Arthur Schultze
The Teaching of Mathematics in Secondary Schools is Arthur Schultze’s practical study of how mathematical instruction can develop disciplined reasoning instead of mere recall. Based on lectures delivered to prospective teachers at New York University, the book addresses the decisions that arise in actual classrooms: what to teach, why to teach it, how to present it, and how to help students attack unfamiliar problems for themselves.Schultze begins by diagnosing ineffective instruction. Overloaded courses, excessive emphasis on information, weak motivation, and demonstrations learned by rote can leave students unable to reason independently. He argues that the value of mathematics lies chiefly in habits of exact thought, logical organization, persistence, and the capacity to move from known relations toward a solution.The book compares deductive, inductive, heuristic, laboratory, and other methods without treating any single procedure as universally sufficient. General pedagogy is tied to concrete mathematical questions. Definitions, axioms, postulates, and the foundations of geometry are examined because a teacher’s own conceptual clarity determines whether classroom explanation becomes meaningful or mechanical.A substantial geometrical section considers original exercises, strategies for attacking theorems, parallel lines, the circle, regular polygons, impossible constructions, solid geometry, and applied problems. Algebraic chapters discuss the curriculum, negative numbers, factoring, equations, graphs, irrational and complex numbers, and the relation between symbolic work and reasoning. Trigonometry receives attention within the broader secondary program.The author’s historical setting is visible in examples, curriculum assumptions, and the language of mental discipline. Contemporary classrooms have different technologies, standards, and research on learning. Even so, the central challenge remains recognizable: students must do more than reproduce a finished proof or procedure.
