrate the criteria in the particular area of proving, drawing on examples and nonexamples of authentic classroom activity from the research literature. Our choice of proving as a context for our discussion was motivated by its central role in mathematicians’ work and the widespread agreement among researchers and curriculum frameworks on the significance of proving in students’ learning of mathematics as early as the elementary school. The fact that proving is also a hard-to-teach and hard-to-learn activity complexifies efforts for conceptualizing and enacting instruction that promotes authentic mathematics in the area of proving, and this further motivated our focus on it in this chapter. Our discussion demystifies the nature of teaching and learning authentic mathematics and identifies productive directions for future research.