Fraction Comparison Task Research Articles

Disentangling magnitude processing, natural number biases, and benchmarking in fraction comparison tasks: A person-centered Bayesian classification approach

Research on fraction comparison shows that students often follow biased comparison strategies, in particular such strategies that build on their knowledge of natural numbers. On the other hand they also apply successful comparison strategies such as benchmarking or fraction magnitude processing. Which strategies are applied or even combined depends on the students’ knowledge and on the task type. To investigate these complex relationships, we developed a balanced 2 × 2-dimensional itemset (congruent vs. incongruent items; benchmarking vs. non-benchmarking items) and a Bayesian classification of individual students’ performance (solution patters, response time, and individual distance effect), which we applied to an assessment of N = 350 sixth graders. We could show that the classification of the students with respect to possible solution strategies matched our hypotheses: We could replicate existing patterns and found additional composite strategies such as ‘benchmarking or bias‘ with a bias only in solution rates of non-benchmark items. In further analyses we found ‘benchmarking or suppressed bias-strategies (i.e., a bias in problem solving time of non-benchmarking items). Our study extends previous knowledge on individual strategies in fraction comparison and proposes a new person-centered approach to classify individual student profiles even with small profile sizes.

Testing the whole number interference hypothesis: Contributions of inhibitory control and whole number knowledge to fraction understanding.

The present study tests two predictions stemming from the hypothesis that a source of difficulty with rational numbers is interference from whole number magnitude knowledge. First, inhibitory control should be an independent predictor of fraction understanding, even after controlling for working memory. Second, if the source of interference is whole number knowledge, then it should hinder fraction understanding. These predictions were tested in a racially and socioeconomically diverse sample of U.S. children (N = 765; 337 female) in Grades 3 (ages 8-9), 5 (ages 10-11), and 7 (ages 12-13) who completed a battery of computerized tests. The fraction comparison task included problems with both shared components (e.g., 3/5 > 2/5) and distinct components (e.g., 2/3 > 5/9), and problems that were congruent (e.g., 5/6 > 3/4) and incongruent (e.g., 3/4 > 5/7) with whole number knowledge. Inhibitory control predicted fraction comparison performance over and above working memory across component and congruency types. Whole number knowledge did not hinder performance and instead positively predicted performance for fractions with shared components. These results highlight a role for inhibitory control in rational number understanding and suggest that its contribution may be distinct from inhibiting whole number magnitude knowledge. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number of decimal digits. The decimal advantage continued to be present for both magnitude comparison and estimation tasks, indicating that the greater accuracy with decimals was not limited to decimals with equal numbers of decimal digits, though unequal numbers of decimal digits did impact performance with decimals on both magnitude comparison and number line estimation tasks. Implications for understanding numerical development and education are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

The developmental relationship between nonsymbolic and symbolic fraction abilities

A fundamental research question in quantitative cognition concerns the developmental relationship between nonsymbolic and symbolic quantitative abilities. This study examined this developmental relationship in abilities to process nonsymbolic and symbolic fractions. There were 99 6th graders (Mage = 11.86 years), 101 10th graders (Mage = 15.71 years), and 102 undergraduate and graduate students (Mage = 21.97 years) participating in this study, and their nonsymbolic and symbolic fraction abilities were measured with nonsymbolic and symbolic fraction comparison tasks, respectively. Nonsymbolic and symbolic fraction abilities were significantly correlated in all age groups even after controlling for the ability to process nonsymbolic absolute quantity and general cognitive abilities, including working memory and inhibitory control. Moreover, the strength of nonsymbolic–symbolic correlations was higher in 6th graders than in 10th graders and adults. These findings suggest a weakened association between nonsymbolic and symbolic fraction abilities during development, and this developmental pattern may be related with participants’ increasing proficiency in symbolic fractions.

Unraveling the role of math anxiety in students' math performance.

Math anxiety (MA; i.e., feelings of anxiety experienced when being confronted with mathematics) can have negative implications on the mental health and well-being of individuals and is moderately negatively correlated with math achievement. Nevertheless, ambiguity about some aspects related to MA may prevent a fathomed understanding of this systematically observed relationship. The current study set out to bring these aspects together in a comprehensive study. Our first focus of interest was the multi-component structure of MA, whereby we investigated the relationship between state- and trait-MA and math performance (MP) and whether this relation depends on the complexity of a math task. Second, the domain-specificity of MA was considered by examining the contribution of general anxiety (GA) and MA on MP and whether MA also influences the performance in non-math tasks. In this study, 181 secondary school students aged between 16 and 18 years old were randomly presented with four tasks (varying in topic [math/non-math] and complexity [easy/difficult]). The math task was a fraction comparison task and the non-math task was a color comparison task, in which specific indicators were manipulated to develop an easy and difficult version of the tasks. For the first research question, results showed a moderate correlation between state- and trait-MA, which is independent of the complexity of the math task. Regression analyses showed that while state-MA affects MP in the easy math task, it is trait-MA that affects MP in the difficult math task. For the second research question, a high correlation was observed between GA and MA, but regression analyses showed that GA is not related to MP and MA has no predictive value for performance in non-math tasks. Taken together, this study underscores the importance of distinguishing between state and trait-MA in further research and suggests that MA is domain-specific.

Fraction mapping and fraction comparison skills among grade 4 Chinese students: An error analysis.

Mapping fraction symbols to magnitudes is easier for students to master than comparing fraction magnitudes. Fraction mapping assesses students' understanding of part-whole interpretations of fractions; fractions represent the parts of an object or set of objects. Fraction comparison assesses students' understanding of measurement interpretations of fractions; a fraction is a single numerical quantity, not a combination of two whole numbers. To examine and compare the types of errors made by emergent fraction learners on fraction mapping and comparison tasks. Grade 4 Chinese students (N=1036; 577 boys; Mage =9.9years). We examined performance and identified errors on fraction mapping and comparison tasks. For mapping, students converted pictorial representations into fraction notation. For comparison, they chose the larger of two symbolic fractions. Consistent with curriculum expectations, most students successfully mapped pictorial representations to fraction notation. In contrast, few students were able to accurately compare fraction magnitudes. Within each task, students' errors were consistent across trials, suggesting that they applied systematic but incorrect procedures. However, errors were not consistent between tasks and the correlation between mapping and comparison performance was weak. Emergent fraction learners can acquire part-whole knowledge of fractions without acquiring measurement interpretations of fractions. Moreover, misconceptions about different interpretations of fractions need not overlap. Awareness of the types of errors that students make can assist educators in identifying misconceptions early so that students do not build their fraction knowledge on erroneous beliefs.

The Role of the Inhibition of Natural Number Based Reasoning and Strategy Switch Cost in a Fraction Comparison Task

Previous research amply showed the importance of a good fraction understanding but also people’s lack of fraction understanding. It is therefore important to investigate the cognitive processes that underlie reasoning with fractions. The present study investigated the role of inhibition and switch costs in fraction comparison tasks. Participants solved a fraction comparison task that alternated between 4 items congruent and 4 items incongruent with natural number reasoning. This allowed to not only investigate congruency switch effects, but also inhibition, given that inhibition was experimentally increased by the prolonged exposure to incongruent trials. Based on data of seventh graders, the present study showed that inhibition does not only play a role in learners’ general mathematics achievement, but also in specific areas of mathematics, such as fractions. Moreover, a switch cost was found in the lower accuracy rates and higher reaction times needed to correctly solve switch items compared to non-switch items.

A Study on Congruency Effects and Numerical Distance in Fraction Comparison by Expert Undergraduate Students

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean engineering undergraduate students answered a computerized fraction comparison task, while their answers and response times were recorded. Task items were selected according to a number of mathematically and/or cognitively relevant characteristics: (a) whether the fractions to be compared shared a common component, (b) the numerical distance between fractions, and (c) the applicability of two strategies to answer successfully: a congruency strategy (a fraction is larger if it has larger natural number components than another) and gap thinking (a fraction is larger if it is missing fewer pieces than another to complete the whole). In line with previous research, our data indicated that the congruency strategy is inadequate to describe participants’ performance, as congruent items turned out to be more difficult than incongruent ones when fractions had no common component. Although we hypothesized that this lower performance for congruent items would be explained by the use of gap thinking, this turned out not to be the case: evidence was insufficient to show that the applicability of the gap thinking strategy modulated either participants’ accuracy rates or response times (although individual-level data suggest that there is an effect for response times). When fractions shared a common component, instead, our data display a more complex pattern that expected: an advantage for congruent items is present in the first experimental block but fades as the experiment progresses. Numerical distance had an effect in fraction comparison that was statistically significant for items without common components only. Altogether, our results from experts’ reasoning reveal nuances in the fraction comparison task with respect to previous studies and contribute to future models of reasoning in this task.

Intuitive errors in learners' fraction understanding: A dual-process perspective on the natural number bias.

Although a good rational number understanding is very important, many learners struggle to understand fractions. Recent research attributes many of these difficulties to the natural number bias - the tendency to apply natural number features in rational number tasks where this is inappropriate. Previous correlational dual process studies found evidence for the intuitive nature of the natural number bias in learners' response latencies. However, the reported correlations do not ascertain the causality that is assumed in this ascription. In the present study we therefore experimentally elicited intuitive responses in a fraction comparison task in educated adults by restricting reaction time. Results show that the natural number bias has an intuitive character. Findings also indicate that the detection of conflict between the natural number-based answer and the correct answer seems to work at an intuitive level.

The Interplay Between the Natural Number Bias and Fraction Magnitude Processing in Low-Achieving Students

Research has identified two core difficulties many students have with fractions: First, they often struggle with processing fraction magnitudes, and second, they rely on natural number concepts in fraction problems (“Natural Number Bias”, NNB). Yet, the relation between these two difficulties is not well understood. Moreover, while most studies of the NNB relied on analyses of the whole samples, there is empirical evidence that the occurrence of the NNB differs between student subgroups. In the present study, we investigate individual students’ profiles of the occurrence of the natural number bias and their ability to process fraction magnitude, using a dynamic assessment that utilizes continuous diagrams on touchscreen devices. We analyze data of 234 low-achieving 6th-grade students from Germany who completed a symbolic fraction comparison task, and a fraction magnitude estimation task with continuous circle and tape diagrams. A cluster analysis on the comparison task revealed three distinct clusters: a Typical Bias cluster (better performance on symbolic fraction comparison items congruent to natural number-based reasoning), a Reverse Bias cluster (better performance on items incongruent to natural number-based reasoning), and a No Bias cluster (similar performance on congruent and incongruent items). Only students in the No Bias cluster but not students in the other clusters demonstrated a distance effect in symbolic fraction comparison, suggesting fraction magnitude processing. Linear mixed models on the percent absolute error in the magnitude estimation task revealed significantly lower percent absolute error for students in the No Bias cluster compared to students in the other two clusters. Moreover, students in the No Bias cluster were significantly slower to solve both fraction comparison and fraction magnitude estimation tasks than students in the other clusters. These results suggest that the occurrence of the natural number bias and the ability to process fraction magnitude are closely related. The continuous representations used in our digital assessment tools appeared to be suitable for assessing both the natural number bias and fraction magnitude processing.

Inhibiting the Whole Number Bias in a Fraction Comparison Task: An Event-Related Potential Study.

IntroductionPeople often use heuristics derived from natural number tasks to solve fraction comparison tasks. For instance, one may falsely consider a fraction with a larger natural number to be the larger in magnitude, as in the case of 1/5 vs 1/4. We hypothesized that inhibitory control was needed to overcome this type of bias.MethodsTo test the hypothesis, Event-related potentials (ERP) were collected when participants were conducting fraction comparison tasks that designed with the negative priming paradigm. Twenty-eight adult participants performed three types of fraction comparison tasks: congruent items, incongruent items and neutral items.ResultsWe found a negative priming effect in terms of response time. Consistently, ERP results demonstrated larger N1 and N2 amplitudes and a smaller P3 amplitude in the test trial than in the control trial.ConclusionThese findings indicated that adults still need to inhibit the “larger natural number-larger fraction” misleading strategy when solving fraction comparison tasks with common components.

Various ways to determine rational number size: an exploration across primary and secondary education

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.

Inhibition and cognitive load in fractions and decimals.

Prior research with adults and children suggests that inhibitory control may have a role to play in learning counterintuitive fractions and decimals that are inconsistent with whole number knowledge. However, there is little research to date with primary school-aged children at the early stages of fraction and decimal instruction that addresses this relationship. Understanding this association has the potential to inform instructional practices concerning the learning of counterintuitive maths concepts. This study examined the relationship between inhibitory control and counterintuitive fractions and decimals in the presence of varying cognitive load in 8- to 10-year-old children. Children aged 8-10years (N=95) completed a fraction and decimal magnitude comparison task with pairs that were either consistent (controls) or inconsistent (counterintuitive) with whole number magnitudes. Cognitive load was manipulated by presenting trials with simple integrated text (no additional load), with integrated text accompanied by supportive illustrations (low load), or with illustrations containing information that needed to be integrated to arrive at an answer (high load). Participants also completed measures of response and semantic inhibition. Inhibitory control uniquely contributed to performance in counterintuitive fractions and decimals only under conditions of high cognitive load, where low semantic inhibition predicted longer response times. Our results indicate a more nuanced relation between inhibitory control and counterintuitive fractions and decimals than presumed by previous research. They suggest that the role of inhibitory control when reasoning about counterintuitive fractions and decimals is not constant, and it is only drawn on at high levels of cognitive load.

Exploring unfamiliar paths through familiar mathematical territory: Constraints and affordances in a preservice teacher’s reasoning about fraction comparisons

Preservice elementary teachers (PSTs) have been described as having difficulties with fractions, relying on standard procedures, and experiencing math anxiety. We are interested in productive ways in which PSTs can and do use their prior knowledge when exploring unfamiliar paths through familiar mathematical territory. We conducted interviews with PSTs in which we challenged them with various fraction comparison tasks and encouraged them to develop new strategies. In this paper, we present a case study focused on one PST who made considerable progress in her reasoning about fraction comparisons during such an interview. We use Greeno’s (1991) environment metaphor to conceptualize number sense as situated knowing in a conceptual domain. This perspective helps us account for both cognitive and affective factors. We highlight 4 themes concerning features of the mathematical environment that Jennifer (pseudonym) appeared to inhabit during the interview: (a) the interview context created a safe space that emphasized the interviewer’s interest in Jennifer’s ideas, as opposed to correct answers; (b) Jennifer used her prior knowledge of parts and wholes to ground her arguments meaningfully and as building blocks to invent new strategies; (c) she used her prior knowledge of cross multiplication as an established path that provided reassurance and facilitated her exploration of unfamiliar paths; (d) it was beliefs and affective factors, not deficiencies in knowledge, that constrained Jennifer’s exploration of unfamiliar paths through familiar mathematical territory. We discuss the implications of these findings for research concerning PSTs’ mathematical thinking and learning and for mathematics teacher education.

Middle Schoolers’ Biases and Strategies in a Fraction Comparison Task

The present study uses a short, computerized task to investigate individual differences among middle school students in terms of the intuitions and strategies that they use to compare fractions. To tap into their intuitions about fractions, students were presented with pairs of fractions on the screen for a limited time of 10 s. Fraction pairs to be compared were controlled as to whether fractions shared or not a common component (numerator or denominator), as well as whether the greater fraction was the one with greater components or not. Data from about 500 students were processed with a clustering analysis based on these four item types, revealing distinct patterns of answers that we interpret in terms of strategies for answering the task. Half of the sample followed a simple larger-component-larger-fraction strategy as suggested by a congruency-based implementation of the natural number bias, although several clusters showed opposite patterns. These results show that a short, simple task can provide valuable information about students’ individual differences in comparing fractions, useful for both research and practice in mathematics education. We discuss implications of these results for research and practice related to mathematics cognition and education.

Eye gaze patterns reveal how we reason about fractions

ABSTRACTFractions 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.

Development of fraction comparison strategies: A latent transition analysis.

The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a "whole number bias," indicating that larger numbers in numerators and denominators produce larger fraction values. However, some children instead chose fractions with smaller numerators and denominators, demonstrating a partial understanding that smaller numbers can yield larger fractions. Latent transition analysis showed that most children eventually adopted normative comparison strategies. Children who exhibited a partial understanding by choosing fractions with smaller numbers were more likely to adopt normative comparison strategies earlier than those with larger number biases. Controlling for general math achievement and other cognitive abilities, whole number line estimation accuracy predicted the probability of transitioning to normative comparison strategies. Exploratory factor analyses showed that over time, children appeared to increasingly represent fractions as discrete magnitudes when simpler strategies were unavailable. These results support the integrated theory of numerical development, which posits that an understanding of numbers as magnitudes unifies the process of learning whole numbers and fractions. The findings contrast with conceptual change theories, which propose that children must move from a view of numbers as counting units to a new view that accommodates fractions to overcome whole number bias. (PsycINFO Database Record

The role of attentional networks in the access to the numerical magnitude of fractions in adults / El rol de las redes atencionales en el acceso a la magnitud numérica de fracciones en adultos

This study explored the relationship between the efficiency of alerting, orienting and executive control networks (attentional executive function) and accessing numerical magnitudes in adults, during a fraction comparison task, presenting pairs of fractions either with (CC) or without (WCC) common components. The results indicate that CC fractions are processed in a componential style, are affected by a whole number bias (WNB) and that performance in this block does not depend on the efficiency of attentional networks. By contrast, pairs of WCC fractions are processed in a holistic way, the WNB interacts with the numerical distance between fractions and performance in this block correlates with the efficiency of attentional orienting and executive control systems. Taken together, our findings show that the efficient selection of information and inhibitory control of intervening variables play an important role in processing fractions, particularly when the task requires the use of more holistic strategies to access the numerical magnitude of the fraction.

The role of visual representations in college students' understanding of mathematical notation.

Developing understanding of fractions involves connections between nonsymbolic visual representations and symbolic representations. Initially, teachers introduce fraction concepts with visual representations before moving to symbolic representations. Once the focus is shifted to symbolic representations, the connections between visual representations and symbolic notation are considered to be less useful, and students are rarely asked to connect symbolic notation back to visual representations. In 2 experiments, we ask whether visual representations affect understanding of symbolic notation for adults who understand symbolic notation. In a conceptual fraction comparison task (e.g., Which is larger, 5 / a or 8 / a?), participants were given comparisons paired with accurate, helpful visual representations, misleading visual representations, or no visual representations. The results show that even college students perform significantly better when accurate visuals are provided over misleading or no visuals. Further, eye-tracking data suggest that these visual representations may affect performance even when only briefly looked at. Implications for theories of fraction understanding and education are discussed.

Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude

We report on interview results from a classroom teaching experiment in a Number and Operations course for prospective elementary teachers. Improving the number sense of this population is an important goal for mathematics teacher education, and researchers have found this goal to be difficult to accomplish. In earlier work, we devised a local instruction theory for the development of number sense, which focused on whole-number mental computation. In this study, the local instruction theory was applied to the rational-number domain, with the help of a framework for reasoning about fraction magnitude, and it guided instruction in the content course. We interviewed seven participants pre- and post-instruction, and we found that their reasoning on fraction comparison tasks improved. The participants made more correct comparisons, reasoned more flexibly, and came to favor less conventional and more sophisticated strategies. These improvements in number sense parallel those that we found previously in mental computation. In addition to the overall results, we highlight two cases of improvement that illustrate ways in which prospective elementary teachers’ reasoning about fraction magnitude can change.