When pre-school children count an array of objects containing one that is broken in half, most count the halves as two separate objects. Two studies explore this predisposition to count discrete physical objects (DPOs) and investigate its robustness in the face of various manipulations. In Experiment 1, 32 children aged three-four years counted arrays of intact and broken objects, comprising familiar objects known to be separable (e.g. a lolly and its stick) or non-separable (e.g. a toothbrush). The meaning of presenting a broken object was made explicit as some children saw a "naughty Ted" causing the breakage. The DPO bias was robust in the face of these familiarity and context manipulations. Experiment 2 tested whether the DPO bias could be overcome by teaching children a strategy for counting two parts as one whole and also considered whether children's prior knowledge of cardinality was associated with the bias. Only eight children (33%) benefited from the strategy teaching. At a second post-test two to three days later, half of these had reverted to their DPO bias. Cardinality knowledge was not associated with improvement. The robustness of the bias to count DPOs is discussed in terms of innate predispositions and abstracted representations. (Contains 3 tables and 1 figure.)