Abstract
Four subsystems of geometrical inquiry in school mathematics that assist in
the transition from concrete manipulation to abstract manipulation were identified: (1)
analysis with intuitive and concrete manipulation; (2) elementary geometrical analysis
without numerical and algebraic methods; (3) elementary geometrical analysis using
numerical and algebraic methods; and (4) coordinate geometrical analysis. After
identifying the educational values to treat oscillatory activity in geometrical
subsystems, we developed two teaching materials—“folding a golden rectangle” and
“making a square by cutting a rectangle”. The characteristics of two teaching materials
is to start from concrete manipulation and then consider the problem using numerical
and algebraic methods, and finally consider it with only elementary geometrical
properties without numerical and algebraic methods. It will enable students to
consider the problem concretely and appreciate both the convenience of consideration
within subsystem (3) and the beauty of elementary geometrical structure derived from
consideration within subsystem (2).
