The small angle approximation sin[theta approximately theta] is central to all treatments of the simple pendulum as a harmonic oscillator and is typically asserted as a result that follows from calculus. Here, however, we show that the geometry of the pendulum "itself" offers a route to understanding the origin of the small angle…

Research on geometric transformations suggests that early learners possess intuitive understandings grounded in motion metaphors, transitioning to mappings. The processes through which students transition between these two conceptions are not fully understood. We propose that Vygotskian hybridizing (related to Vygotsky's articulation of everyday…