We demonstrate the utility of effective Hamilonians for studying strongly correlated systems, such as quantum spin systems. After defining local relevant degrees of freedom, the numerical Contractor Renormalization (CORE) method is applied in two steps: (i) building an effective Hamiltonian with longer ranged interactions up to a certain cut-off using the CORE algorithm and (ii) solving this new model numerically on finite clusters by exact diagonalization and performing finite-size extrapolations to obtain results in the thermodynamic limit. This approach, giving complementary information to analytical treatments of the CORE Hamiltonian, can be used as a semi-quantitative numerical method to study frustrated magnets (as the S=1/2 kagome lattice) or doped systems. ; Comment: Proceedings of the conference on 'Effective Models for Low-Dimensional Strongly Correlated Systems', Peyresq, September 2005. 11 pages, 10 Figures