How do we get students to make sense — make genuine sense — of numerical operations?
One of the fourth grade teachers invited me into her class to help launch a major topic of the grade: multiplying multi-digit numbers. It may have been the first day of the unit, but, of course, the students had had previous experiences with multiplication including multiplying a two digit number by a one digit number, and multiplying multiples of ten. We hadn’t had time to plan the lesson together in advance, but she knows I’m always game to jump in — during the lesson, and also looking at student work afterwards.
The class started by estimating some products. While they haven’t learned how to find the exact answer for these problems, they have worked enough with multiplying by multiples of 10 that this should not be such a stretch.
The first problem was 28 x 15 ≈ ?
We elicited answers from the students.
“240.”
I asked students to give a ‘me too’ sign (thumb and pinkie extended) if they had the same estimate, and to raise their hand to offer a different one. Nobody agreed with this estimate. Many hands went up.
“360.”
Again, I prompted students to signal agreement, or to raise their hand to suggest another estimate. No agreement. Only hands.
“400.”
“420.”
Within 30 seconds, we had amassed a giant list of estimates for 28 x 15:
- 240
- 360
- 400
- 420
- 2000
- 1200
- 450
Typically, when I elicit answers for a problem, I might get three different answers? Four? I was surprised that students kept offering more and more answers. Some of them seemed to be increasing precision, but we went from 420 (the exact answer) to 2000?
We had too many estimates to deal with. I wanted to ask students to justify their thinking, but I thought first we’d pare the list down.
“Are there any estimates we can rule out?”
Leah raised her hand. “2000 seems way too big,” she said.
“Are you sure?” I asked.
She nodded meekly, and I saw other kids hesitantly agree.
“Before we cross it out, we need a reason.” Pairs of eyes leapt from the board to the floor. Silence.
“Well, I think 240 is actually too small…” I said. We needed to start somewhere.
“Oh!” Isaac raised his hand in the back. “28 x 10 is 280, and 15 is bigger than 10.”
Finally, we were getting somewhere! I recorded 28 x 10 = 280 on the board, and crossed out 240 with a dramatic flourish.
“So how do we feel about 2000 now?” I asked. Many students were still avoiding eye contact with me.
“28 x 100 is 2800, and 2000 feels closer to that, so times 15 won’t be close,” Olive added. We could build off of these ideas.
Jack pointed out that we can add 5 more groups of 28 to the 10 groups.
of 
Well, it turned out that 420 was the exact right answer. (I’m glad this wasn’t a middle school class.)
Interestingly, there was a white sticky note icon on the slide that said “what is the rule for multiplying by multiples of 10? Round your numbers to make it easier to multiple!” Not a single student referred to this, and I think, in general, they didn’t even register. This is a reminder not to overcrowd slides. #TeamBlankspace
Plotting Our Estimates for 19 x 19
About how much is 19 x 19?
We elicited answers from students, and they were… all over the place.
- 2,000
- 1,030
- 170
- 190
- 180
- 370
- 400
- 380
- 200
A range of 170 to 2000! I know that converging on consensus — assuming that the majority of students were close to the correct answer — is deeply flawed, but still decided to show the answers on a line plot. What patterns to do we see?
I drew a line, and labeled 0 and 2000. Then, I drew a tic mark for 1,000, “which is halfway between 0 and 2000.” This was followed by 500 — “halfway between 0 and 1000” — and smaller tic marks for each multiple of 100 between 0 and 1000.
I plotted the points with some help from the students.
“What do you notice?”
“I see a clump,” Levi said.
“Where?” I asked.
“Well, two clumps,” Levi corrected. Around 200 and around 400.” I circled the clusters.
Interrogating the clusters: why 200 and 400?
400 is a very reasonable estimate. 19 is close to 20, and 20 x 20 is 400.
So why 200?
I polled the students about whether they thought the answer was closer to 200 or 400. The class was roughly split in 3: those that thought the product is closer to 200, those who thought the product was closer to 400, and those that looked panicked and hoped I wouldn’t call on them.
“Will someone who got either an estimate close to either 200 or 400 explain their thinking?” I kept it open, and thankfully got lucky that the first student wanted to talk about 200.
Isaac started. “Well, 10 x 10 is 100, and 9 x 9 is 81. 100 + 81 = 181. So 180 is really close.” Ah, reverse engineering the estimate from his ‘exact’ answer! I was glad that he spoke first, because it revealed how all of the students got close to 200: they had only used two partial products.
For example, we could model 19 x 19 using an area diagram.
And Isaac did not decompose 19 x 19 into all four parts. He only used the green (10 x 10) and the orange (9 x 9). He missed the two blue areas of 10 x 9. I was a little surprised, since he had decomposed the previous problem with greater accuracy.
I annotated Isaac’s thinking on the board, and noticed a number of students nodding along. Many of these students are high status kids in math class, in part because they attend math programs and tutoring outside of school. They all seemed to agree that this made sense.
I asked for a student to explain their estimate of close to 400. Olive meekly spoke up.
“19 is close to 20,” she began. “And 20 x 10 is 200. But I need to get to 20 x 20 and that’s 400.” I added her thinking to the board.
“Why 20 times 20?” I pressed her.
“Because there are two 19s, and they’re both close to 20.”
I polled the class again: which estimate is closer? More students were persuaded by Olive’s argument, and we had significantly decreased the number of students projecting profound feelings of dread.
Can an area diagram help us?
We had explored area diagrams earlier in the year, to solve two digit by one digit multiplication problems. Here’s Olive’s work from November.
I decided to use an area diagram to anchor the end of this discussion. I highlighted the parts of the diagram that matched Isaac’s thinking.
I realized that, in our previous unit, we had usually dealt with decomposing a problem into two parts. That could work here, but it might also be a nightmare. For example, breaking up 19 x 19 into 10 x 19 and 9 x 19 means that we have to do 9 x 19, which students would probably do by solving 9 x 10 and 9 x 9. Layers of decomposing!
In a last poll, students seemed more convinced of 400 as a reasonable estimate, but also… maybe they were persuaded by the authority in my voice. I don’t know.
The lesson continued with some independent work. The classroom teacher also asked them to estimate 19 x 29 as an exit ticket.
What Next?
The teacher and I met in the back of the room to review student thinking from the exit tickets. The students ate their morning snack.
“What do I do?” She asked me. “There are a ton of kids that seem to be familiar the standard algorithm but they don’t quite remember it, so they’re way off.” I wish I had a neat and tidy answer.
I grabbed a piece of lined paper to brainstorm potential ideas for the next day’s lesson. Unfortunately, I wouldn’t be there due to scheduling conflicts.
I suggested that we focus on multiplying by multiples of 10. Maybe that would help them with a sense of quantity? We drafted the following problem string, deliberately using 7 x 3 as an anchor problem because a few students still need work with 7s facts:
And then where do we go from there?
“What if we have a story problem that asks them to justify their thinking? Then we can dig even deeper into assessing reasonableness.”
There are 21 stickers on a sheet.
[name] buys 12 sheets.
Will she have enough to give one sticker to each of the 200 kids in grade 4?
“Focus on justifying without figuring out the problem exactly,” I advised the classroom teacher, scrawling this note on the paper. We had only a minute or two left of snack time. Here was our final planning sheet. The note at the top (algebra with letters for missing variables) was a reminder that there always questions on our state test that ask students to solve problems like 
, and it’s important to build in more problems like this to the curriculum so that students are familiar with the structure.
“And then continue with the second part of the planned lesson from the curriculum,” I advised her. We identified an activity from the curriculum that seemed to fit smoothly with the plan we had made.
The classroom teacher looked over the problems thoughtfully. “I think it’s important to keep using area diagrams,” the classroom teacher said. I wholeheartedly agreed.
And then?
This last year, I found myself primarily coaching in the fall, and then increasing the number of intervention groups until I was primarily doing intervention in the spring. The students I saw for intervention were ones that were struggling to access any content in class, and I viewed this as very important and challenging work. It also meant that I had increasingly less time to collaborate with teachers.
I know that I benefit from collaboration with others when looking at student work and planning lessons. This experience left me wondering how we can continue to collaborate, even when I’m not in the room to teach or observe.
A week later, the classroom teacher found me in the hallway with a stack of exit tickets. “Can we look at these together?” We made a plan to meet for a few minutes later that day. It would be another rushed session, with mostly incoherent plans scrawled onto a sticky note, but this is also the real work in schools. We usually don’t have time to plan with beautiful, expansive protocols. We have a few minutes to try to get into kids heads, and think about what to do with it. I think a lot of my work is figuring out how to access student thinking, and helping teachers do the same.
