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American Mathematical Monthly analysis Archimedes Aristotle assumption classes taught classroom considered deductive deductive reasoning Descartes described devising and carrying differences discovery Doctor of Philosophy Education elements Euler evaluation experimental F-Ratio Given by Schools GROUPS School H GROUPS heuristic method High School hypothesis importance Indiana University inductive intuition Kinney and Purdy known Learning Leibniz lem solving looking back Luchins mathematical problems mathematicians mathematics education matics mean gains method of attack method of solution Morris Kline National Council Newton ninth grade algebra Pappus past experience pattern of solution Paul Tannery plan of solution Poincare Polya Princeton Princeton University principle prob problem material problem situation problem solving performance procedures Psychology pupils questions recognized sample Science significant solu Solve It 2d solve problems solver Stanford University step subjects Teachers of Mathematics teaching textbook method tion trans understanding University Press unknown verbal problems York