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SMP 8: Look For and Express Regularity in Repeated Reasoning

Max Ray

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.”

George was an amazing student to have in class because his big strength as a mathematical thinker was looking for shortcuts. His favorite activity was coming up with formulas. When we were exploring sums of interior angles in polygons, he was the first to say, “You don’t have to draw all the triangles every time—you know how many there will be from the number of sides.” When we explored surface area and volume, he looked for better ways to find areas or volumes than counting squares and cubes. He was the first to come up with formulas that let us find the surface areas and volumes of all kinds of shapes. George thought memorizing formulas was a waste of time, so he was always sure to explain why his shortcuts made sense based on our previous investigations—so we could understand them instead of having to memorize them.

NOW MEET SHANA. Here’s a typical Shana story, involving a problem about making dollhouse furniture:

You have 31 furniture legs that can be used to make tables (4 legs) or stools (3 legs). How many different ways can you use up all the legs?

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:

-Encourage students to share multiple methods for solving the same problem and then request that students use someone else’s method to solve a similar problem.

-After sharing and naming multiple methods for a problem, tweak the parameters and ask students “Which method would you use to solve the problem if it were like this?” For example, I might use Shana’s method if I was working on the problem and I needed to explain how my answer worked to a second grader. But I might use a different method if the problem involved 321 furniture legs!

-Playfully and explicitly add an element of time to students’ problem solving, once they have had experience developing methods for a particularly problem structure. For example, if students have painstakingly drawn pictures and calculated the sum of the interior angles of rectangles, rhombuses, pentagons, and hexagons, and shared their different methods for doing so, ask them to think about ways they could get more efficient. Then ask them to make a bid for how many polygons they could find interior angle sums for in 5 minutes. Pass out the requested number of puzzles to each team, and start the clock. While you’ll have to pay attention to group dynamics and support students to be good team members even under pressure, you’ll also likely hear very good ideas about mathematical shortcuts that didn’t come up when students didn’t have any reason to need more efficient methods.

Ray-Riek, M. (2013). Powerful Problem Solving: Activities for Sense Making with the Mathematical Practices. Heinemann.