What's in Dots-Per-Box? Conceptual Bootstrapping With Stripped-Down Visual Analogs

Abstract

This study explores student ability to differentiate and make mappings between quantities in different domains before and after a relevant curricular experience. Twenty 7th-grade students were given pre- and postinterviews that probed their understanding of 3 intensive and extensive quantity pairs (dot crowdedness and total dots; sweetness and total sugar; density and total weight) and their ability to use the former quantities to model the quantities of the latter two domains. The 2-week curriculum intervention focused on using an integrated dots-per-box model to represent various everyday quantities like sweetness of Kool-Aid. Prior to instruction there were disparities in student understanding of the 3 domains, and few students could make relevant mappings between the quantities in the model and the domains of sweetness and density. Instruction led to improvement in student differentiation and modeling of sweetness, a quantity directly considered in instruction. There was no change in student differentiation of density, a quantity not considered in instruction, although some students improved in their modeling of density. These findings are interpreted as showing that conceptual bootstrapping involves a multistepped mapping process that is facilitated through classroom dialogue, problem solving, and instruction.