Many of those who put in extra time did so because they enjoyed the work. A friend of mine who is an excellent piano player has told me that when he was a child, he couldn't wait until he got home so he could play the piano for several hours. Over the years, I have noticed that many outstanding math students get very excited when telling me about one of their mathematical discoveries. They didn't do the work that led them to these discoveries because they thought it would make them better at doing the work as much as they did that work because they enjoyed doing the work. I see two consequences to this observation.

The first one is that teachers need to put more emphasis on interesting, challenging, intriguing problems; if they do, the mandated test scores will take care of themselves. Drills and practice just are not going to motivate more practice. They will drive children away from our subject and convince students that they are not good at math, and that they don't even want to be good at it because it is boring. (I just googled "boring math class" and got 46,000 hits.)

The second consequence is related to recent work by Keith Devlin about the potential impact of elctronic games on middle school education. I heard him speak at NCTM and read his book, Mathematics Education for a New Era: Video Games as a Medium for Learning. Among other things, he points out that people willingly play video games for hours on end, and that when they do, they get immediate feedback. They also expect to fail at what they are doing for quite awhile.

I can recall several open ended questions and puzzles I posed to my students that kept them thinking for years. Several students inquired when they were seniors if I would explain to the what was wrong with the "proof" I showed them three years before that all triangles are isosceles. Likewise there are several puzzle-like questions I have asked that students sometimes solved many months after the question was first posed. This sort of persistence only happens when students are surprised and puzzled by something they care about.

A teacher's main job is to pull students in to the subject being taught, to ask them interesting questions, and to let them learn. That means the teacher's main job is to provide students with interesting tasks that students can relate to. Some of these tasks should be problems that relate to their world, and not just school. Some of these tasks should be puzzles and open-ended questions that might take weeks to solve. Perhaps just posing a situation with a photograph or a short video and asking them to ask questions about it will engage them. I am particularly taken with the idea that we need to let students ask some of the questions as part of our regular routine, instead of only requiring them to answer the questions we ask.

Two weeks ago I had the pleasure of being at a talk by Dan Meyer. He raised some of these same questions about getting students involved. In particular, he pointed out that we often break a problem down into so many little parts (scaffolding I think it is called) that the student isn't sure what the real point of the question is and never has to figure out how to solve it. Part of being interested is to figure out what needs to be done. We are so afraid that students will fail that we drag them through every step of the problem in hopes of keeping them from being frustrated. Being frustrated is part of the fun and a valuable part of the experience.

If students work hard because they want a good grade, or because they want to please a parent or teacher, or because they have bought into the belief that hard work will get them a good job, they will stop working as soon as they have made progress toward their goal. If students work hard because they find the work challenging and stimulating, they will keep working no matter what. I think the latter motive for doing work is worth cultivating.

A teacher's main job is to pull students in to the subject being taught, to ask them interesting questions, and to let them learn. That means the teacher's main job is to provide students with interesting tasks that students can relate to. Some of these tasks should be problems that relate to their world, and not just school. Some of these tasks should be puzzles and open-ended questions that might take weeks to solve. Perhaps just posing a situation with a photograph or a short video and asking them to ask questions about it will engage them. I am particularly taken with the idea that we need to let students ask some of the questions as part of our regular routine, instead of only requiring them to answer the questions we ask.

Two weeks ago I had the pleasure of being at a talk by Dan Meyer. He raised some of these same questions about getting students involved. In particular, he pointed out that we often break a problem down into so many little parts (scaffolding I think it is called) that the student isn't sure what the real point of the question is and never has to figure out how to solve it. Part of being interested is to figure out what needs to be done. We are so afraid that students will fail that we drag them through every step of the problem in hopes of keeping them from being frustrated. Being frustrated is part of the fun and a valuable part of the experience.

If students work hard because they want a good grade, or because they want to please a parent or teacher, or because they have bought into the belief that hard work will get them a good job, they will stop working as soon as they have made progress toward their goal. If students work hard because they find the work challenging and stimulating, they will keep working no matter what. I think the latter motive for doing work is worth cultivating.