Over the last 30 years, numerical estimation has been largely studied. Recently, Castronovo and Seron (2007) proposed the bi-directional mapping hypothesis in order to account for the finding that dependent on the type of estimation task (perception vs. production of numerosities), reverse patterns of performance are found (i.e., under- and over-estimation, respectively). Here, we further investigated this hypothesis by submitting adult participants to three types of numerical estimation task: (1) a perception task, in which participants had to estimate the numerosity of a non-symbolic collection; (2) a production task, in which participants had to approximately produce the numerosity of a symbolic numerical input; and (3) a reproduction task, in which participants had to reproduce the numerosity of a non-symbolic numerical input. Our results gave further support to the finding that different patterns of performance are found according to the type of estimation task: (1) under-estimation in the perception task; (2) over-estimation in the production task; and (3) accurate estimation in the reproduction task. Moreover, correlation analyses revealed that the more a participant under-estimated in the perception task, the more he/she over-estimated in the production task. We discussed these empirical data by showing how they can be accounted by the bi-directional mapping hypothesis ( Castronovo & Seron, 2007 ).
Numerical estimation of reachable and controllability sets for a two-level open quantum system driven by coherent and incoherent controls