The development of proof construction in middle childhood.

In attempting to solve a problem in the fewest possible moves, one must determine moves that are individually necessary and jointly sufficient. This amounts to constructing a demonstration or proof of the solution. The development of proof construction was explored through two problems-a hidden figure task and a variant of the game "Mastermind" (J. Piaget (1987). Possibility and necessity: Vol. 2. The role of necessity in cognitive development. Minneapolis: Univ. of Minnesota Press). In Experiment 1, children 11 to 12 years of age distinguished necessary from nonnecessary moves and recognized when sufficient evidence had been established to determine a solution to a greater extent than children 7 to 8 years of age. In a second experiment, the original hidden-figure task was modified with the aim of reducing its information-processing demands. The performance of 8- to 9-year-olds on the modified problems was comparable to, and in certain respects surpassed, the performance of the oldest children on the original problem in Experiment 1. Seven-year-olds did not perform well on the modified problems even with training. Results are discussed in terms of theory and research on children's understanding of necessity and sufficiency.