Making sense by measuring arcs: a teaching experiment in angle measure

I discuss a teaching experiment that sought to characterize precalculus students’ angle measure understandings. The study’s findings indicate that the students initially conceived angle measures in terms of geometric objects. As the study progressed, the students formed more robust understandings of degree and radian measures by constructing an arc length image of angle measures; the students’ quantification of angle measure entailed measuring arcs and conceiving multiplicative relationships between a subtended arc, a circle’s circumference, and a circle’s radius. The students leveraged these quantitative relationships to transition between units with a fixed magnitude (e.g., an arc length’s measure in feet) and various angle measure units, while maintaining invariant meanings for angle measures in different units. These results suggest that quantifying angle measure, regardless of unit, through processes that involve measuring arc lengths can support coherent angle measure understandings.