An equal-armed balance at equilibrium--the bar is horizontal--tips into disequilibrium upon displacing a weight. Equilibrium is restored by reversing that move--putting the weight back where it was, or doing the same on the other side. Piaget adopted the idea of equilibration to describe how the intellect, in relating to the world, develops. Equilibrium arises as: our mind adjusts its structures in response to the outer world (accommodation), so our internalized views can take in this outer world (assimilation). That is the process Piaget calls equilibration. Upon undergoing disequilibrium, the intellect employs these equilibrating moves, changing its structures in the process. When the intellect resolves a disturbing problem no matter how it is encountered, the intellect tries to reverse the disturbing feature: how did the familiar situation get to this disturbing one; how might that change be reversed? These equilibrating processes are encouraged as means of teaching and learning in this paper's math and science examples. The clinical interviewing methodology of Piaget and Inhelder, as adapted by Eleanor Duckworth in the research pedagogy of clinical exploration in the classroom, provides the neutral, safe conditions requisite for these learners and teachers in undergoing disequilibrium, struggling with uncertainty, and constructing new understandings. In beginning to teach through exploration, the author and an undergraduate experimented with free fall motion. Experiencing disequilibrium, the student reconstructed her understanding of time as concurrently continuous and divisible. Seeking to enact methods of Piaget and Duckworth while engaging her, the teacher also experienced disequilibrium.