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Past self-organizing models of collectively moving "particles" (simulated bird flocks, fish schools, etc.) have typically been based on purely reflexive agents that have no significant memory of past movements. We hypothesized that giving such individual particles a limited distributed memory of past obstacles they encountered could lead to significantly faster travel between goal destinations. Systematic computational experiments using six terrains that had different arrangements of obstacles demonstrated that, at least in some domains, this conjecture is true. Furthermore, these experiments demonstrated that improved performance over time came not only from the avoidance of previously seen obstacles, but also (surprisingly) immediately after first encountering obstacles due to decreased delays in circumventing those obstacles. Simulations also showed that, of the four strategies we tested for removal of remembered obstacles when memory was full and a new obstacle was to be saved, none was better than random selection. These results may be useful in interpreting future experimental research on group movements in biological populations, and in improving existing methodologies for control of collective movements in computer graphics, robotic teams, particle swarm optimization, and computer games.