Look at your hands and your fingers, 1 to 5 on one hand and 6 to 10 on another. Doesn’t it look like 5 might be closer to 0? It wasn’t anything I contemplated recently until my students brought it up during Open Middle/Clothesline math session. Make a number that is closer to 200 than 300. Where is 250?
“I think, it is closer to 200, because you count 250 first.”
“Because a teacher told if it’s 50, you round down.”
This made me pause, “What about numbers from 0 to 10, Is 5 closer to 0 or to 10?”
“Closer to zero.”
“Raise your hand, if you think it’s closer to 0.” – More than half of the hands go up.
I did not expect this at all.
“I am not sure. Is 5.1 still 5? Where does the number begin and end?”
Developing the Question and the Claims
In the next few days I’ve been obsessed with this question and with the possible ways to approach it, and I think this question made it into most of the twitter chats. Kent Haines wrote about the rounding conversation in his blog.
In another conversation about natural numbers, David Butler pushed my thinking in an unexpected direction when we started talking about calendar.
I wondered if that might have been the reason for my students’ uncertainty. Ordinal and Cardinal number duality causing a lot of confusion!
Students explored the line in groups to test their ideas and to discuss them.
S: “You start here (points at 0), you count 1, 2, 3, 4, 5, 6. But when I go to 10 if I count from 5, it’s 1, 2, 3, 4, 5.”
Me: ”Why does it change?”
S: “You don’t count 5 because you already counted it.”
Me: “Do you count your starting point?”
Me: “So what happens if you count from 10?”
S: “It will still be 5!”
Me: “Try it.”
S: “1, 2, 3, 4, 5.. (stops)…. 6? It changes. When you walk up, 5 is closer to 10. When you walk down, 5 is closer to 0.”
We were thoroughly confused.
I didn’t want to stop just yet. The discussions were great, and I was looking for some structures to prompt students to verbalize their conjectures and to listen to each others’ arguments. After the recent online session with Kristin Gray at Mount Holyoke College, I’ve decided to start with the Talking Point and this time limit it to one statement to focus our conversation. Talking points helped students to find like-minded peers to do more proving together.
Developing Arguments and Revisiting Claims
We ended up making four teams as there was always one more popular claim. Each team got their own large number line and any other manipulatives or measuring tools that they could find in the classroom. Our most popular choices were meter sticks, snap cubes and balances. I also handed each group a set of cards with numbers 0 to 10 and some tape to arrange their numbers on the number line when and as needed. Some added fractions. Some took rulers to be precise.
Groups presented to each other and asked questions. Many students changed their opinions in the process.
Counting Argument: Five is in the middle.
S: “You go 1, 2, 3, 4, 5 then 1, 2, 3, 4, 5.”
Me: “Do you count zero?”
Me: “Why no? Shouldn’t you?”
S: “You don’t count zero because it’s a number that you don’t use in counting.
Semiotic Argument: Five is closer to zero.
Calculations argument: Five is in the middle.
Change argument: It depends.
Snap cubes argument: Five is in the middle.
“The number of blocks between zero and 5 is 4, and the number of blocks between 5 and 10 is also 4. The number of blocks stays the same so it is the same distance.”
Hotel argument: Five is in the middle.
“Say you wanted to stay for 5 nights and then go to a different hotel for 5 more nights. And then you leave. Days are spaces and nights are numbers.”
In my final move, I asked students to take the numbers away, then find the middle and stand there. Then together we put the numbers back. Five was landing on the middle spot. We proceeded to our journal reflection.
This question appeared to be much trickier than it sounded at first and it brought forward more questions. What is the middle? Are numbers “spaces or points?” Are fractions numbers? How do you count? Do you count zero even if it is nothing?
I thought about a difference between the models of number line and a hundreds chart. I never considered that essential difference before. I wondered if developmentally ordinal understanding of numbers comes first and it is a an abstraction leap to the cardinal numbers. I also wondered about historical development of number concepts and at which point numbers stopped being counters. My twitter and classroom conversations made me think about our zero-less calendar. We count years, days and months as they begin. Our whole calendar system is based on the intuition of ordinality. Zero is hidden in the moment of midnight, too dark and fleeting to count it.
And a large number line is a keeper.