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).     

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.     

The mental representation of numerical fractions: real or integer?

Numerical fractions are commonly used to express ratios and proportions (i.e., real numbers), but little is known about how they are mentally represented and processed by skilled adults. Four experiments employed comparison tasks to investigate the distance effect and the effect of the spatial numerical association of response codes (SNARC) for fractions. Results showed that fractions were processed componentially and that the real numerical value of the fraction was not accessed, indicating that processing the fraction's magnitude is not automatic. In contrast, responses were influenced by the numerical magnitude of the components and reflected the simple comparison between numerators, denominators, and reference, depending on the strategy adopted. Strategies were used even by highly skilled participants and were flexibly adapted to the specific experimental context. In line with results on the whole number bias in children, these findings suggest that the understanding of fractions is rooted in the ability to represent discrete numerosities (i.e., integers) rather than real numbers and that the well-known difficulties of children in mastering fractions are circumvented by skilled adults through a flexible use of strategies based on the integer components.     

Eye gaze patterns reveal how we reason about fractions

ABSTRACT

Fractions are defined by numerical relationships, and comparing two fractions’ magnitudes requires within-fraction (holistic) and/or between-fraction (componential) relational comparisons. To better understand how individuals spontaneously reason about fractions, we collected eye-tracking data while they performed a fraction comparison task with conditions that promoted or obstructed different types of comparisons. We found evidence for both componential and holistic processing in this mixed-pairs task, consistent with the hybrid theory of fraction representation. Additionally, making within-fraction eye movements on trials that promoted a between-fraction comparison strategy was associated with slower responses. Finally, participants who performed better on a non-numerical test of reasoning took longer to respond to the most difficult fraction trials, which suggests that those who had greater facility with non-numerical reasoning attended more to numerical relationships. These findings extend prior research and support the continued investigation into the mechanistic links between numerical and non-numerical reasoning.     

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。     

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.     

两位数的整体与局部加工

关于两位数的加工方式有整体加工说和局部加工说,实验证据主要来自数字数量控制/主动加工任务。本研究主要考察在数字数量自动加工任务中两位数的加工方式。实验一要求被试完成数量大小比较和物理大小比较两个任务,实验二只要求被试完成物理大小比较任务。结果是在数量比较任务和物理比较任务中都存在显著的个位十位一致性效应和数量物理一致性效应,这表明在两位数的数量主动和自动加工任务中均存在整体加工和局部加工两种方式。     

3~5岁幼儿一位数大小比较的信息加工模式

主要探讨我国幼儿对数量大小比较的信息加工模式。实验1探讨幼儿对一位数大小比较的发展状况及其心理表征特点,被试为3岁、4岁与5岁幼儿各20人,要求被试对1-9两两进行大小比较,然后对不同年龄幼儿对比成绩进行比较与聚类分析。实验2进一步探讨幼儿对数字的语义编码情况及其与数的大小比较的关系,被试与实验1相同,要求被试对1-9每个数字作出大、中或小的编码,然后分析数字的语义编码成绩与大小比较成绩的关系。实验3采用因果设计,探讨幼儿关于数字的语义编码对他们关于数的大小判断的影响,被试为30名4岁幼儿,随机分成训练组与控制组,对训练组被试进行数字语义编码训练,然后比较两组被试大小比较的成绩。结果表明：(1)幼儿一位数大小比较直接受其对数的语义表征的影响;(2)随着年龄的增长,幼儿对数的表征逐步表现出离散聚类模式,相应地,对一位数大小比较的信息加工过程就表现为由无序的、随机的过程逐步发展成为层次编码比较的过程。     

Is adding 48 + 25 and 45 + 28 the same? How addend compatibility influences the strategy execution in mental addition

A recent study revealed that adults frequently start to add two two-digit numbers from the larger one, suggesting that addend magnitudes are compared at an early stage of processing. However, several studies showed that symbolic number comparison involves compatibility effects: Such numerical comparison is easier when the larger number also contains the larger unit (48_25) than in the opposite, incompatible case (45_28). In this context, whether the compatibility between tens and units across operands affects the execution of arithmetic-solving strategies remains an open question. In this study, we used two kinds of verbal protocols to assess how addend compatibility influences the implementation of magnitude-based strategies. We observed that participants started their computations from the larger operand more frequently when solving compatible additions than they did when solving incompatible ones. The presence of a compatibility effect extends the view that multidigit number processing is componential rather than holistic, even in an arithmetic task that did not explicitly require a number magnitude comparison. Further, the findings corroborate the notion that number magnitude is used in mental calculation and influences the way calculation strategies are implemented.     

Holistic representation of unit fractions

The present study examined the processing of unit fractions and the extent to which it is affected by context. Using a numerical comparison task we found evidence for a holistic representation of unit fractions when the immediate context of the fractions was emphasized, that is when the stimuli set included in addition to the unit fractions also the numbers 0 and 1. The holistic representation was indicated by the semantic congruity effect for comparisons of pairs of fractions and by the distance effect in comparisons of a fraction and 0 and 1. Consistent with previous results (Bonato, Fabbri, Umilta, & Zorzi, 2007) there was no evidence for a holistic representation of unit fractions when the stimulus set included only fractions. These findings suggest that fraction processing is context-dependent. Finally, the present results are discussed in the context of processing other complex numbers beyond the first decade.     

Negative numbers are generated in the mind

The goal of the present study was to disentangle two possible representations of negative numbers--the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from--100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.     

Mental representations in fraction comparison

In this study, we investigated the mental representations used in a fraction comparison task. Adults were asked to quickly and accurately pick the larger of two fractions presented on a computer screen and provide trial-by-trial reports of the types of strategies they used. We found that adults used a variety of strategies to compare fractions, ranging among just knowing the answer, using holistic knowledge of fractions to determine the answer, and using component-based procedures such as cross multiplication. Across all strategy types, regression analyses identified that reaction times were significantly predicted by numerical distance between fractions, indicating that the participants used a magnitude-based representation to compare the fraction magnitudes. In addition, a variant of the problem-size effect (e.g., Ashcraft, 1992) appeared, whereby reaction times were significantly predicted by the average cross product of the two fractions. This effect was primarily found for component-based strategies, indicating a role for strategy choice in the formation of mental representations of fractions.     

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.     

Developmental changes in the whole number bias

Many students’ knowledge of fractions is adversely affected by whole number bias, the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction's magnitude (ratio of numerator to denominator). Although whole number bias appears early in the fraction learning process and under speeded conditions persists into adulthood, even among mathematicians, little is known about its development. Performance with equivalent fractions indicated that between fourth and eighth grade, whole number bias decreased, and reliance on fraction magnitudes increased. These trends were present on both fraction magnitude comparison and number line estimation. However, analyses of individual children's performance indicated that a substantial minority of fourth graders did not show whole number bias and that a substantial minority of eighth graders did show it. Implications of the findings for development of understanding of fraction equivalence and for theories of numerical development are discussed.     

整数构成对分数加工的影响

本研究通过两个实验考察了整数构成对分数加工的影响。实验一重复了Bonato等(2007)的研究,要求25名被试比较1/5与1/1、1/2、1/3、1/4、1/6、1/7、1/8、1/9等分数的大小,结果发现被试采用了成分加工策略,即通过比较分母判断分数的大小。实验二改变了分数的整数构成,让24名被试比较1/5与1/1、2/4、3/9、4/16、4/24、3/21、2/16、1/9等分数的大小。这些分数与实验一中的分数实数值相等但整数构成不同。结果表明被试既采取了成分策略,又采取了整体策略,即比较分数的实数值,不过成分策略比整体策略的作用更大。这些结果表明,分数加工具有成分和整体两种策略,具体使用哪种策略与分数的整数构成密切相关。     

The mental representation of integers: an abstract-to-concrete shift in the understanding of mathematical concepts

Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of pairs of integers. Experiment 1 demonstrated an inverse distance effect: When comparing numbers across the zero boundary, people are faster when the numbers are near together (e.g., −1 vs. 2) than when they are far apart (e.g., −1 vs. 7). This result conflicts with a straightforward symbolic or analog magnitude representation of integers. We therefore propose an analog-x hypothesis: Mastering a new symbol system restructures the existing magnitude representation to encode its unique properties. We instantiate analog-x in a reflection model: The mental negative number line is a reflection of the positive number line. Experiment 2 replicated the inverse distance effect and corroborated the model. Experiment 3 confirmed a developmental prediction: Children, who have yet to restructure their magnitude representation to include negative magnitudes, use rules to compare negative numbers. Taken together, the experiments suggest an abstract-to-concrete shift: Symbolic manipulation can transform an existing magnitude representation so that it incorporates additional perceptual-motor structure, in this case symmetry about a boundary. We conclude with a second symbolic-magnitude model that instantiates analog-x using a feature-based representation, and that begins to explain the restructuring process.     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Holistic face processing is mature at 4 years of age: evidence from the composite face effect

Although it is acknowledged that adults integrate features into a representation of the whole face, there is still some disagreement about the onset and developmental course of holistic face processing. We tested adults and children from 4 to 6 years of age with the same paradigm measuring holistic face processing through an adaptation of the composite face effect [Young, A. W., Hellawell, D., & Hay, D. C. (1987). Configurational information in face perception. Perception, 16, 747-759]. In Experiment 1, only 6-year-old children and adults tended to perceive the two identical top parts as different, suggesting that holistic face processing emerged at 6 years of age. However, Experiment 2 suggested that these results could be due to a response bias in children that was cancelled out by always presenting two faces in the same format on each trial. In this condition, all age groups present strong composite face effects, suggesting that holistic face processing is mature as early as after 4 years of experience with faces.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

The representation of negative numbers: exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.