This report considers how the mathematical knowledge children develop on their own outside of formal school instruction can be used to increase the distribution and level of mathematical knowledge attained by students in grades K-3. Included are preliminary results of an investigation of the counting and calculating abilities brought to instruction by students in grades K-3 in Chicago (Illinois) and similar research from a number of developing countries. This demonstrates the dangers both of ignoring the knowledge children develop outside of formal school instruction and of becoming too dependent upon it. It is argued that a key means by which elementary school quality can be improved is to begin with the knowledge students develop on their own and transform it though pedagogic and curricular intervention into a set of portable intellectual skills. The point is also made that children around the world bring more knowledge to school with them today than students 50 years ago did, yet curricular materials fail to account for this. This knowledge depends not so much on where they live, what their sex is, or who their parents are, but on the level and complexity of mathematical experience their out-of-school activities provide. (MNS)