References

• We ask Euclid, “can angles contain curves?”, as he is working on definitions 8 and 9 of the Elements. This imaginary activity begins in 290 BCE. The Mathomat kids have travelled to Alexandria in Egypt for the interview. Asking Euclid about curved angles is interesting, because Euclid’s two most important advances in our understanding of the definition of angle (contained in definitions 8 and 9 of the Elements) both involved the use of rectilinear (straight sided) angles. Heath (1956)**believes that Euclid in fact wanted to restrict his definition of angle to a purely straight sided concept, but that he gave in to the common idea at the time that angles could be curved, and he altered definition 8 to allow for curves. In our imaginary interview, Euclid sets the Mathomat kids that task of drawing two different real-world situations by representing them with the two separate notions of angle that are contained in each of definitions 8 and 9 of the Elements. The Mathomat kids use their Mathomat templates for these two modelling tasks.

• We then find Proculus, reflecting on the 900 years of intellectual achievements in the classical age. We ask him the question, “what exactly is being measured when referring to the size of angles?” The Mathomat kids have travelled forwards in time to 390 AD for this imaginary interview, and are in Constantinople. Proculus begins his response by setting the Mathomat kids the task of finding, and making drawings of, three different real-world situations involving angles. Two of these examples are similar to the ones explained earlier by Euclid, but the third is very different. After the Mathomat kids have used their Mathomat templates to make these drawings, Proculus explains how they need to consider all three at once in order to understand what it is they are measuring.

• Renaissance breakthrough, angles can be greater than 180 degrees! Encountering a plague virus lockdown in middle ages Europe the Mathomat kids decide to hold an imaginary Zoom video conference meeting with three middle ages mathematicians; Tartaglia in Venice in 1540, Peletier in Paris in 1570, and with Clavius in Rome in 1600. During this meeting the three mathematicians explain that the mathematicians and philosophers in the classic age (including Euclid and Proculus) thought that angles could not equal or exceed 180 degrees in size. The Mathomat kids table the three drawings that they made with Proculus, and are set the task of showing how these can be modified so as to represent angles of 0°, 180° and 360°.

*The three questions used in the time travel and the teacher’s challenge during the lesson debrief, can be found in a 2004 study of two sixth grade classes in the USA by Jane Keiser. This study found that the 2000-year-old struggle by mathematicians of the past with the concept of angle created a rich framework for sixth grades students to explore the concept of angle. Consistent with the approach adopted in the Connected Mathematics Program in the USA Keiser’s students were encouraged to discuss a number of different concepts for angle, and to negotiate a meaning for them through collaborative discourse. Keiser found that the three questions referred to in this Mathomat activity could be used to frame the historical struggle with angle.

**Heath, T. (1956). The thirteen books of Euclid’s elements. Translated from the text of Hieberg with introduction and commentary (2nd ed. Vol. 3). New York: Dover.