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Purpose The purpose of this paper is to summarize some of the key findings and approaches used in documenting the authors’ longitudinal studies of mathematical learning and understanding. In particular, it focuses on “folding back,” a theoretical construct originally developed by Susan Pirie and Tom Kieren, to show how, over the last two decades, the authors have taken up, built-upon, and elaborated this construct in relation to Pirie and Kieren’s wider theorizing and in relation to classroom practice. Design/methodology/approach The paper documents the various methodologies and methods the authors have used to elaborate theory and contribute to extending teaching practice in a number of related research studies. Findings This paper describes the role of folding back in the growth of students’ mathematical understanding, initially at the level of the individual, more recently at that of the collective – and currently with a specific consideration of the role of the teacher. It notes that the longitudinal nature of the work has allowed it to respond to shifting perspectives in the field of mathematics education and to become a more nuanced and powerful analytic and teaching tool. Originality/value The paper discusses the significance of a longitudinal, shared program of research, deeply rooted in mathematics classrooms, that builds theory systematically and over an extended period of time.

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