Final answer:
Reversibility is the achievement that enables children in the concrete operational stage to solve conservation tasks, understanding that objects can return to their original state and the quantity remains constant despite appearance changes.
Step-by-step explanation:
The achievement that allows children in the concrete operational stage to work out conservation problems in their heads is reversibility. This is the understanding that objects can be changed and then returned back to their original form or condition. In Piaget's theory of cognitive development, reversibility helps children realize that the quantity of something remains the same despite its appearance changing, such as pouring water from one glass to another of a different shape. This cognitive ability is a significant advancement from the earlier preoperational stage where children are unable to comprehend such concepts.