We present thermal observations of 44 Jovian Trojan asteroids with diameters D ranging from 5 to 24 km. All objects were observed at a wavelength of 24 μm with the Spitzer Space Telescope. Measurements of the thermal emission and of scattered optical light, mostly from the University of Hawaii 2.2 m Telescope, together allow us to constrain the diameter and geometric albedo of each body. We find that the median R-band albedo of these small Jovian Trojans is about 0.12, much higher than that of "large" Trojans with D>57 km (0.04). Also the range of albedos among the small Trojans is wider. The small Trojans' higher albedos are also glaringly different from those of cometary nuclei, which match our sample Trojans in diameter, however, they roughly match the spread of albedos among (much larger) Centaurs and trans-Neptunian objects. We attribute the Trojan albedos to an evolutionary effect: the small Trojans are more likely to be collisional fragments and so their surfaces would be younger. A younger surface means less cumulative exposure to the space environment, which suggests that their surfaces would not be as dark as those of the large, primordial Trojans. In support of this hypothesis is a statistically significant correlation of higher albedo with smaller diameter in our sample alone and in a sample that includes the larger Trojans. This correlation of albedo and radius implies that the true size distribution of small Trojans is shallower than the visible magnitude distribution alone would suggest, and that there are approximately half the Trojans with D>1 km than previously estimated.