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The literature has amply shown that primary and secondary school students have difficulties in understanding rational number size. Many of these difficulties are explained by the natural number bias or the use of other incorrect reasoning such as gap thinking. However, in many studies, these types of reasoning have been inferred from comparing students' accuracies in multiple-choice items. Evidence that supports that these incorrect ways of reasoning are indeed underlying is scarce. In the present work, we carried out interviews with 52 seventh grade students. The objective was to validate the existence of students' incorrect ways of thinking about fraction size that were previously inferred from patterns of correct and incorrect answers to multiple-choice items, by looking at students' verbalizations, and examine whether these ways of thinking are resistant to change. Students' verbalizations support the existence of the different incorrect ways of thinking inferred from previous studies in fraction size. Furthermore, the high levels of confidence in their incorrect reasoning and the fact that they were reluctant to change their answer when they were confronted with other reasoning suggest that these ways of thinking may be resistant to change. [ABSTRACT FROM AUTHOR]

Research on rational numbers suggests that adults experience more difficulties in understanding the numerical magnitude of rational than natural numbers. Within rational numbers, the numerical magnitude of fractions has been found to be more difficult to understand than that of decimals. Using a number line estimation (NLE) task, the current study investigated two sources of difficulty in adults' numerical magnitude understanding: number type (natural vs rational) and structure of the notation system (place-value-based vs non-place-value-based). This within-subjects design led to four conditions: natural numbers (natural/place-value-based), decimals (rational/place-value-based), fractions (rational/non-place-value-based), and separated fractions (natural/non-place-value-based). In addition to percentage absolute error (PAE) and response times, we collected eye-tracking data. Results showed that participants estimated natural and place-value-based notations more accurately than rational and non-place-value-based notations, respectively. Participants were also slower to respond to fractions compared with the three other notations. Consistent with the response time data, eye-tracking data showed that participants spent more time encoding fractions and re-visited them more often than the other notations. Moreover, in general, participants spent more time positioning non-place-value-based than place-value-based notations on the number line. Overall, the present study contends that when both sources of difficulty are present in a notation (i.e., both rational and non-place-value-based), adults understand its numerical magnitude less well than when there is only one source of difficulty present (i.e., either rational or non-place-value-based). When no sources of difficulty are present in a notation (i.e., both natural and place-value-based), adults have the strongest understanding of its numerical magnitude. [ABSTRACT FROM AUTHOR]

The present cross-sectional study investigated 953 fifth to tenth grade students’ understanding of the dense structure of rational numbers. After an inductive analysis, coding the answers based on three types of items on density, a TwoStep Cluster Analysis revealed different intermediate profiles in the understanding of density along grades. The analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our profiles showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a natural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers. [ABSTRACT FROM AUTHOR]

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively different profiles regarding students' understanding is scarce. The current study investigated the different ways students reasoned in arithmetic operation items with fractions and decimals. A cross-sectional study with 1,262 participants from 5th to 10th grade was performed. A TwoStep Cluster Analysis revealed eight different student reasoning profiles. We found that the natural number bias is first overcome in addition and subtraction, and later in multiplication and division. Moreover, differences regarding representation were only found in addition and subtraction items, indicating that natural numbers interfered more strongly in fractions than in decimal numbers. Finally, results showed that some students' difficulties with rational number multiplications and divisions had other explanations than the natural number bias. [ABSTRACT FROM AUTHOR]

Although a good rational number understanding is of crucial importance for learners' general maths achievement, many learners have misconceptions about fractions. An often described misconception is that a fraction's numerical magnitude increases when its denominator, numerator, or both increase. The present intervention study investigated how robust this misconception is, after learners have specifically been instructed on the incorrectness of their idea by means of a refutation text, as opposed to a condition where they received an expository text. Results of 76 fourth graders showed a (long-term) learning effect, but no larger learning gains in the group that received a refutation text compared to the group that received an expository text. Results further pointed to large individual differences in the learning gains. While there is a subgroup of learners that benefits from the refutation text, another subgroup was found that tend to overuse and misapply the information given in this text. [ABSTRACT FROM AUTHOR]

Background Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. Aims In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. Sample Participants were 140 sixth graders from six different primary schools. Method Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. Results Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. Conclusion The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement. [ABSTRACT FROM AUTHOR]

The natural number bias is known to explain many difficulties learners have with understanding rational numbers. The research field distinguishes three aspects where natural number properties are sometimes inappropriately applied in rational number tasks: density, size, and operations. The overall goal of this study was to characterize the development of the natural number bias across the span between 4th and 12th grade. To achieve this goal, a comprehensive test was constructed to test 4th to 12th graders' natural number bias. This test was administered to 1343 elementary and secondary school students. Results showed that an overall natural number bias could be found. This bias appeared to be equally strong in tasks with decimal numbers and tasks with fractions. Moreover, the natural number bias was weakest in size tasks, somewhat stronger in operations tasks, and by far the strongest in density tasks. An overall decrease of the strength of the natural number bias-but no disappearance except for size tasks-could be found with grade. [ABSTRACT FROM AUTHOR]

The foundations for more advanced mathematics involve a good sense of rational numbers. However, research in cognitive psychology and mathematics education has repeatedly shown that children and even adults struggle with understanding different aspects of rational numbers. One frequently raised explanation for these difficulties relates to the natural number bias, i.e., the tendency to inappropriately apply natural number properties to rational number tasks. This contribution reviews the four main areas where systematic errors due to the natural number bias can be found, i.e., their size, operations, representations and density. Next, we discuss the major theoretical frameworks from which rational number understanding is currently investigated. Finally, an overview of the various papers is provided. [ABSTRACT FROM AUTHOR]

Abstract: When school students compare the numerical values of fractions, they have frequently been found to be biased by the natural numbers involved (e.g., to believe that 1/4 > 1/3 because 4 > 3), thereby considering fractions componentially as two natural numbers rather than holistically as one number. Adult studies have suggested that intuitive processes could be the source of this bias, but also that adults are able to activate holistic rather than componential mental representations of fractions under some circumstances. We studied expert mathematicians on various types of fraction comparison problems to gain further evidence for the intuitive character of the bias, and to test how the mental representations depend on the type of comparison problems. We found that experts still show a tendency to be biased by natural numbers and do not activate holistic representations when fraction pairs have common numerators or denominators. With fraction pairs without common components, we found no natural number bias, and holistic representations were more likely. We discuss both findings in relation to each other, and point out implications for mathematics education. [Copyright &y& Elsevier]

Rational numbers and particularly fractions are difficult for students. It is often claimed that the ‘natural number bias’ underlies erroneous reasoning about rational numbers. This cross-sectional study investigated the natural number bias in first and fifth year secondary school students. Relying on dual process theory assumptions that differentiate between intuitive and analytic processes, we measured accuracies and reaction times on fraction comparison tasks. Half of the items were congruent (i.e., natural number knowledge leads to correct answers), the other half were incongruent (i.e., natural number knowledge leads to incorrect answers). Against expectations, students hardly made errors on incongruent items. Longer reaction times on correctly solved incongruent than on correctly solved congruent items indicated that students were indeed hampered by their prior knowledge about natural numbers, but could suppress their intuitive answers. [ABSTRACT FROM AUTHOR]

Abstract: A major source of errors in rational number tasks is the inappropriate application of natural number rules. We hypothesized that this is an instance of intuitive reasoning and thus can persist in adults, even when they respond correctly. This was tested by means of a reaction time method, relying on a dual process perspective that differentiates between intuitive and analytic reasoning. We measured fifty-eight educated adults’ accuracies and reaction times in a variety of rational number tasks. In half of the items (congruent), the correct response was compatible with natural number properties (thus intuitive reasoning led to a correct answer). In contrast, in the incongruent items, intuitive reasoning would lead to an incorrect answer. In comparing two numbers, there were hardly any natural-number-based errors but correct responses to incongruent items took longer. Regarding the effect of operations, more mistakes were made in incongruent items, and correct responses required longer reaction time. Incongruent items about density elicited considerably more errors than congruent items. These findings can be considered as evidence that the natural number bias is an instance of intuitive reasoning. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Many studies have addressed the natural number bias in fraction comparison, focusing on the role of congruency. However, the congruency effect has been observed to operate in the opposite direction, suggesting that a deeper explanation must underlie students' different reasoning. We extend previous research by examining students' reasoning and by studying the effect of a gap condition in students' answers. A cross-sectional study was conducted on 438 students from 5th to 10th grade. Results showed that the gap effect could explain differences between congruent and incongruent items. Moreover, students' use of gap thinking decreased towards the end of Secondary Education. [ABSTRACT FROM AUTHOR]

Students have a great deal of difficulties learning about rational number concepts, as they are confounded by misapplying reasoning about natural numbers to fractions and decimals, referred to as a natural number bias. For example, students often think that the number of digits of a decimal, or the size of the component numbers of fractions, is enough information to determine the magnitude of rational numbers. As well, students have trouble understanding that there is an infinite number of numbers between any two fractions or decimals. Using multigroup latent class analysis, the present study examines the structure of 611 Finnish and Flemish students' rational number knowledge in order to determine the similarities and differences between these two sub-samples. Results reveal that, while the Flemish students performed somewhat better, there were only relatively minor differences in the structure of the two sub-samples' rational number knowledge. In general, it appears that the natural number bias affects these Finnish and Flemish students' knowledge of the size and density of fractions and decimals in similar ways. [ABSTRACT FROM AUTHOR]

In this study we longitudinally followed 201 upper elementary school learners in the crucial years of acquiring rational number understanding. Using latent transition analysis we investigated their conceptual change from an initial natural number based concept of a rational number towards a mathematically more correct one by characterizing the various intermediate states learners go through. Results showed that learners first develop an understanding of decimal numbers before they have an increased understanding of fractions. We also found that a first step in learners' rational number understanding is an increased understanding of the numerical size of rational numbers. Further, only a limited number of learners fully understand the dense structure of rational numbers at the end of elementary education. [ABSTRACT FROM AUTHOR]

Background: Previous research indicated that in several cases learners' errors on rational number tasks can be attributed to learners' tendency to (wrongly) apply natural number properties. There exists a large body of literature both on learners' struggle with understanding the rational number system and on the role of the natural number bias in this struggle. However, little is known about this phenomenon in learners with dyscalculia.Aims: We investigated the rational number understanding of learners with dyscalculia and compared it with the rational number understanding of learners without dyscalculia.Method: Three groups of learners were included: sixth graders with dyscalculia, a chronological age match group, and an ability match group.Results: The results showed that the rational number understanding of learners with dyscalculia is significantly lower than that of typically developing peers, but not significantly different from younger learners, even after statistically controlling for mathematics achievement.Conclusion: Next to a delay in their mathematics achievement, learners with dyscalculia seem to have an extra delay in their rational number understanding, compared with peers. This is especially the case in those rational number tasks where one has to inhibit natural number knowledge to come to the right answer. [ABSTRACT FROM AUTHOR]

Many difficulties learners have with rational number tasks can be attributed to the "natural number bias", i.e. the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, 2015). McMullen and colleagues found a relevant source of individual differences in the learning of those aspects of rational numbers that are susceptible to the natural number bias, namely Spontaneous Focusing On quantitative Relations (SFOR) (McMullen, 2014). While McMullen and colleagues showed that SFOR relates to rational number knowledge as a whole, we studied its relation with several aspects of the natural number bias. Additionally, we 1) included test items addressing operations with rational numbers and 2) controlled for general mathematics achievement and age. Results showed that SFOR related strongly to rational number knowledge, even after taking into account several control variables. Results are discussed for each of the three aspects of the natural number bias separately. [ABSTRACT FROM AUTHOR]

Although rational numbers are an essential part of mathematical literacy, they cause many difficulties for students. A major cause is the natural number bias. We examined this natural number bias in secondary school students in two related studies. In Study 1, 8th graders judged the correctness of algebraic expressions that address the effect of operations. The higher accuracy level on congruent items than on incongruent items yielded clear evidence for the natural bias. However, this bias was only significant in multiplication and division items. Additional interview data showed that students doubted more about the applicability of natural number principles in items with addition and subtraction. In Study 2 we additionally confronted 10th and 12th graders with the same tasks. The results of the second study showed that the natural number bias unexpectedly did not decrease towards the end of secondary education and remained present in multiplication and division items. [ABSTRACT FROM AUTHOR]

Rational numbers are amongst the most difficult topics in the elementary and secondary school curriculum and teaching them requires an appropriate knowledge base of teachers to properly deal with students' difficulties. We investigated prospective teachers' content knowledge (CK) and pedagogical content knowledge (PCK) on rational numbers, the relationship between CK and PCK, and differences in CK and PCK among prospective elementary teachers (trained as general classroom teachers) and lower secondary teachers (trained as subject-specific classroom teachers). The results revealed gaps in prospective teachers' CK and PCK, a positive correlation between CK and PCK, and a better CK but not PCK for secondary compared to elementary school teachers. [ABSTRACT FROM AUTHOR]

Abstract: It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number based prior knowledge, certain intermediate states of understanding emerge. This paper presents a study of Greek and Flemish 9th grade students who solved a test about the infinity of numbers in an interval. The Flemish students outperformed the Greek ones. More importantly, the intermediate levels of understanding—where the type of the interval endpoints (i.e., natural numbers, decimals, or fractions) affects students’ judgments—were very similar in both groups. These results point to specific conceptual difficulties involved in the shift from natural to rational numbers and raise some questions regarding instruction in both countries. [Copyright &y& Elsevier]