The air is full of ideas on how to teach maths better. But many of these ideas are single features, such as "use manipulatives" or "ask high-level questions". Other are based solely on intuitions or anecdotal evidence. "Making Sense" attempts to break new ground by describing a system of instruction that fits together the individual pieces and provides the theoretical arguments and empirical evidence to support the system. It presents several key principles for teaching mathematics for understanding that teachers can use to reflect on their own teaching, make more informed decisions, and develop more effective systems of instruction. "Making Sense" reflects current research-based ideas on how to design classrooms that help students learn mathematics with understanding. It is based on the author's work in four different research programmes, all of which investigated the effects of specific instructional approaches. Out of their ongoing discussions emerged a consensus about what features are essential and what features are optional, which they share in this book. They also provide glimpses into their individual projects and in the classrooms from which they have drawn many of their ideas. By describing the essential features of classrooms that support students' mathematical understanding and by offering pictures of several classrooms that exhibit these features, "Making Sense" provides a valuable framework within which elementary teachers can reflect on their own practice and think again about what it means to teach for understanding.