"This book is about teaching students to do the actions of math-ing, the kinds of mental reasoning that mathematicians do. Pam Harris contends that the way math is often taught--using formulas and algorithms that must be memorized without truly understanding why these shortcuts work--keeps the majority of students from a deep understanding of math and therefore limits their possibilities. Harris argues in this book that teaching to algorithms creates distortions about the nature of mathematics that undermine present and future success. Conversely, teaching math that is free of distortions will reach more students more effectively and result in deeper understanding and longer retention. The ability to grapple simultaneously with an increasing quantity of numbers, operations, and spatial relationships is the hallmark of a mathematically developing mind. Harris has developed the hierarchy of mathematical reasoning that helps teachers to help students build on their math understanding in a coherent way that will open the gates to higher math. This book progresses through the most basic domains of math development (counting and adding) and builds to the more complex domains (proportional reasoning and functional reasoning). By understanding these domains, and knowing how students progress through them, teachers will be better equipped to prepare students for higher math"-- Provided by publisher.