Individual differences in the effect of relevant concreteness on learning and transfer of a mathematical concept

• Concrete-relevant features supported middle-school students' knowledge transfer to a mathematical concept.
• The effect of relevant concrete features on transfer depended on student age and reasoning ability.
• Younger students (sixth-graders) benefited most from concrete-relevant features on far-transfer performance.

The concreteness of training materials influences learning and—perhaps more importantly—transfer. Building on prior research finding abstract representations best facilitated transfer to a game task, we conducted a similar study using training figures varying in concreteness but directly assessed transfer to modular arithmetic problems. Training figures: (a) were purely abstract, (b) were abstract but with features relevant to the transfer task, or (c) included additional concrete-relevant features. We hypothesized that concreteness—or number of relevant features—would be positively correlated with learning and transfer—especially among younger and/or lower-ability students. Although there was no overall difference in initial learning, the concrete-relevant and abstract-relevant features independently facilitated near-transfer, where concrete-relevant features supported lower-reasoning students. For far-transfer, eighth-graders benefited from the abstract-relevant features, whereas sixth-graders required additional concrete-relevant features. These findings suggest that concreteness interacts with learner and task characteristics to produce learning and transfer outcomes.