### NCTM Denver 2013: Lester's Whatever Happened to Problem Solving in the Math Curriculum?

Annual Meeting - Saturday, April 20, 9:30 am

Frank K. Lester - Indiana University

Frank Lester was the editor of the 1234-page Second Handbook of Research on Mathematics Teaching and Learning. This was an interesting talk, not so much for any specific content but for how it was put together. Lester began the talk by demoing Algebra Touch, an iOS app that promotes fluency with symbol manipulations in solving equations. He asked, "What will the math classroom of the future be like?"

 Frank K. Lester

Then Lester went into problem solving, something he feels has slowly slipped out of most mathematics curricula. Problem solving, says Lester, is "What you do when you don't know (or aren't sure) what to do." That leads to the teacher's role in the classroom:

From here, Lester showed some of his favorite problem solving tasks. The first is probably familiar to most of you:

A snail is at the bottom of a well that is 10 meters deep and it wants to get out. Every day it climbs up 4 meters. It then slides back 2 meters when it rests at night. If it does this day after day, how many days will it take the snail to reach the top of the well?

As part of the discussion, Lester related his problem solving heuristics, harkening back to Polya's How to Solve It. Suddenly a talk that began with a discussion of technology and the future was using a (good?) problem that felt like it was from the 1980s and (good) strategies that were from the 1940s. Lester's choice of heuristics to apply here were "Draw a picture/diagram" and "Be skeptical of your solutions," since many initially reason that the snail reaches the top of the well in 5 days.

Lester then looked at finding the square root of 12,345,678,987,654,321. His heuristic -- one he called a "super heuristic" -- was to look for a pattern. I couldn't help but feel like this was a trivial problem with a trivial answer.

Things got better with the next problem: "On a European river cruise, 2/3rds of men are married to 3/5ths of the women. How many men and how many women are on the cruise?"

Lester joked that this problem predated talk of same-sex marriage, and I found it to be a bit out of touch. Lester said the problem could be adapted to involve pairings of animals or objects. Another heuristic here was "make reasonable guesses, not as final answers, but to get you started." After discussion of this problem, we moved on to one more: "A club has 500 members. At the Spring dance, tickets for new members were $14 but$20 for longtime members. All of the new members attended but only 70% of the longtime members attended. How much ticket revenue was collected?" It seems like there isn't enough information, but solving this plays off the fact that $14 is 70% of$20. That might elicit some interesting reasoning, but again I think this trivializes the problem.

Lester returned to technology at the end, mentioning strategy games like Math Dice and Rush Hour. He advised that teachers have an important role to play when students play games. Prior to the play, teachers need to help students be clear about the rules for playing, model how to play, and discuss special situations. I think that depending on the game and the goals, this could turn into too much guidance. During student game play, teachers need to watch students play, attend to their thinking, help and point out misunderstandings. It's important, says Lester, to not suggest strategies for playing. Kids should be left to figure those out for themselves. After gameplay, reflection is important, just as with other classroom activities.

Lester's slides can be found at the conference planner.