Although convection is agreed to exist within the newly formed neutron stars in core collapse supernovae, its role remains unclear. Much of the uncertainty concerning the role of convection may be attributed to the fact that the main tools for the investigation of convection in these supernovae are multi-dimensional codes. While these codes provide insight into the role of convection in supernova, their computationally intensive nature makes it impractical to run these codes repeatedly under different physical assumptions and to explore the physics of the late time evolution of the iron core collapse. A one dimensional algorithm for modeling time-dependent convection was developed by treating the convectively unstable zones as two streams. Models using this algorithm were run using various parameters. A series of models were run including different types of neutrinos and different convective effects. Without convection, the shock stalled at a radius of 2 x 10⁷ cm. No qualitative differences that would aid the supernova explosion were observed when convection was included. In models containing all neutrino species, an entropy spike developed in the immediate post shock region. In contrast to interpretations which attribute this spike to radiative heating by neutrinos, the spike in the current models are believed to be caused by shock heating. Comparison of models containing different neutrino species suggests that a "heating dilemma" exists, in that for a shock evolving quasi-statically whose post-shock region is in hydrostatic equilibrium, increased post shock entropy is associated with a smaller shock stall radius. Therefore higher entropies behind the shock in these conditions correspond to model failure rather than to successful explosions. This correlation also appears to produce an inverse relationship between root mean square radiated neutrino energy and shock stall radius