We propose a novel adaptive filtering scheme named metric-combining normalized least mean square (MC-NLMS). The proposed scheme is based on iterative metric projections with a metric designed by combining multiple metric-matrices convexly in an adaptive manner, thereby taking advantages of the metrics which rely on multiple pieces of information. We compare the improved PNLMS (IPNLMS) algorithm with the natural proportionate NLMS (NPNLMS) algorithm, which is a special case of MC-NLMS, and it is shown that the performance of NPNLMS is controllable with the combination coefficient as opposed to IPNLMS. We also present an application to an acoustic echo cancellation problem and show the efficacy of the proposed scheme.