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Critical Learning Phases

The Critical Learning Phases, identified by Kathy Richardson focus on number concepts. The development of number concepts is the foundation and heart of the mathematics program for young children. What children know and understand about number and number relationships impacts every other area of mathematical study. Children cannot analyze data, determine functional relationships, compare measures of area and volume, or describe relative lengths of sides unless they can use numbers in meaningful ways. Number concepts are the foundation that children must have in order to achieve high standards in mathematics as a whole.

Children who understand number concepts know that numbers are used to describe quantities and relationships, and are useful tools for getting information about the world they live in. They see relationships between numbers and can take numbers apart and put them back together without counting. They understand the structure of numbers and think of numbers as made up of groups of tens and ones, or hundreds, tens and ones, and so forth. They know what happens to the numbers when they add, subtract, multiply, or divide. They use symbols to represent numerical ideas and can explain and interpret what the symbols mean. They are able to think with numbers, and to use numerical ideas to analyze situations and solve a variety of problems. They work with numbers with facility and ease and demonstrate proficiency with computation.

When children receive instruction before they have these foundational ideas in place they may memorize the steps that give an “illusion of learning,” but they will not grasp the underlying logic of the mathematics they need for future growth and understanding. The Assessing Math Concepts (AMC) assessments  provide the tool for teachers to identify the stage of development, or Critical Learning Phase, a student has achieved. These insights provide teachers with the most comprehensive information available to provide targeted instruction to meet each student at their specific level.

The AMC assessments follow these stages of students’ growth through these phases as they learn the core concepts for numbers up to 100. The AMC assessments determine the level of thinking each student has reached. Knowing this level guides instruction so that each student is working at the most appropriate level for maximum growth and understanding.

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How Children
Learn Number

A Guide to the
Critical Learning Phases

by Kathy Richardson

Grades: K-5

A "must-have" book for every educator wanting to know how young children develop an understanding of number concepts. Through her years of research and extensive work with young children, Richardson has identified the stages, or Critical Learning Phases, children go through as they develop mathematical understanding. This resource will help all math educators understand how children make sense of numbers and what mathematics they are ready to learn at each phase of development.


Number Relationships

Addition and Subtraction

Place Value

Multiplication and Division


"At a time when policy makers expect children to learn mathematics concepts and skills at earlier and earlier ages, this book should give teachers the courage to spend the time necessary for children to develop numerical understandings strong enough to provide a solid foundation for, rather than illusions of, learning."

Cathy Humphreys, Doctoral Student, Stanford University
Co-author of Connecting Mathematical Ideas with Jo Boaler
Co-author of A Collection of Math Lessons, Grades 6-8 with Marilyn Burns

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