Dissociation between magnitude comparison and relation identification across different formats for rational numbers

ABSTRACT

The present study examined whether a dissociation among formats for rational numbers (fractions, decimals, and percentages) can be obtained in tasks that require comparing a number to a non-symbolic quantity (discrete or else continuous). In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of ratio represented by the number. The magnitude task was performed more quickly using decimals (for both quantity types), whereas the relation task was performed more accurately with fractions (but only when the image showed discrete entities). The pattern observed for percentages was very similar to that for decimals. A dissociation between magnitude comparison and relational processing with rational numbers can be obtained when a symbolic number must be compared to a non-symbolic display.     

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.     

Parts and 'holes': gaps in rational number sense among children with vs. without mathematical learning disabilities

Many middle-school students struggle with decimals and fractions, even if they do not have a mathematical learning disability (MLD). In the present longitudinal study, we examined whether children with MLD have weaker rational number knowledge than children whose difficulty with rational numbers occurs in the absence of MLD. We found that children with MLD failed to accurately name decimals, to correctly rank order decimals and/or fractions, and to identify equivalent ratios (e.g. 0.5 = 1/2); they also 'identified' incorrect equivalents (e.g. 0.05 = 0.50). Children with low math achievement but no MLD accurately named decimals and identified equivalent pairs, but failed to correctly rank order decimals and fractions. Thus failure to accurately name decimals was an indicator of MLD; but accurate naming was no guarantee of rational number knowledge - most children who failed to correctly rank order fractions and decimals tests passed the naming task. Most children who failed the ranking tests at 6th grade also failed at 8th grade. Our findings suggest that a simple task involving naming and rank ordering fractions and decimals may be a useful addition to in-class assessments used to determine children's learning of rational numbers.     

Making Reading More Difficult: A Degraded Text Microworld for Teaching Reading Comprehension Strategies

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students’ understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The results showed that the idea of discreteness (i.e., that fractions and decimals had “successors” like natural numbers) was robust in all age groups; that students tended to believe that the intermediate numbers must be of the same type as the interval endpoints (i.e., only decimals between decimals and fractions between fractions); and that the type of interval endpoints (natural numbers, decimals, or fractions) influenced students’ judgments of the number of intermediate numbers in those intervals. We interpret these findings within the framework theory approach to conceptual change.     

Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective

ABSTRACT

Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities (tulips, roses) are aligned with addition, whereas certain functional relations between entities (tulips, vases) are aligned with division. Similarly, discreteness vs. continuity of quantities (marbles, water) is aligned with different formats for rational numbers (fractions and decimals, respectively). These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook analyses revealed semantic alignments for arithmetic word problems, but not for rational numbers. Nonetheless, Russian college students showed semantic alignments both for arithmetic operations and for rational numbers. Since Russian students exhibit semantic alignments for rational numbers in the absence of exposure to examples in school, such alignments likely reflect intuitive understanding of mathematical representations of real-world situations.     

Understanding the real value of fractions and decimals

Understanding fractions and decimals is difficult because whole numbers are the most frequently and earliest experienced type of number, and learners must avoid conceptualizing fractions and decimals in terms of their whole-number components (the "whole-number bias"). We explored the understanding of fractions, decimals, two-digit integers, and money in adults and 10-year-olds using two number line tasks: marking the line to indicate the target number, and estimating the numerical value of a mark on the line. Results were very similar for decimals, integers, and money in both tasks for both groups, demonstrating that the linear representation previously shown for integers is also evident for decimals already by the age of 10. Fractions seem to be "task dependent" so that when asked to place a fractional value on a line, both adults and children displayed a linear representation, while this pattern did not occur in the reverse task.     

Understanding the real value of fractions and decimals

Understanding fractions and decimals is difficult because whole numbers are the most frequently and earliest experienced type of number, and learners must avoid conceptualizing fractions and decimals in terms of their whole-number components (the “whole-number bias”). We explored the understanding of fractions, decimals, two-digit integers, and money in adults and 10-year-olds using two number line tasks: marking the line to indicate the target number, and estimating the numerical value of a mark on the line. Results were very similar for decimals, integers, and money in both tasks for both groups, demonstrating that the linear representation previously shown for integers is also evident for decimals already by the age of 10. Fractions seem to be “task dependent” so that when asked to place a fractional value on a line, both adults and children displayed a linear representation, while this pattern did not occur in the reverse task.     

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.     

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college‐educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse primes for the equation that immediately followed it (e.g., 4 × 3/4 = 3 followed by 3 × 8/6 = 4). Students with relatively high math ability showed relational priming (speeded solution times to the second of two successive relationally related fraction equations) both with and without high perceptual similarity (Experiment 2). Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers (e.g., 4 × 3/4 = 3 followed by 3 × 4/3 = 4). Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers.     

Early Developmental Trajectories Toward Concepts of Rational Numbers

Difficulties with rational numbers have been explained by a natural number bias, where concepts of natural numbers are inappropriately applied to rational numbers. Overcoming this difficulty may require a radical restructuring of previous knowledge. In order to capture this development, we examined third- to fifth-grade students' understanding of the size of rational numbers across a 1-year period. On three occasions, students answered a test about the size of decimals and fractions. Latent transition analysis allowed profiling five different rational number size concepts and mapping students' developmental trajectories during the early learning of this topic. Development from natural number-based reasoning to mathematically correct understanding was almost nonexistent during this time period. Instead, students appear to move through intermediary concepts. Locating these temporary and mediating number concepts supports the idea of a slow and gradually developing conceptual change in rational numbers.     

Comparative visual search: a difference that makes a difference

In this article we present a new experimental paradigm: comparative visual search. Each half of a display contains simple geometrical objects of three different colors and forms. The two display halves are identical except for one object mismatched in either color or form. The subject's task is to find this mismatch. We illustrate the potential of this paradigm for investigating the underlying complex processes of perception and cognition by means of an eye-tracking study. Three possible search strategies are outlined, discussed, and reexamined on the basis of experimental results. Each strategy is characterized by the way it partitions the field of objects into "chunks." These strategies are: (i) Stimulus-wise scanning with minimization of total scan path length (a "traveling salesman" strategy), (ii) scanning of the objects in fixed-size areas (a "searchlight" strategy), and (iii) scanning of object sets based on variably sized clusters defined by object density and heterogeneity (a "clustering" strategy). To elucidate the processes underlying comparative visual search, we introduce besides object density a new entropy-based measure for object heterogeneity. The effects of local density and entropy on several basic and derived eye-movement variables clearly rule out the traveling salesman strategy, but are most compatible with the clustering strategy.     

Relational framework improves transitive inference across age groups

Transitive inference is a complex task, conducive to the use of multiple strategies. We investigated whether transitive inference accuracy can be improved by biasing strategy choice towards a proposition-based approach that relies on the extraction of relations among stimuli. We biased strategy choice by using familiar stimuli with known relations that tap prior knowledge. Semantic information led to increased accuracy for younger and older adults, and increased awareness of stimulus relations. Increased age was associated with reduced awareness. Awareness accounted for the variability in performance accuracy to a greater extent than age, as aware older and younger adults showed similar accuracies on all conditions. The current work indicates that age differences in performance can be minimized by providing semantically meaningful stimuli that bias participants to use a relational proposition-based approach.     

Eye movements and bisection behavior in spatial neglect syndrome: representational biases induced by the segment length and spatial dislocation of the stimulus

The present study explored behavioral and eye-movement measures in unilateral neglect patients in response to online bisection task (unfilled gap line). Two different tasks supported the bisection performance, a pointing and a grasping strategy. It was explored whether these different strategies may influence subjects' behavioral and eye-movement measures in response to different segment features: segment length (from shorter to longer) and segment spatial dislocation (from right to left spatial location). Consistent spatial biases were found for both bisection responses, fixation count, and duration, as well as for the first fixation count in case of pointing task. An "extreme-left" gradient effect was suggested and discussed, with patients' behavioral and eye measures more impaired. On the contrary, the patients' performance overlaps with the controls' one in case a grasping task. The direct link of visual pointing and grasping strategy, respectively, with the two cortical ventral and dorsal pathways was adduced to explain our results.     

Eye movements and bisection behavior in spatial neglect syndrome: representational biases induced by the segment length and spatial dislocation of the stimulus

The present study explored behavioral and eye-movement measures in unilateral neglect patients in response to online bisection task (unfilled gap line). Two different tasks supported the bisection performance, a pointing and a grasping strategy. It was explored whether these different strategies may influence subjects’ behavioral and eye-movement measures in response to different segment features: segment length (from shorter to longer) and segment spatial dislocation (from right to left spatial location). Consistent spatial biases were found for both bisection responses, fixation count, and duration, as well as for the first fixation count in case of pointing task. An “extreme-left” gradient effect was suggested and discussed, with patients’ behavioral and eye measures more impaired. On the contrary, the patients’ performance overlaps with the controls’ one in case a grasping task. The direct link of visual pointing and grasping strategy, respectively, with the two cortical ventral and dorsal pathways was adduced to explain our results.     

类比推理的眼动研究:揭示个体类比推理策略发展的有效手段

近年来,研究者利用眼动技术具有高时间精度的优势,探明不同年龄群体完成类比推理过程的眼动模式特点并得出其在进行类比推理时所使用的策略。基于类比推理的眼动研究发现了三种典型的类比推理策略——项目优先策略、结构匹配策略和语义限制策略。成人更多表现为项目优先策略,儿童更多表现为语义限制策略。未来研究可以优化类比推理眼动指标,尤其是全局扫视路径的计算方法,并重点关注特殊群体的类比推理眼动模式以及关注类比推理策略与其他认知能力的交互作用。     

Never getting to zero: Elementary school students' understanding of the infinite divisibility of number and matter

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infinitely divisible in the sense that they can be repeatedly divided without ever getting to zero. Their conceptualizing number as infinitely divisible was strongly related to their having a model of fraction notation based on division and to their successful judgment of the relative magnitudes of fractions and decimals. In addition, their understanding number as infinitely divisible was strongly related to their understanding physical quantities as infinitely divisible. These results support a conceptual change account of knowledge acquisition, involving two-way mappings between the domains of number and physical quantity.     

A validation of eye movements as a measure of elementary school children's developing number sense

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded during task solution reflect children's use of the number line. By means of a cross-sectional design with 66 children from Grades 1, 2, and 3, we show that eye-tracking data (a) reflect grade-related increase in estimation competence, (b) are correlated with the accuracy of manual answers, (c) relate, in Grade 2, to children's addition competence, (d) are systematically distributed over the number line, and (e) replicate previous findings concerning children's use of counting strategies and orientation-point strategies. These findings demonstrate the validity and utility of eye-tracking data for investigating children's developing number sense and estimation competence.     

视觉信息分布调节信息整合策略：来自眼动的证据

以空白单元格定位为实验任务,操纵序列点阵的分布,考察视觉短时记忆与视知觉的信息整合机制。实验一发现任务正确率与点阵1的圆点数呈近似U形关系。实验二发现在拐点前后被试的眼动模式从偏向于注视圆点位置转为偏向于注视空格位置。因此,视觉信息的分布模式调节整合策略,当处于视觉短时记忆中的圆点数在其容量之内时,采用图像—知觉整合策略,反之,则采用转换—比较策略。     

Encoding Strategies in Primary School Children: Insights From an Eye-Tracking Approach and the Role of Individual Differences in Attentional Control

The authors explored different aspects of encoding strategy use in primary school children by including (a) an encoding strategy task in which children's encoding strategy use was recorded through a remote eye-tracking device and, later, free recall and recognition for target items was assessed; and (b) tasks measuring resistance to interference (flanker task) and inhibition of attention to task-irrelevant stimuli (distractibility). Results revealed that the ability to inhibit distraction and resist interference undergoes developmental changes between the ages of 7–10 years. At the same time, children's capability to strategically focus on task-relevant aspects also continues to improve in primary school years. Although there were substantial relationships between encoding strategies and later recognition, encoding strategies appeared to be unrelated to basic aspects of attentional control.     

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.