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Bronx Baseball Bat & Ball Set

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Hoover JD, & Healy AF (2017). Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later. Psychonomic Bulletin & Review, 24, 1922–1928. doi: 10.3758/s13423-017-1241-8 [ PubMed] [ CrossRef] [ Google Scholar] Baron J, Scott S, Fincher K, & Metz SE (2015). Why does the Cognitive Reflection Test (sometimes) predict utilitarian moral judgment (and other things)? Journal of Applied Research in Memory and Cognition, 4, 265–284. doi: 10.1016/j.jarmac.2014.09.003 [ CrossRef] [ Google Scholar]

Travers E, Rolison JJ, & Feeney A (2016). The time course of conflict on the Cognitive Reflection Test. Cognition, 150, 109–118. doi: 10.1016/j.cognition.2016.01.015 [ PubMed] [ CrossRef] [ Google Scholar] Response times were log transformed for the analyses although the means and standard deviations reported here are untransformed values (in seconds). Again, a 3 × 2 mixed factorial analysis of variance was conducted. In this case, as mentioned earlier, the analysis was limited to the response times from incorrect reasoners, defined as those who were incorrect on the standard problem (ignoring the correctness of their response on the control problem). There was a main effect of condition such that incorrect reasoners responded faster to the control question than to the standard question (see Table 1), F(1, 224) = 38.833, p< .001, η p 2 = .15. Thompson VA, Prowse Turner JA, & Pennycook G (2011). Intuition, reason, and metacognition. Cognitive Psychology, 63, 107–140. doi: 10.1016/j.cogpsych.2011.06.001 [ PubMed] [ CrossRef] [ Google Scholar] Lichtenstein S, & Fischhoff B (1980). Training for calibration. Organizational Behavior and Human Performance, 26, 149–171. doi: 10.1016/0030-5073(80)90052-5 [ CrossRef] [ Google Scholar]Klayman J, Soll JB, González-Vallejo C, & Barlas S (1999). Overconfidence: It depends on how, what, and whom you ask. Organizational Behavior and Human Decision Processes, 79, 216–247. doi: 10.1006/obhd.1999.2847 [ PubMed] [ CrossRef] [ Google Scholar]

The question really belongs more to the science of the mind than it does to mathematics and logic— it is about the assumptions we make, rather than whether or not we have the ability to solve the question. Bago B, & De Neys W (2017). Fast logic? Examining the time course assumption of dual process theory. Cognition, 158, 90–109. doi: 10.1016/j.cognition.2016.10.014 [ PubMed] [ CrossRef] [ Google Scholar] Lichtenstein S, & Fischhoff B (1977). Do those who know more also know more about how much they know? Organizational Behavior and Human Performance, 20, 159–183. doi: 10.1016/0030-5073(77)90001-0 [ CrossRef] [ Google Scholar]Israeli-American psychologist Daniel Kahneman examines what he calls the machinery of the mind — two distinct systems in our brain that dictate how we think and make decisions — in his book, Thinking Fast, and Slow. For recognition, all incorrect reasoners’ responses were included in the analyses because their answers came in the form of a multiple-choice forced response. A mixed effects logistic regression (with subject as the random variable) was again conducted due to the dichotomous dependent variable (with or without “more than”). Once again, incorrect reasoners usually recognized the standard problem, but not the control, as containing “more than” (see Table 1), with this effect of condition significant, b = 3.00, odds ratio ( OR) = 20.07, χ 2 = 50.34, p< .001, 95% confidence interval ( CI) [8.76, 45.94].

Johnson ED, Tubau E, & De Neys W (2016). The Doubting System 1: Evidence for automatic substitution sensitivity. Acta Psychologica, 164, 56–64. doi: 10.1016/j.actpsy.2015.12.008 [ PubMed] [ CrossRef] [ Google Scholar] Keren G (1997). On the calibration of probability judgments: Some critical comments and alternative perspectives. Journal of Behavioral Decision Making, 10, 269–278. doi: 10.1002/(SICI)1099-0771(199709)10:3<269::AID-BDM281>3.0.CO;2-L [ CrossRef] [ Google Scholar] B. T., & Stanovich KE (2013). Dual-process theories of higher cognition: Advancing the debate. Perspectives on Psychological Science, 8, 223–241. doi: 10.1177/1745691612460685 [ PubMed] [ CrossRef] [ Google Scholar] Finally, latency responses to both the standard and control versions were collected. Correlations between response time and response confidence were expected, in line with previous work (e.g., Johnson et al., 2016; Kelley & Lindsay, 1993; Thompson, Prowse Turner, & Pennycook, 2011; Thompson et al., 2013). Arkes HR, Christensen C, Lai C, & Blumer C (1987). Two methods of reducing overconfidence. Organizational Behavior and Human Decision Processes, 39, 133–144. doi: 10.1016/0749-5978(87)90049-5 [ CrossRef] [ Google Scholar]Mata A, & Almeida T (2014). Using metacognitive cues to infer others’ thinking. Judgment and Decision Making, 9, 349–359. [ Google Scholar] Mata A, Ferreira MB, Voss A, & Kollei T (2017). Seeing the conflict: An attentional account of reasoning errors. Psychonomic Bulletin & Review, 24, 1980–1986. doi: 10.3758/s13423-017-1234-7 [ PubMed] [ CrossRef] [ Google Scholar]

For the recall response, a mixed effects logistic regression (with subject as the random variable) was conducted due to the dichotomous dependent variable (with or without “more than”), considering only those participants who wrote down an answer that could be coded as with or without the “more than” phrase (e.g., “don’t know” responses were excluded). Incorrect reasoners usually recalled the standard problem, but not the control, as containing “more than” (see Table 1), with this effect of condition significant, b = 3.02, odds ratio ( OR) = 20.44, χ 2 = 21.69, p< .001, 95% confidence interval ( CI) [5.74, 72.75]. De Neys W, & Bonnefon J-F (2013). The ‘whys’ and ‘whens’ of individual differences in thinking biases. Trends in Cognitive Sciences, 17, 172–178. doi: 10.1016/j.tics.2013.02.001 [ PubMed] [ CrossRef] [ Google Scholar] The analysis of variance was conducted on proportions to enable comparison with the findings of De Neys et al. (2013), who used an analysis of variance on percentages to compare standard and control problems. However, because the dependent variable is dichotomous (0 or 1), a mixed effects logistic regression (with subject as the random variable) is more appropriate and yielded comparable results. Specifically, participants were significantly more likely to respond accurately to isomorphic control questions than to standard variants, b = −3.73, odds ratio ( OR) = 41.57, χ 2 = 153.54, p< .001, 95% confidence interval ( CI) [0.01, 0.04]. That is, participants’ odds of answering the isomorphic control questions correctly was about 42 times more likely than answering the standard variants correctly. B. T. (2008). Dual-processing accounts of reasoning, judgment, and social cognition. Annual Review of Psychology, 59, 255–278. doi: 10.1146/annurev.psych.59.103006.093629 [ PubMed] [ CrossRef] [ Google Scholar]

The two systems that led to the choices.

Fischhoff B, Slovic P, & Lichtenstein S (1977). Knowing with certainty: The appropriateness of extreme confidence. Journal of Experimental Psychology: Human Perception and Performance, 3, 552–564. doi: 10.1037/0096-1523.3.4.552 [ CrossRef] [ Google Scholar] Koriat A, Lichtenstein S, & Fischhoff B (1980). Reasons for confidence. Journal of Experimental Psychology: Human Learning and Memory, 6, 107–118. doi: 10.1037/0278-7393.6.2.107 [ CrossRef] [ Google Scholar] Aczel B, Szollosi A, & Bago B (2016). Lax monitoring versus logical intuition: The determinants of confidence in conjunction fallacy. Thinking & Reasoning, 22, 99–117. doi: 10.1080/13546783.2015.1062801 [ CrossRef] [ Google Scholar] Mata A, Schubert A-L, & Ferreira MB (2014). The role of language comprehension in reasoning: How “good-enough” representations induce biases. Cognition, 133, 457–463. doi: 10.1016/j.cognition.2014.07.011 [ PubMed] [ CrossRef] [ Google Scholar] As for the response confidence scores, the opinion judgments are reported here as proportions rather than percentages. There was a positive relationship between standard question confidence and standard question opinion judgments for incorrect reasoners, r(225) = .480, p< .001, such that incorrect reasoners who were less confident in their response were also less likely to think other reasoners could answer the standard question correctly. This strong relationship lends support to the notion that opinion judgments and response confidence scores are reflecting similar cognitive processes.

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