'Grid cells' in the dorsocaudal medial entorhinal cortex (dMEC) are activated when a rat is located at any of the vertices of a grid of equilateral triangles covering the environment. dMEC grid cells have different frequencies and phase offsets. However, cells in the dentate gyrus (DG) and hippocampal area CA3 of the rodent typically display place fields, where individual cells are active over only a single portion of the space. In a model of the hippocampus, we have shown that the connectivity from the entorhinal cortex to the dentate granule cells could allow the dentate granule cells to operate as a competitive network to recode their inputs to produce sparse orthogonal representations, and this includes spatial pattern separation. In this paper we show that the same computational hypothesis can account for the mapping of EC grid cells to dentate place cells. We show that the learning in the competitive network is an important part of the way in which the mapping can be achieved. We further show that incorporation of a short term memory trace into the associative learning can help to produce the relatively broad place fields found in the hippocampus.