Openness in real-world mathematical problems: a systematic literature review

Völlinger, Nils; Schukajlow, Stanislaw; Verschaffel, Lieven; Krawitz, Janina

Abstract

Open real-world problems play an important role in the teaching and learning of mathematics. Research on open problems comprises a variety of research methodologies, theoretical conceptualizations, and constructs. This systematic literature review aims to synthesize and structure these various approaches by examining the current state of research on open problems in mathematics education. With a full-text analysis of 67 peer-reviewed journal articles, we investigated the descriptive characteristics of studies in this field, the conceptual frameworks used to define the openness of real world mathematical problems, and the structure of problems in terms of their initial, intermediate, and goal states. Additionally, we analyzed the constructs employed in the studies. We identified four sets of conceptual frameworks: (1) psychological frameworks of problem-solving, (2) mathematical problem-solving frameworks, (3) mathematical modelling frameworks, and (4) suspension of sense-making frameworks. The findings demonstrated that most studies have focused on problems with open initial and intermediate states, whereas problems with open goal states have received less attention. Furthermore, cognitive constructs were addressed more frequently than affective-motivational ones. The findings of this review reveal the need for a theoretically grounded conceptualization of the notion of openness in empirical studies. Important future research directions include investigating problems with open goal states, analyzing instruction to improve student performance in solving open problems, and addressing affective-motivational constructs in studies with open problems. This review contributes to a deeper understanding of openness as an important task characteristic and provides a comprehensive synthesis of the available research on open real-world mathematical problems.