Thomas P. Carpenter

"With the collaboration of a number of dedicated teachers and their students, Susan Empson and Linda Levi have produced a volume that is faithful to the basic principles of CGI while at the same time covering new ground with insight and innovation." --Thomas P. Carpenter This highly anticipated follow-up volume to the landmark Children's Mathematics: Cognitively Guided Instruction addresses the urgent need to help teachers understand and teach fraction concepts. Fractions remain one of the key stumbling blocks in math education, and here Empson and Levi lay a foundation for understanding fractions and decimals in ways that build conceptual learning. They show how the same kinds of intuitive knowledge and sense making that provides the basis for children's learning of whole number arithmetic can be extended to fractions and decimals. Just as they did in Children's Mathematics and Thinking Mathematically, Empson and Levi provide important insights into children's thinking and alternative approaches to solving problems. Three themes appear throughout the book: building meaning for fractions and decimals through discussing and solving word problems the progression of children's strategies for solving fraction word problems and equations from direct modeling through relational thinking designing instruction that capitalizes on students' relational thinking strategies to integrate algebra into teaching and learning fractions. With illuminating examples of student work, classroom vignettes, "Teacher Commentaries" from the field, sample problems and instructional guides provided in each chapter, you'll have all the tools you need to teach fractions and decimals with understanding and confidence.

In Children's Mathematics: Cognitively Guided Instruction, Thomas Carpenter, Megan Franke, and Linda Levi helped hundreds of thousands of teachers understand children's intuitive problem-solving and computational processes and how to use that knowledge to enhance students' understanding of arithmetic. In Thinking Mathematically, the same author team shows how Operations and Algebraic Thinking can be viewed as a unified field by understanding how children's intuitive strategies naturally draw upon the properties of operations and other algebraic concepts. This book shows how teachers can recognize and support children's use of the properties of operations and other algebraic concepts in a manner that deepens students' understanding of arithmetic and provides a solid foundation for learning algebra. This book also shows how teachers can increase their own knowledge of mathematics in the process of interacting with their children and reflecting about their practice. Thinking Mathematically provides numerous examples of classroom dialogues that indicate how properties of operations and other algebraic ideas emerge in children's thinking and what problems and questions help to elicit them. Special features of the book help teachers develop their own understanding of mathematics along with their students': Teacher Commentaries capture the voices of a number of teachers, providing realistic portrayals of what happens in class. End-of-chapter Challenges offer a variety of problems and activities for teachers to increase their own knowledge of mathematics and to help their students develop algebraic thinking. Additional Online Resources provide rich illustrations of ideas in the book, including extended interactions with individual children or classroom episodes--all clearly linked to the text.