Looking at Figure 1a, I say duck, you say rabbit, so let’s call the whole thing off, because it can’t be both. Looking at Figure 1b, though, I say two rows of three X’s, you say three columns of two X’s, so let’s not call the whole thing off, because our disagreement could be reconciled in the form of a mutually valuable insight into the commutative property of multiplication, where the two perceptual orientations are complementary construals of six X’s (ie, 2× 3= 3× 2). Abrahamson and Wilensky (2007) used this example to introduce an educational design framework–learning axes and bridging tools–centered on fostering conceptual insight through setting up students to experience then reconcile ambiguous perceptual constructions of instructional materials. Engaging with these materials, students are to experience different meanings that are each valid in their own right yet initially appear incompatible with each other. The learning goal requires finding a new way of thinking that would accommodate or resolve the conflict, whereby the alternative perceptions become complementary or dialectic rather than contradictory. The educational design principle of learning through reconciling competing perceptual constructions has been applied also to the case of ratio and proportion (eg, Abrahamson, Lee, Negrete & Gutiérrez, 2014). The objective of the current article is to investigate the application of the framework to geometry, in particular to designing activities where students engage in task-oriented embodied investigations into voluminous objects. The idea is that students build these objects themselves, moving from 2D images to 3D structures.
