Students' dynamic geometric reasoning about quantum spin-1/2 states

Quantum states are traditionally cognitively managed exclusively with algebra rather than geometry. One reason for emphasizing algebra is the high dimensionality of quantum mathematical systems; even spin-1/2 systems require a 2-d complex number space for describing their quantum states, which can be hard to visualize. Using ``nested phasor diagrams,'' which use nesting to increase the dimensionality of graphic space, we taught undergraduate students to represent spin-1/2 states graphically as well as algebraically. In oral exams, students were asked to identify which spin-1/2 states, expressed numerically, would generate the same set of probabilities as each other (i.e., they are the same except for a different overall phase factor). Video records of oral exams show that no students (N=13) performed this task successfully using an algebraic method; instead, all students solved the problem graphically. Furthermore, every student who succeeded used a certain gesture to solve the problem.