The questions and common complaints students make can often mask a hidden script of assumptions about teaching and learning that are often themselves the problem for the student and teacher alike. While these questions hinge on the psychological and social side of learning, this area is as important as anything, and we have few tools for responding, in mathematics classrooms, to student frustration and confusion about the ``rules''.
As teachers, we may feel we give repeated encouragement and advice to our classes, but in many ways, students are on their own in their struggle to understand what it means to really know something, and what they must do, individually, to achieve this understanding. They have very little other than tests by which to assess learning. Generally, students are unaware of the sheer amount of effort one must put into mathematics classes, whether this means doing all the homework, keeping up with the lectures, reading the text, or even coming to class. Students evidently feel there is some sort of initiation process behind the scenes in mathematics, where survivors eventually ``catch on'' to the methods of mathematics by undergoing frustrating, even painful experiences.
Students learn best when they ask the questions; unfortunately, they seldom ask how to study or learn; rather, their questions to the teacher frequently surface as aggressive/defensive questions. These ``trigger'' questions are questions which in some sense are generic; they have been kicked around with other students and have undergone peer filtering; they are only rough approximations for expressions of something students are dissatisfied with or concerned about. And the teacher may not always have the time or the focus on the individual student to respond sensitively.
The purpose of this paper is to translate these imprecise questions and complaints into practical activities. In this way, it becomes possible to engage in a dialogue with students which helps them find their own solutions and gain control over the process of learning. The answers here do not give models of direct answers to the questions; the actual responses we make are apt to come out spontaneously, filtered through our prior experiences with the student. If the responses are viewed as put-downs, the process is the reverse of what we intend. Rather these responses are intended to suggest ways of thinking about the questions, as a heuristic which helps students articulate better questions and which provides the potential for the instructor to interact productively with students from time to time.
Learning is a communicative process: it goes on through dialogue and cross-checking, and in class not much in this regard is going on. Improving channels of communication is more necessary than ever as mathematics classes move to interactive modes. If one can map out for oneself ways of using these kinds of questions as a vehicle for meaningful exchange with students, then one has an interactive learning tool, ready at hand. We found that when we dealt seriously and practically with these questions, after the initial surprise to the class, students became absorbed in the ensuing discussion, and follow-up questions were more focused and useful.