Piaget's theory of intellectual growth describes the child as the architect of his own thinking. The child's initial thought patterns are perception-oriented interpretations of the surrounding world. As his experiences continue they are transformed internally by the child's previous storage level of understanding. This introspection leads to a gradual but spectacular change in the basis of thinking, there is a move from semilogic to rationality. Each act of understanding involves an element of invention which has had no existence before either in the external world or in the subject's mind. The concept of invariance is central to Piaget's theory. It concerns the individual's recognition that a change in an external dimension does not imply a change in quantity unless something has been added or removed. The fieldwork section of this study provides an examination of some of the known necessary conditions for conserving behaviours. The New Zealand sample followed the pattern outlined by Piaget; many young subjects were included in the group of non-conservers, a middle group of borderline conservers were more likely to conserve quantity than volume, and the third group of subjects were assured conservers and included fewer younger subjects in their number. These findings are used to illustrate some aspects of the current literature on the necessary conditions for invariance. The child, who has access to a wide range of recognizable operational variety in his environment, whose experience has undergone some internalized ordering, who has confidence to explore his world, whose present equilibrated structures can make an adjustment to a mismatch, is likely to develop conserving behaviours. The significance of the semiotic function rather than the specific dominance of language is discussed in the study. Piaget recognizes the extending role of language but demonstrates that invariance can be established without language but not without operations. The sufficient conditions for conserving are not yet definitive, they do arise from the child's active involvement with his environment. The present studies are based on post hoc explanations concerning particular aspects of invariance. The concept is a comprehensive understanding but the current measures are centred on specifics of the whole. Some of the sufficient conditions include the preliminary recognition of permanency of objects which contributes to the identity element in conservation, Identity and equivalence by compensation and reversibility are the accepted (sufficient) conditions for invariance in contemporary terms. The final section examines some of the implications for educators. Teachers have the task of providing individual children with operational challenge. The failure of many studies which attempt to influence a child's conserving responses by language and operational training is salutory.