Collective Mathematical Understanding as Improvisation

Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie–Kieren Theory

“…see Kieren, Mason, Davis, & Pirie, 1993;Kieren et al, 1999;Kieren, Pirie, & Reid, 1994;Manu & Martin, 2004;Martin, 1999;Martin, Pirie, & Kieren, 1996;Pirie & Kieren, 1989, 1992a, 1992bPirie and Martin, 2000;Thom, 2003;Towers, 1998;Towers, 2001aTowers, , 2001b other researchers have sought to apply the theory to new fields of inquiry. These include teacher preparation (Borgen, 2006;Glanfield, 2004) and the analysis of teachers' growth in understanding school mathematics (Berenson, 2002;Berenson, Cavey, Clark, & Staley, 2001;Borgen, 2006;Cavey & Berenson, 2005;McDougall & Nason, 2005), teacher actions (Warner & Schorr, 2004;Warner, Schorr, Samuels, & Gearhart, 2005), the growth of mathematical understanding in the context of bilingualism (Manu, 2005a(Manu, , 2005bManu & Martin, 2004), the nature of collective mathematical understanding (Droujkova, Berenson, Slaten, & Tombes, 2005;Martin, Towers, & Pirie, 2006;Thom, 2004;Towers & Davis, 2002;, and workplace training (Martin & LaCroix, in press;Martin, LaCroix, & Fownes, 2006). Pirie and Kieren (1991) defined folding back as A person functioning at an outer level of understanding when challenged may invoke or fold back to inner, perhaps more specific local or intuitive understandings.…”

Cognitive Activities in Solving Mathematical Tasks: The role of a Cognitive Obstacle

“…Collaborative learning is realized through the interaction and co-action in such options of solving tasks (Francisco, 2013;Martin et al, 2006), which contributes to a more complete understanding of the content which must be mastered. These capacities become mutually complementary in situations of cooperative mathematical tasks solving, that requires a higher level of cognitive effort.…”

A framework for analyzing the collaborative construction of arguments and its interplay with agency

“…Martin, Towers and Pirie (2006) describe collective mathematical understanding as the kind of learning and understanding that takes place when a group of students work together on a mathematical task. Martin, Towers, and Pirie state that collective understanding is "a phenomenon that is bound in the social context of the learning environment, and as such cannot be described by merely attending to the actions of individual learners" (p. 150).…”

“…They posit that when students are provided opportunities to build collective mathematical understanding, co-acting occurs. Co-acting is a process through which an individual's mathematical ideas and actions are adopted, built upon, and internalized by others, thereby becoming shared understandings rather than being limited to the individual (Martin et al, 2006). Through the process of working jointly on problem solving, students share ideas and ways of solving problems; thus individual understanding becomes shared.…”