Braithwaite, D. W., & Goldstone, R. L. (2011). Effects of grounded and formal representations on combinatorics learning. *Proceedings of the Thirty-Third Annual Conference of the Cognitive Science Society*. (pp. 3431-3436). Boston, Massachusetts: Cognitive Science Society.

Two experiments examined the differential effects of ground- ed and formal representations on learning of mathematics. Both involved combinatorics, using outcome listing and com- binatorics formulas as examples of grounded and formal rep- resentations, respectively. Experiment 1 compared perfor- mance on near and far transfer problems following instruc- tions involving listing or formulas. Instruction in formulas led to more near transfer, while far transfer performance did not differ by condition. Experiment 2 compared performance fol- lowing four types of instruction: listing only, formulas only, listing fading (listing followed by formulas), and listing intro- duction (formulas followed by listing). The listing fading condition led to performance on par with the formulas only condition, and for near transfer problems, significantly higher than the listing introduction and pure listing conditions. The results support the inclusion of grounded representations in combinatorics instruction, and suggest that such representa- tions should precede rather than follow formal representations in the instructional sequence.