How Children Learn Number Concepts: Addition & Subtraction

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

Wow, what a powerful chapter!  I continue to be amazed at how much is packed into each chapter of this book and how easy it is to read and follow.

She really sets the tone of this chapter by adding Parts of Numbers to the title.  You just can’t know addition and subtraction without understanding parts of numbers.  Take a look at this statement on the very first page of the chapter:

“When children know the parts of numbers through 10, they automatically know the basic facts.”

What could be more important in Kindergarten and 1st Grade than to help children understand numbers and their relationships?  If they slip through without that understanding, they will likely struggle throughout their mathematical career.  Again, that’s why this book is so perfect–it’s a roadmap of the critical phases.

The first critical learning phase related to addition and subtraction involves…drum roll…subitizing!  Children need to recognize quantities in a variety of different configurations so they are not just memorizing a picture and associating it with a number.  For example, the dot patterns on dice. I made a set of cards showing the numbers 2-5 in various patterns.  Of course, you would start with the smaller numbers first and gradually work your way up to the 5s.  Click on the picture to grab your copy.

Once students recognize small groups, they’ll begin to see those smaller numbers in larger numbers.  Look at the pictures above.  Think about the smaller numbers you see in each configuration.  Do you see 4 and 1 in the first picture?

In the next phase, children learn to combine parts.  This is not just one step, but a series of stages.  A key milestone is understanding that amounts are not changed when the parts are moved around.  For example, if a student is working with 5 counters, they recognize that it will still be 5 counters whether they group the counters as 4 and 1 or as 2 and 3. As they continue to work with numbers, children start to use combinations they know to help with those they don’t.  They might realize that knowing 3 + 3 can help them solve 3 + 4.  She makes the point, and I think it’s an important one, that it is better for children to make that discovery on their own, rather than trying to teach them.  The teacher’s job is to carefully craft experiences to lead children to make the discoveries on their own.

After combining comes decomposing.  This involves finding missing parts.  We’ve all seen versions of missing part games.  You show a child 5 counters and then hide some.  The child has to tell you how many are hidden.

Three ranges of numbers are identified for combining and decomposing: numbers to 6, numbers to 10, and numbers to 20.  I found it kind of interesting that the first range was numbers to 6.  I think we routinely think of the first range as numbers to 5.  She included a cautionary note about combinations for 5, and explained why the first range is to 6.  She said that teachers are often fooled into thinking that students know the parts of 5, when in fact they have only memorized the pairs.  If students truly know the combinations for 5, they should be able to use that knowledge to determine combinations for 6.  If they can’t, it’s a signal that they don’t really understand the parts of 5.

Finally, it takes time!!  And, not surprisingly, children learn about number parts more quickly when they have lots of concrete experiences.

Great chapter with lots of powerful information!  As with the other chapters, I’ve created a checklist you can use with your kiddos.  I find it’s helpful for me to make the checklists–it helps me organize my thoughts from the chapter.

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

Next week, Chapter 4, Understanding Place Value: Tens and Ones.


  1. Amy B

    I thought this chapter was GREAT! I even read it while watching Meet The Fockers…haha!!! Alot of what I read I had heard before when talking about the upcoming common core. Kindergarten works with up to 5 and first grade works with problems up to 10. One of the first pages said “When children know the parts of numbers through 10, they automatically know the basic facts.” Makes perfect sense when you see it in writing.
    I am getting so much out of this book and now know how important it is for children to fully understand and be fluent in either facts to 5 if I am in Kindergarten or facts to 10 if I am in first grade.
    Amy Burton

    • Donna Boucher

      I know what you mean, Amy, about having read most of this before, but I just love how she’s organized it into such an incredible, step-by-step resource!

      Double check the common core, because I think K is to 10 and 1st is to 20. So basically, they should come out of K knowing the parts of 10 and be ready to tackle the rest of the facts to 20 in 1st. That’s how I read it. Now, that DOESN’T mean flashing flash cards in 1st grade! It still needs to be very concrete learning.


  2. Amy B

    You are right…K is up to 10 and 1st is up to 20…but fluently it’s up to 5 in K and up to 10 in 1st. I know most teachers just think of doing flashcards when they have to teach math facts…me included, I use them alot! Next year I want to really see if my kids can compose and decompose numbers before working on the fluency piece.
    And, I LOVE how she has organized this book too!!!! It is incredible! Our coach is talking about holding a book study on it at school because I have raved so much about it!!!! 🙂
    Happy Memorial Day and thanks for the sale too!!!

    • Donna Boucher

      Ah, yes, I see the fluently part now. This would definitely be a great book study for a campus. While I’m loving the book study now, I wish we were still in school, so we could go back and try things out. I’m thinking we’ll need to revisit the book in the fall as we put it into practice!


  3. Storie

    Just found your blog & started following. I can’t wait to look around more to see what else is here. Do you have any experience with Origo math materials? If so, I would love to chat with you about them. Stop by my blog sometime.
    Stories by Storie

    • Donna Boucher

      Hey Storie! Glad you found your way here! Be sure to link up your blog on my Blog Hop page. Yes, I am quite familiar with Origo. They produce GREAT stuff. Their professional development, if you’re ever lucky enough to attend a session, is amazing. I really like their Think Tanks and Zupelz for math workstations.

    • Miriam C

      HI, I have read your blog a couple of times and I find it amazing. I teach first grade and we started implementing the Kathy Richardson centers for math this year. These work great with students. Do you know of another resource that has the same impact as KR?

  4. Penny Messick

    I love your blog and I’m one of your newest followers! I have chosen your blog to receive the One Lovely Blog award. I would be honored if you would visit my blog to check out the details! Thanks!


  5. Sitting Behind Homeplate

    I’m really enjoying these Sunday blogs (even thought I get them on Monday). I’m teaching first grade next year and plan to read through the book this summer and prepare myself and students. This past year I was our RTI coach and the two years before our taks tutor and the problem I had trying to fill the gaps was the lack of number sense. Thanks for the summaries.

    • Donna Boucher

      Exactly! I taught primarily 3rd-5th and saw the same thing–huge gaps due to lack of number sense. The kiddos just kept getting farther and farther behind. I think that’s why I’m so fascinated with early numeracy now.

  6. Tracie Gones

    I’m a few days behind, but that’s something I love about this kind of thing! I can participate even if I’m a few days behind. 🙂 After reading your comment to Connie, it sounds like we are very similar. I currently teach 4th grade and experience huge gaps in number sense, making it all but impossible to teach multiplication, division, and early algebra. I almost didn’t think this book would be relevant since it is for teachers of younger students. But I can definitely see how I can use some of the ideas to help my struggling 4th graders. My district is also contemplating the idea of having building math coaches, probably in 2013-14. I am hoping to do this and originally thought I would want to be a coach in one of our 4th-6th grade buildings because that is the grade level I enjoy teaching. But this book has made me seriously consider that I could be a coach in a K-3rd building. It would be so rewarding to help kids really gain that number sense so that they can be successful in the upper grades.
    Thank you for all the work you are doing to host this book study. I am downloading and saving all your freebies to share with my primary friends, and possibly use myself in a year or two!

    • Donna Boucher

      Tracie, I’ll just say that I have loved working with the primary teachers. And by working with them, I KNOW I’m also impacting the upper grades.

  7. Tammy

    This looks like a great chapter. Thank you for the checklist. It will be very helpful.
    ❀ Tammy
    Forever in First


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