Yesterday morning, I presented a warm-up problem to my students - a "real-life" math problem no less from my own life. My intention was that it be a WARM-UP not half of my class time. The understanding that I received from allowing this math experience to morph into something more was well worth while.
Here's the situation: I have those universal desks that defy group work. They just don't position well for sharing materials or being able to get into your seat without climbing. The legs of the desks and chairs have tennis balls to make them quieter and less damaging to the floor. These little bumpers prevent the legs from getting too close to each other. They get loose and become rolling objects. And they collect clumps of yucky stuff from the custodian's broom that I can't even describe....
I had the idea to begin removing the tennis balls and replace them with those little felt pads that you can by in any store. Here's where the math comes in. If the pads come in packages of 12, how many packages will I need to buy, and how much is this little project going to set me back?
I gave the problem to my students to solve in pairs, and as I walked around the room to see how they were approaching the problem, I noticed many of them were immediately drawing factor trees! HUH? Why did the students think finding the prime factorization, the GCF, or the LCM of the number of desks and chairs and pads in a package was the way to go?
I asked several of the students why they were doing the factor trees and the responses were generally to "find" the GCF or LCM. How does that relate to this situation? They couldn't say how or why. My feeling on this was that it was the most recent mathematical concept we worked on last week. After some questioning, most of them abadoned this idea and started again.
Another common mistake was adding up the number of chairs and the number of desks (10 + 32) and then dividing by 12 (the number of pads in one package) to tell me I needed to buy 4 packages of 12 pads to complete my job. Even some of my top-notch students made this mistake! So I asked them how many pads that was (48), and then had them count by 4's as I pointed to the desks in the vicinity. They quickly realized that this was not enough because they didn't multiply by the number of legs on the furniture.
The next thing I did was ask several groups to come to the document camera to show and explain their work. This gave them a chance to put into words a logical step-by-step explanation. Boy, this is a tough one for many of them. To be able to go back and explain your thinking means you have to keep track of your thinking in some manner. Most of their papers looked like this:
We had an opportunity to talk about how to organize the work so that it could be easily followed by another person. This is how one student edited his work:
Today, we'll add the writing component to this. Perhaps, this is the most difficult aspect of the task. Writing (as I am doing right now...) takes time, thought, and effort to pull all the pieces together into a coherent form. To get the process (which happens quickly in the brain) down onto paper requires slowing that process down. Most of the students I work with have not developed that patience for the reader. They would rather that I be a telepath and cryptographer when it comes to their work!
Through all of this I didn't get discouraged because I realized that I provided a great learning opportunity for my students. I also see my skills with questioning and facilitating math tasks improving every day. It is a challenge to give up that "stage" of lecturing on a full-time basis. I am starting to see the glimmer of light at the end of the tunnel....it might be as small as a pinprick, but still there nonetheless.
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