This paper aims to address the major challenges of Federated Learning (FL) on edge devices: limited memory and expensive communication. We propose a novel method, called Partial Variable Training (PVT), that only trains a small subset of variables on edge devices to reduce memory usage and communication cost. With PVT, we show that network accuracy can be maintained by utilizing more local training steps and devices, which is favorable for FL involving a large population of devices. According to our experiments on two state-of-the-art neural networks for speech recognition and two different datasets, PVT can reduce memory usage by up to 1.9$\times$ and communication cost by up to 593$\times$ while attaining comparable accuracy when compared with full network training.