Smaller, Bigger, or More Precise: Refining Our Internal Truth Detectors

Yesterday, I got this text:

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It was from Mrs. G, the teacher I worked with on Tuesday.  I really wanted to check in with these kiddos and talk with them about these questions.  Today, I stopped by to ask them to tell me more.

First, I checked in with Ms. G. She told me that these questions came about when the class was doing a choral count.  She said the class was counting by hundredths. When they got to nine hundredths, someone suggested the next number might be one whole. Then, the students had a conversation about how it wouldn’t be one whole. It would be ten hundredths or one tenth.  They continued counting until sixteen hundredths. At this point, Max said, “wait! When we are doing this, are the numbers getting smaller or larger?” This question prompted a different question from Seth, “If whole number places can go on and on forever, can decimal places too?”

At this point, Ms. G wrote a long decimal up on the board and asked, “can I do that?”  Charles said, “I think that number would still be between 2 tenths and 3 tenths.”

Max responded, “wait! Can you just keep putting places because once you get ten of them, it is going to go into the next place and once you get ten more it will go into the next place and on and on.”

After Mrs. G caught me up, I turned my attention to the kids. I asked them if we could talk about this number again.

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Max immediately jumped in to the conversations. He said, “yesterday, Seth asked if decimals go on forever and I was the only one who said, ‘no.’ I don’t think they would go on forever because eventually they would all add up to one.”

I wasn’t sure if I understood what Max was saying so I asked him to explain it. Listen.

 

“That is why I don’t think that you could count forever in decimals because I think eventually this will add up to one like I did here. I think that wouldn’t work. Eventually, no matter how big the number is, if you’re still adding, eventually, even if it takes years, it will eventually make one whole.”

At this point, the students went back to their Social Studies lesson. Yes, it is true. I totally interrupted Social Studies to revive a math lesson. I love Social Studies, but sometimes, I think it is okay to Drop Everything and Do Math.

Mrs. G and I went over to the kidney shaped table to reflect a little. Mrs. G took some time to share the back story of these questions. She explained what happened the day before. Listen.

I wondered, what is Max’s claim?  Is he claiming that decimal numbers DON’T go on forever or is he claiming that all unit decimals (is this a thing?) will eventually add up to one whole? Mrs. G and I wondered how language was impacting our conversations with Max.

Mrs. G said, “I think I kept saying “adding” a place value.  Can we keep “adding decimal place values”? Max is hearing the word ‘adding’. Maybe he is thinking about counting as adding.  We decided to ask Max a few more questions about his claim, but try to use more precise language this time.

When Max sat down, I said, “I want to try to understand the question you are asking.”

He said, “Well. I only half understand it myself.”  Have I mentioned, yet, how much I absolutely love this kid?

I tried to rephrase Max’s claim without using the word ‘adding’. Listen to the conversations:

I have listened to this clip several times and I wish I had done something differently. When Max says, “so you are just adding place values. You are not adding the numbers one by one.”, I wish I had not said anything. I wish I would have waited and let the magnitude of his statement settle into the silence. 

Max goes on to rephrase his claim. He says, “no matter how small the number is, you are eventually going to get to one whole, no matter how long the number is, even if you give up, if you didn’t give up, eventually it will go back to one whole.”  

Now that I think I understand Max’s claim, I am wondering how it fits in with Seth’s original question about whether or not decimals can go on forever.  Listen as Max invites Seth into our conversation:

So, at this point, I am still wondering about what Max is disagreeing with. When he is talking to Seth, he says, “And I asked, without it going into wholes? Did you mean adding?” These words make me wonder if Max is still talking about the cumulative addition of unit decimals, as opposed to the literal writing or naming of a decimal number.

I told Max that I was still unsure. As we talked, I wrote the number below. Listen.

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Next, Max catches me off guard. He is thinking so fast, I have a hard time keeping up with him.  I was trying to see if we all agreed that I could keep writing digits forever. However, I got lazy and just started writing zeros. Well, that added a whole new layer to the conversation.  Max didn’t miss a beat. Listen.

This kid is thinking so fast and so deep that I can’t keep up.  I started using the word “adding” again which didn’t help with clarity.  Fortunately, Max persevered and straightened me out, at least as far as the whole “zeros question” goes.

I was still unsure about whether he thought decimals could go on forever. He keeps bringing in these other nuanced constraints: “without it going into a whole”, and “you have to count with decimals by one.”

I asked him, “what if I didn’t write zeros. Couldn’t I just keep writing digits forever?”

He said, “That is not correct. You’ve got to do one and then another plus one to make a zero. You can’t just add ten numbers at a time or seven numbers at a time.”

At this point, I tried to sift through what I thought were two different claims- one about writing/naming decimals and one about counting/adding decimals. Listen:

I am not totally sure we all ended up on the same page about understanding our claims, but this conversation with Seth and Max was one of the highlights of my career.  I could probably spend the rest of my day just reflecting on this conversation.  These boys pushed me to think differently and to try to truly understand them. What if we all did more of this?  What if we dropped everything and did math? What if we dropped everything and listened to understand each other’s thinking?  I am so grateful for these boys and their amazing thoughts. I tried to conclude our conversation by letting them know how much I appreciate them. Listen

 

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