  Six specific modes of reasoning about ratio and proportion have been
delineated as a means of operationalizing expert practice. These modes stem
from consideration of how physicists reason in context, are informed by prior
work in physics and mathematics education, and have grain size matched to the
steps in reasoning needed to solve problems commonly used in physics
instruction. A suite of assessment questions has been developed and validated
to probe student facility with the reasoning modes. Responses to open-ended and
multiple-choice versions of the assessment questions have been collected from
more than 3000 students at Western Washington University, Rutgers University,
and New Mexico State University. Results have been used to identify specific
reasoning difficulties, to document differences in performance between student
populations, and to explore the effect of question context on student
reasoning. We find that students enrolled in university physics courses have
difficulty interpreting and applying ratios in context, and in many cases lack
facility with the reasoning underlying basic arithmetic operations of division
and multiplication.
