In order to investigate the importance of cooperative effects in the cooling of systems into a supercooled liquid or glassy state, a kinetic Ising model is considered with all transition rates thermally activated, so that cooling can lead to the system being trapped in a long-lived nonequilibrium glassy state. A difference in energy between the states with up and down spin permits the definition of a fictive temperature T-f that defines the state of the system, and in this paper the effects of different cooling rates on the evolution of T-f are examined. Two types of model system are considered, one in which the transitions of the different spins are independent and the other in which they need to be facilitated by the presence of adjacent up spins (the Fredrickson-Andersen model), so that cooperative effects are involved. It is found that the requirement of facilitation not only slows down the transition rates but also affects the apparent activation energy deduced from the variation of the final fictive temperature with the cooling rate, and a theoretical analysis of the results is presented.