Monday, September 23, 2013

Manilla folders [Number Sense]

I must share this short story with you about the number sense I've been witnessing for the past few weeks with many of my students. Be warned: not for the faint of heart.

Today, a few 7th grade boys came in after school today and wanted to help with some things around my room: tidy up desks, organize whiteboards, etc. There was a box of 100 manilla folders that I had just brought back from the office. I needed 80 of the folders and would later return the remaining folders to the office.

Here's my exchange with the student we'll refer to as Albert:
Me: I need 80 folders from that box. Albert, think of a quick way you could get those 80.
***Let's pause. How would you (the reader) quickly get 80 folders from this box?
Albert: I could count in 5's.
Me: Okay. Any quicker than that?
Albert: By 10's.
I get what Albert is trying to do. He doesn't want to count to eighty by ones. To that point, I would say his initial two responses made sense and are practical, in the mind of a 7th grader. I thought maybe I'm asking the question incorrectly, so I try it again.
Me: Right. That would be a good way to organize the folders to keep track of them as you count. Albert, I'd like you to think of a way to quickly get those folders out of the box. 
I can see the look on his face is one of confusion. Not that look like he's trying to figure it out, but that look like he has no idea what I'm asking. So after a minute, I mistakenly ask another question (in hindsight, I wish I would've stopped the conversation and let him do his thing).
Me: How many folders are in the box?
Albert: 100.
Me: How many do I need?
Albert: 80.
Me: Is there a way to get me the 80 without counting all 80?
Albert has no idea. I like this question because now it's specific. His challenge is to get me 80 folders without actually counting all 80. After a minute. He needs some prompting.
Me: If the box has 100 and I need 80, how many will be left?
Albert: 30.
Me: So 80 plus 30 is 100?
Albert: No wait, 20. 
Me: Okay, so I will send 20 back to the office. How can I get 80 folders out of the box without counting all 80?
Albert has no idea. This exact exchange continued for another round. I'd like to say that Albert eventually came to an efficient way on his own, but he didn't. I tossed 100 up on the whiteboard. We subtracted 80 and wrote 20. I thought after Albert saw the 20 on the board, he might realize to count 20 folders from the box, take out the remaining folders and switch them with the 20.

This is quite common. The number sense (or lack thereof) my students have (or don't yet) is quite fascinating. I have a lot of work ahead of me this year. One thing is for sure. I can get better at asking shorter questions. I can get better at looking for these learning opportunities. I can get better at not looking for one answer when asking questions of students. As Max Ray would say, "2 > 4."



  1. I see this very, very often in the college students. I am absolutely confident that most people wouldn't believe that these folks have graduated high school and passed all kinds of math classes without ... understanding subtraction. It's going to be a priority for my mad animations ;)

  2. I had an analogous experience in a first grade classroom.

    A student was moving a pile of Legos from one place in the room to another. He picked up one Lego, carried it across the room, and put it down. He went back to the pile, picked up one more Lego, carried across the room, and put it down. He went back to the pile ...

    I watched for a while. Then I asked him if he's like a basket to carry all the Legos at once. He looked at me quizzically. It would be faster, I said. Why would I want to do it faster, he replied, and continued walking back and forth.

    I truly think this is a similar phenomenon. My first grader was attached to his method. Efficiency was not on his mind. Your 7th grader was locked into the task of counting 80 folders. He couldn't separate the abstract idea from the physical task.

    It takes stepping back from the task to see other possibilities.

    We adults perform the same way sometimes.

    1. Interesting story Seth. I can totally see that happening. I see those similar tendencies with my 3-year old boy too.

  3. Sounds like the folder kiddo might not be seeing numbers as "parts and wholes," per ( is the online supplement to our lessons about it -- "finding parts" has the folder situation.)

  4. (and I dumped google's blogger because the captcha crap gets REALLY OLD!!!!)

    1. Thanks for checking in. Parts-whole theory... yes, numeracy does matter!

  5. A professor asked some college students this question: If a baseball player is batting .310 and he goes 1 for 3 in a game, will his average go up or down? Some had no idea.

    This discussion was the impetus for our posting the fraction problem...

    ...which we then "advertised" with a tweet and to which you responded...

    "@Five_Triangles At first I didn't like this. I like it a little more seeing the importance of common numerator. Why'd you hashtag it CCSS?"

    We're responding to your tweet about this fraction problem on your blog because it's impossible in 140 characters, and also related to the topic of this blog post.

    Your solution, finding a common numerator, is obviously the most clever, but we don't anticipate that many students will see it, and so we were thinking in terms of a more brute force method that, hopefully, builds some number sense.

    For instance, comparing 11/19 and 22/37, while the numerator doubles, the denominator doesn't quite double, so how does that affect the second fraction's value? It's related to the baseball average problem.

    We use hashtags to reach a wider audience, of course, but we also continue to critique Common Core's lack of vision: it uses the term "number sense" twice, but fairly abstractly. We have little faith that elementary school teachers (and textbook authors) will now, under the Common Core regime, be able to instill the kind of number sense that you, we, and others bemoan continues to go missing.

    Maybe we'll next post this question: Compare 97/98, 98/99 and 99/101.

    1. Thanks for the detailed post. I understand and appreciate your reasoning. Personally, I'm working with middle school students who still have no idea how a fraction even works or what it represents and this will be a work in progress for me this year. When I saw your challenge posted, I immediately thought of my students and how they'd have no idea. I'd bet that most adults wouldn't have an idea as well and the fractions seemed impractical. Where would a common person work with such fractions? Again, on a personal note, I'd like to see students master representations, visuals, and applications of common fractions such as halves, thirds, fourths, fifths, eighths, tenths, and a few more. We agree that building number sense is important. I appreciate your problem as a higher order thinking challenge that would most likely challenge the top percent of a class. Your last examples seem a little more practical, especially if a teacher can help students compare fractions like 1/2 and 99/100.
      Thanks again for sharing. I enjoy seeing your posts.