Knowing right from wrong in mental arithmetic judgments: calibration of confidence predicts the development of accuracy

Children's confidence on mathematical equivalence and fraction problems

Metacognition is how people think about their own thinking, and it includes children's confidence in their problem-solving solutions. We assessed children's metacognition in two areas of mathematics that are often plagued by misconceptions and often studied separately-mathematical equivalence (e.g., 3 + 4 = 5 +__) and fraction magnitude (e.g., 1/3 = 2/__). Specifically, we quantified children's metacognitive skills across these topics, examined whether these skills are topic-specific, and determined how these skills covary with domain-general executive functioning. A total of 80 elementary school children (Mage = 7.70 years, SD = 0.69) provided trial-by-trial performance data and confidence judgments on equivalence and fraction problems as well as assessments of executive function. Children's metacognitive skills were especially impaired when using incorrect strategies based on misconceptions, and we did not find robust evidence for cross-topic associations. We found modest associations between children's metacognitive skills and their working memory and inhibitory control. Findings have theoretical and practical implications for understanding which children struggle with these important mathematics concepts and why.

The structure of metacognition in middle childhood: Evidence for a unitary metacognition-for-memory factor

It has been debated whether children's metacognitive monitoring and control processes rely on a general resource or whether metacognitive processes are task specific. Moreover, findings about the extent to which metacognitive processes are related to first-order task performance are mixed. The current study aimed to uncover the relationships among children's monitoring (discrimination between correct and incorrect responses), control (accurate withdrawal of wrong answers), and performance across three memory-based learning tasks: Kanji learning, text comprehension, and secret code learning. All tasks consisted of a study phase, a test phase, monitoring (confidence judgments), and control (maintaining/withdrawing responses). Participants were 325 children (151 second graders [Mage = 8.12 years] and 174 fourth graders [Mage = 10.20 years]). Confirmatory factor analyses showed that a model in which monitoring and control loaded on a joint factor and performance on a separate factor provided the best fit to the data. Fourth graders had better monitoring and control accuracy than second graders. However, the factor structure of metacognition was similar for both age groups, contradictory to the assumption that metacognition generalizes across tasks as children grow older. After accounting for task-specific processes, monitoring and control skills for language-based memory tasks appear to be generalizable in middle childhood. In sum, children's monitoring and control for three separate memory tasks appear to reflect a unitary metacognition-for-memory factor related to, but distinguishable from, performance.

Relation of life sciences students' metacognitive monitoring to neural activity during biology error detection

Metacognitive calibration-the capacity to accurately self-assess one's performance-forms the basis for error detection and self-monitoring and is a potential catalyst for conceptual change. Limited brain imaging research on authentic learning tasks implicates the lateral prefrontal and anterior cingulate brain regions in expert scientific reasoning. This study aimed to determine how variation in undergraduate life sciences students' metacognitive calibration relates to their brain activity when evaluating the accuracy of biological models. Fifty undergraduate students enrolled in an introductory life sciences course completed a biology model error detection task during fMRI. Students with higher metacognitive calibration recruited lateral prefrontal regions linked in prior research to expert STEM reasoning to a greater extent than those with lower metacognitive calibration. Findings suggest that metacognition relates to important individual differences in undergraduate students' use of neural resources during an authentic educational task and underscore the importance of fostering metacognitive calibration in the classroom.

Modality independent or modality specific? Common computations underlie confidence judgements in visual and auditory decisions

The mechanisms that enable humans to evaluate their confidence across a range of different decisions remain poorly understood. To bridge this gap in understanding, we used computational modelling to investigate the processes that underlie confidence judgements for perceptual decisions and the extent to which these computations are the same in the visual and auditory modalities. Participants completed two versions of a categorisation task with visual or auditory stimuli and made confidence judgements about their category decisions. In each modality, we varied both evidence strength, (i.e., the strength of the evidence for a particular category) and sensory uncertainty (i.e., the intensity of the sensory signal). We evaluated several classes of computational models which formalise the mapping of evidence strength and sensory uncertainty to confidence in different ways: 1) unscaled evidence strength models, 2) scaled evidence strength models, and 3) Bayesian models. Our model comparison results showed that across tasks and modalities, participants take evidence strength and sensory uncertainty into account in a way that is consistent with the scaled evidence strength class. Notably, the Bayesian class provided a relatively poor account of the data across modalities, particularly in the more complex categorisation task. Our findings suggest that a common process is used for evaluating confidence in perceptual decisions across domains, but that the parameter settings governing the process are tuned differently in each modality. Overall, our results highlight the impact of sensory uncertainty on confidence and the unity of metacognitive processing across sensory modalities.

Learning with certainty in childhood

Learners use certainty to guide learning. They maintain existing beliefs when certain, but seek further information when they feel uninformed. Here, we review developmental evidence that this metacognitive strategy does not require reportable processing. Uncertainty prompts nonverbal human infants and nonhuman animals to engage in strategies like seeking help, searching for additional information, or opting out. Certainty directs children's attention and active learning strategies and provides a common metric for comparing and integrating conflicting beliefs across people. We conclude that certainty is a continuous, domain-general signal of belief quality even early in life.

Confidence in COVID problem solving: What factors predict adults' item-level metacognitive judgments on health-related math problems before and after an educational intervention?

The advent of COVID-19 highlighted widespread misconceptions regarding people's accuracy in interpreting quantitative health information. How do people judge whether they accurately answered health-related math problems? Which individual differences predict these item-by-item metacognitive monitoring judgments? How does a brief intervention targeting math skills-which increased problem-solving accuracy-affect people's monitoring judgments? We investigated these pre-registered questions in a secondary analysis of data from a large Qualtrics panel of adults (N = 1,297). Pretest performance accuracy, math self-efficacy, gender, and math anxiety were associated with pretest item-level monitoring judgments. Participants randomly assigned to the intervention condition, relative to the control condition, made higher monitoring judgments post intervention. That is, these participants believed they were more accurate when answering problems. Regardless of experimental condition, those who actually were correct on health-related math problems made higher monitoring judgments than those who answered incorrectly. Finally, consistent with prior research, math anxiety explained additional variance in monitoring judgments beyond trait anxiety. Together, findings indicated the importance of considering both objective (e.g., problem accuracy) and subjective factors (e.g., math self-efficacy, math anxiety) to better understand adults' metacognitive monitoring.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11409-022-09300-3.

Children’s confidence using incorrect strategies on mathematical equivalence problems

Children often struggle to solve mathematical equivalence problems correctly. The change-resistance theory offers an explanation for children's difficulties and suggests that some incorrect strategies represent the overgeneralization of children's narrow arithmetic experience. The current research considered children's metacognitive abilities to test a tacit assumption of the change-resistance theory by providing a novel empirical examination of children's strategy use and certainty ratings. Children were recruited from U.S. elementary school classrooms serving predominantly White students between the ages of 6 and 9. In Study 1 (n = 52) and Study 2 (n = 147), children were more certain that they were correct when they employed arithmetic-specific incorrect strategies relative to other incorrect strategies. These findings are consistent with the change-resistance theory and have implications for the development of children's metacognition.

Connecting teachers' classroom instructions with children's metacognition and learning in elementary school

Many children have difficulties with accurate self-monitoring and effective regulation of study, and this may cause them to miss learning opportunities. In the classroom, teachers play a key role in supporting children with metacognition and learning. The present study aimed to acquire insights into how teachers' cognitive and metacognitive strategy instruction, as well as teacher-directed and child-centered instructional practices are related to children's self-monitoring accuracy, regulation of study, and learning performance. Twenty-one teachers and 308 children (2nd and 4th grade elementary school) participated. Teachers instructed a secret code task, children had to learn the match between letters of the alphabet and corresponding symbols. Teachers were observed and audio-recordings were made of their instructions. Then, children were asked to (a) make restudy selections, (b) complete a test, and (c) self-monitor test performance. Although teachers both addressed cognitive and metacognitive strategies, they more often instructed children about cognitive strategies. Further, teaching practices were more often teacher-directed than child-centered. Although there were no relations between teachers' instructions for metacognitive strategies and children's outcome measures, teaching cognitive strategies was positively associated with children's performance and self-monitoring accuracy. However, teaching cognitive strategies did not predict effective restudy selections. Rather, child-centered instructions (i.e., giving children autonomy to regulate their own learning) positively predicted children's restudy, and further, children's self-monitoring was more accurate in classrooms where teachers more often used child-centered instructional practices. This seems to imply that not only the content of the instructions itself, but particularly the way these are given, affects children's metacognition.

Exploring psychometric properties of children' metacognitive monitoring

Two independent data sets assessing children's metacognitive monitoring abilities were used to explore the psychometric properties of classical and often-used monitoring measures in primary school age. Theoretically, monitoring is an overarching skill that helps individuals evaluate task mastery, strategy use, and correctness of performance. Monitoring skills are increasingly targeted when addressing individual differences in scholastic achievement and intervention approaches to foster students' self-regulated learning early on. In such contexts, knowledge about central psychometric properties is essential. Results of both studies revealed high internal consistency of prospective and retrospective monitoring judgments. When equivalent item sets (in terms of item difficulty) were considered (Study 1), split-half reliabilities were also satisfying. However, analyses revealed that the monitoring judgments' reliability depends on the reliability of the first-order task (recognition memory test). Retesting children of Study 2 after six months revealed considerable fluctuations in the monitoring measures. Among the included monitoring measures, reliabilities of within-person correlations (Gammas) between performance and confidence and recognition response times and confidence were poorest. Results are discussed in the context of the underlying theoretical construct and implications for research and practice.

Comparing metacognitive monitoring between native and non-native speaking primary school students

Metacognitive monitoring is a significant predictor of academic achievement and is assumed to be related to language competencies. Hence, it may explain academic performance differences between native and non-native speaking students. We compared metacognitive monitoring (in terms of resolution) between native and non-native speaking fourth graders (~ 10 year olds) in two studies. In Study 1, we matched 30 native and 30 non-native speakers and assessed their monitoring in the context of a paired-associates task, including a recognition test and confidence judgements. Study 1 revealed that recognition and monitoring did not differ between native and non-native speaking children. In Study 2, we matched 36 native and 36 non-native speakers and assessed their monitoring with the same paired-associates task. Additionally, we included a text comprehension task with open-ended questions and confidence judgments. We replicated the findings of Study 1, suggesting that recognition and monitoring do not necessarily differ between native and non-native speakers. However, native speaking students answered more open-ended questions correctly than non-native speaking students did. Nevertheless, the two groups did not differ in monitoring their answers to open-ended questions. Our results indicate that native and non-native speaking children may monitor their metacognitive resolution equally, independent of task performance and characteristics. In conclusion, metacognitive monitoring deficits may not be the primary source of the academic performance differences between native and non-native speaking students.

Childhood growth in math and reading differentially predicts adolescent non-ability-based confidence: An examination in the SECCYD

Non-ability-based confidence is one of the most pervasive human psychological biases. It is a part of a family of confidence judgments, including overconfidence and metacognitive calibration accuracy, defined by a discrepancy between self-perception of ability and actual ability. Across many domains, most people exhibit some degree of miscalibration in their confidence. Some people may be overconfident and others are underconfident. Despite the prevalence of non-ability-based confidence, relatively little research has investigated how non-ability-based confidence develops and why some people are more or less confident than others despite sharing the same level of ability. We use a longitudinal dataset to explore the childhood predictors of adolescent non-ability-based confidence. Achievement growth in math and reading in childhood was modeled and used to predict adolescent non-ability-based confidence in math and reading. Results show that the initial level of achievement predicts lower non-ability-based confidence in math. On the other hand, a faster rate of achievement growth across childhood predicts greater non-ability-based confidence in reading. These results highlight how previous experiences inform people's self-perceptions over and above their true abilities. Discussion focuses on the factors that shape non-ability-based confidence over the lifespan and the limitations of the current findings.

Differential brain activity as a function of social evaluative stress in early adolescence: Brain function and salivary cortisol

Understanding individual differences in neural responses to stressful environments is an important avenue of research throughout development. These differences may be especially critical during adolescence, which is characterized by opportunities for healthy development and increased susceptibility to the development of psychopathology. While the neural correlates of the psychosocial stress response have been investigated in adults, these links have not been explored during development. Using a new task, the Minnesota Imaging Stress Test in Children (MISTiC), differences in activation are found in fusiform gyrus, superior frontal gyrus, insula, and anterior cingulate cortex when comparing a stressful math task to a nonstressful math task. The MISTiC task successfully elicits cortisol responses in a similar proportion of adolescents as in behavioral studies while collecting brain imaging data. Cortisol responders and nonresponders did not differ in their perceived stress level or behavioral performance during the task despite differences in neuroendocrine function. Future research will be able to leverage the MISTiC task for many purposes, including probing associations between individual differences in stress responses with environmental conditions, personality differences, and the development of psychopathology.

The neural basis of metacognitive monitoring during arithmetic in the developing brain

In contrast to a substantial body of research on the neural basis of cognitive performance in several academic domains, less is known about how the brain generates metacognitive (MC) awareness of such performance. The existing work on the neurobiological underpinnings of metacognition has almost exclusively been done in adults and has largely focused on lower level cognitive processing domains, such as perceptual decision‐making. Extending this body of evidence, we investigated MC monitoring by asking children to solve arithmetic problems, an educationally relevant higher‐order process, while providing concurrent MC reports during fMRI acquisition. Results are reported on 50 primary school children aged 9–10 years old. The current study is the first to demonstrate that brain activity during MC monitoring, relative to the control task, increased in the left inferior frontal gyrus in children. This brain activity further correlated with children's arithmetic development over a 3‐year time period. These data are in line with the frequently suggested, yet never empirically tested, hypothesis that activity in the prefrontal cortex during arithmetic is related to the higher‐order process of MC monitoring.

Metacognition across domains: Is the association between arithmetic and metacognitive monitoring domain-specific?

Metacognitive monitoring is a critical predictor of arithmetic in primary school. One outstanding question is whether this metacognitive monitoring is domain-specific or whether it reflects a more general performance monitoring process. To answer this conundrum, we investigated metacognitive monitoring in two related, yet distinct academic domains: arithmetic and spelling. This allowed us to investigate whether monitoring in one domain correlated with monitoring in the other domain, and whether monitoring in one domain was predictive of performance in the other, and vice versa. Participants were 147 typically developing 8-9-year-old children (Study 1) and 77 typically developing 7-8-year-old children (Study 2), who were in the middle of an important developmental period for both metacognitive monitoring and academic skills. Pre-registered analyses revealed that within-domain metacognitive monitoring was an important predictor of arithmetic and spelling at both ages. In 8-9-year-olds the metacognitive monitoring measures in different academic domains were predictive of each other, even after taking into account academic performance in these domains. Monitoring in arithmetic was an important predictor of spelling performance, even when arithmetic performance was controlled for. Likewise, monitoring in spelling was an important predictor of arithmetic performance, even when spelling performance was controlled for. In 7-8-year-olds metacognitive monitoring was domain-specific, with neither correlations between the monitoring measures, nor correlations between monitoring in one domain and performance in the other. Taken together, these findings indicate that more domain-general metacognitive monitoring processes emerge over the ages from 7 to 9.

More than number sense: The additional role of executive functions and metacognition in arithmetic

Arithmetic is a major building block for children's development of more complex mathematical abilities. Knowing which cognitive factors underlie individual differences in arithmetic is key to gaining further insight into children's mathematical development. The current study investigated the role of executive functions and metacognition (domain-general cognitive factors) as well as symbolic numerical magnitude processing (domain-specific cognitive factor) in arithmetic, enabling the investigation of their unique contribution in addition to each other. Participants were 127 typically developing second graders (7- and 8-year-olds). Our within-participant design took into account different components of executive functions (i.e., inhibition, shifting, and updating), different aspects of metacognitive skills (i.e., task-specific and general metacognition), and different levels of experience in arithmetic, namely addition (where second graders had extensive experience) and multiplication (where second graders were still in the learning phase). This study reveals that both updating and metacognitive monitoring are important unique predictors of arithmetic in addition to each other and to symbolic numerical magnitude processing. Our results point to a strong and unique role of task-specific metacognitive monitoring skills. These individual differences in noticing one's own errors might help one to learn from his or her mistakes.

Error monitoring in decision-making and timing is disrupted in autism spectrum disorder

Individuals with autism spectrum disorder (ASD) have difficulties in social interactions. The cognitive domains that support these interactions include perceptual decision-making, timing, and error-monitoring, which enable one to appropriately understand and react to the other individual in communicative settings. This study constitutes a comprehensive exploration of decision-making and interval timing in ASD as well as the first investigation of error-monitoring abilities of individuals with ASD regarding their performance in the corresponding domains. We found that children with ASD fared similar to typically developing (TD) children in their first-order task performance in two-alternative forced choice perceptual decision-making and temporal reproduction tasks as well as the secondary tasks (signal detection and free finger tapping tasks). Yet, they had a deficit in error-monitoring in both tasks where their accuracy did not predict their confidence ratings, which was the case for the TD group. The difference between ASD and TD groups was limited to error-monitoring performance. This study attests to a circumscribed impairment in error-monitoring in individuals with ASD, which may partially underlie their social interaction problems. This difficulty in cognitively evaluating one's own performance may also relate to theory of mind deficits reported for individuals with ASD, where they struggle in understanding the mental states and intentions of others. This novel finding holds the potential to inform effective interventions for individuals with ASD that can target this error-monitoring ability to have broad-ranging effects in multiple domains involved in communication and social interaction. Autism Res 2019, 12: 239-248 © 2018 International Society for Autism Research, Wiley Periodicals, Inc. LAY SUMMARY: Decision-making, timing, and error-monitoring are three of many abilities that underlie smooth social interactions. To date, these domains have been only investigated separately, but given their interactive role in social interactions that are impaired in ASD, we conducted the first study to investigate them together. Children with ASD were as successful as typically developing children in their task performances, but unlike them, were unaware of their errors in both decision-making and timing tasks. This deficit that is limited to error-monitoring can contribute to unraveling the unique cognitive signature of ASD and to formulating interventions with positive implications in multiple domains.

Is Math Anxiety Always Bad for Math Learning? The Role of Math Motivation

The linear relations between math anxiety and math cognition have been frequently studied. However, the relations between anxiety and performance on complex cognitive tasks have been repeatedly demonstrated to follow a curvilinear fashion. In the current studies, we aimed to address the lack of attention given to the possibility of such complex interplay between emotion and cognition in the math-learning literature by exploring the relations among math anxiety, math motivation, and math cognition. In two samples-young adolescent twins and adult college students-results showed inverted-U relations between math anxiety and math performance in participants with high intrinsic math motivation and modest negative associations between math anxiety and math performance in participants with low intrinsic math motivation. However, this pattern was not observed in tasks assessing participants' nonsymbolic and symbolic number-estimation ability. These findings may help advance the understanding of mathematics-learning processes and provide important insights for treatment programs that target improving mathematics-learning experiences and mathematical skills.

Unskilled but subjectively aware: Metacognitive monitoring ability and respective awareness in low-performing students

Two studies were conducted to further examine the unskilled-and-unaware effect and to test whether low-performing students are indeed unaware of their (expected) lower metacognitive monitoring abilities. Postdicted judgments of performance and second-order judgments (SOJs) were solicited to test students' metacognitive awareness. Given that global and local judgments tend to differ (the confidence-frequency effect), we investigated whether students' (un)awareness pertains to both types of judgments. A first study focusing on global judgments was conducted in a regular exam setting with 196 undergraduate education students. A second study with 115 undergraduate education students examined both global and local judgments. Local judgments were analyzed on an average level and according to different signal detection theory categories (hits, correct rejections, misses, and false alarms). In both studies, students were grouped in four performance quartiles. The results showed that low-performing students highly overestimated their performance (they were functionally overconfident). However, their SOJs indicated that they were less confident in their judgments than the other students, and thus seemed to be aware of their low ability to estimate their own performance (they were not subjectively overconfident). This was observed for global as well as for averaged local SOJs. Moreover, an analysis of the local judgments revealed that students' SOJs varied depending not only on whether their judgments were accurate but also on whether or not they thought they knew the answer to an item. In sum, SOJs provide valuable information about students' metacognitive awareness.