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In robots, nonprehensile manipulation operations such as pushing are a useful way of moving large, heavy or unwieldy objects, moving multiple objects at once, or reducing uncertainty in the location or pose of objects. In this study, we propose a reactive and adaptive method for robotic pushing that uses rich feedback from a high-resolution optical tactile sensor to control push movements instead of relying on analytical or data-driven models of push interactions. Specifically, we use goal-driven tactile exploration to actively search for stable pushing configurations that cause the object to maintain its pose relative to the pusher while incrementally moving the pusher and object towards the target. We evaluate our method by pushing objects across planar and curved surfaces. For planar surfaces, we show that the method is accurate and robust to variations in initial contact position/angle, object shape and start position; for curved surfaces, the performance is degraded slightly. An immediate consequence of our work is that it shows that explicit models of push interactions might be sufficient but are not necessary for this type of task. It also raises the interesting question of which aspects of the system should be modelled to achieve the best performance and generalization across a wide range of scenarios. Finally, it highlights the importance of testing on non-planar surfaces and in other more complex environments when developing new methods for robotic pushing.