MAKING INFINITY CONCRETE

BY PROGRAMMING NEVER-ENDING PROCESSES

Ken Kahn, Evgenia Sendova, Ana Isabel Sacristan, Richard Noss

London Knowledge Lab, Institute of Education, kenkahn@toontalk.com

Great minds ranging from Zeno to Galileo found the concept of infinity puzzling and difficult to understand. A deep understanding of infinity had to wait until Cantor’s discoveries in the second half of the nineteenth century. Cantor’s theorems that the set of all integers, the set of all positive integers, the set of all positive even numbers and even the set of all rational numbers all have the same cardinality (while the set of real numbers between 0 and 1 is of a higher cardinality) are not usually introduced to students until advanced high school or university mathematics courses.

The general assumption that infinity is too abstract for young children is hardly surprising. There are deep mathematical waters to be navigated, and it may seem impossible to make much headway without an appropriate formalism with which to express ideas. The existing formalisms are static and difficult for most students (see, for example, Monaghan, 2001; Tall, 2001; Tsamir, 2001).

This paper describes an attempt to help children approach these ideas, 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. 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.

References

Monaghan, J. (2001). ‘Young Peoples’ Ideas of Infinity’, Educational Studies in Mathematics, 48, 239–257

Tall, D. (2001). ‘Natural and Formal Infinities’ Educational Studies in Mathematics, 48, 199–238

Tsamir, P. (2001) ‘When ‘The Same’ is not perceived as such: The case of infinite sets’, Educational Studies in Mathematics, 48, 289–307