Yesterday during the second nutrition break, I sat down to eat lunch with two teachers from our school. AJ, our Instructional Coach, was one of these teachers. During lunch, he was speaking about a math problem that he did with a group of educators. Here’s the problem:

Peter mows a whole lawn in 3 hours. Paul mows a whole lawn in 2 hours. If Peter and Paul start on the opposite sides of the lawn, how long will it take them to mow the whole field?

My initial response to AJ was, “I need a lawnmower.” 🙂 Then with just a couple of minutes of lunch left, I started to think about this problem. How could I figure it out? As the bell rang and I got up to leave, AJ said to me, “Let me know when you get an answer.” I took about four steps, turned around and said, “Is the answer 2 1/2 hours?” No Aviva — keep on thinking! My first thought was that they needed to split the mowing, so half of 3 hours is 1 1/2 hours and half of 2 hours is 1 hour. One and one-half hours plus two hours equals 3 1/2 hours. Then the other teacher point out to me that Paul could mow the entire lawn in 2 hours, so how does 3 1/2 hours make sense? Oh yes! He’s right. I said that the answer must be around 1 1/2 hours and left it at that.

Here’s the problem though: I couldn’t leave this problem alone. I kept on thinking about it. I even mentioned to my students that our instructional coach asked me a really hard math problem, and it was all that I could think about. They wanted to give it a try as well, but I really wanted to be able to solve it first. I said that I’d keep thinking about it, and I did.

Then on the drive into school this morning, I started to think about fractions. If Peter mows the lawn in 3 hours, then he mows 1/6 of the lawn every half hour. If Paul mows the lawn in 2 hours, then he mows 1/2 of the lawn every half hour. Quick! I needed to get to school to draw a diagram. As soon as I arrived, I ran into the staffroom, grabbed a piece of paper and a pen, and started drawing. Another teacher even arrived early and asked me what I was doing, and I had to share the math problem with him as well. Through my calculations, I managed to figure out that Peter and Paul mowed 5/6 of the lawn together in 1 hour. So how long did it take them to mow the other 1/6? I had a diagram. I knew the fractional parts, but I couldn’t make them align. Bring on Twitter. Here was my early morning tweet (with coffee spilled on the paper of course 🙂 ):

It didn’t take long for my wonderful Twitter PLN to chime in. Here’s what another educator said:

With this tweet, AJ stopped by our classroom just before first nutrition break and said, “Do you want to go over this problem together?” Yes! I said that I thought that the answer was between 1 hour and 10 minutes and 1 hour and 15 minutes, but I couldn’t explain why. So AJ patiently sat down with me during the break, and we went through this problem together. Now I get it!

Why does all of this matter? Because in the past, I would have heard this problem, maybe thought about it for a minute, and then completely forgotten about it. That didn’t happen this time. Why? Because as much as I want my students to be better thinkers, I think that I want to be a better thinker as well. I want to work through hard problems. I want to struggle. I want to be willing to ask for help. And I want to meet with success. 

This year, I’ve had so many students tell me that “their brains hurt.” This is good! How often do I allow my “brain to hurt?” How often do I think through problems, and how often do I share that thinking with others? AJ’s problem was more than just a math problem. It was a realization for me that as an educator, I need to constantly engage in “challenging thinking.” How do you do this, and how do you encourage others to “think hard?” I’d love to hear your stories!

Aviva