Final answer:
The child who advanced in understanding from incorrectly assessing quantity based on arrangement to correctly stating that quantities are equal despite changes in appearance has moved from the preoperational stage to the concrete operational stage.
Step-by-step explanation:
When a child was shown a line of five cookies and a line of five cups and incorrectly said there were more cups than cookies because the line of cups was longer, but later is able to correctly state that there is an equal number of each, the child has advanced from the preoperational stage to the concrete operational stage. In the preoperational stage, children do not possess the understanding of conservation, believing that the change in the arrangement of objects can affect quantity. However, upon entering the concrete operational stage, children develop the ability to think logically about concrete events and master the concept of conservation, understanding that even when the appearance of something changes, its quantity remains the same.