Final answer:
Children can store symbolic information in their memories typically by the time they are preschoolers, around 3 to 5 years old, after they have passed the sensorimotor stage of development that ends around the age of 2.
Step-by-step explanation:
Jerome Bruner, a developmental psychologist like Jean Piaget, contributed to our understanding of cognitive development in children. While the provided information does not explicitly mention Bruner's research on the exact age at which children can store symbolic information in their memories, it highlights that preschool-age children, which include those between 3 to 5 years old, are capable of symbolic thinking and pretend play. This indicates that by this age, children can understand and use symbols – a critical aspect of storing symbolic information.
Moreover, Piaget's stages of cognitive development, specifically mentioning the sensorimotor stage lasting until about 2 years old, show that it is after this period that more complex symbolic thinking begins to solidify.
In summary, although Bruner's precise age guideline is not provided here, based on Piaget's cognitive development timeline and typical cognitive milestones, it is reasonable to conclude that children begin to store symbolic information in their memories at some point after the sensorimotor stage, which ends around the age of 2. As preschoolers, by ages 3 to 5, children are actively engaging in symbolic play and are clearly able to store and use symbolic information.