  Research-validated multiple-choice questions comprise an easy-to-implement
instructional tool for scaffolding student learning and providing formative
assessment of students' knowledge. We present findings from the implementation
of a research-validated multiple-choice question sequence on the basics of
two-state quantum systems, including inner products, outer products,
translation between Dirac notation and matrix representation in a particular
basis, and change of basis. This study was conducted in an advanced
undergraduate quantum mechanics course, in both online and in-person learning
environments, across three years. For each cohort, students had their learning
assessed after traditional lecture-based instruction in relevant concepts
before engaging with the multiple-choice question sequence. Their performance
was evaluated again afterwards with a similar assessment and compared to their
earlier performance. We analyze, compare, and discuss the trends observed in
the three implementations.
