Recently, my colleague, Dan, asked me to help him help his students understand how to model division of fraction situations.

When I walked into Dan’s classroom, he was asking his students a question about decimals.  He wrote this number on the board:

Dan asked his students to divide this number by 100.  He said this was a short routine they had been doing during transition time for the last couple of days. Dan said they enjoyed being able to quickly do a problem that looked really hard at first glance.

I asked, “Is the answer going to be bigger or smaller?”

Many of the students said “smaller because you are dividing.”

I asked, “can you think of a time when you would use a number like this?”

One student said, “When you are counting sand.”

Me: “Why?”

“Because grains of sand are sooooooo small.”

Another student said, “I think I know what the answer would be, but I don’t know how to say it.  How would you say that number?”

Me, thinking out loud: “Hmmm. Let’s see, tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths??  I am not really sure. Maybe fifteen thousand eight hundred ninety-two ten millionths??? I need to ask my math friends for some help with this.”

I told the kids about the Math Twitter Blogosphere while I typed my question on twitter. I am so glad I asked for help. I hadn’t even considered scientific notation until I started reading some of the twitter feedback.  I wondered “if scientific notation is the aspirin, is 0.0015892 the headache?” I haven’t had time to dive into the twitter feed. I am looking forward to revisiting it. I would also like to revisit the comment above about sand.  I want to ask, “tell me more about how 0.0015892 could represent sand.”

At first, I thought, this student was wrong.  He is thinking that really small numbers represent really small objects.  0.0015892 can’t represent sand. Then I realized maybe I don’t know enough about what, exactly, this student was thinking. Can 0.0015892 represent sand?  What if .0015892 represented the size of a grain of sand in relation to the rock that it was once a part of?  Could a grain of sand be .0015892/10,000,000 of a rock? How do I get the students to consider the  relationship of a unit this size?

My colleague Dan and I are pretty similar.  We tend to live “outside the box.”  In fact, at times, we humbly admit that we need help getting back in the box.  The boundaries of the box are often elusive to us.  We don’t even know how we wandered outside of them.  We were probably following something shiny. For example, when I arrived to teach in Dan’s classroom yesterday, I asked him to video tape the lesson for me.  This morning, I anxiously sat down to watch myself teach and reflect on the lesson.  This is what I found:

I love this video.  At first, I didn’t love it. I thought, “Dan!  What the heck did you do?” Then, I laughed.  Next, I tried to edit the video. Can I slow it down to “normal pace”? Then, I laughed some more.  There is no sound.  I can’t read lips. What is the point of slowing it down? Finally, I settled into leaving it the way it was.  I wondered, what can I learn from this super fast, upside down video?

At first, I thought, “bummer.  All I see is me in front of the class the whole time.  Was I being the ‘sage on the stage’?”

Then, I looked a little closer and noticed something interesting.  There are a lot of students bobbing their heads and turning around.  There are a lot of students moving their mouths. This part of the lesson is when I was doing a Number Talk from the book Making Number Talks Matter. I am trying to record student thinking on the board.

You can’t see what I am recording because my Google Slide presentation for the next part of the lesson is taking up most of the space on the white board.  Here is the perfect example of me not even being slightly aware of the boundaries of the box.  As I watch myself now, I am thinking (kind of yelling), “TURN OFF YOUR SLIDESHOW!”  “Hello??? You are monopolizing most of your board space with a presentation that you aren’t even using right now!”

See. I can learn stuff from watching a super fast upside down lesson.

I remember this part of the lesson.  In fact, If I blow up a screenshot from the video and rotate it I can see what I did.

See. Doesn’t that help you?

What do you mean, “no”.  It helps me.

Down on the bottom, I see the number 99 and I am remembering the great conversation we had about whether 50/99 was more than one half or less than one half.   Some students said less than one half.  Some said more than one half.  I highly suggest you take a minute to figure it out for yourself before you move on.  I asked the students if anyone wanted to defend either of the solutions.  One boy said he wanted to defend less than one half.

He said “half of 100 is 50 and 99 is less than 100 so 50/99 would be less than one half.”

I remember thinking really carefully about how I should record what he said.  I chose to use words.  Immediately, one of the other students said, “but half of 99 is 48.5”. At this point, I chose to just stand there and wait.  Then, I said, “can you tell us more?”

“Well, half of 99 is 48.5 and 50 is more than 48.5 so 50/99 has to be more than 1/2”

Then, the first boy chimed back in.

“I was wrong. It is more than one half, but it is 49.5, not 48.5.”

Me: “What do you mean?”

“half of 99 is 49.5, not 48.5.”

other boy:  “oh, yeah.”

Me: “So, why did you change your mind?”

“because 49.5  would be half and 50 is more than 49.5.”

At this point, I am noticing that no one has actually referenced the units. They aren’t talking about the fraction in it’s entirety. The students are referring to the relationship of 1/2 to the numerator or to the denominator, but no one has actually said, “49.5 ninety ninths or 50 one hundredths”  You can’t see this because the video is upside down and super fast. I remember it because I was trying really hard to listen “to” and not “for” answers. I was also trying really hard to cultivate my students’ math intuition.

I remember thinking, “how do I get them to think about the unit without telling them to think about the unit.”

Finally, I said, “how many hundredths would be equal to one half?”

Silence.

“one?”

“ten?”

“two?”

“fifty?”

Wow. This is most definitely NOT what I anticipated for answers.  I thought I would hear a loud chorus of “fifty.”

So. I recorded those answers on the board, somewhere up in the top corner, smooshed in, next to the magnetic marble tracks. (Get back in the box Sarah!!).

I asked, “which of these answers is correct?”

Several students said “fifty.”

Me:  “Why?”

“because fifty is half of one hundred.”

“fifty cents if half of one dollar.”

A lot of “oh yeahs….”

“Does anyone still think it is something other than 50?”

Silence and heads nodding “no”.

I asked the students, “Does anyone want to share why they changed their minds?  Some of you originally thought the answer would be 1/100 or 2/100 or 10/100.  Can you tell us why you changed your mind?”

Silence.

Okay. What if I try this, “Does anyone want to share why you think someone might have gotten one of these other answers the first time (1/100, 2/100, 10/100)?”

One boy said, “I could see why someone might say 2/100 because 2 x 50 is 100 and 50 is half of 100.”

Several students nodded in agreement.

I silently did a giant happy dance.  I think I am doing it!! I think I am helping my students develop their math intuition.  Wow. I need to ask those kinds of questions a lot more.

At this point, we had to move on.  This was a nice segue to the division of fractions exploration that I had planned for today. (I could write a whole other blog post about the second part of the lesson.)

We definitely need to offer these students more opportunities to explore and discuss the relative magnitude of numbers. We also need to cultivate more opportunities for them to explore and communicate the units they are working with.  At first glance, this post may seem like a series of upside down and super fast examples that are unrelated, but I actually think they are the beginning of a journey towards deeper understanding of when and why the unit matters.