One main reason to quantize gravity is the hope that the singularities of classical General Relativity will be absent within the framework of a quantum theory of gravity. One candidate for such a theory is the well developed framework of Loop Quantum Gravity. During the last years remarkable results concerning the big bang singularity have been achieved within symmetry reduced cosmological models of Loop Quantum Cosmology. In this talk we will address the question whether these results are characteristic also for the full, unreduced theory. As a first step towards answering that question we will present an explicit analysis of the geometric operator corresponding to the classical inverse scale factor occurring in cosmological models which has been performed within the full theory. The results will be discussed.