Learning-to-learn, a progressive acceleration of learning while solving a series of similar problems, represents a core process of knowledge acquisition. To investigate its underlying brain mechanism, we trained a recurrent neural network (RNN) model on a series of arbitrary sensorimotor mapping problems. The network displayed an exponential speedup in learning across problems. The neural substrate of a sensorimotor task schema emerges as low-dimensional neural representations of task variables that are shared across problems. Its reuse limits the connection weight changes required to learn new problems, thus facilitating their learning. Since the population trajectory of a recurrent network produces behavior, learning is determined by changes in the network’s vector field which governs its dynamics. We propose a novel analysis of vector field changes, which showed that novel stimuli in new problems can distort the schema representation. Weight changes eliminate such distortions and improve the invariance of the reused representations in future learning. The accumulation of such weight changes across problems underlies the learning-to-learn dynamics. Taken together, these findings elucidate the neural substrate of a visuomotor mapping schema, why its reuse dramatically improves learning efficiency, and how its progressive refinement gives rise to learning-to-learn. In doing so, they offer experimentally verifiable predictions, and present novel methods to analyze learning in RNNs by linking changes in neural activity and network structure. Therefore, they are of value to a broad audience in neuroscience, cognitive science, and machine learning.