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Critical Learning Phases

“For each major mathematical idea, there are certain understandings that must be in place to ensure that children are not just imitating procedures or saying words they don’t really understand. These understandings must be in place, or at least developing, in order for a child to benefit from particular instructional experiences. These mathematical insights are predictable and universal. I have labeled these understandings, Critical Learning Phases.”

- Kathy Richardson

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Illusion of Learning

Kathy Richardson’s research into how children develop an understanding of the core math concepts for mathematics reveals certain essential ideas that are milestones in the process. Richardson has identified these stages as Critical Learning Phases. For each major mathematical idea, there are certain understandings that must be in place to ensure the student is truly learning and not simply imitating procedures. 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. AMC provides 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.

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