Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the twofast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow. Bursting electrical oscillations are common in nerve cells and endocrine cells. These consist of episodes of electrical activity followed by periods of quiescence. Since each electrical impulse is itself an oscillation, this is an example of a multi-time scale oscillation, which has been analyzed successfully using a decomposition of the system of equations into fast and slow subsystems. In this article, we compare two alternate fast/slow analysis techniques for the study of the type of bursting oscillations that typically occur in pituitary lactotrophs and somatotrophs. We show the relationships between the key elements of both types of analysis.