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Constantinos Xenofontos describes working with intending teachers
For the autumn semester 2014, I developed, and taught, a module for undergraduate students reading for a bachelor's degree in primary education, entitled "Mathematical Problem Solving in the Elementary SchoolBeing a young lecturer at a university in Cyprus, I was delighted that I was given the opportunity to decide on the content of the module, prepare it from scratch, and apply ideas from my recently completed PhD thesis (Xenofontos, 2011). Drawn from my previous work in the field (see Andrews and Xenofontos, 2014; Xenofontos and Andrews, 2012, 2014), the module aimed to enhance students' competence and selfefficacy beliefs both as problem solvers and as future teachers of problem-solving. The sessions combined lectures and practical collaborative activities, and students were asked to keep a reflective diary after each meeting. For their diary entries, students had to position themselves in relation to the content of the lesson, and comment on their feelings and knowledge.
Central to the module's philosophy was the idea that any mathematical task can be a problem for some individuals, and at the same time, an exercise for others. In other words, "being a 'problem is not a property inherent in a mathematical task. It is a particular relationship between the individual and the task that makes it a problem for that person" (Schoenfeld, 1985: 74). Following the ideas of Borasi (1986) and Blum and Niss (1991), the students were exposed to both purely mathematical and applied problems, for which they worked collaboratively in small groups of three or four to resolve them. Here, I present four of the geometry problems I used, and discuss the students' experiences, as depicted in their reflective diaries. A common feature of these tasks is that they do not involve advanced mathematics; in fact, the mathematics hidden behind these is rather elementary, like simple area formulas. Yet, finding [a] solutions] require[s] advanced thinking in mathematics (for a more detailed discussion about the various perspectives on advanced mathematical thinking, see Harel and Sowder, 2005, and Tall, 1991). Although Cypriot pupils are introduced to Euclidean geometry during high-school, (some study the topic more intensively than others, depending on whether they've chosen to specialise in mathematics) tasks of this type are rarely...