Welcome to the first in a series of Book Study Mondays, a virtual book study of How Children Learn Number Concepts, by Kathy Richardson. I hope you’ll join me each Monday to read my reflections and contribute your own. The more the merrier!
Use these links to visit any of the chapters:
Chapter 1, Understanding Counting
Chapter 2, Understanding Number Relationships
Chapter 3, Understanding Addition and Subtraction: Parts of Numbers
Chapter 4, Understanding Place Value: Tens and Ones
Chapter 5, Understanding Place Value: Numbers as Hundreds, Tens, and Ones
Chapter 6, Understanding Multiplication and Division
I loved this book from the moment I took it out of the shipping box. It just looks like a readable book. Do you know what I mean? It’s small, thin, and looks friendly. What I have found is that in addition to be readable, it is usable. This is one book that you will want to have within reach at all times. After reading Chapter 1, I just wanted to go out, grab a Kindergarten child, and put what I read into action.
Kathy begins in the Forward by defining Critical Learning Phases as “understandings that must be in place if children are going to be successful in the study of mathematics.” And while I mentioned Kindergarten in the last paragraph, the audience of this book is teachers in grades Pre-K through 4. So where did the Critical Learning Phases come from? Over forty years of Kathy’s classroom experience working with kiddos from pre-school through 6th grade. This book reads like a wonderful road map! So let’s delve in to Chapter 1.
Chapter 1, Understanding Counting
As adults, we take many of our abilities for granted. It’s easy to forget that at one time we didn’t know how to organize objects to count them or count by twos instead of ones. It is habit to us at this point. The chapter starts out with an overview of the difference between truly counting and reciting numbers. It distinguishes between how adults see numbers and how children see them. As with each chapter (I peeked ahead…), Kathy lists the critical learning phases and then discusses each in just enough detail. The counting critical phases include four broad groups: counting objects, knowing one more/one less, counting objects by groups, and using symbols. Each broad category contains a number of phases.
A common thread through many of the counting critical phases is that children are beginning to make sense of numbers. It helps to remember that the symbol 5 means nothing to a child unless they understand the quantity it represents. Reciting numbers in sequence is just a memorization task. The book points out that when children begin to adjust estimates or catch the fact that a mistake in their counting doesn’t make sense, they are beginning to make sense of numbers.
Another point that I found interesting is that the critical phases are not all or nothing. A child might be able to keep track of an unorganized pile up to a certain number, but then struggle as the numbers get larger.
I found the section on counting objects by groups particularly fascinating. She first points out that when kids count by groups without actually counting objects, it can lead to misconceptions (back to that CRA idea). What can result is students pointing to single objects and counting by, say, 2s, because they don’t completely understand the idea behind counting by groups. We actually reinforce that misconception when we ask kids to count by nickels or dimes, because they point to one object but count by 5s or 10s. So what does that mean in the classroom? To me it means children need lots of concrete practice counting objects in groups prior to recitation.
Maybe it’s just me, but I automatically thought of having a checklist to keep track of students’ progress along these phases. Use the links below to grab yours!
I hope you enjoyed this chapter as much as I did! See you next Monday night for Chapter 2.