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Although the application of deep learning to automatic speech recognition (ASR) has resulted in dramatic reductions in word error rate for languages with abundant training data, ASR for languages with few resources has yet to benefit from deep learning to the same extent. In this paper, we investigate various methods of acoustic modeling and data augmentation with the goal of improving the accuracy of a deep learning ASR framework for a low-resource language with a high baseline word error rate. We compare several methods of generating synthetic acoustic training data via voice transformation and signal distortion, and we explore several strategies for integrating this data into the acoustic training pipeline. We evaluate our methods on an indigenous language of North America with minimal training resources. We show that training initially via transfer learning from an existing high-resource language acoustic model, refining weights using a heavily concentrated synthetic dataset, and finally fine-tuning to the target language using limited synthetic data reduces WER by 15% over just transfer learning using deep recurrent methods. Further, we show improvements over traditional frameworks by 19% using a similar multistage training with deep convolutional approaches.