Knowledge building in mathematics: Supporting collaborative learning in pattern problems

Abstract

While it has been suggested that patterning activities support early algebra learning, it is widely acknowledged that the shift from perceiving patterns to understanding algebraic functions-and correspondingly, from reporting empirical patterns to providing explanations-is difficult. This paper reports on the collaborations of grade 4 students (n= 68) from three classrooms in diverse urban settings, connected through a knowledgebuilding environment (Knowledge Forum), when solving mathematical generalizing problems as part of an early algebra research project. The purpose of this study was to investigate the underlying principles of idea improvement and epistemic agency and the potential of knowledge building-as supported by Knowledge Forum-to support student work. Our analyses of student-generated collaborative workspaces revealed that students were able to find multiple rules for challenging problems and revise their own conjectures regarding those rules. Furthermore, the discourse was sustained over 8 weeks and students were able to find similarities across problem types without the support of teachers or researchers, suggesting that these grade-4 students had developed a disposition for evidence use and justification that eludes much older students.

Towards a Methodology for the Characterization of Teachers’ Didactic-Mathematical Knowledge

“…They suggest working with concrete materials but at the same time, state that if the students do not have background knowledge of algebra, they will face great difficulties. In other words, the teachers do not perceive or do not believe that the work with patterns is a powerful vehicle for the comprehension of the relations of dependency between amounts that underlie mathematical functions as a concrete and transparent way for young students to start working with notions of abstraction and generalization (Moss & Beatty, 2006). The teachers manifest a conflict between their beliefs about secondary algebra, and elementary algebraic reasoning.…”

“…Patterns offer both, a powerful vehicle for the comprehension of the relations of dependency among quantities that lie beneath mathematical functions, as well as a concrete and transparent way for young students to start working with notions of abstraction and generalization (Moss & Beatty, 2006). In general, the importance of patterns in mathematics has been pointed out by many authors.…”

“…The activity proposed was taken from Moss & Beatty (2006), and falls within a functional perspective for the development of algebraic reasoning. The teachers must be able to identify patterns, find the underlying regularity and expressing it as an explicit function or rule through a generalization process.…”

Translating Euclid: Designing a Human-Centered Mathematics

“…New theories of mathematical cognition (Bransford, Brown & Cocking, 1999;Brown & Campione, 1994;Greeno & Goldman, 1998;Hall & Stevens, 1995;Lakatos, 1976;Lemke, 1993;Livingston, 1999) and math education (Boaler, 2008;Cobb, Yackel & McClain, 2000;Lockhart, 2009;Moss & Beatty, 2006), in particular, stress collaborative knowledge building (Bereiter, 2002;Scardamalia & Bereiter, 1996;Schwarz, 1997), problem-based learning (Barrows, 1994;Koschmann, Glenn & Conlee, 1997), dialogicality (Wegerif, 2007), argumentation (Andriessen, Baker & Suthers, 2003), accountable talk (Michaels, O'Connor & Resnick, 2008), group cognition (Stahl, 2006) and engagement in math discourse (Sfard, 2008;Stahl, 2008a). These approaches place the focus on problem solving, problem posing, exploration of alternative strategies, inter-animation of perspectives, verbal articulation, argumentation, deductive reasoning and heuristics as features of significant math discourse (Maher, Powell & Uptegrove, 2010;Powell, Francisco & Maher, 2003;Powell & López, 1989).…”

Book review: Exploring thinking as communicating in CSCL

“…They consider patterns of several configurations of blocks that grow step by step according to a rule (see also Moss and Beatty 2006). They develop recursive and quadratic expressions for the count of blocks and number of unduplicated sides in the patterns.…”