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<p>In this study, we tested a developmental progression of students’ understanding of angles involving five stages of quantification: gross, intensive, extensive, ratio, and rate. The Quantifications of Angularity Instrument was used to categorize 152 middle-school students by their stage of angle quantification. Developmental hypotheses were tested using quantitative methods. Findings from the study provide evidence that students’ understanding of angles progresses through these five developmental stages and that students develop non-circular notions of angle prior to circular notions of angle. Implications for instruction on angles and future research are discussed.</p>

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A developmental progression for students’ quantification of angularity

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月18日

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  • Sara Brooke Mullins,
  • Jesse L. M. Wilkins

摘要

In this study, we tested a developmental progression of students’ understanding of angles involving five stages of quantification: gross, intensive, extensive, ratio, and rate. The Quantifications of Angularity Instrument was used to categorize 152 middle-school students by their stage of angle quantification. Developmental hypotheses were tested using quantitative methods. Findings from the study provide evidence that students’ understanding of angles progresses through these five developmental stages and that students develop non-circular notions of angle prior to circular notions of angle. Implications for instruction on angles and future research are discussed.