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This work aims to enable on-device training of convolutional neural networks (CNNs) by reducing the computation cost at training time. CNN models are usually trained on high-performance computers and only the trained models are deployed to edge devices. But the statically trained model cannot adapt dynamically in a real environment and may result in low accuracy for new inputs. On-device training by learning from the real-world data after deployment can greatly improve accuracy. However, the high computation cost makes training prohibitive for resource-constrained devices. To tackle this problem, we explore the computational redundancies in training and reduce the computation cost by two complementary approaches: self-supervised early instance filtering on data level and error map pruning on the algorithm level. The early instance filter selects important instances from the input stream to train the network and drops trivial ones. The error map pruning further prunes out insignificant computations when training with the selected instances. Extensive experiments show that the computation cost is substantially reduced without any or with marginal accuracy loss. For example, when training ResNet-110 on CIFAR-10, we achieve 68% computation saving while preserving full accuracy and 75% computation saving with a marginal accuracy loss of 1.3%. Aggressive computation saving of 96% is achieved with less than 0.1% accuracy loss when quantization is integrated into the proposed approaches. Besides, when training LeNet on MNIST, we save 79% computation while boosting accuracy by 0.2%.