  University students taking introductory physics are generally successful
executing mathematical procedures in context, but often struggle with the use
of mathematical concepts for sense making. Physics instructors note that their
students experience difficulty with basic algebraic reasoning, a foundation on
which more advanced mathematical thinking rests. However, little systematic
research has been done to measure and categorize such difficulties in this
population. This paper describes an investigation of trends in student
reasoning with proportions, quantification, and symbolizing in introductory
calculus-based physics. Although the assessment items used to probe student
thinking require mathematical reasoning typically taught at the middle school
level in mathematics courses, we find success rates of about 50% among
calculus-based physics students. For many of these students, numerical
complexity and the level of abstraction of the quantities interfere with basic
arithmetic reasoning. We argue that the algebraic thinking of physicists stems
from an idiosyncratic cognitive blend, which is not addressed in prerequisite
algebra courses. We suggest that for more students to understand and adopt the
mathematical thinking characteristic of physics, instructors and education
researchers must explore how to make mathematization a more explicit part of
the curriculum.
