lthough the problem is equivalent to solving three simultaneous linear equations in three unknowns (Figure 2), it is not necessary for students to use algebra to make sense of the structure. Given the opportunity, this exercise can provide an interesting context for students to make predictions and conjectures to develop their powers of mathematical reasoning. Traditionally, teachers would have approached the problem in a prescriptive method, where the methods are explained to learners one step at a time, possibly via a memorised rule: mnemonics or songs. In this example, the teacher could have introduced a memorised rule such as, orally saying, “Add the two near (clenching the fists and brought together), subtract the (square) number opposite (arms go apart) and divide by two (halving action). This approach as Swan (2015, p. 3) argues is “mechanistic… imitating a routine or procedure without any depth of thought” or as “rules without reasons” (Skemp, 1976, p. 2), where learners would have derived at the answer via teacher’s prescriptive method rather than at their own level of understanding. This potentially risks a mathematics lesson becoming monotonous and tiresome for learners and in turn made me re-evaluate what constitute a good maths teacher, not just assessing our performance, the children’s behaviour, the subject content of the lesson or what the intended learning is, but also being able to strike the balance for managing both proscriptive method, where learners are challenged and arrive at understanding through discussion and prescriptive method, where learners watch, listen and imitate until fluency is attained, without ruining what could be considered a rich problem for solving activities like the Arithmagon activity .

that is to have an ability to explain a topic in a number of different ways. Each student has a particular preferred method for learning, even if they are not aware of it. Skemp (1976) identifies learning theory points to different types of understanding – relational and instrumental. Instrumental understanding is more algorithmic; the pupil will learn a sequence of steps for solving a particular problem, and will simply apply this to any similar problem, ending up with the right answer, but with little or no understanding of how or why the method works. Relational understanding, on the other hand, refers to a much wider level of comprehension; the pupil can see why a method works, has some gras