Being sloppy about slope: The effect of changing the scale

What is the slope of a (linear) function? Due to the ubiquitous use of mathematical software, this seemingly simple question is shown to lead to some subtle issues that are not usually addressed in the school curriculum. In particular, we present evidence that there exists much confusion regarding the connection between the algebraic and geometric aspects of slope, scale and angle. The confusion arises when some common but undeclared default assumptions, concerning the isomorphism between the algebraic and geometric systems, are undermined. The participants in the study were 11^th-grade students, prospective and in-service secondary mathematics teachers, mathematics educators and mathematicians-a total of 124 people. All participants responded to a simple but nonstandard task, concerning the behavior of slope under a non-homogeneous change of scale. Analysis of the responses reveals two main approaches, which we have termed 'analytic' and 'visual', as well as some combinations of the two.