Nearly three hundred years after historians of science began analyzing the works of astronomers in medieval India, it remains quite unclear how those astronomers actually looked at celestial objects in the sky, and how (or whether) they used what they saw there to create and modify their quantitative theoretical models. The long shadow cast by Ptolemaic astronomy has blurred the details of a wide variety of ancient and medieval scientific practices. It is frequently taken for granted by historians that the basic theory-building process that Ptolemy described (which may not have been identical to the one he followed) was or should have been the default approach for all serious astronomers: hypothesize a self-consistent geometric model for celestial motions and then establish its parameters from observational data. But it is still far from certain whether and to what extent medieval Indian astronomy fits into this methodological mold. This talk examines the existing textual sources and proposed reconstructions of Indian astronomers' scientific methods, and identifies some crucial sources of controversy.