The Myth of Shortcuts

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.