Children's understanding of equality--inequality relationships for liquid quantities: the role of minimal-difference...

When one of two same-amount liquid quantities was actually, or proposed to be, altered by a very tiny (smidgen) amount, kindergartners and 2nd graders failed to recognize the resulting inequality, whereas 3rd and 4th graders were increasingly successful. Alterations larger than a smidgen, but still small, improved the judgments of 2nd, 3rd, and 4th graders but left kindergartners still thinking that equality was unchanged. Alterations in larger amounts that provided noticeable perceptual discrepancies led to nearly perfect inequality judgement across all grade levels. A rigorous evaluation of children's understanding of equality-to-inequality changes must include minimal-amount differences in a quantity along with the solicitation of verbal reports.