During the processing of the first problem, Sarah notes that the second answer is incorrect. This puzzles other students. Together we take a critical look at the words in a word problem and see how those words connect to the symbols used in the number sentences provided.
TRACY LEWIS: Did this mathematician answer the question?
TRACY LEWIS: Let's take a look at number 1. They said that there were 82 people. How did they answer this question? Victoria?
STUDENT: By... they actually used numbers, and words, and they used a picture to figure out. They told us how they did it, and they used base 10 blocks.
TRACY LEWIS: Okay. So we know what they did because, you're looking at the base 10 blocks. I counted the tens place, then the ones place, I got 82. So. Did this mathematician break these two numbers up? Where is the 63? Where is the 63? They said the added the tens place and the ones place. I see this number, 63, and I see this number, 19. Where is it? Where is is located? Sarah?
STUDENT: This number problem down here isn't correct. Because 92 minus 85 ... it can't go over 92, and 140 is way over.
TRACY LEWIS: Okay, so you're making an observation about this problem. Can you hold that until we finish processing the first one?
TRACY LEWIS: Okay, I definitely want to hear what you're saying. My question is, where is the 63? Shivani?
STUDENT: Right there.
TRACY LEWIS: Where is right there? Give me a ...
STUDENT: Right here.
TRACY LEWIS: Ah! So here's the 63. So this is where this mathematician put 63. And... can we say that this is the 19?
TRACY LEWIS: So we know, very clearly, that they counted the tens, all the tens. Are there eight tens up here? One, two, thre, four, five, six, seven...Where did the other ten come from? Where'd the other ten come from?
TRACY LEWIS: Because they told me their answer is 82, but I only counted seven 10s. Can you tell what they did? Isaiah?
STUDENT: The other ten came from the 19.
TRACY LEWIS: Okay, but I counted that ten. I only got seven 10s. Where did that other one come from? Ka'Lon.
STUDENT: It came from the 1 from the 19, �cause they add 10 plus 9 equals 19, and 63 plus 19 equals 82.
TRACY LEWIS: 16 plus 19 equals 82.
TRACY LEWIS: 63! Okay. Aaron?
STUDENT: What they did is they took the, 1 from the 3, and they added it to the 9 in the 19.
TRACY LEWIS: They took the 1 from the 3 in the 63, and they added it to the 9. So Ka'Lon mentioned something about adding. He said he wished that this mathematician would have put in an addition symbol.
TRACY LEWIS: Can we figure out that they know that they were supposed to add?
STUDENT: And there's another way..
TRACY LEWIS: Hold on, Ka'Lon. I need someone to answer my question. We need to share the air. How do you know that you're supposed to add? Is there any clue in here that tells us that we're actually supposed to add? Iyanna?
STUDENT: Because it says how many people are going on the field trip.
TRACY LEWIS: It says how many people are going. Are parents people?
TRACY LEWIS: Are second graders people?
TRACY LEWIS: Okay. So let's take a look at what this mathematician did for the second part. Because Sarah jumped ahead and said, Oh! Ms. Lewis, this number sentence down here is not right!
TRACY LEWIS: So let's see what it says. The apple farm is 92 miles from the school. They have traveled 58 miles so far. Let's take a look at what this person did. They said, 90 plus 2 plus 50 plus 8 equals...
TRACY LEWIS: 140. Okay. I added the 90, that the 2, or maybe they meant then, to the 50 then the 8 I got 140. So...
STUDENT: It should be minus, and then, instead of the 0, they should have put the 2 in, so it would be 92, minus 58.
TRACY LEWIS: Oh, so you're noticing that they did break the 92 down into the tens and the ones, but you're thinking that this should be a subtraction? Does anybody else think that? What are you thinking? I'm not sure. What do you think? Iyanna?
STUDENT: I'm thinking, how did she figure out which one she's supposed to subtract?
TRACY LEWIS: Very good question. How did this... this mathematician thought we were supposed to add. So this mathematician took 92 and they took 58, they broke it down and they added it all together. But Sarah's saying, hm, I think they were supposed to subtract. Let's see if we can figure it out as we look at another sample. Okay?
TRACY LEWIS: Yes, Aaron.
STUDENT: I know why we're supposed to subtract, because it says how many more miles do they have to go?
TRACY LEWIS: So you're looking at the words that say, how many more.
STUDENT: How many more is the code!
TRACY LEWIS: How many more is the...
TRACY LEWIS: Oh! It's a code! It's a code for what? How many more is a code for...what? It's a code for... I've heard from you, I've heard from you...I've heard from you...D'wone?
STUDENT: It's a code for the... number sentence?
TRACY LEWIS: It's a code for the number sentence? It's supposed to tell you something about the number sentence. Let's take a look at the next poster.
I intended for students to not only look at how their peers decomposed numbers to come up with answers, but also look at how they documented it. Did they use a complete number sentence and words to clearly explain their thinking? If not, my goal here was for students to link the words in the problem to the symbols that would complete the number sentence. This resulted in the lesson taking a bit of a turn. I needed to know what my students were thinking in regards to this new dilemma presented by Sarah and supported by a few others (Aaryn and Nia). In this segment of the lesson Sarah is very quick to point out the error made by another student in how they understood the problem. She expresses very clearly where the mathematician went wrong in their understanding and how the answer they provided just does not make sense. She states her observations and conclusions very quickly.
During math lessons early in the year we spent a lot of time decomposing numbers to help answer questions and learn regrouping. While some students agree with Sarah's position, others continue to be puzzled by how she knew to subtract. Aaryn points out that the words in the word problem "how many more" are a clue to subtract.
The key idea I wanted my students to gain was "DOES THIS MAKE SENSE, GIVEN WHAT YOU KNOW AND HAVE BEEN TAUGHT." Iyanna who is a very strong reader, struggles in math and often shuts down. The process of re-engaging a task allows students like Iyanna time and space to make sense of a problem and get clarification.
After Sarah shares her thinking I am very careful about the statements and questions I use. "...but, you are thinking this should be a subtraction problem? Does anybody else think that? What are you thinking? I'm not sure. What do you think?" I am intentionally opening the door for students to also say exactly where they are in their process with this problem. There is no pressure. In fact Iyanna boldly states where she is in her thinking. It is the place where she is stuck. Keeping in mind that my goal here is to get struggling students to ask more questions and secure students in math to share their thinking. Through this process we can all learn more. As students speak up, and mumble on the carpet, I have a window into how they approached the problem. Now we can begin to attack misconception and break this poor habit of looking for number and just adding them together.
On way to break this habit is the use of thinking of certain phrases as code words. Nia speaks up and says, "How many more is the code!" D'wone, another struggling mathematician understand that it is a code for the number sentence. However, he is not 100% clear on what exactly it is supposed to tell him.
It is very tempting to jump in and make this a lesson about the words, "How many more" in a word problem. This group of students have found it useful to equate this phrase to subtraction, however I could have really taught this as a "comparison" or strategies for finding the difference by adding. I choose to maintain my course and allow student space and time to explore and discover errors, knowing that they will get to do this with their own work soon.