Research Papers

Published

October 2022

by

Mathematician & Educator

What types of physical tools do you expect students and educators to use in the learning and teaching of geometric constructions? Although Euclid’s Elements remains silent on prescribing particular tools, do you expect learners and teachers to keep to the so-called Platonic restrictions? That is, are they limited to exploring geometric constructions with an unmarked straightedge and compass only? If so, then why? Certainly, the compass and straightedge are the most well-known set of tools in the learning and teaching of geometry, with learners using them over millennia. Thus, our connection to these tools is partially driven by tradition and familiarity. In addition, there are some high-school curricula that impose the use of these two tools alone, and thus our subscription to the Platonic restriction is partly driven by such curriculum constraints.

However, there are non-trivial problems with students and teachers using compasses as a practical tool of geometry. For example, many learners (especially younger learners) find compasses difficult to operate from a kinesiological perspective due to the fine motor skills required. We all have experienced the unpleasantness of a compass slipping when being rotated and producing wobbly and unsatisfying arcs. As a result, the compass as a drawing tool is somewhat inefficient and the produced drawings can be frustratingly inaccurate. Furthermore, traditional compasses pose safety risks as they can be weaponized by using the pointed leg. Thus there is a need to reconsider the compass and to offer an alternative in the learning and teaching of geometric constructions.

The purpose of this short article is to respond to the above need by discussing the potential of a circle arc template to replace the compass in the learning and teaching of geometric constructions within schools. I will discuss a circle arc template and, through a reflection on a series of questions I have received, probe its design and capabilities against a traditional compass. As such, I hope this article will be of value to the GeT: A Pencil community by providing an alternative construct that can be explored in the classroom.

Figure 1 is a photograph of a circle arc template. Essentially it is a clear piece of polycarbonate with parts removed in it to make a PAC-MAN type shape. There is a positionable centre point in the centre of the template. Learners can place the centre point of the template over a given point on the page and draw around the inside to form an arc of fixed radius up to 300 degrees. Learners operate the circle arc template in a familiar way much like a ruler, that is, one hand keeps the template steady and the other hand holds the pencil to trace out an arc. Let me reflect on, and respond to, some of the most common questions that I have received about the template when I have shown it to colleagues, teachers and students.

A:No. All of the Euclidean constructions can be performed under this restriction together with an unmarked straightedge. In fact, the idea of the fixed radius of the template is similar to the concept of a “rusty compass” in geometric construction, where the angle of the opening of legs of the compass cannot be adjusted (and thus circles and arcs with a fixed radius can only be drawn). The capability of a rusty compass and an unmarked straightedge to act as sufficient tools for all of Euclid’s constructions has been known for more than 450 years (Mackay, 1886).^{1}

A:No. All of the Euclidean constructions can be performed under this restriction together with an unmarked straightedge. More than a century ago, Severi (1904)^{2}showed that if students have an arc of any circle (however short in arc length) together with its centre point and an unmarked straightedge then all of Euclid’s constructions are possible. Observe that the circle arc template in Figure 1 satisfies Severi’s conditions, when used in conjunction with an unmarked straightedge.

A:Not necessarily. In simple constructions such as bisecting a given angle, the steps can be identical, however, in other simple circumstances, such as constructing an equilateral triangle on a given line segment the steps are different, and owe much of their strategies to those for the rusty compass. Thus, there are some familiarities and some differences, and employing such a circle arc template has the ability to foster students to think beyond traditional problem solving pathways. Some basic constructions with the circle arc template can be found in this YouTube playlist (Tisdell, 2021).^{3}

A:Yes, software in the learning and teaching of geometric construction is becoming more prevalent in schools. However, access to digital resources is not always readily available due to cost, complexity of the infrastructure and the challenge of connecting with it. Furthermore, in my opinion, the choice between the physical and the digital forms a false dichotomy. They can be developed together. For example, what roles and possibilities could a “digital” circle arc template or rusty compass have within educational geometry software?

A:Many students have expressed their familiarity with such templates and communicated their ease of use to me. Especially in younger learners (ages 7-11), I have observed much more accuracy in their geometrical drawings using the circle arc template than when using a traditional compass. This suggests that using a circle arc template enables learners to devote more attention to the relationship between geometric objects and principles rather than concentrating on producing an accurate drawing.

A:In addition to the YouTube playlist (Tisdell, 2022) mentioned above, there is a recent publication (Tisdell and Bee Olmedo, 2022)^{4}that expands on the ideas herein.

Although I am excited by the circle arc tool and secretly hope it deserves a “round” of applause, I’d also love to hear your thoughts, ideas and experiences in using non-traditional tools for the learning and teaching of geometric constructions. I welcome any questions and feedback from the GeT: A Pencil community.

- Mackay, J.S., Solutions of Euclid's Problems, with a rule and one fixed aperture of the compasses, by the Italian geometers of the sixteenth century. Proceedings of the Edinburgh Mathematical Society, 1886, 5: 2‒22. Click here
- Severi, F., Sui problemi determinati risolubili colla riga e col compasso (Estratto da una Lettera al Prof. F. Enriques). Rendiconti del Circolo matematico di Palermo, 1904, 18(1): 256‒259. Click here
- Tisdell, C. C. Learning geometry with a circle arc template. YouTube Playlist, 2022. Click here
- Tisdell, C. C. and Bee Olmedo, D. Beyond the compass: Exploring geometric constructions via a circle arc template and a straightedge. STEM Education, 2022, 2(1): 1-36. Click here

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