How Children Learn Number Concepts: Multiplication & Division

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

“Multiplication is not only an important computational skill, it is the foundation for understanding much of the mathematics children will encounter as they move on to higher-level mathematics.”

Right off the bat this quote sets the tone for the chapter, and I think it’s safe to say that many teachers have not really thought of multiplication in this light.  If you skimmed through the section on Multiplicative Thinking, be sure to go back and take some time with these pages.  Multiplicative thinking is different, and much more complex, than additive thinking.  Students are now required to understand that the numbers in a multiplication problem, the factors, take on different meanings.  One represents the number of groups and the other represents the numbers in each groups.  That is huge!  Again, a recurring theme in this book, students can appear to understand multiplication if they have memorized their basic multiplication facts.  But students need lots of concrete experiences with the principles of multiplication and division to truly understand the meaning of the facts they have memorized.

3 groups of 4

So, obviously, the key to understanding multiplication and division is understanding the concept of equal groups.  Students go through several stages of understanding as they experience equal groups in many different contexts.  Richardson comments that we must focus not only on the total, but also on the number of groups.  In the picture above, the common question is, “How many stars?”  Questions that also need to be asked are, “How many groups?  How many stars in each group?”  These types of questions and experiences help children understand the meaning of each number in the number sentence 3 x 4 = 12.  They also help kiddos make the bridge to understanding multiplication word problems, which add another layer of complexity to the process.

Arrays are another common model for multiplication, and I found it fascinating to hear that the row and column structure of arrays is not always obvious to students.  Another reminder that we need to expose students to a wide range of representations.

So division comes naturally if students know multiplication, right?  Apparently not!  Richardson states that students often don’t recognize the relationship between multiplication and division, especially when division results in leftovers (a remainder).  It goes without saying that students who have not had enough experience with equal groups will stumble over division.

Finally, with a strong foundation in single-digit multiplication and division, students are prepared for multi-digit operations.  I LOVED the story on the bottom of page 186 and the top of 187.  A third grade teacher panicked because she felt she’d spent too much time on single-digit multiplication and division and hadn’t left enough time for covering multi-digit multiplication.  Of course you can probably guess the happy ending.  Her kiddos had such a strong foundation working with equal groups that the transition to multi-digit multiplication was a breeze.  Now, one of the keys to this easy transition is the ability to decompose numbers into their parts.  Richardson devotes several pages to showing the representation of multi-digit multiplication using arrays.  The picture below, which shows 18 x 6, is a very simple example.  This is NOT an easy concept for many teachers to understand.  Take some time with it.

So, that’s the book!!  As with any new knowledge, be sure to revisit this book and study it more in-depth.  Reflect on how the information you have been digesting will look in your classroom.

Finally, with so many great books about math instruction out there, I suspect you haven’t seen the last of Book Study Mondays.  🙂

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


  1. Rikki

    I love your ideas…

    • Donna Boucher

      Thank you so much, Rikki!! 🙂

  2. Amy

    This was my favorite chapter. I agree with everything in it. I always have my students draw their pictures with the first number being the number of groups and the second number the amount in each group. I have some teacher friend that think the order is not important. I feel it is very important in learning the meaning of multiplying.

    This was a great book. Thanks so much for sharing it with us.

  3. Teacher Adriana

    Thanks for sharing I loved every minute!

  4. Barb Kelly

    I read all your posts on this book and have ordered the book. I am also a math coach who was given k and 1st this year for the first time and am learning so much. Thank you for all your ideas, you are my go to person for math knowledge and ideas.

    • Donna Boucher

      I appreciate your sweet comments, Barb! It’s an excellent book–very easy to read and one you’ll refer to over and over.


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