The word “παιδεία” (education) comes from the ancient Greek verb “παιδεύω”, which means εκπαιδεύω (educate, train). “παιδεύω” comes from the noun “παις”, which means παιδί (child). The word “παιγνίδι” (toy) also has the same root.
The correlation is obvious as the children undergo education, even with games and toys.
It is interesting to note that the word “παιδεύω” nowdays means to give someone a hard time, to torture.