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Human intelligence is characterized by a remarkable ability to infer abstract rules from experience and apply these rules to novel domains. As such, designing neural network algorithms with this capacity is an important step toward the development of deep learning systems with more human-like intelligence. However, doing so is a major outstanding challenge, one that some argue will require neural networks to use explicit symbol-processing mechanisms. In this work, we focus on neural networks' capacity for arbitrary role-filler binding, the ability to associate abstract "roles" to context-specific "fillers," which many have argued is an important mechanism underlying the ability to learn and apply rules abstractly. Using a simplified version of Raven's Progressive Matrices, a hallmark test of human intelligence, we introduce a sequential formulation of a visual problem-solving task that requires this form of binding. Further, we introduce the Emergent Symbol Binding Network (ESBN), a recurrent neural network model that learns to use an external memory as a binding mechanism. This mechanism enables symbol-like variable representations to emerge through the ESBN's training process without the need for explicit symbol-processing machinery. We empirically demonstrate that the ESBN successfully learns the underlying abstract rule structure of our task and perfectly generalizes this rule structure to novel fillers.