Problem-posing triggers or where do mathematics competition problems come from?
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Problem-posing triggers or where do mathematics competition problems come from? Igor’ Kontorovich 1
# The Author(s) 2020
Abstract
Spurred by Kilpatrick’s (1987) “Where do good problems come from?”, this study explores problem-posing triggers of experienced problem posers for mathematics competitions. Triggers are conceived as instances of noticing, where an impulse draws a poser’s attention and “triggers off” a mathematical re-action, one of the outcomes of which is a creation of a problem that gets accepted to some mathematics competition. The data were collected from 26 problem posers residing in nine countries, and who had experience in creating problems for national, regional, and international competitions. Three types of triggers emerged from the analysis: (i) Instances where the participants extracted mathematical phenomena from activities that are replete with modern elementary mathematics. These triggers were described in emotionally loaded terms that resonated with appreciation, surprise, challenge, and feeling of innovation. (ii) Cases where the participants abstracted mathematical phenomena from common everyday-life tasks in which mathematical optimization was beneficial. These situations emphasized participants’ desire to compensate for self-disappointment with an initially made decision and the value of finding a better alternative. (iii) Situations where the participants were asked to pose a problem ‘here and now’. The posers were unanimous in their dislike of such situations. They elaborated on how difficult it is to pose ‘here and now’ and highlighted the low quality of the resulting problems. The findings are situated in the body of knowledge on expert problem posing and educational literature on school students and teachers. Keywords Problem-posing triggers . Discipline of noticing . Expert problem posing . Mathematics competitions
* Igor’ Kontorovich [email protected]
1
The University of Auckland, Auckland, New Zealand
Kontorovich, I.
1 Introduction Problem posing has been drawing the keen attention of the mathematics education community for almost four decades (e.g., Brown & Walter, 1983; Ellerton, 1986; Walter, 1978). In his seminal paper entitled “Problem formulating: Where do good problems come from?”, Kilpatrick (1987) presented an ambitious agenda for providing opportunities for school students to create their own mathematical problems. Since then, the potential of problem posing for promoting problem solving, advanced mathematical thinking, conceptual understanding, creativity, and positive attitudes towards mathematics has been extensively explored (see Cai, Hwang, Jiang, & Silber, 2015 and Singer, Ellerton, & Cai, 2013 for comprehensive reviews). However, Singer et al. (2013) note that despite the accumulated body of valuable insights on the development of problem posing in students and teachers (who are usually novices to this activity), the issue of insufficient quality of their problems remains contentious and requires further investigation. In a similar vein, Cai
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