Eye movements reflect and shape strategies in fraction comparison

The comparison of fractions is a difficult task that can often be facilitated by separately comparing components (numerators and denominators) of the fractions—that is, by applying so-called component-based strategies. The usefulness of such strategies depends on the type of fraction pair to be compared. We investigated the temporal organization and the flexibility of strategy deployment in fraction comparison by evaluating sequences of eye movements in 20 young adults. We found that component-based strategies could account for the response times and the overall number of fixations observed for the different fraction pairs. The analysis of eye movement sequences showed that the initial eye movements in a trial were characterized by stereotypical scanning patterns indicative of an exploratory phase that served to establish the kind of fraction pair presented. Eye movements that followed this phase adapted to the particular type of fraction pair and indicated the deployment of specific comparison strategies. These results demonstrate that participants employ eye movements systematically to support strategy use in fraction comparison. Participants showed a remarkable flexibility to adapt to the most efficient strategy on a trial-by-trial basis. Our results confirm the value of eye movement measurements in the exploration of strategic adaptation in complex tasks.     

Mental scanning of sonifications reveals flexible encoding of nonspeech sounds and a universal per-item scanning cost

A mental scanning paradigm was used to examine the representation of nonspeech sounds in working memory. Participants encoded sonifications – nonspeech auditory representations of quantitative data – as either verbal lists, visuospatial images, or auditory images. The number of tones and overall frequency changes in the sonifications were also manipulated to allow for different hypothesized patterns of reaction times across encoding strategies. Mental scanning times revealed different patterns of reaction times across encoding strategies, despite the fact that all internal representations were constructed from the same nonspeech sound stimuli. Scanning times for the verbal encoding strategy increased linearly as the number of items in the verbal representation increased. Scanning times for the visuospatial encoding strategy were generally slower and increased as the metric distance (derived metaphorically from frequency change) in the mental image increased. Scanning times for the auditory imagery strategy were faster and closest to the veridical durations of the original stimuli. Interestingly, the number of items traversed in scanning a representation significantly affected scanning times across all encoding strategies. Results suggested that nonspeech sounds can be flexibly represented, and that a universal per-item scanning cost persisted across encoding strategies. Implications for cognitive theory, the mental scanning paradigm, and practical applications are discussed.     

小学生分数比较中的加工模式:基于反应时和口语报告的研究

选取北京和四川两地53名小学六年级学生分别完成同分子、同分母与异分子异分母三类分数比较任务,收集被试口语报告的策略作为直接证据,并以分数大小和分数距离对反应时的回归分析结果作为间接证据,共同探究被试在分数比较任务中的加工模式,结果发现,（1）在三类分数比较中,被试均采用成分加工模式而非整体加工模式;（2）口语报告的策略与反应时回归分析的结果并不完全吻合,从侧面证明了原有研究方法的不稳定性。     

How does coping help people resist lapses during smoking cessation?

OBJECTIVES: To determine whether types of coping strategies have differential effects on preventing lapses and lowering urge levels and to investigate mechanisms by which coping strategies prevent lapses during smoking cessation. DESIGN: Sixty-one respondents performed ecological momentary assessment using palm-top computers and tape recorders to report their coping strategies and urge levels before and after temptations to smoke. Multilevel linear regression models were used to compare the effects of individual strategy types with the average strategy. MAIN OUTCOME MEASURES: Lapses versus resisted temptations and changes in urge levels. RESULTS: Number of strategies significantly predicted resisting smoking and change in urge levels. Compared with the effect of the average strategy, movement/exercise was marginally worse at preventing lapses, and food/drink was marginally related to higher postcoping urge levels. CONCLUSION: Although using multiple coping strategies helps people resist the urge to smoke, no particular coping strategy works better than any other. Coping strategies prevent lapses by reducing high urge levels during temptations.     

Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds?

This study tested whether 10- and 12-year-olds who can correctly compare the magnitudes of fractions with common components access the magnitudes of the whole fractions rather than only compare the magnitudes of their components. Time for comparing two fractions was predicted by the numerical distance between the whole fractions, suggesting an access to their magnitude. In addition, we tested whether the relative magnitude of the denominator interferes with the processing of the fraction magnitude and, thus, needs to be inhibited. Response times were slower for fractions with common numerators than for fractions with common denominators, indicating an interference of the magnitude of the denominators with the selection of the larger fraction. A negative priming effect was shown for the comparison of natural numbers primed by fractions with common numerators, suggesting an inhibition of the selection of the larger denominator during the comparison of fractions. In conclusion, children who can correctly compare fractions with common components can access the magnitude of the whole fractions but remain sensitive to the interference of the relative magnitude of the denominators. This study highlights the fact that beyond the interference of natural number knowledge at the conceptual level (called the “whole number bias” by Ni & Zhou, 2005), children need to manage the interference of the magnitude of the denominators (Stroop-like effect).     

Use of a self-regulated strategy intervention to improve word problem-solving skills of students with mild disabilities

We investigated the effects of self-regulated strategy instruction on the ability of four elementary-school students with mild handicaps (2 with learning disabilities [LD] and 2 with mild mental retardation [MMR]) to perform 4 types of addition and subtraction word problems. We used a multiple baseline design and a self-instructional strategy to address story problem solving. Results indicated that the strategy was effective in increasing the number of correctly solved word problems. Gains were maintained across a 6 and 8 week follow-up check. The study supports previous research indicating that cognitive strategies are effective for students with LD and suggests that cognitive strategies are appropriate for students with MMR.     

Fractions but not negative numbers are represented on the mental number line

The present study is the first to directly compare numerical representations of positive numbers, negative numbers and unit fractions. The results show that negative numbers and unit fractions were not represented in the same way. Distance effects were found when positive numbers were compared with fractions but not when they were compared with negative numbers, thus suggesting that unit fractions but not negative numbers were represented on the number line with positive numbers. As indicated by the semantic congruity effect, negative numbers were perceived to be small, positive numbers were perceived as large, while unit fractions were perceived neither as large nor small. Comparisons between negative numbers were faster than between unit fractions, possibly due to the smaller differences between the holistic magnitudes of the unit fractions. Finally, comparing unit fractions to 1 was faster than comparing them to 0, consistent with the idea that unit fractions are perceived as entities smaller than 1 (Kallai & Tzelgov, 2009). The results are consistent with the idea of a mental division between numbers that represent a quantity (positive numbers and unit fractions) and those that do not (negative numbers).     

Solving Difference Problems: Wording Primes Coordination

In this study, we investigate the effect of problem text on solving three types of difference problems by first graders. COMPARE problems were significantly harder than EQUALIZE and WON'T GET problems. In terms of solution strategies, EQUALIZE problems were generally solved by an ADD-ON strategy; and the WON'T GET problems, by a MATCH strategy, which reflected the coordination of two mental number lines as the problem representation. No one strategy was used more often for the COMPARE problems, indicating that the COMPARE wording did not facilitate such coordination. These results suggest the coordination of mental number lines as the crucial component in solving difference problems.     

An integrated theory of whole number and fractions development

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores, that fraction magnitude representations account for substantial variance in mathematics achievement test scores beyond that explained by fraction arithmetic proficiency, and that developing effective strategies plays a key role in improved knowledge of fractions. Theoretical and instructional implications are discussed.     

Better control with less effort: The advantage of using focused-breathing strategy over focused-distraction strategy on thought suppression

It has been suggested that unwanted thoughts usually intrude during mind wandering due to a shortage of mental resources. However, strategies for suppressing such thoughts have never been examined from a mind wandering perspective. Here, we compare the effectiveness of two types of attention distraction strategies that either redirect users’ attention to their own breathing (focused-breathing strategy, FBS) or to a mental image (focused-distraction strategy, FDS) as related to working memory capacities. Eighty-two undergraduates were randomly assigned into a FBS or FDS group. They completed a concentration task and a thought suppression task, in which mind wandering and thought intrusions were each measured. Our results support the hypothesis that mind wandering is positively correlated to thought intrusions and shows that FBS is more effective than FDS in reducing mind wandering and thought intrusions. Moreover, in contrast to FDS, the effect of FBS is independent of users’ mental resources.     

Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.     

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction–decimal mixed pairs and decimal–fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.     

Finger-digit compatibility in Arabic numeral processing

Finger-digit response compatibility was tested by asking participants to identify Arabic digits by pressing 1 of 10 keys with all 10 fingers. The direction of the finger-digit mapping was varied by manipulating the global direction of the hand-digit mapping as well as the direction of the finger-digit mapping within each hand (in each case, from small to large digits, or the reverse). The hypothesis of a left-to-right mental number line predicted that a complete left-to-right mapping should be easier whereas the hypothesis of a representation based on finger counting predicted that a counting-congruent mapping should be easier. The results show that when all 10 fingers are used to answer, a mapping congruent with the prototypical finger-counting strategy reported by the participants leads to better performance than does a mapping congruent with a left-to-right oriented mental number line, both in palm-down and palm-up postures of the hands, and they demonstrate that finger-counting strategies influence the way that numerical information is mentally represented and processed.     

Finger–digit compatibility in Arabic numeral processing

Finger–digit response compatibility was tested by asking participants to identify Arabic digits by pressing 1 of 10 keys with all 10 fingers. The direction of the finger–digit mapping was varied by manipulating the global direction of the hand–digit mapping as well as the direction of the finger–digit mapping within each hand (in each case, from small to large digits, or the reverse). The hypothesis of a left-to-right mental number line predicted that a complete left-to-right mapping should be easier whereas the hypothesis of a representation based on finger counting predicted that a counting-congruent mapping should be easier. The results show that when all 10 fingers are used to answer, a mapping congruent with the prototypical finger-counting strategy reported by the participants leads to better performance than does a mapping congruent with a left-to-right oriented mental number line, both in palm-down and palm-up postures of the hands, and they demonstrate that finger-counting strategies influence the way that numerical information is mentally represented and processed.     

Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fractions

This study investigated adults' ability to compare the magnitude of fractions without common components (e.g., 5/7 and 3/8), and the representation accessed in that process. We hypothesized that the absence of common components would enhance access to the magnitude of the fractions (i.e., a holistic representation) rather than a direct comparison of the numerators or the denominators. This hypothesis was tested in four between-subject conditions. Two types of experimental pairs were used that differed in the congruity of the magnitude of the denominator and the magnitude of the fraction. Each type of experimental pair was presented either alone or with filler pairs that introduced variability into the congruity of the components. In all four conditions, accuracy was above chance and the effect of the distance between the fractions on response times was significant, indicating an access to the magnitude of the fractions. Nevertheless, the variability of the congruity of the components had also a significant effect on performance, suggesting that the relative magnitude of the components was also processed. In conclusion, the representation of the fraction magnitude is hybrid, rather than purely holistic, in a magnitude-comparison task on fractions without common components.     

Girls who use "masculine" problem-solving strategies on a spatial task: proposed genetic and environmental factors.

This study investigated strategy and performance differences between right-handed boys and girls on a mental rotation task. Based on predictions from Casey and Brabeck's (1990) theory of sex differences, the study was also designed to identify a target group of right-handed girls with the optimal combination of genetic and environmental factors (high math/science achievers with nonright-handed immediate relatives). They were predicted to show strategies and performance more similar to those of the boys than to those of both the low math/science achieving girls and the high math/science girls with all right-handed immediate relatives (predicted to have the nonoptimal genotype). Strategy preference was measured using selective interference, whereby subjects solved mental rotation items concurrently with either verbal or visual-spatial interference tasks. Group comparisons were made on the amount of decrement in mental rotation performance as a result of the two types of interference tasks. This provided a basis for comparing the groups on the use of visual-spatial or verbal strategies on the mental rotation task. The boys: (1) did not show a significant advantage over the girls on the mental rotation items, but (2) did depend more on visual-spatial strategies than the girls, and (3) depended less on verbal strategies than the girls. The target girls: (1) outperformed the low math/science achieving girls on the mental rotation items and did not show a significant advantage over the other high math/science group, (2) depended more on visual-spatial strategies than both the other two groups of girls, and (3) depended less on verbal strategies than the low math/science girls, while showing no significant difference compared to the nonoptimal high math/science girls. Examining within-group differences, the boys preferred visual-spatial strategies, while the girls in both the nontarget groups preferred verbal ones. However, for the target girls, no within-subject strategy differences were found. The present findings support the theory that, like the boys, the target girls depend more on visual-spatial strategies than do other girls. It is possible that the target girls use a combination of visual-spatial and verbal strategies when solving mental rotation tasks.     

Response selection strategies and realism of confidence judgments

Two studies were conducted to examine how response selection strategy is related to confidence ratings and to performance on general knowledge questions. In both studies subjects were asked to answer 80 general knowledge questions and to rate their confidence in the correctness of the answer selected. A pilot study, in which subjects thought aloud while answering general knowledge questions, was carried out to identify different response selection strategies. In the first study, 40 subjects were asked to indicate which of four strategies (immediate recognition, inference, intuition, or guessing) they used for selecting an answer. In Study 2, think aloud reports from 20 subjects were coded into the same four strategies. The distribution of strategies differed between the studies, but there were very similar relations among strategy, confidence, and correctness of answer in the two studies. Response selection strategy was related to correctness of answer when confidence was partialed out. More specifically, immediate recognition was associated with higher proportion correct than with the other strategies. It was also found that ratings of how difficult the knowledge questions were to fellow students of the subjects were on a much more realistic level than the confidence ratings were. It is concluded that people could improve their confidence judgments by taking into account (a) how difficult a question is to other people, and (b) the response selection strategy used for answering the question.     

The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks.