摘要：The Manifesto of our conference, Children’s Mathematical Education, calls our attention to the ways that our efforts are tools for both the development of the child and for solving critical problems in a global society. Indeed, these two ways of thinking about our work are always interwoven, since individual children are always present and at the same time future members of our society. That is, our global society is nothing other than ourselves. Education in general, and mathematics education in particular, is central to the basic existence and aims of social life. I start today by reminding us that this manifesto is, if anything, gentle in its call for principals, education officials, the general community, and, most importantly, parents and teachers, to consider how and why fundamental mathematical concepts are at the heart of both personal and social development. Toward this aim, I ask us to think about two concepts that undergird most of the intellectual work of teachers and curriculum workers in mathematics education, and to think critically about the ways in which they impact on our beliefs about what we should do and what might be changeable: the support of sense-making by pupils, and the overarching aim of facility with representations.
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