n one our first Subject Studies session on the PGCE course, Welcome to PGCE Maths (UCL IOE, 2018), we were introduced to fellow student teachers by being numbered from 1-30, and via series of exercises we partnered up with our corresponding thoughts. The exercises helped us realise the beginnings of misunderstandings that can occur amongst students in a mathematics lesson. For example, in one of the exercises we were asked to Introduce yourself to your double, does one pair up with their double 10 or one pair up with their mirror number 5? Such choice of language as Skemp (1976) illustrates can create a faux amis situation, in which words which sound the same, or very alike, but whose meanings are different. This in effect can cause inconvenient mistakes that is considered possibly one of the reasons for the difficulties in mathematics education today. A thought I thought lesser of when considering a career in teaching. It made me appreciate the challenges that teachers face daily, not only the choice of language and words when teaching the Mathematics Curriculum but also the pedagogy to successfully teach it. On another session, we were introduced to polygons with a circle number placed at each vertex and a square number in between so that each square number is the sum of the two adjacent circle numbers, known as an Arithmagon problem (UCL IOE, 2018) – Figure 1. Although finding the missing square numbers in a triangular arithmagon when the circle numbers are given is relatively straightforward (involving addition). Finding the circle numbers however, when the square numbers are given will requires a number of logical reasoning before we can derive answers. The exercise leads us to consider how we might approach a response to the individ>