A few minutes after publishing my last blog post, “A Third Grade Dilemma: Dividing by Zero,” I received an e-mail about it from my father. (We are the cutest family. Not up for debate.) His love language is sharing knowledge, wonder, and curiosity — about anything, but he has a special interest in mathematics. He wrote the following:
| James B Orlin <xxxxx@xxx.xxx> | Sat, Apr 6, 2024 at 9:50 AM |
| To: Jenna Orlin Laib <xxxx.xxxxx@xxxxx.xxx> | |
| Hi Jenna, I enjoyed your post today on your blog. I also have a suggestion on explaining division by 0, albeit it is a bit advanced. Division has two stories, as Ben points out in his soon-to-be-published book. (He learned the second story long after becoming a math teacher.) The second story can be used to illustrate division by fractions, and it can make it clear that one cannot divide by 0. Here is an illustration of the second story of division. Suppose that you have $10 in coins. And you want to divide it into piles of $2.00 each. How many piles would you create? Suppose you want to divide it into piles of 1/4 of a dollar each. How many piles would you create? Now suppose you want to divide it into piles each of which has no money in it. How many piles would you create? Love, Dad | |
He’s absolutely right! I remember learning this in my first few years as a teacher. I typically credit Marilyn Burns’ Do the Math module on concepts of division, but the seeds may have been planted earlier. CGI? Liping Ma? But, sometimes, we know something so much better when we have to teach it. The Do the Math module is brilliantly designed, in that it gives elementary students participating in intervention time to internalize both of these. The first five lessons focus on sharing cookies using The Doorbell Rang, a classic sharing (partitive) situation. The next five lessons focus on making packages of pencils, a grouping (quotative) situation. Having spent ten days anchored on these two situations means that students have a choice of contexts to draw upon when they’re trying to visualize, say, 
Or 
I didn’t have time to respond in the moment — we were on our way out the door for my son’s karate class, and my daughter had tickets to the school musical that afternoon — but we dropped by my father’s house later to say hello. (Okay, and play with his dog.)
Connecting Division Stories to Understandings and Algorithms
My father was right that a quotative, or grouping story, would have worked much better to illustrate why
I wrote about these two frameworks for thinking about division in a post called, “Windows and Mirrors: Fair Shares for Eid al-Adha” (August 2, 2021). In it, I posed two different contexts that came out of my family’s celebrations of Eid al-Adha. (We celebrated Eid Al-Fitr last week, marking the end of Ramadan! Eid al-Adha 2024 is in June.)
I told my father about how his idea about division contexts had been in the forefront of my mind last fall, as I planned a district-wide PD for fourth grade teachers last fall. While reviewing the next unit of study, “Multiple Towers and Cluster Problems,” I noticed that some of the lessons explicitly called for students to think about division as taking away groups.
I wanted to draw attention to this during the PD. We wanted to use both sharing and grouping situations throughout the unit, but there were times when one situation might illustrate the computational work much better. Are there some algorithms that match better with one particular kind of division story?
Fourth Grade Professional Development
To launch the session, I asked teachers to try out the end of unit assessment.
As participants compare and contrasted their work with colleagues, I monitored for whether they had written sharing or grouping contexts. Most fourth grade teachers gave a sharing context. Here’s one from fourth grade teacher Stephanie, that I used as an example, five practices style:
I considered how her operational work matched her story. What would the 8 x 10 represent? And the 8 x 1? I could envision distributing 10 pieces of candy to each teacher, to start, and then realizing that, with the leftovers, each teacher was given another piece.
We looked at work from fourth grade teacher Jess next.
She used short division to solve the problem. It’s similar to long division, but instead of going through the process of multiplying an subtracting, students consider how many ‘leftovers’ there will be, and record those leftovers as a superscript.
I also considered how we could match Stephanie’s work to Jess’ story. Now, the 8 x 10 would represent 10 groups of 8 kids, and then we can make 1 more group of 8 kids. Both contexts will work for Stephanie’s work, but Jess’ story felt like a more natural fit. It’s less to keep track of. We’re just replicating our group making process, over and over and over again.
I illustrated how thinking of division as making groups works well with the thinking that is emphasized during this unit. In the first section of the unit, students focus on multiplication by splitting and decomposing a factor. In the next section, students think of division as the inverse of multiplication, and form equal groups to solve division problems. For example, here are ways to use multiplication to solve 91 ÷ 8, focusing on making groups.
There’s nothing to distribute. We’re just joining groups.
Launching the Division Unit: Swapping Stories
I co-taught this unit in a fourth grade class at my school. Before launching the division section, the classroom teacher and I gave every student a quick assessment on the multiplication investigation. We asked them to solve 18 x 7, and draw a representation to match. Later, we reviewed the student work samples together, and felt stuck.
“Some kids just get it,” the classroom teacher said, showing Owen’s work as an example.
“…but other kids?” She continued.
We noted that the student drew an array, and knew to decompose a factor, but then seemed to count each little box individually. Her records of 18 x 3 and 18 x 4 aren’t accurate.
This student also decomposed factors, but nothing seemed connected to an understanding of multiplication and arrays.
I met with Marilyn Burns, one of my favorite people, that afternoon, armed with this dilemma. I noted that we’d given them a decontextualized problem, and they’d attempted to make it visual, but not all of the connections were… mathematically sound.
We looked at the first division lesson together. It launched with a sharing division story. I told Marilyn about the PD that I had led, and how it felt like grouping situations match partial quotients better.
So what if we gave students a problem that they could make a nice visual for? One where they would draw something that would also highlight some of this unit’s thinking about multiplication and division as inverse operations, and dealing with groups?
Marilyn and I ended up writing this problem:
Marilyn blogged about the lesson in a post entitled, “Mulling About Teaching Division.”
During the lesson, I used several pieces of student work to anchor our class discussion. This was one of them, from fourth grader Adelaide. The fact that Adelaide wrote repeated 4s (instead of drawing dots) seemed to help students make deeper connections to multiplication.
What Furthers Understanding?
I drew little diagrams for my father on the back of an old envelope. We examined them together, hunched over his kitchen island. My father thinks a lot about math, but it’s not often he has reason to think about something like how different division contexts can illuminate different ideas about the operation.
It’s the same for other operations. Take away stories (5 kids were playing and 3 went home) and comparison stories (how much longer is a 20 inch string than a 17 inch string) reveal different things about subtraction. These stories connect to different algorithms and strategies.
Which is all to say: as happens frequently, my father was totally right, and I love that he found this particular facet of elementary mathematics interesting. These nuances are important when our guiding question is “what furthers mathematical understandings and skills?” And, as teachers, we use our pedagogical content knowledge to help us determine what to use and when.
How is math a metaphor for our real world, and how is our real world a metaphor for math?
I care about improving my explanations to students, as in “A Third Grade Dilemma: Dividing by Zero,” and also improving the problems I pose to students, like the picnic table problem that Marilyn and I wrote. Knowing about these two kinds of division contexts helps both. It’s all in pursuit of deepening and clarifying mathematical understandings.
Jenna Laib // Embrace the Challenge 