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What Educators are Saying about Kathy Richardson’s
How Children Learn Number Concepts: A Guide to the Critical Learning Phases

How do we help children develop the foundation necessary for future mathematical success? With clarity and eloquence, Kathy Richardson answers this question by illuminating the complexity of the intellectual work young children must do in order to build a robust sense of number. She identifies critical phases in the development of understandings in core topics such as counting and place value and provides examples that bring to life their importance for future mathematics learning. 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

I have used Kathy Richardson’s Assessing Math Concepts inter-views with students over many years. I think they are very powerful tool for identifying crucial missing links in mathematical knowledge. Now, with this new book, I have the background knowledge in one place to read and study. It is a great addition to Kathy's work.

Susan Friel, Ed.D
Professor of Mathematics Education
The University of North Carolina at Chapel Hill

Through her extensive research of interviewing thousands of children, Kathy Richardson discovered children go through predictable stages on their way to learning number concepts. She identifies these stages as "phases" and “critical” because of their importance. During this transition time toward implementation of the Common Core Standards in Mathematics, How Children Learn Number Concepts: A Guide to the Critical Learning Phases is a must have for teachers working with young children in number. She unpacks content as well as the learning journey for children acquiring number and number concepts. The Guide is clear and concise while giving guidance to empower me, as a teacher, to understand and in that understanding to meet the growth needs of students.

Sheri Willebrand, President
California Mathematics Council

I hope every teacher of young children, every parent, every admin-istrator and every educational policy maker will read this book. Kathy has studied children’s thinking about mathematics for decades and has now written a book that will help us all understand the critical stages of development in young children’s thinking that are essential to building a solid foundation for numerical reasoning.

Educational policymakers will come to understand the complexity of the ideas that young children encounter in thinking about number, and they, and we, will better understand that pushing abstract mathematical ideas too early in the grades can actually undermine children’s efforts to come to know and understand numbers and our number system.

As a grandparent, I learned a lot from this book about where to focus while helping my grandchildren with early number concepts. This book is a wonderful gift to teachers and parents alike who want to help young children make sense of early number concepts.

Ruth Parker, CEO of Mathematics Education
Collaborative (MEC)

This book might well become essential reading for any teacher desiring to understand a child’s developing sense of number. Richardson has taken complex ideas about number development and translated those ideas for teachers using practical, common sense terms in detailed description. Since Common Core was first published we have heard talk of progressions, trajectories and learning continua, often hard to translate for practical implementation by teachers. But Richardson transforms the complexity by introducing ”Critical Learning Phases”; obviously gleaned from observed behaviors of many children. In this easy to decipher text she offers teachers a treasure trove of ideas to help alleviate the confusion many teachers feel about teaching math. Richardson steers the reader towards some practical understandings. And she rightly warns us how “Sometimes the indicators that reveal whether or not a child understands the mathematics are overlooked, because in certain situations, he or she appears to know. So, assumptions are made that children know more about the concept than they do.” This warning nicely captures why it would be wise for all K, 1, and 2 teachers to read Richardson’s thoughts and practical interpretations on number related learning.

Hal Melnick, PhD
Bank Street College of Education NYC
Leadership in Mathematics Education

Kathy Richardson has provided us a wonderful explanation of how children acquire early skills and conceptual understanding of number. This book offers teachers and interventionists the crucial background information to understand what children REALLY know and what we ASSUME they know.

It is only with this magnified examination that we can “move” children along in their mathematical competencies or help them fill the gaps that have long brought them misconceptions. Making use of the information in this book should help teachers of young children be much more knowledgeable about the real meaning of master.

Patsy Kanter
Author, Every Day Counts, Calendar Math and Partner Games,
Summer Success: Math, USDE Helping Your Child Learn Math

This is a wonderful, in-depth explanation of how children learn beginning number concepts and how they should be taught if children are to have some depth, rather than superficial understanding, of the mathematics behind them. I think this book would be very valuable for the experienced teachers of young students as well as new teachers, in helping to shape their understanding of early mathematics learning. This book could act as a guide in their teaching, choice of curriculum, and assessment of students’ basic of number concepts.

The organization by Critical Learning Phases is logical and progressive. This path through the chapters provides a step-by-step understanding that learning about number is like building with blocks — as the building grows, each building block below is important for those that are added on top.

Paul Giganti, Jr.
Coordinator of Public Programs for Graduate School of Education
University of California, Berkeley
Author, Each Orange Had 8 Slices