Fraction Magnitude Comparison Research Articles

Comparing fraction magnitudes: Adults’ verbal reports reveal strategy flexibility and adaptivity, but also bias

Many studies have used fraction magnitude comparison tasks to assess people’s abilities to quickly assess fraction magnitudes. However, since there are multiple ways to compare fractions, it is not clear whether people actually reason about the holistic magnitudes of the fractions in this task and whether they use multiple strategies in a flexible and adaptive way. We asked 72 adults to solve challenging fraction comparisons (e.g., 31/71 vs. 13/23) on a computer. In some of these comparisons, using benchmarks (i.e., reference numbers such as 1/2) was potentially beneficial. After each trial, participants provided verbal reports of their strategies. We found that participants used a large variety of strategies. The majority of strategies were holistic and relied on fraction magnitudes, and most of these strategies were based on benchmarks. Participants sometimes used gap comparison (i.e., comparing the differences between each fraction’s numerator and denominator), a heuristic that is not always valid and that does not rely on fraction magnitudes. Participants used strategies flexibly: they used many different strategies, they used highly efficient strategies most often, and they adapted their strategy use to features of the items. However, participants sometimes used gap comparison on items for which it did not yield the correct response, and this lack of adaptivity partly explained the “natural number bias” observed in this study.

Relational thinking: An overlooked component of executive functioning.

Relational thinking, the ability to represent abstract, generalizable relations, is a core component of reasoning and human cognition. Relational thinking contributes to fluid reasoning and academic achievement, particularly in the domain of math. However, due to the complex nature of many fluid reasoning tasks, it has been difficult to determine the degree to which relational thinking has a separable role from the cognitive processes collectively known as executive functions (EFs). Here, we used a simplified reasoning task to better understand how relational thinking contributes to math achievement in a large, diverse sample of elementary and middle school students (N = 942). Students also performed a set of ten adaptive EF assessments, as well as tests of math fluency and fraction magnitude comparison. We found that relational thinking was significantly correlated with each of the three EF composite scores previously derived from this dataset, albeit no more strongly than they were with each other. Further, relational thinking predicted unique variance in students' math fluency and fraction magnitude comparison scores over and above the three EF composites. Thus, we propose that relational thinking be considered an EF in its own right as one of the core, mid-level cognitive abilities that supports cognition and goal-directed behavior. RESEARCH HIGHLIGHTS: Relational thinking, the process of identifying and integrating relations, develops over childhood and is central to reasoning. We collected data from nearly 1000 elementary and middle schoolers on a test of relational thinking, ten standard executive function tasks, and two math tests. Relational thinking predicts unique variance in math achievement not accounted for by canonical EFs throughout middle childhood. We propose that relational thinking should be conceptualized as a core executive function that supports cognitive development and learning.

Number line unidimensionality is a critical feature for promoting fraction magnitude concepts

Children’s ability to estimate fractions on a number line is strongly related to algebra and overall high school math achievement, and number line training leads to better fraction magnitude comparisons compared with area model training. Here, we asked whether unidimensionality is necessary for the number line to promote fraction magnitude concepts and whether left-to-right orientation and labeled endpoints are sufficient. We randomly assigned second- and third-graders (N = 148) to one of four 15-min one-on-one, experimenter-led trainings. Three number line trainings had identical scripts, where the experimenter taught children to segment and shade the number line along the horizontal dimension. The number line conditions varied only in the vertical dimension of the training number line: pure unidimensional number line (17.5 cm horizontal line), hybrid unidimensional number line (17.5 × 0.6 cm rectangle), and square number line (17.5 × 17.5 cm). In the area model condition, children were taught to segment and shade a square (17.5 × 17.5 cm) along both dimensions. The conditions significantly differed in posttest fraction magnitude comparison accuracy (a transfer task), controlling for pretest accuracy, reading achievement, and age. In preregistered analyses, the hybrid unidimensional number line condition significantly outperformed the square area model condition and the square number line condition. In exploratory analyses accounting for training protocol fidelity, these results held and the pure unidimensional number line also outperformed the area model condition on fraction magnitude comparisons. We argue that unidimensionality is a critical feature of the number line for promoting fraction magnitude concepts because it aligns with a key concept—that real numbers, including fractions, can be ordered along a single dimension.

Accuracy of Area Model and Number Line Representations of Fractions for Students with Learning Disabilities

AbstractAssessment results are used to investigate relations between performance on a fraction number line estimation task and a circular area model estimation task for students with LD in Grades 6–8. Results indicate that students’ abilities to represent fractions on number lines and on circular area models are distinct skills. In addition, accurate fraction magnitude estimation using number lines was more strongly related to other fractions skills (e.g., fraction magnitude comparison) than accurate fraction magnitude estimation using circular area models. Our findings call for greater integration of the number line into early fractions instruction, and highlight the importance of explicitly teaching students to make connections between different visual representations of fractions. Implications for research and practice are presented.

Is a Picture Worth 1,000 Words? Investigating Fraction Magnitude Knowledge Through Analysis of Student Representations

The present study examines the features and quality of visual representations (VRs) created by middle school students with learning disabilities and difficulties in mathematics in response to a released fraction item from the National Assessment of Educational Progress (NAEP). Relations between VR quality and scores on other measures of fraction knowledge are also investigated. Results show that students used circular area models most frequently to represent the NAEP item, but used bar models most accurately. Based on results, bar models may be the most efficient and effective area model VRs for use in fractions instruction. Representation quality was associated with problem-solving accuracy, as well as with performance on fraction number line estimation and fraction magnitude comparison. Implications for practice are discussed.

Who uses more strategies? Linking mathematics anxiety to adults’ strategy variability and performance on fraction magnitude tasks

ABSTRACTAdults use a variety of strategies to reason about fraction magnitudes, and this variability is adaptive. In two studies, we examined the relationships between mathematics anxiety, working memory, strategy variability and performance on two fraction tasks: fraction magnitude comparison and estimation. Adults with higher mathematics anxiety had lower accuracy on the comparison task and greater percentage absolute error (PAE) on the estimation task. Unexpectedly, mathematics anxiety was not related to variable strategy use. However, variable strategy use was linked to more accurate magnitude comparisons, especially among adults with lower working memory performance or those who use mathematics less frequently, as well as lower PAE on the estimation task. These findings shed light on the role of strategy variability in fraction problem solving and demonstrate a link between mathematics anxiety and fraction magnitude reasoning, a key predictor of general mathematics achievement.

The number line is a critical spatial-numerical representation: Evidence from a fraction intervention.

Children's ability to place fractions on a number line strongly correlates with math achievement. But does the number line play a causal role in fraction learning or does it simply index more advanced fraction knowledge? The number line may be a particularly effective representation for fraction learning because its properties align with the desired mental representation and take advantage of preexisting spatial-numeric biases. Using a pretest-training-posttest design, we examined second and third graders' fraction learning in 3 conditions: number line training, area model training, and a non-numerical control. Children who received number line training improved at representing fractions with number lines, and children who received area model training improved at representing fractions with area models. Critically, only the number line training led to transfer to an untrained fraction magnitude comparison task. We conclude that the number line plays a causal role in children's fraction magnitude understanding, and is more beneficial than the widely used area model. (PsycINFO Database Record

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.

Strategy use and strategy choice in fraction magnitude comparison.

We examined, on a trial-by-trial basis, fraction magnitude comparison strategies of adults with more and less mathematical knowledge. College students with high mathematical proficiency used a large variety of strategies that were well tailored to the characteristics of the problems and that were guaranteed to yield correct performance if executed correctly. Students with less mathematical proficiency sometimes used strategies similar to those of the mathematically proficient students, but often used flawed strategies that yielded inaccurate performance. As predicted by overlapping waves theory, increases in accuracy and speed were related to differences in strategy use, strategy choice, and strategy execution. When asked to choose the best strategy from among 3 possibilities-the strategy the student originally used, a correct alternative, and an incorrect alternative-students with lower fraction knowledge rarely switched from an original incorrect strategy to a correct alternative. This finding suggests that use of poor fraction magnitude comparison strategies stems in large part from lack of conceptual understanding of the requirements of effective strategies, rather than difficulty recalling or generating such strategies. (PsycINFO Database Record

THE RELATION BETWEEN LEARNERS’ SPONTANEOUS FOCUSING ON QUANTITATIVE RELATIONS AND THEIR RATIONAL NUMBER KNOWLEDGE

IntroductionRational Number KnowledgeThere is a broad agreement in the literature a good understanding of rational numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction number line estimation, 0-5 fraction number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded fifth graders' rational number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, number knowledge, family income, and family education.Despite the critical importance of a good rational number knowledge, a large body of literature reported children and even adults have a lot of difficulties dealing with various aspects of rational numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for rational numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with rational number tasks are often - at least in part - attributed to the number (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of whole number bias), which is the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to rational numbers in the classroom, they have already formed an idea of what a number is. This idea is based on their experiences (both in daily life and in school) with natural numbers. Once the learners are then instructed about rational numbers, the properties of natural numbers are not always applicable anymore, leading to problems and misconceptions with rational numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in rational number tasks where reasoning purely in terms of natural numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in rational number tasks where reasoning in terms of natural numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural number bias reports three main aspects elicit such systematic errors. The first aspect relates to the density of the set of rational numbers. While natural numbers are characterized by a discrete structure (one can always indicate which number follows a given number; for example after 13 comes 14), rational numbers are characterized by a dense structure (you cannot say which number comes next, because between any two given rational numbers are always infinitely many other rational numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of rational numbers. Research indicates errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural numbers, longer decimals are larger, shorter decimals are smaller, or that a fraction's numerical value always increases when its denominator, numerator, or both increase (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …

Supplemental Material for Strategy Use and Strategy Choice in Fraction Magnitude Comparison

Supplemental Material for Strategy Use and Strategy Choice in Fraction Magnitude Comparison

The representation of fraction magnitudes and the whole number bias reconsidered

Two experiments investigated the whole number bias in the representation of fraction magnitudes with adults. A fraction magnitude comparison task was used where half of the comparisons were consistent with whole number ordering and the other half were not. Distance effects were found in Experiment 1 indicating that participants were comparing the magnitude of the whole fraction rather than just the parts. However, accuracy and response time also depended on the comparisons' consistency with whole number ordering. Experiment 2 manipulated the distance between the fraction pairs and showed that the whole number effect was strongest when the distance between the fraction pairs was very small. The results suggest that even skilled adults do not always have direct access to a fraction's magnitude on the number line. When the magnitudes are especially close together, adults may rely on alternative implicit or explicit strategies, such as examining the whole number parts, to evaluate the comparison.

Adaptive processing of fractions — Evidence from eye-tracking

Recent evidence indicated that fraction pair type determined whether a particular fraction is processed holistically, componentially or in a hybrid manner. Going beyond previous studies, we investigated how participants adapt their processing of fractions not only to fraction type, but also to experimental context. To examine adaptation in fraction processing, we recorded participants' eye-fixation behaviour in a fraction magnitude comparison task.Participants' eye fixation behaviour indicated componential processing of fraction pairs with common components for which the decision-relevant components are easy to identify. Importantly, we observed that fraction processing was adapted to experimental context: Evidence for componential processing was stronger, when experimental context allowed valid expectations about which components are decision-relevant.Taken together, we conclude that fraction processing is adaptive beyond the comparison of different fraction types, because participants continuously adjust to the experimental context in which fractions are processed.

Learning and instruction

Two experiments investigated the whole number bias in the representation of fraction magnitudes with adults. A fraction magnitude comparison task was used where half of the comparisons were consistent with whole number ordering and the other half were not. Distance effects were found in Experiment 1 indicating that participants were comparing the magnitude of the whole fraction rather than just the parts. However, accuracy and response time also depended on the comparisons' consistency with whole number ordering. Experiment 2 manipulated the distance between the fraction pairs and showed that the whole number effect was strongest when the distance between the fraction pairs was very small. The results suggest that even skilled adults do not always have direct access to a fraction's magnitude on the number line. When the magnitudes are especially close together, adults may rely on alternative implicit or explicit strategies, such as examining the whole number parts, to evaluate the comparison. © 2014 Published by Elsevier Ltd.