Final answer:
By their 12th birthday, children should have reached the formal operational stage of cognitive development according to Piaget, characterized by the ability to think abstractly and handle hypothetical situations.
Step-by-step explanation:
By the time children reach their 12th birthday, according to Jean Piaget's theory of cognitive development, they should have reached the formal operational stage of cognitive development. This is the fourth and last stage in Piaget's theory, which begins from about age 11 into adulthood. At this stage, children develop the ability to think abstractly and deal with hypothetical situations. They can use abstract thinking to problem solve, look at alternative solutions, and test these solutions. They go beyond concrete thinking and are now able to contemplate ideas that are not immediately visible or tangible.