1.0 Introduction

This workshop is informed by the collective voice of 69 Australian mathematics teachers from grades three to eight who participated in my PhD survey. Their 259 descriptions of enacted angle teaching, along with responses about definitions, concept images, tool use, and lesson planning resources, provide a thick description of current classroom practice. The workshop aims to build on this analysis by offering a coherent, intuitive, and toolsupported approach to teaching angle.

2.0 The Angle Teaching Paradox

Survey results reveal a paradox. Many teachers attempt to connect students’ experiences of turn with the geometric structure of angle. This is consistent with the reform agenda embedded in the Australian Curriculum (ACARA, 2025). Yet teachers also express pessimism about the success of this approach. This can be seen in answers to two questions in the survey that were based on prior research by Mitchelmore and White (2000) for the first question and Goos et al., (2020) for the second:

• As shown in Figure 1, teachers report substantial difficulty with protractor use at early secondary school.

Figure 1

Estimated Percentage of Students Whose Difficulty with Protractors Makes the Teaching of Angle Measurement Problematic.

Note: Twenty-three survey participants answered this question and the question for Figure 2.

Figures One and Two indicate that about 50% of students taught at grade levels 6 to 8 by survey participants are thought not to be able to conceptualise angle at a level that is required in the secondary school mathematics curriculum. This paradox frames the need for a more effective approach to angle teaching.

3.0 Angle Learning Theory

Understanding angle requires that students pay attention to its attribute. If this is done successfully students will be engaged with a number of different concepts that result in a sense of the spatial structure that makes angle measurable. Teacher education texts such as Elementary and Middle School Mathematics (Van de Walle et al., 2017), Teaching Mathematics: Foundations to Middle Years (Siemon et al., 2017), and Teaching Primary Mathematics (Booker et al., 2023) emphasise the need for this spatial understanding.

The Attribute of Angle

Terms such as spread (Van de Walle, 2017, p. 505) or openness (Mitchelmore & White, 2003, p. 7) between an angle’s rays function as conceptual metaphors. A metaphor is a relatively concrete and easy to understand word that is used to understand something more abstract and difficult to grasp. Research into how the brain works (Denesi, 2023) has found that conceptual metaphors are essential for building mathematical knowledge. They are an engine which students use to integrate fragmented knowledge into a usable notion of the multifaceted concept of angle.

Developing a Conceptual Protractor

Students need a conceptual protractor, which gives them an ability to visualise and transform angle structure in different situations, before they can use physical protractors effectively. Tools such as the Rotagram (Giles, 1979) support this development by presenting angle in its bare structural form: vertex and rays. The Rotagram is shown in Figure 3.

• The dynamic action and minimal design of the Rotagram creates a physical model that students can use to assist them in developing a conceptual protractor.

The Rotagram is integrated as a tool into the lesson sequences for angle that are recommended for teachers by Booker at al., (2023) and Siemon et al., (2017). 

Figure 3

The Rotagram

4.0 Findings from the Teacher Survey

Rotatingarm tools were mentioned by only two of the 65 survey participants who participated in tool preference questions, despite the recognised importance of these tools in the teaching of angle. No survey participants mentioned using a Rotagram. None of the descriptions of their teaching, or their classroom definitions, of angle by participants gives a formal definition for the attribute of angle (such as openness or spread).

Survey responses reveal a fragmented collective concept of angle:

22% used a multifaceted definition involving turn and geometric structure (aligned with ACARA).

16% focused solely on plane angle.

21% taught geometry without defining angle at all.

When teachers’ angle concept definitions were compared with their concept images, a high degree of disconnect emerged. This fragmentation mirrors the pessimism teachers expressed about student understanding.

5.0 Why Textbooks Fall Short

Textbooks frequently used by survey participants were reviewed in my thesis. They rely heavily on procedural learning involving what Stacey (2003) describes as:

• “asking students to follow procedures without reasons”, resulting in “a syndrome of shallow teaching” (p. 119).

This pattern is consistent with analysis in other Australian (Shield & Dole, 2013) and international (Smith et al. 2016) research. Textbooks do not support reformbased geometry teaching or the development of conceptual protractor thinking.

6.0 The workshop

The workshop aims to bridge the gap between teacher education publications and classroom resources by introducing ROTOMAT and an intuitive learning trajectory for angle.

Introducing the Rotomat Tool

Rotomat has the same basic function as the Rotagram designed by Geoff Giles, it has a clear rotating disc which is attached to a transparent panel. Rotomat is designed to integrate with the Mathomat template. Its larger, 100 mm diameter rotating disk provides a smooth, accurate action. Rotomat is illustrated in Figure 4.

Figure 4

The Rotomat

Explorations with Rotomat

In this workshop we begin by discussing strategies for introducing students to angle so that they engage with its attribute. This will involve the use of language that invokes spatial reasoning about angle.

Rotagram as a tool can be overlaid on traditional static angle representations and then used to compare those with other angle drawings. This involves students in connecting the dynamic experience of turn with the geometric structure of angle in static angle drawings. The minimal design and dynamic action of Rotomat demonstrates the notion of openness between angle sides in a tactile and compelling way. In the workshop we use this to consolidate student learning from a number of different classroom lessons involving angle.

When students are being introduced to protractors and measurement of angle in formal units such as degrees, Rotomat can be used to connect with earlier learning experiences. In the workshop we explore a number of ways to use Rotomat to transfer, and transform, the static angle representations that they experience in textbooks onto the Mathomat protractor. The ability of Rotomat to physically model the spread between an angle’s rays can be used to visualise degree measurement with protractors as a form of subtended arc, as recommended by Van de Walle (2017).

7.0 Conclusion

The survey reveals a fragmented teacher understanding of angle and widespread pessimism about student learning. By strengthening language use, modelling the conceptual protractor, and adopting intuitive approaches, teachers can significantly improve student understanding.

Rotomat, packaged with Mathomat, offers practical tools to support this shift. With more coherent teaching and assessment, students can develop robust angle understanding that supports later geometry and trigonometry learning — and teachers can move toward a shared, multifaceted conceptualisation of angle.

8.0 References

Australian Curriculum Assessment and Reporting Authority (ACARA). (2025). Australian curriculum version 9: Mathematics. ACARA Retrieved from https://www.australiancurriculum.edu.au/

Booker, G., Bond, D., & Seah, R. (2023). Teaching Primary Mathematics (6th ed.). Pearson Australia.

Denesi, M. (2023). Poetic logic and the origins of mathematical imagination. Springer. https://doi.org/10.1007/978-3-031-31582-4

Giles, G. (1979). ROTAGRAM Workcard booklet series. Development of Ideas in Mathematical Education (DIME Projects).

Goos, M., Vale, C., & Stillman, G. (2020). Teaching secondary school mathematics. Routledge.

Mitchelmore, M., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational studies in mathematics (41), 209-238.

Mitchelmore, M., & White, P. (2003). Teaching about angles. Stage 2. Bankstown, NSW: Australian Government Department of Education Science and Training

Shield, M., & Dole, S. (2013). Assessing the potential of mathematics textbooks to promote deep learning. Educational studies in mathematics, 82(2), 183-199. https://doi.org/10.1007/s10649-012-9415-9

Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2017). Teaching mathematics: foundations to middle years (3rd ed. ed.). Oxford University Press UK.

Smith, J., Males, L. M., & Gonulates, F. (2016). Conceptual limitations in curricular presentations of area measurement: one nation’s challenges. Mathematical thinking and learning, 18(4), 239-270. https://doi.org/10.1080/10986065.2016.1219930

Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2017). Teaching mathematics: foundations to middle years (3rd ed. ed.). Oxford University Press UK.

Stacey, K. (2003). The need to increase attention to mathematical reasoning. In H. Hollingsworth, J. Lokan, & B. McCrae (Eds.), Teaching mathematics in Australia: Results from the TIMMS 1999 video study (pp. 119-122). Australian Council for Education Research.

Van de Walle, J., & Karp, K. (2021). Elementary and middle school mathematics; Teaching developmentally (10th ed.). Pearson.

 

Beyond the classical straight edge: Explaining the potential of the finite double-edged straight edge. 

Fabian Schmitt (School of Mathematics and Statistics, UNSW, Sydney, 2052)
Christopher C Tisdell (School of Mathematics and Statistics, UNSW, Sydney, 2052)
Appeared in Parabola, volume 62 issue 1, 2026