Multiplication as Coordinated Measurement

• In Event: 45.082 - Multiplicative Models of Situations Across Disciplines

Abstract

Objectives: A significant body of research demonstrates that many topics related to multiplication pose persistent challenges for students, teachers, and theorists. Much of this research has examined the limitations of introducing students to multiplication as an abbreviated form of repeated addition (e.g., Greer 1992; Thompson & Saldanha, 2003), and various researchers have sought to specify what makes multiplication distinct from addition (Confrey, 1994; Davydov, 1992; Schwartz, 1988; Steffe, 1988, 1994; Thompson & Saldanha, 2003). The present study examines the affordances of an approach that emphasizes both whole numbers and fractions as outcomes of measurement and that characterizes multiplication as measuring products simultaneously with units and groups of units. This coordinated measurement perspective extends a measurement meaning for multiplication previously articulated by Davydov for whole numbers.

Theoretical Framework: Our theoretical framework coordinates mathematical analysis with a cognitive perspective. Elsewhere we have demonstrated how a coordinated measurement meaning for multiplication can provided a common underlying structure that supports a coherent view of diverse situations where one typically multiplies or divides, with either whole numbers or fractions. Our cognitive perspective is informed by the Knowledge-in-Pieces epistemological perspective (e.g., diSessa, 1993, 2006), which likens knowledge systems to ecologies consisting of many fine-grained elements related in complex ways. We see such a complex ecology supporting when and where people perceive and interpret situations in terms of coordinated measurement.

Methods & Data Sources: As part of a larger study, following an IRB-approved protocol, we recruited 6 future middle grades mathematics teachers from a content course that emphasized the coordinated measurement meaning for multiplication and length-based representations of quantities (e.g., number lines and strip diagrams). We conducted a series of 6 one-on-one, video-recorded interviews with each participant. During the interviews, participants solved paper-and-pencil tasks involving a range of topics related to multiplication, including whole-number multiplication and division, fraction multiplication and division, proportional relationships, and linear equations. We transcribed the interviews verbatim and analyzed talk, gesture, and inscription line-by-line to infer how each future teacher reasoned.

Results: We focus on 3 of the 6 participants, who collectively provided contrasting cases. The first future teacher provided evidence that she attended consistently to coordinated measurement and to a psychological operation called units coordination (e.g., Steffe, 1988, 1994) as she demonstrated proficient reasoning about a wide range of tasks. The second provided evidence that she attended consistently to measurement but inconsistently to units coordination and, as a consequence, encountered difficulties on some tasks. The third provided evidence that she attended inconsistently to measurement but consistently to coordinating units and, as a consequence, encountered different difficulties than the second participant. An ecology of cognitive resources appeared to support when and where the future teachers attended to measurement and to units coordination.

Significance: Steffe (1988, 1994), among others, has argued that units coordination is the critical constraint on students’ reasoning with topics related to multiplication. Our results suggest that attention to measurement is a separate, also consequential, capacity supported by a complex substrate of knowledge.

Authors

• Andrew G. Izsak, Tufts University

• Sybilla Beckmann, University of Georgia