We study the lattice of closed ideals of bounded operators on two families of Banach spaces: the Baernstein spaces B_p for 1<p<∞ and the Schreier spaces S_p for 1≤p<∞. Our main conclusion is that there are 2 to the continuum many closed ideals that lie between the ideals of compact and  strictly singular operators on each of these spaces, and also 2 to the continuum many closed ideals that contain projections of infinite rank. 

Counterparts of results of Gasparis and Leung using a numerical index to distinguish the isomorphism types of subspaces spanned by subsequences of the unit vector basis for the higher-order Schreier spaces play a key role in the proofs, as does the Johnson-Schechtman technique for constructing 2 to the continuum many closed ideals of operators on a Banach space.