Effective robotic autonomy in unknown environments demands proactive exploration and precise understanding of both geometry and semantics. In this paper, we propose ActiveSGM, an active semantic mapping framework designed to predict the informativeness of potential observations before execution. Built upon a 3D Gaussian Splatting (3DGS) mapping backbone, our approach employs semantic and geometric uncertainty quantification, coupled with a sparse semantic representation, to guide exploration. By enabling robots to strategically select the most beneficial viewpoints, ActiveSGM efficiently enhances mapping completeness, accuracy, and robustness to noisy semantic data, ultimately supporting more adaptive scene exploration. Our experiments on the Replica and Matterport3D datasets highlight the effectiveness of ActiveSGM in active semantic mapping tasks.