# Investigating Elementary School Students’ Reasoning about Dynamic Angles

8-2022

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## College/School

College of Science and Mathematics

Mathematics

Nicole Panorkou

Joseph DiNapoli

Mika Munakata

Teo Paoletti

## Abstract

Angle measurement is a significant topic in almost all areas of mathematics learning and also in many disciplines outside mathematics education, such as engineering and architecture. According to the literature, there are three common conceptions of angles – as union of rays, rotations, and wedges. Researchers argued that students must consider these three angle concepts together to construct a meaningful understanding of angles. However, the curriculum standards for mathematics often present these angle conceptions separately to students, probably resulting in a fragmented understanding of the angle concept. In addition to this problem, the research literature documents multiple alternative conceptions that students exhibit when they engage with static representations of angles, which is the prevalent way of the current teaching and learning of the concept. Consequently, this dissertation study aimed to explore how students may reason about angles when they engage in tasks that present angles dynamically and bridge the three conceptions. Specifically, this dissertation examined (a) the forms of reasoning that students exhibit as they engaged in dynamic digital tasks that bridged the three conceptions of angles, (b) the characteristics of the design (tasks, tools, and questioning) that supported particular forms of students’ reasoning, and (c) how the design evolved to support students’ reasoning for angles.

Prior research on dynamic measurement and quantitative reasoning guided the design of tasks in GeoGebra to prompt the students to examine how angles are generated and change dynamically. A design experiment methodology was followed to engineer particular forms of reasoning about angles in these dynamic situations and explore how the specific design supports these forms of reasoning. Video-recorded data were collected from four third-grade students working on the tasks individually. Two phases of data analysis were conducted – ongoing analysis and two levels of retrospective analysis. The ongoing analysis as each design experiment unfolded showed how students’ prior knowledge and in-the-moment reasoning about angles influenced the modification of the design. The first level of retrospective analysis conducted at the end of each design experiment illustrated four categories of student reasoning, namely reasoning about the three angle conceptions, constructing multiplicative comparisons between angles, reasoning about an angle as a discrete or continuous quantity, and measuring angles using multiplicative reasoning. The second level of retrospective analysis at the end of all design experiments crosscompared all students’ reasoning and demonstrated the specific characteristics of the design that supported students’ reasoning about angles in those four categories. These findings can be foundational for supporting students’ conceptual understanding of angles in both research and practice.

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