This lecture examines the failure of the United States to produce literate, mathematically competent high school graduates and criticizes "cyclic faddism" as a ploy of desperate educators who keep going back to methods that have been tried and found wanting. Bright spots in education are cited, including the achievement of Jaime Escalante's Hispanic high school students and the performance of the United States in the International Mathematical Olympiad. Suggestions for overhauling school organization include the development of longitudinal teaching teams and development of special accelerated educational opportunities for intellectually talented youth. Public attitudes toward intellectual precocity and brightness are explored. The educational system's tendency to ignore prior academic information about highly able students when making placement decisions is criticized. A model developed by the Study of Mathematically Precocious Youth, known as diagnostic testing followed by prescribed instruction, is described. Virginia's participation in the talent searches of the Center for the Advancement of Academically Talented Youth is noted. An introduction to the lecture, by John M. Nagle, outlines the purpose of the lecture series and offers a brief biography of the lecturer, Julian C. Stanley. (Contains 11 references.) (JDD)