Led Astray by a Right Triangle: Misconception, Epiphany, and Redemption - Redemption

Frank J. Swetz (The Pennsylvania State University)


While my speculative adventure into traditional Chinese problem solving techniques and the allures of the right triangle faltered, nevertheless, I came away from the experience with several valuable insights, including:

  1. An enhanced appreciation of the early importance of gnomon observations in the development of mathematical thought. Such observations provided an intuitive recognition of angle and, in particular, the importance of the right angle and the concept of perpendicularity (orthogonality). They also resulted in the eventual realization and utilization of the ratio, height of gnome to its shadow length, suggesting the strong possibility that gnomon observations and measurements provided an impetus to further investigate the properties of right triangles leading to a formulation of the Xian thu proof.
  2. The facility of the ancient geometric-algebraic approach to problem solving, which reflects modern pedagogical theories for mathematics discovery and learning: physical and visual interaction, and manipulation followed by data collection and hypothesis formation and testing. Mathematical concepts were initially obtained through physical experiences. Semi-concrete modeling and visually performed solution methods were a logical step in the development of algebraic thought and, indeed, in most mathematical concept formation.
  3. It is necessary to appreciate historical materials in their intellectual and societal contexts. Remain cautious that what appears to be so mathematically obvious in a modern light may have been unrecognized by our ancestors.

And finally, there is still much to learn about the evolution of mathematical concepts and the early techniques of problem solving. The journey of curiosity and inquiry still beckons.