## Abstract

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.

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## Notes

- 1.
I use the term

*understanding*to refer to a student's system of schemes and conceptual operations. - 2.
These studies did not investigate the teachers’ degree measure understandings.

- 3.
For the purpose of this study, I define connected angle measure understandings as meanings that can be used to interpret angle measures in consistent ways regardless of the angle measure unit. Connected understandings of angle measure also enable students to flexibly convert between units of angle measure while maintaining these common meanings.

- 4.
Values and numbers can be specified or unspecified. For instance, an individual can anticipate making or determining a measure and consider the meaning of this anticipated result independent of a specified value.

- 5.
Judy and Zac are pseudonyms.

- 6.
The results of these teaching sessions are part of a manuscript in preparation.

- 7.
My use of the term

*calculational*is consistent with the notion of a*calculational orientation*(Thompson, Philipp, Thompson, & Boyd, 1994), which describes an orientation towards identifying procedures, executing calculations, and working with numbers/expressions devoid of contextual reference. - 8.
I note that Zac did refer to a vague area or space between two rays during the interview, but this only occurred once.

- 9.
To improve readability, I refer to the perimeter of the curved part of a protractor—a half circle—as the circumference of the protractor.

- 10.
The students used the terms

*radians*and*radii*(or radiuses) interchangeably when referencing any measure that involved measuring relative to a circle’s radius, even in the arbitrary case. - 11.
As an alternative to (c) and (d), the students also reasoned about multiplicatively comparing the arc length with the radius to determine the arc length in radii.

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## Acknowledgments

National Science Foundation (NSF) grant number EHR-0412537, led by the principal investigator Marilyn Carlson, supported this work. All opinions expressed are solely those of the author and do not necessarily reflect the views of the NSF. I thank Marilyn Carlson, Andrew Izsák, and the reviewer team for providing feedback on previous versions of this article. I also thank Pat Thompson for his substantive conversations about angle measure and quantitative reasoning. These conversations have undoubtedly influenced my thinking on these topics and shaped the study.

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Moore, K.C. Making sense by measuring arcs: a teaching experiment in angle measure.
*Educ Stud Math* **83, **225–245 (2013). https://doi.org/10.1007/s10649-012-9450-6

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### Keywords

- Angle measure
- Teaching experiment
- Quantitative reasoning
- Student thinking
- Trigonometry
- Multiplicative reasoning