Ball frames and responds to these dilemmas, providing a view of underlying pedagogical complexities and the conditions needed in order to work toward current educational visions. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. But I, more experienced with young children, am quite convinced. The things that children wonder about, think, and invent are deep and tough. Somehow, what I do with children should be honest, both to who they are and to what I am responsible to help them learn.
Intellectual honesty implies twin imperatives of responsiveness and responsiblity.
Because mathematical knowledge is socially constructed and validated, sense making is both individual and consensual. Drawing mathematically reasonable conclusions involves the capacity to make mathematically sound arguments to convince oneself and others of the plausibility of a conjecture or solution. It also entails the capacity to appraise and react to others' reasoning and to be willing to change one's mind for good reasons. First, constructing a classroom pedagogy on the discipline of mathematics would be in some ways inappropriate, even irresponsible. Mathematicians focus on a small range of problems, working out their ideas largely alone.