Oops, scratch that! Monitoring one’s own errors during mental calculation

Aesthetic Considerations in the Development of Plate Tectonics

Aesthetic considerations played a substantial and positive role in the development and acceptance of plate tectonics, the modern theory of the earth's major geological features and the unifying framework of the earth sciences. Here I give an overview of how aesthetics influenced plate tectonics and take a detailed look at a handful of examples from this history where elegance and simplicity tipped the balance in favour of a given hypothesis. I discuss some implications of this case study for extant accounts of aesthetics in science and argue that the positive role aesthetics played can be explained by recent accounts that posit an indirect and contingent link between knowledge and psychological mechanisms underlying aesthetic appreciation.

Dual-Process Theory and Two Types of Metacognitive Monitoring and Control Processes

Metacognitive monitoring and control processes are important parts of our cognitive system. In this article, they are considered in the light of the dual-process theory and interpreted as occurring at the level of Type 1 and Type 2 information processing. Associative connection is the main factor that allows us to divide these processes into two types. Accordingly, metacognitive monitoring of the first type occurs when feelings of rightness/error automatically appear along with a certain judgment. The second type occurs when a controlled inference is made about whether a judgment is true or false. Metacognitive control of the first type occurs when the decision to reject, revise or accept the received judgment is associated with the feelings of rightness/error and automatically appears when one of these feelings appears. The second type takes place when a person rejects (or they are unclear) the results of the first type of metacognitive control and deliberately decides what to do with the received judgment - reject, revise or accept.

Not feeling right about uncertainty monitoring

De Neys proposed a "switch" model to address what he argued to be lacuna in dual-process theory, in which he theorized about the processes that initiate and terminate analytic thinking. We will argue that the author neglected to acknowledge the abundant literature on metacognitive functions, specifically, the meta-reasoning framework developed by Ackerman and Thompson (2017), that addresses just those questions.

Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect "online" effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.

Do performance-monitoring related cortical potentials mediate fluency and difficulty effects on decision confidence?

It remains unclear to what extent the performance-monitoring (PM) processes involved in perceptual decisions are also involved in more complex reasoning decisions. To address this problem, we examined whether PM-related cortical potentials known to correlate with confidence ratings in the context of simple perceptual decisions also correlate with confidence ratings in a more complex reasoning task. In an EEG experiment, 49 participants had to quickly decide whether an equation (e.g. 9 * 7 = 65) was correct or incorrect and then report their confidence in that decision. Task difficulty and response fluency were varied to manipulate confidence. Pooling responses across correct as well as incorrect trials, we analyzed amplitudes of Error-Related Negativity (pooled ERN) and Error Positivity (pooled Pe). We found that pooled ERN but not pooled Pe had a positive significant correlation with decision confidence and that unpooled ERN correlated with confidence for correct, but not for error trials. Further analysis indicated that although pooled ERN correlated with task difficulty, neither pooled or unpooled potentials mediated the effects of difficulty or fluency on confidence. These results suggest that pooled ERN might reflect implicit performance monitoring and that confidence for simple perceptual and complex reasoning decisions may rely on different neural mechanisms.

Exploring the determinants of confidence in the bat-and-ball problem

People often fail to solve deceptively simple mathematical problems, a tendency popularly demonstrated by the bat-and-ball problem. The most prominent explanation of this finding is that, to spare cognitive effort, people substitute the difficult task with an easier one, without being aware of the substitution. Despite this latter assumption, recent studies have found decreased levels of post-decision confidence ratings when people gave the answer of an easier calculation, suggesting that people are sensitive to their errors. In the current study, we investigated a mechanism that might be responsible for such a decrease in people's confidence ratings when they make errors: their attempts to make certain that their answer is correct (verification) and the perceived level of task difficulty (verifiability). We found that these two factors predicted people's confidence, suggesting that people's self-assessment of the perceived task difficulty and of their attempt to verify their response might determine their confidence. Implication for current models of post-decision confidence on reasoning problems is discussed.