Fostering embodied coherence: A study of the relationship between learners’ physical actions and mathematical cognition

When students first learn properties of number (e.g., order, magnitude, and basic arithmetic), they do so with respect to natural numbers. Learning fractions, however, necessitates recalibration of previously developed integer schemas (Gelman & Williams, 1998; Fuson, 2009). For example, when one multiplies two positive integers, their product is greater than either of them, but when one multiplies proper fractions, the reverse is true. Faced with such contradictions many students discard their emerging intuitions about number and instead rely on memorizing rules and procedural algorithms (Freudenthal, 1985). Consequently, they fail to develop a grounded, connected understanding of fractions and, perhaps worse yet, become discouraged from or unable to engage with more advanced mathematics (Ma, 2000; NCTM, 2000; Wilensky, 1993).To date, pedagogical efforts to improve students’ understanding of fractions complement the introduction of fraction notation and algorithms with activities in which students are guided to create or transform objects such as diagrammatic figures or blocks in order to see, appreciate, and articulate relationships between unit wholes and constituent unit parts. The rationale is to ground otherwise rote operations on symbols in direct actions on objects. While such experiences have proven effective for many learners, national assessments suggest that more work is still needed.My dissertation adopts an embodied cognition perspective (Barsalou, 1999; Glenberg, 1997; Lakoff & Nunez, 2000) in order to examine the challenges and opportunities that students encounter when participating in artifact-mediated fraction instruction. In adopting this theoretical framework, I attempt to identify possible tensions between: (a) learners’ physically embodied, multi-modal, goal-oriented actions; (b) meanings that the learners assign to novel semiotic forms the instructor introduces as symbolizing these actions (Abrahamson, 2009); and (c) the conceptual foundations that we attempt to build learners’ initial understanding of fractions upon.Taking a design-based research approach (Brown, 1992), I developed Water Works, a novel activity sequence involving the iteration of measured volumes of water into a vessel. In its pedagogical rationale, Water Works bears some similarity to the Elkonin–Davydov approach to teaching natural numbers as well as curricular designs favoring the number line (e.g., Carraher, 1993; Kalchman, Moss, & Case, 2000). The design begins by presenting students with a “whole” (e.g., one-cup) and labeled unit parts (e.g., ½, ¼ cups, each marked with a corresponding symbol). Students are guided to engage in a set of activities of filling the one-cup. They are to make sense of part-to-whole fractional relationships and fraction arithmetic in terms of observable and reversible physical actions (Piaget, 1971).Vitally, Water Works has been designed to foster coherence between learners’ pre-existing schemas for whole number and emerging understanding of fraction arithmetic by preserving one-to-one correspondence between learners’ physical actions and the resulting mathematical outcomes. For example, the arithmetic operation 4 x ¼ is enacted as four physical iterations of a ¼ cup of water. The resulting product—both multiplicative and literal—is 1 cup of water that can visually be perceived as 4 times greater in magnitude than the original ¼ unit measure. Over the course of seven months, students from two 4th grade general-education public-school classrooms (n = 40) participated in a scripted sequence of individually administered problem-solving tasks and written-assessments. Qualitative data from videotaped clinical interviews and quantitative comparisons of written assessments are used to develop a model of artifact-mediated cognitive interaction, and implications are drawn for scaling up this design