Abstract
This study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through-investigations (PPI) in a dynamic geometry environment (DGE). The first type of expertise—MO—involved being a candidate or a member of the Israeli International Mathematical Olympiad team. The second type—MM—was comprised of mathematics majors who excelled in university mathematics. We conducted individual interviews with eight MO participants who were asked to perform PPI in geometry, without previous experience in performing a task of this kind. Eleven MMs tackled the same PPI task during a mathematics test at the end of a 52-h course that integrated PPI. To characterize connections between creativity and expertise, we analyzed participants’ performance on the PPI tasks according to proof skills (i.e., auxiliary constructions, the complexity of posed tasks, and correctness of their proofs) and creativity components (i.e., fluency, flexibility and originality of the discovered properties). Our findings demonstrate significant differences between PPI by MO participants and by MM participants as reflected in the more creative performance and more successful proving processes demonstrated by MO participants. We argue that problem posing and problem solving are inseparable when MO experts are engaged in PPI.
Highlights
The research presented in this paper was motivated by several observations concerning research associated with mathematical creativity, expertise, problem solving and problem posing and the relationships between them.The first observation concerns research on expertise in mathematics
While expertise is commonly addressed as superior performance in a particular domain, in the research literature the notion of mathematical expertise acquires a broad range of meanings, as expressed in different groups of target populations varying from school students who excel, to professional mathematicians
PPI tasks employed in this study allow both manipulation of givens and goals and this activity is supported by the use of dynamic geometry environment (DGE), which is naturally associated with investigations in geometry (Yerushalmy et al 1990)
Summary
The research presented in this paper was motivated by several observations concerning research associated with mathematical creativity, expertise, problem solving and problem posing and the relationships between them. PPI tasks are open from the start and from the end (Leikin 2019), since solvers are encouraged to choose what they examine and how, and the outcomes usually constitute an individual space of posed problems, which are based on the discovered properties. These collections differ among different individuals in terms of the number, types and complexity of the posed problems. The openness of the PPI tasks and their complexity determines the power of these tasks as tools for the investigation of creativity and problem-solving expertise
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