טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentKontorovich Igor
SubjectHow do Experts Pose Problems for Mathematics Competitions?
DepartmentDepartment of Education in Science and Technology
Supervisor Professor Boris Koichu
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

Mathematics education research has accumulated an extended body of knowledge on problem posing by school children and mathematics teachers who, as a rule, are novices in problem posing. At the same time, empirical evidence on problem posing conducted by experts, such as professional mathematicians and mathematics educators who create new problems for various mathematical and educational needs, barely exists. This is in spite of the established practice of using research on experts as a source of ideas for fostering mathematical competences in novices. The purpose of the present research was to identify and characterise mental mechanisms that are involved in expert problem posing for mathematics competitions. 

Twenty six experts from nine countries participated in the research. The research consisted of three studies: a preliminary study, a theoretical study and a central empirical study. The preliminary study was focused on the identification of the experts' beliefs, motivation and values regarding mathematics, competition movement and problem posing. The theoretical and the central studies were focused on the experts' problem-posing practices.

The findings show that, when posing problems, the experts are driven by the desire to fulfil their pedagogical agendas and satisfy their internal needs: an intellectual need for enriching their mathematical knowledge and a socio-psychological need for belonging to the mathematics competition movement, for recognition and appreciation. Four mental mechanisms were identified in expert problem posing: (1) the mechanism of detecting problem-posing triggers; (2) the mechanism of inquiry and problem solving, with particular attention to mathematical phenomena that may underlie new problems; (3) the mechanism of problem formulating, which is responsible for turning the phenomenon into a problem; (4) the mechanism of control. Three scenarios of creating competition problems were distinguished: a problem can be created as a result of developing a problem-posing trigger intentionally chosen by the expert; a problem can be created as a by-product of expert's activity, which is not explicitly related to mathematics competitions; a problem can be created in response to the request to offer a problem of prescribed specifications. The experts prefer the first two scenarios because those enable them to gain new mathematical knowledge and lead to the problems of a better quality.

A theoretical contribution of the research is in the identification of ways of thinking and the characterization of practice of expert problem posers. The research findings have implications for promoting problem-posing competences in novice learners of mathematics.