How Children Learn Number Concepts: Number Relationships

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

Thanks for joining me for Book Study Mondays!  We are doing a virtual book study of Kathy Richardson’s book, How Children Learn Number Concepts.

Use the links to step through the entire series of posts:

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

In the introduction to Chapter 2, we are reminded that when children first learn to count, they are simply labeling the items they are counting.  To move forward in their mathematical understanding, they must realize that larger numbers contain smaller numbers.  Interesting that throughout this chapter Kathy never uses the words compose or decompose, yet that is exactly what she is talking about.  She uses the example that 4 is contained within 6.

The first group of critical phases related to number relationships fall under the category of changing numbers.  This involves counting on and counting back to make a new number.  For example, if you ask a child to change a pile of 5 counters to a pile of 7 counters.  I wonder how often we ask children to do that task in that way?  I know we frequently give students a number and ask them to find one more or one less, but that’s really a totally different task.  The way the task is outlined in the book, you are not telling the child how to change the number (add 1, etc.), you are telling them what number you want them to change the original number to. That is a much more challenging task and requires more mathematical thinking.  When you ask a child to change a larger number to a smaller one, we’re told that children only begin counting back when they understand that smaller numbers are contained within larger ones.  Prior to that understanding, they will count the group beginning at one and remove the extras.  I thought that was really interesting,and I’d love to try it out on some kiddos.  For example, if you ask a child to change 7 to 4, they would count from 1 to 4 and remove the three extra counters.  If they count back from 7 to 4, it signals an entirely different level of understanding. The next two steps in this process involve knowing how many were added or taken away after changing the number and finally knowing how many to add or take away without counting.

Ah, comparing numbers.  From my own experience, I knew that how many more was a difficult concept, but I felt validated when I read it in print.  Big ah ha moment in this chapter–when you ask a child how many more? they are often answering the question how many in the group with more?  I always knew it was a common mistake to simply state the number in the bigger group, but now I feel like I know why.  Another key idea that we need to keep in mind is that this task becomes more or less difficult depending on the size of the numbers used and the differences in those numbers.  So a child might feel comfortable determining that 7 is 2 more than 5, but might struggle with the difference between 6 and 11.  Also, and I think we knew this, children are able to compare groups that are lined up before they can compare groups that aren’t lined up.

Are you loving this book?  It lays out a crystal clear math trajectory, so to speak.  And what an amazing diagnostic tool!  With simple, quick tasks, teachers can pinpoint exactly what phase each student is in and can tailor instruction to each individual student’s needs.

Here’s the checklist for the phases in Chapter 2.  Next week, Understanding Addition and Subtraction.  Have a great week and happy reading!

Checklist, Word document (editable)
Checklist, PDF (better formatting)


  1. Barbara

    I think I have this book! If I do,, I’ll link up. Your major haha moment was mine too (from your post). I always wondered why so many kids told you “how many more” was the actual amount in that group.
    Grade ONEderful
    Ruby Slippers

    • Donna Boucher

      I know! It’s so nice when one of life’s little mysteries is solved.

      BTW, it should be a “You know you’re a teacher when…” when you say you THINK you have this book. Ha ha. I do that all the time. “Hmmmm, that book looks so familiar. I think maybe I own it!”


  2. Amy B

    Hi Donna! Again, I thought this chapter was an easy chapter to read and made perfect sense! I have honestly never thought of asking a child to change a number without telling them how to change it. That is certainly a different type of skill and seems obvious, now that I see it in print, that it should be practiced before using math symbols!!!!

    The “how many more” discussion really peaked my interest because my students ALWAYS have a difficult time with this concept! For the most they can do it if the two sets of objects are in nice neat rows that offer an easy comparison! I always try and relate it to words they might say to their friends… “that’s not fair I have 2 and you have 5…you have more than me!” I even bring kids up to the front of the room and act out “friend” situations that may help explain this concept. Usually, only my good math thinkers that really understand story problems and how to solve them can get this concept when the two sets are not shown in rows.

    I LOVE your sheets for each chapter. Whether I am in K or 1st next year, my goal is to use this book!!!

    Amy Burton

    • Donna Boucher

      Hey Amy,

      Can’t wait to work with my teachers on this next year. I started reading Chapter 3, and it’s going to be great!


  3. mondaymathmessage

    “For example, if you ask a child to change 7 to 4, they would count from 1 to 4 and remove the three extra counters. If they count back from 7 to 4, it signals an entirely different level of understanding.”

    Yes – and we also need to teach children the Number Word Sequence for counting backward, so when children develop the understanding that 4 is part of 7, they know the correct sequence of numbers to count backward.

    • Donna Boucher

      Of course! I think the more they work with concrete objects and connect them to the number names, the better they’ll learn that sequence, don’t you think? Line up, for example, 7 teddy bear counters along a number line. They can count up to 7 and then back down to 4 with the support of the number line, but still have concrete objects to connect the numbers to.



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