Dr. Amy J. Hackenberg taught mathematics to middle and high school students for 9 years in L.A. and Chicago, prior to earning a Ph.D. in mathematics education from the University of Georgia. Amy is currently an associate professor of mathematics education at Indiana University-Bloomington. She conducts research on how middle school students construct fractions knowledge and algebraic reasoning and on how teachers can learn to develop productive student-teacher relationships. In her current project she is investigating how to differentiate instruction for diverse middle school students (see https://idream.sitehost.iu.edu/), studying her own teaching as well as working with practicing teachers. She is the proud co-author of the Math Recovery series book, Developing Fractions Knowledge.Dr. Anderson Norton is a professor in the Department of Mathematics at Virginia Tech. His research focuses on building models of students’ mathematical development. This work has generated interdisciplinary collaborations with psychologists and neuroscientists. Prior to this volume, Norton served as chair of the steering committee for the North American Chapter of the International Group for the Psychology of Mathematics Education, co-editor on a pair of publications bridging psychology and mathematics education, and co-author of the Math Recovery series book, Developing Fractions Knowledge.Dr Robert J. (Bob) Wright holds Bachelor’s and Master’s degrees in mathematics from the University of Queensland (Australia) and a doctoral degree in mathematics education from the University of Georgia. He is an adjunct professor in mathematics education at Southern Cross University in New South Wales. Bob is an inter­nationally recognized leader in assessment and instruction relating to children’s early arithmetical knowledge and strategies, publishing six books, and many articles and papers in this field. His work over the last 25 years has included the development of the Mathematics Recovery Program, which focuses on providing specialist training for teachers to advance the numeracy levels of young children assessed as low-attainers. In Australia and New Zealand, Ireland, the UK, the USA, Canada, Mexico, South Africa and elsewhere, this programme has been implemented widely, and applied extensively to classroom teaching and to average and able learners as well as low-attainers. Bob has conducted several research projects funded by the Australian Research Council including the most recent project focusing on assessment and intervention in the early arithmetical learning of low-attaining 8- to 10-year-olds.

With its beginnings in whole-number arithmetic understanding, Developing Fractions Knowledge is the most comprehensive look at fractions learning to date. It journeys with the practitioner and learner to view fractions as part-whole concepts, measures, and ultimately as numbers with meaning. The assessment and instructional tasks and practical connections to research and Math Recovery principles make it a must-have resource for educators to use again and again!

• IntroductionProfessional Learning and Three Grand Organizers for Arithmetic InstructionFrom Whole Numbers to FractionsAssessing Stages of Units CoordinationTeaching Students at Stage 1: FragmentingTransitioning to Stage 2: Part–Whole ReasoningTeaching Students at Stage 2: Measuring with Unit FractionsTeaching Students at Stage 2: Reversible ReasoningTeaching Students at Stage 3: Fractions as NumbersTeaching Students at Stages 2 and 3: Equal Sharing of Multiple ItemsTeaching Students at Stages 2 and 3: Multiplying FractionsTeaching Students at Stages 2 and 3: Adding and Subtracting FractionsTeaching Students at Stages 2 and 3: Dividing FractionsFrom Fractions to Algebra