Folding back and growing mathematical understanding: a longitudinal study of learning

Abstract

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.

Developing theoretical coherence in teaching and learning: case of neuroscience-framed learning study

“…This is in consonance with Runesson’s (2016) study that explored the use of an alternative theoretical framework in LS. The current findings suggest that contemporary neuroscience information could support the goals of LS to support teachers’ pedagogical practices and to increase student learning (Martin and Towers, 2016; Pang and Lo, 2012). Mirroring earlier learning studies (Holmqvist, 2011; Vikström, 2014), teachers in this study increased their sensitivity to students’ learning through focusing on what and how students learn, as they developed sound, theory-framed pedagogy.…”

“…Baillie et al , 2013; Bussey et al , 2013 ) and LS must be open to other theories to broaden its scope and support its further development (cf. Martin and Towers, 2016).…”

“…This is because translating theoretical principles into educational contexts depends on engaging teachers in reflective practice that is open to testing, and developing learning and pedagogical theories (Elliott, 2015; Kullberg et al , 2016). Martin and Towers’ (2016) recent study further demonstrated how pedagogical principles of “folding back,” as an alternative theoretical framework to variation theory, could be fruitfully explored for its effectiveness as a pedagogical tool and thus alludes to its potential to widen teachers’ repertoire of theoretical lenses they could use to examine student learning.…”

“…Baillie et al, 2013;Bussey et al, 2013) and LS must be open to other theories to broaden its scope and support its further development (cf. Martin and Towers, 2016). Learning and pedagogical theories are relevant when engaging in PD because theories provide the lenses through which teachers understand various aspects of their professional practice (Tan, 2014b;Korthagen, 2010).…”

“…The benefits of teachers using theoretical knowledge to underpin their pedagogies are being acknowledged increasingly in learning study (LS) literature. This praxis serves to bridge the theory-practice gap, and to promote teacher and student learning (Martin and Towers, 2016; Pang and Lo, 2012). Recently, Runesson (2016) made explicit the need to separate the theoretical framework applied (predominantly, variation theory, Lo, 2012) from the LS approach.…”