From integers to fractions: The role of analogy in transfer and long-term learning.

Fractions are the gatekeepers to advanced mathematics but are difficult to learn. One powerful learning mechanism is analogy, which builds fraction understanding on a pre-existing foundation of integer knowledge. Indeed, a short intervention that aligned fractions and integers on number lines improved children's estimates of fractions (Yu et al., 2022). The breadth and durability of such gains, however, are unknown, and analogies to other sources (such as percentages) may be equally powerful. To investigate this issue, we randomly assigned 109 fourth and fifth graders to one of three experimental conditions with different analogical sources (integers, percentages, or fractions) or a control condition. During training, children in the experimental conditions solved pairs of aligned fraction number line problems and proportionally-equivalent problems expressed in integers, percentages, or fractions (e.g., 3/8 on a 0-1 number line aligned with 3 on a 0-8 number line). Children in the control group solved fraction number-line problems sequentially. At pretest and a two-week delayed posttest, children completed a broad fraction knowledge battery, including estimation, comparison, categorization, ordering, and arithmetic. Results showed that aligning integers and fractions on number lines facilitated better estimation of fractional magnitudes, and the training effect transferred to novel fraction problems after two weeks. Similar gains were not observed for analogies using percentages. These findings highlight the importance of building new mathematical knowledge through analogies to familiar, similar sources.

Impacts of number lines and circle visual displays on caregivers' fraction understanding.

Playful fraction picture books, together with math instructional content called "back matter," may promote fraction learning, which is crucial because fractions are difficult and often disliked content. However, open questions remain regarding how different types of back matter may affect caregivers' ability to use fraction picture books as a teaching tool. The current study offers a novel investigation into how back matter affects caregivers' (N = 160) fraction understanding (i.e., equivalence and arithmetic) and subjective beliefs about math using a pretest/posttest design. We contrasted existing back matter text with research-informed back matter text crossed with either circle area or number line visual displays. Caregivers' performance improved from pretest to posttest in the Researcher-Generated + Circles condition (fraction equivalence) and in the Existing + Circles, Researcher-Generated + Circles, and Researcher-Generated + Number Lines conditions (fraction arithmetic). In addition, caregivers were aware of their learning; they predicted improvements in their fraction arithmetic performance over time. These findings suggest that brief interventions, such as back matter in children's picture books, may improve adults' fraction understanding.

Building integrated number sense in adults and children: Comparing fractions-only training with cross-notation number line training.

Mounting evidence points to the predictive power of cross-notation rational number understanding (e.g., 2/5 vs. 0.25) relative to within-notation understanding (e.g., 2/5 vs. 1/4) in predicting math outcomes. Although correlational in nature, these studies suggest that number sense training emphasizing integrating across notations may have more positive outcomes than a within-notation focus. However, this idea has not been empirically tested. Thus, across two studies with undergraduate students (N = 183 and N = 181), we investigated the effects of a number line training program using a cross-notation approach (one that focused on connections among fractions, decimals, and percentages) and a within-notation approach (one that focused on fraction magnitude representation only). Both number line approaches produced positive effects, but those of the cross-notation approach were larger for fraction magnitude estimation and cross-notation comparison accuracy. In a third study (N = 63), we adapted the cross-notation number line training for use in place of typical classroom warm-up activities for middle school students. Similar to the results with undergraduate students, the cross-notation training program yielded positive benefits for middle school students over a typical warm-up activity (fraction arithmetic practice). Together, these results suggest the importance of an integrated approach to teaching rational number notations, an approach that appears to be uncommon in current curricula.

What Drives Preventive Health Behavior During a Global Pandemic? Emotion and Worry.

BACKGROUND & PURPOSE: Primary prevention of COVID-19 has focused on encouraging compliance with specific behaviors that restrict contagion. This investigation sought to characterize engagement in these behaviors in U.S. adults early during the pandemic and to build explanatory models of the psychological processes that drive them. METHODS: US adults were recruited through Qualtrics Research Panels (N = 324; 55% female; Mage = 50.91, SD = 15.98) and completed 10 days of online reports of emotion, COVID-19 perceived susceptibility and worry, and recommended behaviors (social distancing, hand washing, etc.). Factor analysis revealed behaviors loaded on two factors suggesting distinct motivational orientations: approach and avoidance. RESULTS: Changes in approach and avoidance behaviors over the 10 days indicated large individual differences consistent with three types of participants. Discrete emotions, including fear, guilt/shame, and happiness were associated with more recommended behaviors. Fear and COVID-19 worry indirectly influenced each other to facilitate more behavioral engagement. While emotions and worry strongly predicted individual differences in behavior across the 10 days, they did not predict as well why behaviors occurred on one day versus another. CONCLUSIONS: These findings suggest how daily affective processes motivate behavior, improving the understanding of compliance and efforts to target behaviors as primary prevention of disease.

From continuous magnitudes to symbolic numbers: The centrality of ratio.

Leibovich et al.'s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Individual differences in the components of children's and adults' information processing for simple symbolic and non-symbolic numeric decisions.

How do speed and accuracy trade off, and what components of information processing develop as children and adults make simple numeric comparisons? Data from symbolic and non-symbolic number tasks were collected from 19 first graders (Mage=7.12 years), 26 second/third graders (Mage=8.20 years), 27 fourth/fifth graders (Mage=10.46 years), and 19 seventh/eighth graders (Mage=13.22 years). The non-symbolic task asked children to decide whether an array of asterisks had a larger or smaller number than 50, and the symbolic task asked whether a two-digit number was greater than or less than 50. We used a diffusion model analysis to estimate components of processing in tasks from accuracy, correct and error response times, and response time (RT) distributions. Participants who were accurate on one task were accurate on the other task, and participants who made fast decisions on one task made fast decisions on the other task. Older participants extracted a higher quality of information from the stimulus arrays, were more willing to make a decision, and were faster at encoding, transforming the stimulus representation, and executing their responses. Individual participants' accuracy and RTs were uncorrelated. Drift rate and boundary settings were significantly related across tasks, but they were unrelated to each other. Accuracy was mainly determined by drift rate, and RT was mainly determined by boundary separation. We concluded that RT and accuracy operate largely independently.

Numerical landmarks are useful--except when they're not.

Placing landmarks on number lines, such as marking each tenth on a 0-1 line with a hatch mark and the corresponding decimal, has been recommended as a useful tool for improving children's number sense. Four experiments indicated that some landmarks do have beneficial effects, others have harmful effects, and yet others have no effects on representations of common fractions (N/M). The effects of the landmarks were seen not only on the number line task where they appeared but also on a subsequent magnitude comparison task and on correlations with mathematics achievement tests. Landmarks appeared to exert their effects through the encodings and strategies that they promoted. Theoretical and educational implications are discussed.

Relations of different types of numerical magnitude representations to each other and to mathematics achievement.

We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both magnitude comparison and number line estimation tasks. After controlling for non-mathematical cognitive proficiency, both symbolic and non-symbolic numerical magnitude understandings were uniquely related to mathematics achievement, but the relation was much stronger for symbolic numbers. A meta-analysis of 19 published studies indicated that relations between non-symbolic numerical magnitude knowledge and mathematics achievement are present but tend to be weak, especially beyond 6 years of age.

What drives preventative health behaviors one year into a pandemic? A replication and extension.

Objective: There is continued interest in understanding what leads people to engage in CDC-recommended COVID-19 prevention behaviors. We tested whether fear and COVID-19 worry would replicate as the primary drivers of six CDC recommended prevention behaviors. Methods and Measures: We recruited 741 adult participants during the second major peak of the COVID-19 pandemic in the United States (early 2021). Using very similar methods to the original study, participants completed a 10-day daily diary. Mixed effects models identified the strongest predictors of each individual prevention behavior as well as approach and avoidance behavior clusters. Results: At the between-person level, COVID-19 worry, COVID-19 perceived susceptibility, fear, and positive emotions all had positive zero-order associations with the prevention behaviors. However, with all predictors in the same model together, primarily COVID-19 worry remained significant for both the individual behaviors and behavior clusters. At the within-person level, only fear related to assessing oneself for COVID-19 and approach behaviors on the same day, but not the next day. Mediational analyses suggested COVID-19 worry, but not COVID-19 susceptibility, mediated the links between fear and approach/avoidance behaviors. Conclusion: Findings replicated worry about yourself or a loved one getting COVID-19 as the strongest predictor of prevention behaviors.

Metacognitive Cues, Working Memory, and Math Anxiety: The Regulated Attention in Mathematical Problem Solving (RAMPS) Framework.

Mathematical problem solving is a process involving metacognitive (e.g., judging progress), cognitive (e.g., working memory), and affective (e.g., math anxiety) factors. Recent research encourages researchers who study math cognition to consider the role that the interaction between metacognition and math anxiety plays in mathematical problem solving. Problem solvers can make many metacognitive judgments during a math problem, ranging from global judgments such as, "Do I care to solve this problem?" to minor cue-based judgments such as, "Is my current strategy successful in making progress toward the correct solution?" Metacognitive monitoring can hinder accurate mathematical problem solving when the monitoring is task-irrelevant; however, task-relevant metacognitive experiences can lead to helpful control decisions in mathematical problem solving such as checking work, considering plausibility of an answer, and considering alternate strategies. Worry and negative thoughts (i.e., math anxiety) can both interfere with the accuracy of metacognitive experiences as cues in mathematical problem solving and lead to avoidance of metacognitive control decisions that could otherwise improve performance. The current paper briefly reviews and incorporates prior literature with current qualitative reports (n = 673) to establish a novel framework of regulated attention in mathematical problem solving (RAMPS).

Experimental tests of hypothetical lottery incentives on unvaccinated adults' COVID-19 vaccination intentions.

OBJECTIVE: In May 2021, U.S. states began implementing "vaccination lotteries" encouraging COVID-19 vaccination. Drawing from Prospect Theory and math cognition research, we tested several monetary lottery structures and their framing to determine which would best motivate unvaccinated adults. METHOD: In two online experiments, U.S. adults were asked to imagine that their state implemented a vaccination lottery. In Experiment 1, participants (N = 589) were randomly assigned to 1 of 12 conditions varying the monetary amount and number of winners, holding constant a $5 million total payout. In Experiment 2, participants (N = 274) were randomly assigned to one of four conditions in a 2 (Message Framing: Gain versus Loss) by 2 (Numeric Framing: Big versus Small) factorial design; in all conditions, five people would each win $1 million. Participants rated their baseline vaccination willingness (1 = not at all to 4 = very) and postmanipulation COVID-19 vaccination intentions "if their state offered this incentive" (0 = definitely would not to 100 = definitely would). RESULTS: Intentions did not differ across conditions (Experiment 1: F[11, 561] = 1.29, p = .224, ηp² = .03; Experiment 2: Message Framing, F[1, 266)] = .01, p = .940, ηp² = .000; Numeric Framing, F[1, 266] = 1.40, p = .237, ηp² = .01; Interaction, F[1, 266] = 1.40, p = .238, ηp² = .01). When participants were shown a list of 12 lottery structures and asked which they preferred, participants on average preferred options that awarded less money to more people. However, 41.9% of participants across both experiments indicated they would not vaccinate for any lottery-based monetary incentive. CONCLUSIONS: Multiple lottery structures could be equally (un)motivating for unvaccinated adults. Structures that distribute incentives across more people or alternative public health strategies should be considered. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Emotional complexity under high stress: Do protective associations for risk behaviors persist even during a pandemic?

Risk behaviors like substance use and binge eating are often used to cope with negative emotions. Engagement in these behaviors has been exacerbated by the COVID-19 pandemic. Past research suggests that complex emotion conceptualizations captured as emotion differentiation (ability to discriminate between emotional states) and polarity (ability to integrate positive and negative features of emotional experience) may be protective. We examined associations of mean affect intensity, emotion differentiation, and emotion polarity with frequency of daily substance use and binge eating across 10 days in a demographically diverse sample of U.S. adults (N = 353) recruited between March 24 and April 9, 2020, when stay-at-home orders were initiated. Owing to the nested data structure and excessive zero values, analyses were conducted using multilevel zero-inflated negative binomial regression. Consistent with past research, negative affect was positively associated with frequency of substance use and binge eating. Importantly, results indicated that negative emotion differentiation was protective, predicting greater likelihood of not using substances and binge eating at all across the sampling period. These effects remained even after controlling for mean affect intensity, emotion polarity, and positive emotion differentiation. Neither positive emotion differentiation nor emotion polarity were significantly associated with either behavior. Our results suggest that greater complexity in conceptualization of negative emotions facilitates some protection against risk behaviors such as substance use and binge eating, even during periods of high environmental stress. These findings have important implications for optimizing interventions to reduce engagement in risk behaviors. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Attending to what's important: what heat maps may reveal about attention, inhibitory control, and fraction arithmetic performance.

Math proficiency is an important predictor of educational attainment and life success. However, developing mathematical competency is challenging, and some content (e.g., fractions) can be enigmatic. Numerous factors are suspected to influence math performance, including strategy knowledge, attention, and executive functions. In two online studies, we investigated the relationship between adults' fraction arithmetic performance, confidence judgments, inhibitory control (a component of executive functions), and attention to strategy-relevant fraction components. We explored the utility of heat maps (based on mouse clicks) to measure adults' attention to strategy-relevant fraction arithmetic components (operationalized according to each mathematical operation). In Study 1, attending to strategy-relevant fraction components was correlated with inhibitory control, but this finding did not replicate in Study 2. Across both studies, inhibitory control and attention to strategy-relevant fraction components were correlated with arithmetic accuracy. Intraindividual variability in participants' attention to strategy-relevant fraction components was also found. Our findings suggest that heat map questions may be a viable alternative to assess participants' attention during fraction tasks and that attention to specific fraction-arithmetic problem features is related to problem-solving accuracy.

Leveraging Math Cognition to Combat Health Innumeracy.

Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias-a tendency to misapply knowledge about natural numbers to all numbers-is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people's penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and how measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.

Number lines can be more effective at facilitating adults' performance on health-related ratio problems than risk ladders and icon arrays.

Visual displays, such as icon arrays and risk ladders, are often used to communicate numerical health information. Number lines improve reasoning with rational numbers but are seldom used in health contexts. College students solved ratio problems related to COVID-19 (e.g., number of deaths and number of cases) in one of four randomly assigned conditions: icon arrays, risk ladders, number lines, or no accompanying visual display. As predicted, number lines facilitated performance on these problems-the number line condition outperformed the other visual display conditions, which did not perform any better than the no visual display condition. In addition, higher performance on the health-related ratio problems was associated with higher COVID-19 worry for oneself and others, higher perceptions of COVID-19 severity, and higher endorsement of intentions to engage in preventive health behaviors, even when controlling for baseline math skills. These findings have important implications for effectively presenting health statistics. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Task features change the relation between math anxiety and number line estimation performance with rational numbers: Two large-scale online studies.

Math performance is negatively related to math anxiety (MA), though MA may impact certain math skills more than others. We investigated whether the relation between MA and math performance is affected by task features, such as number type (e.g., fractions, whole numbers, percentages), number format (symbolic vs. nonsymbolic), and ratio component size (small vs. large). Across two large-scale studies (combined n = 3,822), the MA-performance relation was strongest for large whole numbers and fractions, and stronger for symbolic than nonsymbolic fractions. The MA-performance relation was also stronger for smaller relative to larger components, and MA relating to specific number types may be a better predictor of performance than general MA for certain tasks. The relation between MA and estimation performance changes depending on task features, which suggests that MA may relate to certain math skills more than others, which may have implications for how people reason with numerical information and may inform future interventions. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Children are not like older adults: a diffusion model analysis of developmental changes in speeded responses.

Children (n = 130; M(age) = 8.51-15.68 years) and college-aged adults (n = 72; M(age) = 20.50 years) completed numerosity discrimination and lexical decision tasks. Children produced longer response times (RTs) than adults. R. Ratcliff's (1978) diffusion model, which divides processing into components (e.g., quality of evidence, decision criteria settings, nondecision time), was fit to the accuracy and RT distribution data. Differences in all components were responsible for slowing in children in these tasks. Children extract lower quality evidence than college-aged adults, unlike older adults who extract a similar quality of evidence as college-aged adults. Thus, processing components responsible for changes in RTs at the beginning of the life span are somewhat different from those responsible for changes occurring with healthy aging.

Perceptions of ease and difficulty, but not growth mindset, relate to specific math attitudes.

BACKGROUND: People report negative attitudes towards fractions and percentages relative to whole numbers (WNs, Sidney, Thompson, Fitzsimmons, & Taber, 2021), and these attitudes may relate to an individual's interpretation of what experiences with these number types signify. Because fractions are challenging, individual differences related to beliefs about challenge, such as endorsement of a growth versus fixed mindset (Dweck, 2006) and interpretations of easy or difficult experiences (Fisher & Oyserman, 2017), could relate to attitudes towards fractions relative to other number types. AIMS: Two studies tested whether gender, math skills, mindset beliefs, and perceptions of difficulty relate to negative math attitudes towards specific number types. SAMPLES: Two samples of college students (Study 1: N = 491; Study 2: N = 415), approximately 19 years of age (17% male, 51% first year students) participated. METHODS: Participants rated attitudes pertaining to WNs, fractions, and percentages, endorsement of a growth mindset, and perceptions of ease and difficulty. RESULTS: Replicating prior work (Sidney, Thompson, Fitzsimmons, & Taber, 2021), college students endorsed more negative attitudes about fractions than WNs and percentages. Self-reported ACT scores related to all number-type attitudes, endorsement of the belief that 'difficult tasks/goals are important' related to fraction attitudes, and endorsement of the belief that 'easy tasks/goals are possible' related to whole number attitudes. Endorsement of a growth mindset did not relate to specific math attitudes. CONCLUSIONS: People struggle to integrate their whole number and rational number representations, and one reason people hold negative attitudes about fractions may be that they view them as difficult and even impossible.

From integers to fractions: The role of analogy in developing a coherent understanding of proportional magnitude.

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions would improve understanding of fractions, but little is known about how the representations develop and interact with one another. To address this, we asked 177 third, fourth, and fifth graders (M = 9.85 years, 87 girls, 69% White, 19% low income) to estimate the position of proportionally equivalent integers and fractions on number lines (e.g., 3 on a 0-8 number line vs. 3/8 on a 0-1 number line, Study 1). With increasing age, children's estimates of implicit and explicit proportions became more coherent, such that a child's estimates of fractions on a 0-1 number-line was a linear function of the same child's estimates of equivalent integers. To further investigate whether preexisting integer knowledge can facilitate fraction learning through analogy, we assigned 100 third to fifth graders (M = 10.04 years, 55 girls, 76% White) to an Alignment condition, where children estimated fractions and integers on aligned number lines, or to a No Alignment condition (Study 2). Results showed that aligning integers and fractions on number lines facilitated a better understanding of fractional magnitudes, and increased children's fraction estimation accuracy to the level of college students'. Together, findings suggest that analogies can play an important role in building a coherent understanding of proportions. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Self-care behaviors and affect during the early stages of the COVID-19 pandemic.

OBJECTIVE: Self-care behaviors aimed at maintaining physical and mental health are often recommended during stressful contexts. We tested emotional predictors of self-care behaviors (healthy eating, exercise, engaging in a hobby, relaxation/meditation, time spent with a supportive person, talking online with friends/family) during the COVID-19 pandemic and their emotional consequences. We hypothesized a reciprocal within-person process whereby positive affect increases self-care behaviors (Hypothesis 1) and self-care behaviors increase positive affect while decreasing negative affect (Hypothesis 2). METHOD: A 10-day daily diary was completed by 289 adult participants in the United States during spring 2020 when counties in 40 out of 50 states had some form of stay-at-home orders. RESULTS: Lagged analyses for Hypothesis 1 suggested that positive affect did not significantly predict residualized change in self-care behaviors; however, more intense negative affect predicted increased self-care behaviors from one day to the next. Concurrent analyses for Hypothesis 2 indicated most self-care behaviors were associated with more positive affect and some with less negative affect on the same day. Lagged analyses for Hypothesis 2 indicated that self-care behaviors largely did not predict residualized change in positive or negative affect from one day to the next. At the between-person level, people who experienced more positive affect engaged in more self-care behaviors across the sampling period. CONCLUSION: Self-care behaviors continue to have mental health benefits during stressful environments such as the COVID-19 pandemic and stay-at-home orders. Negative affect can play an adaptive role during times of stress by facilitating self-care. (PsycInfo Database Record (c) 2022 APA, all rights reserved).