This book is a companion to Children’s Mathematics: Cognitively Guided Instruction1 and is driven by the same guiding principles of teaching and learning mathematics. Learning about the development of children’s mathematical thinking and attending to the details of children’s counting and problem-solving strategies can support you to recognize and build from children’s intuitive understandings of mathematics. Focusing on what children already know and can do mathematically allows you to position them as competent and to support them to communicate their understandings. Centering children’s thinking in your decision making as a teacher allows you to purposefully design or adapt instructional tasks and to intervene in the moment in ways that are responsive to the needs and understandings of each child.
This book will support you to notice and make sense of young children’s mathematical thinking, and provide you with examples of how teachers have created opportunities to build from children’s mathematical ideas in their instruction. Throughout this book we focus on the ways that children count collections of objects and use counting to solve problems. The development of understanding that we illustrate across chapters and examples is not particular to certain ages, but spans the entirety of early childhood. In Chapter 2, we detail the most critical principles that support the development of counting. In Chapter 3, we discuss cases of children acquiring and applying counting principles, focusing on what children do know about counting as they demonstrate emerging understanding of the principles.
In the following chapters, we discuss how children apply their emerging counting skills to solve problems. It turns out that young children are remarkably successful in intuitively applying their counting skills to solve a variety of problems in a number of different contexts. In Chapter 4, we look specifically at how counting can be extended into problem solving, the types of problems children can solve, and the strategies they use to solve them. In Chapter 5, we highlight the spaces in which counting and problem solving emerge in classrooms. In Chapter 6 we discuss how the strategies described in Chapter 4 are extended to solve arithmetic story problems. Chapter 7 examines how teachers engage with children and support them to develop their counting and problem-solving strategies. Chapter 8 draws on what we know about children’s mathematical thinking inside and outside of school and looks at how ideas of children’s thinking and informal practices can be used to bridge children’s worlds of home and school. Chapter 9 returns us to an examination of children’s thinking and looks forward to place value and how the ideas of grouping naturally arise as children extend counting to base-ten number concepts. Chapter 10 draws together some of the major themes developed throughout the book.