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Numerals as triggers of System 1 and System 2 in the ‘bat and ball’ problem

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Abstract

The ‘bat and ball’ is one of the problems most frequently employed as a testbed for research on the dual-system hypothesis of reasoning. Frederick (J Econ Perspect 19:25–42, 2005) is the first to envisage the possibility that different numerical arrangements of the ‘bat and ball’ problem could lead to different dynamics of activation of the dual-system, and so to different performances of subjects in task accomplishment. This possibility has triggered a strand of research oriented to accomplish ‘sensitivity analyses’ of the ‘bat and ball’ problem. The scope of this paper is to test experimentally the specific hypothesis that numerals are responsible for the selective activation of the two systems of reasoning in this task. In particular, we argue that their role goes beyond and cannot be reduced to that of numbers conceived as magnitudes. To test our hypothesis, we devise an experimental setting in which the role of numbers (as magnitudes) is rendered irrelevant. We find experimental results consistent with our hypothesis. We further provide a link between the literature on mathematical problem-solving and that on mathematical cognition research, in particular that branch labeled embodied mathematical cognition.

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Notes

  1. We avoid, in our setting, to provide versions of the task in which magnitudes were identical (e.g. relying on fractions vs decimals) because the experimental outcome could be imputed to the effect of ‘frame’ (see Druckman 2001) and not, strictly, to ‘numerals’, even if it was actually the case.

  2. We do not find in Frederick (2005) specific indications on how much time in CRT accomplishment is allocated to answering the tasks. Since he gives 45 min to perform the three tasks and a questionnaire, it seems reasonable to give 10 min for the ‘bat and ball’ alone.

  3. De Neys et al. (2013) and Bourgeois-Gironde and Vanderhenst (2009) are not attentive to such a factor.

  4. The assumptions ‘incorrect answer = use of System 1’ and ‘correct answer = use of System 2’ can be partially relaxed (see e.g. Oechssler et al. 2009, p. 148) considering also the cases in which the subjects are not able to technically implement the correct resolution process, although not being biased by the problem. Here, however, we prefer to avoid such a further degree of freedom, performing the experiment in order to be comparable with the other experiments in this literature.

  5. We use the Fisher’s exact test because the Chi square test was not suitable to our experimental data, since one cell of the contingency table has a numerosity less than 10 and we have just 1 degree of freedom. The Fisher’s test we implement is one-tailed since, consistently with our hypothesis and the CRT, we are not interested in testing the hypothesis that harder computations entail an increasing activation of System 1. To check the result, we also performed a one-tailed Barnard test for the contingency table; Barnard’s test is considered more powerful than Fisher’s one because it maximizes with respect to the nuisance parameter when calculating p. The result is p = 0.0351.

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Correspondence to Enrico Petracca.

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Mastrogiorgio, A., Petracca, E. Numerals as triggers of System 1 and System 2 in the ‘bat and ball’ problem. Mind Soc 13, 135–148 (2014). https://doi.org/10.1007/s11299-014-0138-8

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