JOURNAL OF EXPERIMENTAL CHILD PSYCHOLOGY 62, 84–101 (1996) ARTICLE NO. 0023 Even More Precisely Assessing Children’s Understanding of the Order-Irrelevance Principle RICHARD COWAN Educational Psychology and Special Educational Needs, Institute of Education, University of London ANN DOWKER Department of Experimental Psychology, University of Oxford, England AND ANASTASIA CHRISTAKIS* AND SHEILA BAILEY² *Child Development and Primary Education and ² Educational Psychology and Special Educational Needs, Institute of Education, University of London There are conflicting views of when children understand that the cardinality of a set is independent of the order in which the objects are counted. Some discrepancies between studies using similar tests may be due to children interpreting the experimenter as questioning the accuracy of their counting. Three studies are reported of children between 3 and 6 years old. In Experiments 1 and 2, more expected a count in a different order to yield the same number when they did not have to do any counting than when they had to count or monitor a puppet’s counting. However, some also expected subtraction not to affect cardinality. In Experiment 3 substantially more children predicted correctly whether a recount would yield the same number than what the number would be. These variations may reflect the power of forms of questioning to orient the child toward principles or practice.  1996 Academic Press, Inc. Children begin to count from a very early age but what they understand about counting is not agreed. Some follow Piaget (1952) in seeing early The authors thank the staff and children of participating schools. The second author gratefully acknowledges the support of the ESRC during the study. The third and fourth authors conducted Experiments 1 and 2 for their MA dissertations under the supervision of the first author. Address correspondence and reprint requests to Dr. Richard Cowan, Institute of Education, University of London, 20 Bedford Way, London WC1H 0AL, UK. Electronic mail may be sent via Internet to tejtjrc@ioe.ac.uk. 0022-0965/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 84