Structuring Complexity of Mathematical Problems: Drawing Connections Between Stepped Tasks and Problem Posing Through Investigations

Abstract

Focusing on geometry proof problems, we describe two types of sets of mathematical problems, each of which includes problems of different levels of complexity. The sets of the first type embrace spaces of problems posed through investigations (PPI) in a Dynamic Geometry environment. These PPI sets can be generated by experts and novices in problem-solving and problem posing, by individuals or by groups of individuals. Each of the PPI sets is connected to a particular geometric figure given in the PPI task and other figures are designed by enriching the given figure through auxiliary constructions performed during PPI. Problems in this type of set are usually semi-structured from the viewpoint of mathematical complexity and, as we suggest in this paper, serve types of sets, suggest some criteria as a rich source for designing Stepped Tasks – the second type of set illustrated in this paper. Stepped Tasks are designed systematically in a top-down structure, from the most complex problem (called the Target problem) to less understanding of mathematical structure to be two core complex problems that can help students to solve the target problem. Designing Stepped Tasks requires understanding of the problem’s structure; this is not simple and usually entails deep mathematical knowledge and problem-solving expertise. In this paper, we illustrate the two types of sets, suggest some criteria for the evaluation of task complexity, and draw connections between these two types of sets of tasks, explaining how Stepped Tasks can be designed using PPI Sets.