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The conditions that make engagement more likely are: Expectations. This goes hand in hand with the Equity Principle from the NCTM, which is near and dear to me. I believe all people can do significant, important mathematics. I really do. I try hard to communicate this to my learners. Responsibility. Learners make their own decisions about when, how and what to learn. THIS IS ANARCHY. I know. It's dangerous, especially when our learners have been trained for dependence and helplessness. Most students are not ready for full freedom immediately, but it is my goal for all of them. It's also my ongoing struggle to get students to understand that I both mean this and it doesn't mean that work is optional. Employment.  Learners need time to try out their learning in authentic situations. This connects with real life mathematics, with project based learning, with discovery learning and more. The students need to be the ones working if they are to be the ones learning. Approximation.  Learners need to be able to make mistakes without fear of punishment. If there's one area that math has completely screwed up on in the past, it is this. I do it, now you do it perfectly. This is crazy. We all know that no one learns anything important this way. And that the mistakes people make are crucial for learning in the first place. Response. Learners need real and meaningful feedback about what they are interested in working on. I now have my students put stickies on any work they turn in with what questions they have for me. Cambourne: "response must be relevant, appropriate, timely, readily available and nonthreatening." Grades are not feedback in this sense, and can only strive to be timely and readily available. (Although that does make grades better so far as it goes.)

Graphing Ask students to line up along a line, which will be the x-axis. (The line at the middle of a basketball court works well, but any line will do.) They should all face in the same direction. The middle student will be 0, and students on his or her right will be 1, 2, 3, etc., while students on the left will be -1, -2, -3, etc. Then, say: "y = 3, everyone takes 3 steps forward" (Pause, so the students can see where they are standing) "Go back to the x-axis" (Pause) "y = -2, everyone takes 2 steps backward" (Pause) "Go back to the x-axis" These initial two maneuvers should help students calibrate their steps with their neighbors, so that they end up on a line parallel to the x-axis with each. Now say: "y = 2x, everyone step 2 times your number -- forward or back, depending on whether 2 times your number is positive or negative!" Presumably, students will be standing on the graph of y = 2x. And so on, with different functions.