By Max Ray (@maxmathforum) of the Math Forum, author of Powerful Problem Solving
MEET GEORGE. He does his homework often, though he skips any problems that will take more than a few seconds to solve. On tests and quizzes, he gets almost every problem he attempts correct, though he’s been known to leave entire sections blank. When I suggested that he come in for extra help on those sections he didn’t know, he explained, “You went over it in class a couple of times and I knew it would be hard for me, so I just went with the stuff I knew instead.” But none of those things are the reason that George was the first, and by far most impressive, of my students to earn the honor of “lazy mathematician.”
NOW MEET SHANA. Here’s a typical Shana story, involving a problem about making dollhouse furniture:
Shana knew that she could solve the problem by directly modeling it with drawings. She guessed different numbers of tables and checked to see if she could use all the remaining legs by making stools. For each guess, she drew that number of tables, then drew four legs on each table, then counted them by ones, then drew stools one at a time, added 3 legs to each stool, and counted up until she got to 31. With enough time, she definitely would have found all the ways to use the 31 legs, though it might have taken her a full week of math classes! Here is a student who is persevering, who has confidence and tries every problem. Why would we want her to be more like George and earn the title of “lazy mathematician” too?
You probably have students like George to Shana. Being a mathematician means balancing both qualities. Sometimes mathematicians need to persevere and to stick with a set of strategies they know will definitely work, given time. Sometimes being painstaking on the path to understanding or solving, putting in the effort even though it is hard, is necessary. But getting stuck doing the same inefficient methods every time is not what being a mathematician is all about either. We also want to look for efficient and elegant solutions. We want our methods to be powerful and generalizable. We want to use what we know and what we’ve done to become faster, more powerful problem solvers.
Here are three classroom routines or activities I’ve used with students to help them appreciate the power of being “lazy,” by which of course I mean looking for and representing regularity in repeated reasoning in order to develop efficient shortcuts or notations:
SMP 8 is one of the hints that the authors of the Common Core give us about how we might teach the content standards through problem solving. The authors are reminding us that mathematicians move from painstaking but reliable problem-solving methods to efficient procedures via looking for regularity in their repeated reasoning. Since a picture is worth 1000 words, a five-minute video must be worth millions of words. In closing, I’ll offer you my take on how helping your students like Shana be more like George can be a way into not just SMP 8, but also the content itself:
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