We will describe our an attempt to help children approach the concepts of the cardinality of infinite sets and one-to-one correspondences, by providing them with an appropriate alternative formalism with which to think and talk about ideas like these. Our hypothesis is that via carefully-designed computational explorations within an appropriately constructed medium, infinity can be approached in a learnable way that does not sacrifice the rigour inherent in the concept. The curious child can learn some deep, interesting, and different mathematics without first having mastered more advanced mathematics.

We will describe how children explored concepts of cardinality of infinite sets by interpreting and constructing computer programs in ToonTalk as part of the WebLabs Project. Children programmed infinite or non-terminating processes that produce infinite sequences including the natural numbers, the even numbers, the integers, and the rational numbers. They show constructively the one-to-one correspondence between the corresponding sets of numbers. Our field studies have supported the hypothesis that children can build useful intuitions of infinity by constructing and manipulating infinite processes and the computational objects that hold the eternally growing sequences produced by these processes.