The Shulchan Aruch writes (658:6) that on the first day of Yom Tov one should not give a lulav to a child before fulfilling the mitzvah onself because the child cannot be makneh the lulav back to an adult and without owning the lulav one cannot be yotzei the mitzvah of "v'lakachten lachem". A second opinion holds this would not be a problem if the child is old enough to understand and make simple monetary transactions. The issue (see Biur Halacha) here is whether da'as acheres makneh - i.e. even though a katan has no formal da'as, a katan can acquire goods if someone with da'as does the hakna'ah - is a din d'oraysa or a din derabbanan. If it is a din d'oraysa, a katan can min hatorah acquire a lulav from a gadol, but has no d'oraysa means of "selling" it back. However, the Ran holds da'as acheres makneh is only derabbanan, and m'derabbanan children can make the simple sale to return the lulav. A third opinion holds
that the adult owner of the lulav should hold the lulav with the child for the na'anuim. The Mishna Berura points out (Sha'as haTziyun 36) that according to this opinion of the Ra'avan the katan never actually acquires the lulav. Even though a gadol would not be yotzei unless he owns a lulav, with respect to the mitzvah of chinuch the we suspend this extrinsic formality. It seems to me that if we define the chiyuv of chinuch as an obligation m'dinei lulav that falls on the father or child, the Ra'avan is difficult to understand. However, if we define chinuch as a general seperate obligation, even though the formal mitzvah of lulav has not been fulfilled, the act of taking the lulav has served the broader goal of educating the katan.