A visit I made to a 1st grade classroom at the end of the year got me thinking about what this might look and sound like with young children. Students were playing Adding within 100. In this game, students turn over 2 cards between 1 and 50, build each number with cubes organized into sticks of 10 and single cubes, and then determine the total. Most of the students I observed worked quickly and confidently, and gave clear explanations in response to my questions about how they knew what to build and how they determined the total. For example, Kayla worked on 48 + 16. She built both numbers easily, moved some cubes around, and then determined the total. When I asked her how she solved the problems she said:
“This is 4 (points to the 4 sticks of ten). That’s this 4 (points to 4 in 48) and this is 8 (points to the tower of 8). I know 8 + 2 is 10 so I took off 2 from this one (points to the tower of 6) and that made another 10.” Kayla then counted her cubes (6 sticks of 10 and 4 ones) and got 64.
I was struck by the understanding students were using to solve these problems. Most were able to use cubes to represent the two-digit numbers without counting by 1s, and many used their knowledge of tens and ones, and of combinations that make ten, to solve these problems. Even so, many seemed to be working in an almost routinized way. They would look at the numbers, quickly build them, move some cubes around to make as many sticks of ten as possible, and then count the cubes by 10s and ones. I was curious how deep an understanding these students had of what was happening in these problems, so I began to ask them to estimate before they built the numbers and solved the problem.
Estelle and Denayah were about to work on 18 + 45.
Me: Do you think there are more than 20 altogether?
Estelle: Yes, because 45 is more than 20.
Me: Are there going to be more than 50?
They both think.
Denayah: The same.
They build the two numbers, then Denayah breaks off 5 cubes from the tower of 8 and adds them to the tower of 5.
Me: How did you know how much to add on?
Denayah: I knew it was 5 here and 5 and 5 make 10 so five more.
Then they figure out the total number of cubes.
Me: Was it more than 50?
Denayah and Estelle: Yes.
I know that the numbers in these problems are pretty large for 1st graders to add, and that they aren’t expected to add them without models and representations (CCSS.Math.Content.1.NBT.C.4.) But, asking them to estimate seemed to interrupt the more routinized approach they were taking and encourage them to think more about the numbers and the operation of addition. Most students recognized when one of the addends was larger than the estimated total; I wonder now how they would have responded to obvious overestimations, like 100.
The thing I’m left thinking about is how powerful a practice estimation can be. As Jennifer Clerkin Muhammad says, in Becoming the Math Teacher You Wish You’d Had (Zager, 2017):
“[Estimation is] supposed to be–not a chore–but something that really helps you! At certain times, it’s all you need, and other times, it’s all you have time for, and it always helps you think about reasonableness.
In fourth grade, with division, sometimes all common sense goes out the window. As soon as paper and pencil get involved, it’s all about getting that answer, and they stop thinking about reasonableness. So I’ve been trying to push the kids to think about what makes sense or not.”
Asking students to estimate is asking them to engage in math in ways that are essential to becoming powerful math thinkers. Expecting math to make sense and applying what they understand about mathematical structures and relationships should be a natural part of a student’s process of solving problems.