Folding back, thickening and mathematical met-befores

Tracking Grade 10 learners’ geometric reasoning through folding back

“…Working with rudimentary facts is not in line with the notion of folding back which provides learners with an opportunity to build a connected understanding of mathematical concepts through returning and thickening their earlier existing images (e.g. Martin & Towers, 2016)…”

“…A teaching experiment methodology research design (Steffe & Thompson, 2000), which researchers use to experience learners' mathematical learning and reasoning, was used to track whether folding back (Martin, 2008;Martin & Towers, 2016;Pirie & Kieren, 1994) supported Grade 10 learners' development of geometric reasoning. Using data from a master's study (Mabotja, 2017), we targeted data from two groups of learners from a class of 54 in which learners grouped themselves: the first group consisted of two members (John and Koena) and the second group consisted of three members (Lesiba, Lebogang and Sipho).…”

“…Therefore it is a cognitive shift that happens when previously held understanding, inner layer understanding, acts are informed by outer layer knowing that ultimately leads to further outer layer understanding (Pirie & Martin, 2000). There are few studies that show that folding back extends learners' existing understanding of mathematics concepts (Delgado, Codes, Monterrubio, & Astudillo, 2014;Martin & Towers, 2016;Wright, 2014). Using Tall's (2008) argument, learners' geometric development builds on experiences that they have met before.…”

“…Using Tall's (2008) argument, learners' geometric development builds on experiences that they have met before. Previous experiences, referred to as met-befores, form connections in the brain and that would affect how learners make sense of new mathematical situations that are either consistent (Martin & Towers, 2016) or inconsistent (Tall, 2008). This article uses parts of qualitative data from the first author's study that focused on exploring Pirie and Kieren's process of folding back, revisiting previously held understandings of a concept for its extended understanding in order to solve the problem of enhancing Grade 10 learners' reasoning in geometry.…”

Mathematical Reasoning Required when Students Seek the Original Graph from a Derivative Graph

“…They take a long time to understand their meaning and reflect on learned concepts. The notion of met-before is crucial for this condition when new learning is affected by the learner's previous experiences (Martin & Towers, 2016;Mcgowen & Tall, 2013). A met-before probably affect the mental structure that we have now resulting from experiences we have met before, for instance, making a relation between continuity and differentiability.…”