When I first moved to Bucks County, I knew the major routes to get around the area. I could, by rote, drive from my house to my in‐laws’ house. I could also drive from my house to the school where I worked. I could flawlessly and efficiently travel those well‐worn paths and arrive promptly at my destination.
One day, I received a simple phone call from my wife: “My parents are making dinner for us tonight. Just come straight from school and meet us there.”
Not a problem. I left work at my usual four o’clock and with traffic arrived a little after 5:30 PM.
“What took you so long? Did you have a meeting after school?”
“No, I left as soon as I could.”
“But it should only take a half hour.”
“That’s impossible. It’s more than that just to our house, then another 40 minutes to your parents.”
“Um, no, dear. There’s a more direct route.”
Turns out I had driven a half hour south only to turn around and drive a nearly parallel route back north to their house. If I’d just gone east instead, I wouldn’t have had to sit through that light four times.
The problem wasn’t that I didn’t know how to get there. I didn’t get lost, I didn’t get confused. I did what I knew how to do. The problem was that I only knew a very specific path and had no idea how the various routes related to each other or where my destination was related to my starting point.
Learning the route is easy. Learning the whole map is hard.
It is a great temptation in teaching to teach students the route instead of the map. It’s faster, simpler, and more often than not produces the right results.
We can’t give in to that temptation, though. I recently taught a lesson about estimation to a group of fifth grade students. They had memorized a multi‐step procedure for transforming a number into its rounded version. I quickly discovered, though, that like the students could do little more than mindlessly play back the recording of the algorithm. Many of them got the steps confused, or missed some, and since they had no idea how the process fit into the greater picture of what they were trying to accomplish, they didn’t recognize that there was a problem. When I asked them to explain what rounding was for, for the most part, their answers were along the lines of, “To get a rounded number.” Several committed the common error when asked to estimate a sum of adding the two original numbers then rounding the answer. Most used the words “rounding ” and “estimating” interchangeably.
All of this could have been avoided if the teachers in second, third, and fourth grade had taken the time to build an understanding of the function and purpose of estimation, to explain that rounding is just one tool in the estimating toolbox, to build in number sense and develop mental models of what is happening when we round. Before introducing the algorithm.
As I found out the hard way driving to my in‐laws’ house, the shortcut is only shorter when it is used in the proper context.