Three on-going studies on (authentic) problem posing for different educational needs

Posing an original mathematical problem is frequently regarded as an instantiation of one's mathematical creativity. However, one of the most robust observations from the research literature on mathematical problem posing is that diagnostic and learning opportunities associated with this activity are, as a rule, much more impressive than problems produced as a direct result of the effort. This is hardly surprising, given that the main research effort has been put into studying students and teachers, who did not have substantial problem-posing experience, but posed problems upon request in different instructional settings. There are also a few reflective publications by mathematics practitioners and educators on what they do when posing mathematical problems. These reflections are helpful for identifying various sources for posing new problems, as well as for mapping expert problem-posing strategies. Still, reflection-based publications, as fruitful as they are, cannot be a substitute for systematic observation-based studies aimed at revealing the nature of problem posing by those who successfully and systematically invent new problems for different educational needs. In particular, surprisingly little empirically-based knowledge exists about posing problems for the actual use at mathematical competitions, exams or mathematics classrooms. This research lacuna may be considered as one of the reasons for which a comprehensive theoretical framework for investigating problem posing as an authentic mathematical activity is still not established (e.g., Christou et al., 2005; Singer et al., 2009). This essay outlines some of the effort of the Technion mathematics education research group towards constructing such a framework. An additional goal of this essay is to invite interested colleagues from different countries to collaborate on pursuing the research questions presented below.