Rational Number Tasks Research Articles

An empirically validated rational number sense framework

AbstractA theoretical model describing Grade 7 students’ rational number sense was formulated and validated empirically (n = 360), hypothesizing that rational number sense is a general construct consisting of three factors: basic rational number sense, arithmetic sense, and flexibility with rational numbers. Data analysis suggested that rational-number tasks can be categorized based on the validated model. The flexibility component reflects thinking about rational numbers in terms of noticing, using, and expressing relations and properties of numbers in patterns, functions, covariation, and complicated computational tasks. It includes utilizing number structure and relational understanding of operations and numbers. Analysis identified three categories of students that represent different rational-number sense profiles. Category 1 students exhibited a limited basic profile that solved mainly traditional school-based tasks. Category 2 students reflected the basic emergent arithmetic sense profile that responded adequately in operation tasks. Category 3 students represented the flexible emergent profile, as they manipulated underlying structures in a variety of situations, indicating an emergent fundamental shift from an arithmetic to an algebraic focus. A discriminant analysis showed that basic and flexible factors could discriminate students best between the three identified profiles of rational number sense.

Profiles in understanding operations with rational numbers

ABSTRACT 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.

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 natural number bias and its role in rational number understanding in children with dyscalculia. Delay or deficit?

BackgroundPrevious 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. AimsWe investigated the rational number understanding of learners with dyscalculia and compared it with the rational number understanding of learners without dyscalculia. MethodThree groups of learners were included: sixth graders with dyscalculia, a chronological age match group, and an ability match group. ResultsThe 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. ConclusionNext 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.

Number sense in the transition from natural to rational numbers.

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. 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. Participants were 140 sixth graders from six different primary schools. Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. 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. 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.

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. …

Who can escape the natural number bias in rational number tasks? A study involving students and experts.

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth-grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.

Inappropriately applying natural number properties in rational number tasks: characterizing the development of the natural number bias through primary and secondary education

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.

Naturally biased? In search for reaction time evidence for a natural number bias in adults

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.

Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.

Engaging Whole-Number Knowledge for Rational-Number Learning Using a Computer-Based Tool

We present data from a two-year teaching experiment involving 8- and 9-year-old children. The purpose of the teaching experiment was to investigate children's fraction learning and the role their whole-number knowledge might play in it. A major source for the children's experiences and experiments was an operator-like computer program called Copycat. In particular, we focus on the accomplishments of Elliot and Shannon as they solved fraction comparison problems. We identify and discuss three cognitive schemes that Elliot and Shannon used. These were an Equal Outputs scheme, limited in effectiveness to unit fraction comparisons; an Equal Inputs scheme that activated strategies for determining a common multiple; and a Scaling scheme in which fractions were scaled up or down using ratios. The major focus of the discussion is the role whole-number knowledge played in the application of these schemes. This study demonstrates an interdependence between development of rational-number knowledge and whole-number knowledge. Rational-number tasks in operator settings can help stimulate and extend children's whole-number knowledge. Facility with whole-number relationships enables students to solve fraction comparison problems.

Brief Reports: Cognitive Restructuring Ability, Teacher Guidance, and Perceptual Distracter Tasks: An Aptitude-Treatment Interaction Study

The aptitude-treatment interaction (ATI) study reported here explored the relationship between cognitive restructuring ability, as measured by the Group Embedded Figures Test (GEFT), and treatments varying in amounts of teacher guidance. It specifically investigated how these two variables affected performance on rational number tasks involving perceptual distracters.

Cognitive Restructuring Ability, Teacher Guidance, and Perceptual Distracter Tasks: An Aptitude-Treatment Interaction Study

The aptitude-treatment interaction (ATI) study reported here explored the relationship between cognitive restructuring ability, as measured by the Group Embedded Figures Test (GEFT), and treatments varying in amounts of teacher guidance. It specifically investigated how these two variables affected performance on rational number tasks involving perceptual distracters. Perceptual distracter problems evolved from the work of the Rational Number Project (Behr & Post, 1981). Such problems contain visualstimulus information inconsistent with the demands of the task that must