Rosen, D. M., Palatnik, A., & Abrahamson, D. (2018). A better story: An embodied-design argument for generic manipulatives. 

In N. Calder, N. Sinclair, & K. Larkin (Eds.), Using mobile technologies in the learning of mathematics (pp. 189-211). New York: Springer.

ABSTRACT: Mathematics education practitioners and researchers have long debated best pedagogical practices for introducing to students new concepts. We worked with interactive technology of our own design to evaluate an instructional methodology whereby students: (1) first learn to move physically in a new way by solving a coordination problem of moving two virtual objects at the same time, one in each hand; and only then (2) ground these new movements in informal narratives about the objects; as well as (3) generalize these movements as formal mathematical rules. We compared students’ learning gains under two conditions of this methodology, where the virtual objects were either generic (non-representational, not signifying specific contexts, e.g., a circle) or situated (representational, signifying specific contexts, e.g., a hot-air balloon). We summarize findings from analyzing the behaviors of 25 Grade 4 – 6 students who participated individually in tutorial trials with one of the authors. The situated objects gave rise to richer stories than the generic objects, presumably because the students could bring to bear their everyday knowledge of these objects’ properties, scenarios, and typical behaviors. However, in so doing, the students treated the objects’ movements only as framed by those particular stories rather than considering other possible interpretations of those movements, even when those other interpretations would have advanced the learning process, such as by creating productive challenges. We argue that these findings can be explained in light of theoretical models that conceptualize knowledge as emerging from goal-oriented sensorimotor interaction with objects in the environment. We caution that designers and teachers should be aware of the double-edged sword of rich situativity: Familiar objects are perhaps more engaging but can also limit the scope of learning. We advocate for our instructional methodology of entering mathematical concepts through the action level.