Protractor Tools for Understanding Geometry

Two Innovative Protractor Designs That Students Should Learn to Operate with Their Eyes Closed

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Consider the deep connection between the science of physics and the mathematics of geometry. Physical objects and events that occur around us exist in physical space. Therefore, as Adler (1967) pointed out, concepts from the physical world and concepts in geometry are deeply interconnected. Concepts in geometry are abstracted from physical experience. But the reverse is also true, to make sense of the physical world around us we project our geometric concepts into physical space. As Adler (1967) pointed out, the things we see in the world are to a certain degree extensions of what we have created in our minds.

The interplay between physical experience and abstract reasoning is reflected in the study of angle in geometry by the mapping of physical turns onto a numerical structure for angle. It is this specific mapping process that structures the Australian Mathematics curriculum (ACARA, 2025) for angle. Research into how students develop a concept of angle by Clements and Burns (2000) was able to describe in precise terms how Adler’s (1967) notion of projection of geometric concepts plays out in students’ minds as they learn about angle. Clements and Burns (2000) described a process in which students learn to replace the physical act of rotation with a mental model for angle. This mental model, or conceptual protractor, can become so strong that students learn to project it into objects and events in the world around them. Students who can be described as conceptual protractor users have insight into the geometric structure of their world, they are able to understand intuitively what the angular structure of objects is, and how to measure angle size.

Unfortunately, many students do not develop their understanding of angle to the level that is expected by the Australian Curriculum (ACARA, 2025; Goos et al., 2020; Seah & Horne, 2021). Having a shallow conceptualisation of a fundamental concept in geometry such as angle places significant limitations on students’ ability to develop numeracy and to engage with school mathematics. The two protractors on this page, Tisdell’s Geometry Tool (TGT) and the ROTAGRAM by Geoff Giles, are innovative designs that provide students with physical models of the conceptual protractor that they need to construct in their heads. They are designs that students should learn to operate with their eyes closed.

John Lawton, May 2025
Publisher at OLM

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