Inquiry-based teaching occurs when teachers focus their attention on what is happening in the classroom and make notes about things they would like to revisit (Schifter, 1996). Such notes can be mental or written out. They typically follow a discrepancy between what has occurred and what might have been expected (Strauss, 1998). The practice of talking with others about plans, conjectures, and the like enhances the learning that such observation affords (Palincsar, Magnusson, Marano, Ford, & Brown, 1998).
Working with the cylinder lesson, we examined some of the moments and focused on some of the skills and principles that effectively bring concepts within reach. We looked at interventions that contribute to students' increasing confidence in pursuing a problem and evaluating strategies and/or answers.
As a result of the opportunity to collaborate on this project, our own discourse has also been enhanced. Because we have talked and listened to each other, each of us now has more well-defined tools in our teaching repertoire, and colleagues with whom we can continue to learn. We have a sense of the indicators that can help us make decisions about interventions that would match our students' strengths and needs. We also have a variety of possible ways of solving a problem available to us if we need them.
Following are some of our reflections on our experiences of this past year. We understand ourselves to be consolidating our knowledge about the rich problem of how to teach mathematics. We have found that there is no one way to facilitate students' mathematical thinking, although there may be more effective ways for a given person to meet the strengths and needs of his or her students. In reviewing these reflections, we are aware that we too are constantly learning. Because of this, our classes remain interesting and our students are (mostly) engaged.
We have also found that bringing these issues up outside of class makes it more likely that we will notice them during class and will have an idea of what to do about them. This practice-centered discourse, then, has led us toward our goal of encouraging students' mathematical thinking.
There are, of course, many unanswered questions. It is for this reason that we view this paper as part of a longer conversation -- one to which we will continue to return and from which we expect to continue to learn.