Wednesday, October 3, 2012

Teaching an Old Word Problem New Tricks

Today, I used the fairly well known 'Locker Problem' with my students. If you haven't heard of it, here's my version:

Imagine 50 closed lockers in a hallway.
One student goes by and opens every one.
Another students goes to every second locker and closes it.
A third student goes to every third locker and if it is open, closes it, and if it is closed, opens it.
A fourth student then does the same but for every fourth locker.
A fifth student does the same but for every fifth.
And so on.
At the end, which lockers will be open and which will be closed? Why these numbers?

I think there is merit in analyzing the best way to deliver the problem. Case in point, the students did question the words 'And so on.' "Do you mean forever?", "How far should we go?", "When do we stop?" were fairly common.

But here, I want to emphasize something that totally surprised me. About how the students solved the problem, and what they chose as their method. The options were wide open. I'm defining wide open here as 7 different ways of solving this.

1) Pen and paper (immediately popular when they were reading the problem)
2) Chart paper (the response was, muted)
3) A long roll of mural paper (the rolling, intriguing, but not much commitment from the crowd)
4) The white board (notable enthusiasm)
5) Cube Links (crickets)
6) 2-sided plastic counters (aha moments for most groups, interest was building)
7) Sticky Notes (game over...close the door...the kids are getting rowdy)

But I didn't close the door. Instead, I offered two groups of students (per class) to take the problem outside, into the hallways, with their sticky notes. Inside the classroom, 2-sided counters were popular, followed by some independent pen and paper enthusiasts.

Having given some time for the groups outside to get going while I watched students inside, I wandered out the door to check on the progress. And there was the surprise. I thought, Sticky Notes outside the class, was one option. My misconception was that I could imagine how students outside were using those Sticky Notes. It wasn't one option. And I had no idea the students were going to do all of this:

Lay numbered Stickys on the Floor

Using O's and C's

Only OPEN Stickys without using Close Stickys

Choosing to skip lockers instead of all being labelled.

Tally Marks on each Sticky as needed.

Moving Stickys up and down, for Open and Closed

How long will 50 remain OPEN?

24 has been through some changes:

Some inside highlights can be seen here:

Here come the counters,

Some planning with Paper and Pencil.

I thought I knew what I was going to see. Today was a pleasant surprise. This is good.

At the end of the day, I still have these questions:

How could I have widened their options (choices) before starting?
Logistics and timing were an issue. How can I manage this better?
What prompting questions, if any, are needed here?
How can I better word the question, "Which lockers will be open and which will be closed"?
An attempt at answering that last one: Can I just say, "What will happen?"


  1. Thanks for this post. Here's my two cents on your questions.

    I love the way you left it so open for them to decide how they wanted to proceed. You could have widened their options by giving them access to anything available in the school or classroom that they thought would help them answer it.

    Timing on these things is tricky. You don't want to rush them, but you don't want them at it for a week. Regardless, it's time well spent. Using personal strategies to solve problems is the only honest answer to, "when will I ever use this?"

    Don't prompt. Let them prompt and come up with the questions.

    How about putting it in the context of the released prisoners, and using the lockers as a way for them to visualize it? Then your question is, "which prisoners get freed?"

  2. Thank you scamdog. I appreciate the feedback.

    'Timing is tricky' was an understatement on Wednesday. Some classes had time to spare during the wrap up. Others were caught by the bell. One solution that comes to mind is less talk from the teacher. Which leads me to your options comment.

    The option to use anything in the class is brilliant. Perhaps have an ongoing analysis of the effectiveness of whiteboard v smart board, chart paper v grid, counters v cube links. Or any combo of options. When do these work better than these? What students like this over that? Once they've seen everything we have in class, there's the answer is the openness. As you say, anything available.

    Don't prompt. Another good suggestion. Also involves less talk from the teacher.

    Interesting theme here. The more I get in their way, the more I get in the way. Less talk. More action.

    Looking forward to sharing more.

  3. I really like how you allowed the students to figure out solutions on their own. This gave them the opportunity to apply previous knowledge and different strategic methods to find a solution. When I am a teacher one day, I will definitely look into problems similar to this and allow students to work together to find solutions on their own.