Reflected Abstraction

In this chapter, we describe reflected abstraction and discuss its implications for empirical research into the teaching and learning of mathematics. We discuss the relationship between reflected abstraction and other theoretical constructs within Piaget’s genetic epistemology, and the mathematics education research literature generally. With this background established, we outline various implications of reflected abstraction for students’ learning of mathematics and teachers’ construction of pedagogical content knowledge. We then consider strategies for promoting reflected abstractions and propose explicit standards of evidence for supporting valid claims about whether an individual has constructed schemes at the level of reflected thought. We conclude by outlining pertinent questions that mathematics education researchers might pursue regarding either the necessary conditions for promoting reflected abstractions or the implications of reflected schemes for students’ mathematical activity and teachers’ instructional practices.