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Recommendations for Teaching Division with Remainders by Joshua Lerner

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Welcome fellow Recovering Traditionalists to Episode 122 where we are going to look at Teaching Division with Remainders Even in The Early Years.

Division is often only thought of something that we teach students in third grade and beyond, yet if you are an early years teacher I’d encourage you to keep listening because there are things we can do in the early grades to help kids have a foundational knowledge that will help them as they get into 3rd grade and really have a focus on division.  Plus, division situations come up often for kids in their life and so they naturally experience division very early on.

In last week’s episode, I talked about how one division problem 26 ÷ 6 can have 4 different answers all depending upon the context and what the context says to do with the remainder.  My training in CGI which taught me that, also introduced me to the idea that division isn’t just dividing an amount into equal parts.  This is often known as “fair sharing” division problems, or Partitive Division.  It is problems like this:

Christina has 26 cookies.  She wants to give them to 6 of her friends so they each get the same amount of cookies.  How many cookies can she give each friend?

This is typically the natural way that kids think about division and the way that most textbooks start teaching kids division.  However, it only gives kids a partial understanding of division.  In Partitive problems, the number of groups you need to divide the total into is known.  But the 2nd type of division problem is when the number of groups is actually not known and instead you know how many go in each group.  This type of problem is known as a Quotative division problem:

Christina has 26 cookies.  She wants to give 6 cookies to each of her friends.  How many friends can she give cookies to?

According to a recent article published by NCTM, Partitive Division isn’t the place we should start if we want to help kids build an understanding of division with remainders.

Now, I usually like to only talk about articles that you can read for free, but teaching division is such a hard thing especially when dealing with remainders.  The article I’m sharing today is only available to NCTM members.  If you are an NCTM member you can get the link to the article at  If you are not an NCTM member, I encourage you to join.  I joined the first year I started teaching and I have learned so much from the journals they publish. I’ll link to their registration page on as well.

In the December 2020 issue is the article Recommendations for Teaching Division with Remainders by Joshua Lerner.  He refers to Division With Remainders as DWR throughout the article and on page 1030, he writes:

“On the basis of our lesson study work, we make the following recommendations for high-quality instruction of DWR: (1) strategically use quotative division contexts, (2) help students understand why the remainder must be smaller than the divisor, and (3) teach how to interpret the meaning of remainders in context only after students have achieved some fluency.”  

As he talks specifically about the first recommendation, strategically use quotative division contexts, he says:

“…we discovered through our work that it may be more effective to use quotative division to introduce DWR. One of the most common mistakes students make when solving DWR problems is attempting to further distribute the remainder among some of the parts (Van de Walle, Karp, and Bay-Williams 2010). Indeed, Lamberg and Wiest (2012) showed that when students were asked to create their own DWR situations, their discomfort with having something “left over” led them to either continue to partition the remainder into fractional parts or simply change the dividend so the problem worked out evenly. However, by using a problem in which the amount in each group is already known, we found there was no need for students to take the left-over amount and try to redistribute it. Student learning focused squarely on the remainder as a quantity to be understood in its own right.”

As I said earlier, I know not every teacher listening teaches division with remainders.  For those of you who do, start looking at the problems your students encounter around division.  For those of you who do not teach DWR, the CGI research shows that kids are able to solve division problems as early as preschool.  They count and model the problems.  In fact, young kids are presented daily with situations in their lives that require them to divide things up…we even naturally do it with kids when situations arise in class.  So I want you to see what types of situations your students are encountering…if they are always partitive (like: I have 12 papers to hand out to your group. There are 4 of you at the table, how many do you think I’m going to give each of you?), try switching it up to be Quotative (like: I have 12 papers.  Each of you is going to get 2 pieces of paper, do you think we have enough papers for everyone in the group?)

Math is about helping us solve problems that occur in our lives, so even though young kids don’t have standards about division, they actually do have problems that come up that are division contexts.  We can encourage kids to begin thinking about sharing and dividing in both types of scenarios before they hit 3rd grade and then most state standards address both types of division in 3rd grade so kids should formally be exposed to both Partitive and Quotative division problems.  Then in 4th grade when the standards say kids need to be able to interpret the remainder (and if you aren’t sure what that means, it’s basically what I talked about in last week’s episode #121), start out with Quotative division problems because those contexts make kids pay attention to the remainder.  Eventually you do want to bring in problems with partitive contexts but be aware, those are the contexts that make you ‘fair share’ and if there is a remainder kids want to share that remainder out and that gives you opportunities to dive into putting the remainder as a fraction.

A reminder that I will link up the article over at and you have to be logged into your membership in order to view it.  But I’ll also put a link to join NCTM on that website as well.

Thank you for joining me today and I hope this helped you build your math mind so you can build the math minds of your students.

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As you start off the school year, I want you to keep in mind what is really important as we're trying to teach mathematics to our students.