Unraveling the small tie problem mystery: Size effects from finger counting to mental strategies in addition

Determining how children solve arithmetic problems when they stop using their fingers is a real challenge. To take it up, the evolution of problem-size effects for tie and non-tie problems was observed when 6-year-olds (N = 65) shift from finger counting to mental strategies. These observations revealed that the problem-size effect remained the same for non-tie problems, whereas it drastically decreased for tie problems. Moreover, the solving strategy for tie problems switched directly from the representation of both operands on fingers to retrieval without transition through the representation of only one operand on fingers. This direct switch could be made possible by the relative ease to commit symmetrical representations to memory (in the case of tie problems) rather than non-symmetrical ones (in the case of non-tie problems). This would explain why, early during development, small tie problems are solved quickly and present null or negligible size effects. All in all, our results and interpretations provide an answer to the long-standing question as to why tie problems have a special cognitive status. Our results also nuance the classical description of the developmental pattern reported in all textbook chapters devoted to numerical cognition according to which a finger strategy where only one operand is represented on fingers constitutes a developmental stage between the representation of two operands on fingers and retrieval. We demonstrate here that it is true only for non-tie problems.