Representations of Fractions: Evidence for Accessing the Whole Magnitude in Adults

Proficiency with fractions serves as a foundation for later mathematics and is critical for learning algebra, which plays a role in college success and lifetime earnings. Yet children often struggle to learn fractions. Educators have argued that a conceptual understanding of fractions involves learning that a fraction represents a magnitude different from its whole number components. However, it is not well understood whether adults represent a fraction's magnitude similarly to whole numbers. This study investigated the distance effect during a comparison task using fraction pairs that discouraged comparing a fraction's components. Accuracy improved and reaction times decreased with greater distance between fraction pairs, showing a distance effect similar to that seen with whole numbers. This study suggests that a representation of a fraction's magnitude is present in the fully developed number system.     

Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The integrated theory of numerical development posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of the magnitudes to which they refer, and this magnitude understanding is central to general mathematical competence. We investigated relations among fraction magnitude understanding, arithmetic and general mathematical abilities in countries differing in educational practices: U.S., China and Belgium. Despite country-specific differences in absolute level of fraction knowledge, 6th and 8th graders' fraction magnitude understanding was positively related to their general mathematical achievement in all countries, and this relation remained significant after controlling for fraction arithmetic knowledge in almost all combinations of country and age group. These findings suggest that instructional interventions should target learners' interpretation of fractions as magnitudes, e.g., by practicing translating fractions into positions on number lines.     

Discriminating between reward-produced memories: Effects of differences in reward magnitude

The greater the dissimilarity between exteroceptive stimuli, the easier it is to discriminate between them. To determine whether a similar relationship holds for memories produced by reward events, rats in three runway investigations received trials in pairs, the number of food pellets (0.045 g) occurring on Trial 1 indicating whether reward or nonreward would occur on Trial 2. In each investigation, discriminative responding on Trial 2 was better the larger the difference in reward magnitude on Trial 1. This finding was obtained under a wide variety of conditions: for example, when the larger of two reward magnitudes on Trial 1 signaled nonreward on Trial 2 (Experiment 1, 10 vs. 2 pellets); when the smaller of two reward magnitudes on Trial 1 signaled nonreward on Trial 2 (Experiment 2, 10 vs. 2 pellets); and when the same magnitude of reward on Trial 1 signaled nonreward on Trial 2 (Experiment 3, either 5 pellets or 0 pellets). The findings obtained here indicate that the greater the dissimilarity between reward magnitudes, the greater the dissimilarity between the memories they produced and, thus, the easier it is to discriminate between them. It is suggested that the present results may provide a basis for understanding findings obtained in other instrumental learning investigations in which reward magnitude is varied.     

The role of numerical and non-numerical ordering abilities in mathematics and reading in middle childhood

This study investigated whether different types of ordering skills were related to mathematics achievement in children (n = 100) in middle childhood, after the effects of age, socio-economic status, IQ, and processing speed were taken into account. The relations between ordering skills and magnitude processing were also investigated, as well as the possibility that some of the shared variance between math and reading is explained by ordering abilities. The ordering tasks included the ordering of familiar numerical and non-numerical sequences, a parental report of children’s everyday ordering skills, and an order working memory task. Three magnitude processing tasks (symbolic and non-symbolic comparison and number line estimation), were also administered, as well as measures of inhibition and spatial working memory. From this set of measures, number ordering, order working memory and number line estimation emerged as the most important predictors of mathematics skills. We found that number ordering mediated the effect of both symbolic and non-symbolic comparison skills on mathematics, further confirming that this task captures some essential skills related to mathematics. Additionally, order working memory mediated the effect of both number comparison and reading skills on math. Finally, whereas non-symbolic comparison and number line estimation are considered important indicators of magnitude processing skills, there was no relationship between these abilities, but there was a correlation between each of these abilities and reading skills, with number line estimation also mediating the effect of reading skills on math. These novel findings could contribute to a better understanding of the basic processes underlying math ability, and why math and reading are strongly related in typical populations and in children with learning difficulties.     

Analogue equivalents in number processing of simple arithmetic sums

This study had two aims. The first was to test the postulate of analogical equivalents in number processing using a stimulus set based on the differences between pairs of numbers, and second, to look for IQ-dependent differences in this processing. Participants were asked to make judgments concerning the differences between pairs of numbers—each number pair being defined according to overall numerical size and level of difference—and to draw the magnitude of the differences using a free-hand line. In agreement with previous findings, results indicated that the magnitude of participants' responses was dependent on the sizes and levels of the differences within the number pairs. In particular, participants' responses to a difference of nine units at the highest level were based on unit lengths smaller in magnitude compared with other number pairs. In contrast, participants' responses to a second test requiring them to estimate the length of a line using an independent number scale showed remarkable accuracy across all lengths. The results of the two tests did not, however, indicate any difference in responses based on IQ. The results gave support to the idea that transformation of numerical quantities to their analogical equivalents occurs during simple arithmetic sums, and that this transformation was flawed as suggested by the number size effect.     

Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison

This study examined the generality of the logarithmic to linear transition in children's representations of numerical magnitudes and the role of subjective categorization of numbers in the acquisition of more advanced understanding. Experiment 1 (49 girls and 41 boys, ages 5-8 years) suggested parallel transitions from kindergarten to second grade in the representations used to perform number line estimation, numerical categorization, and numerical magnitude comparison tasks. Individual differences within each grade in proficiency for the three tasks were strongly related. Experiment 2 (27 girls and 13 boys, ages 5-6 years) replicated results from Experiment 1 and demonstrated a causal role of changes in categorization in eliciting changes in number line estimation. Reasons were proposed for the parallel developmental changes across tasks, the consistent individual differences, and the relation between improved categorization of numbers and increasingly linear representations.     

Children’s mental representation when comparing fractions with common numerators

Researchers debate whether one represents the magnitude of a fraction according to its real numerical value or just the discrete numerosity of its numerator or denominator. The present study examined three effects based on the notion that people possess a mental number line to explore how children represent fractions when they compare fractions with common numerators. Specifically, the effect of the spatial numerical association of response codes (SNARC), the distance effect and the size effect in representing fractions were examined in a sample of 72 sixth graders, who successfully solved the fraction comparison task with a real number (.2) or a fraction (1/5) as the reference. Results showed that in the fraction-reference group (1/5 as the reference), there was a significant reverse SNARC effect and a distance effect between the denominators of the target fractions and the reference fraction; in the real number-reference group, the three effects were also observed. These results revealed that both groups used the mental number line to represent fractions and did not represent their real numerical values but rather the discrete numerosities of denominators when comparing fractions with common numerators. It seems that the way people represent fractions may depend on their strategy choices.     

Children’s Knowledge of Symbolic Number in Grades 1 and 2: Integration of Associations

How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict addition but was predicted by number comparison, suggesting that magnitude associations underlie ordering performance. In contrast, for Grade 2 children (n = 80, M = 90 months), number ordering predicted addition concurrently and at the end of Grade 2; number ordering was predicted by number comparison, sequencing, and inhibitory processing. Development of symbolic number competence involves the hierarchical integration of sequence, magnitude, order, and arithmetic associations.     

The role of keypecking during filled intervals on the judgment of time for empty and filled intervals by pigeons

Pigeons were trained in a within-subjects design to discriminate empty intervals (bound by two 1-sec visual markers) and filled intervals (a continuous visual signal). The intervals were signaled by different visual stimuli and they required responses to different sets of comparison stimuli. In Experiment 1, empty intervals were judged longer than filled intervals. The difference between the point of subjective equality (PSE) for the empty intervals and the PSE for filled intervals increased as the magnitude of the anchor-duration pairs increased. Although there was more pecking during filled intervals than during empty intervals, there was no significant correlation between pecking during filled intervals and the value of the PSE. In Experiment 2, empty intervals continued to be judged longer than filled intervals, even when pigeons were required to refrain from pecking during filled intervals. Keypecking per se does not appear to play an important role in the empty-filled timing difference.     

Modeling the developmental trajectories of rational number concept(s)

The present study focuses on the development of two sub-concepts necessary for a complete mathematical understanding of rational numbers, a) representations of the magnitudes of rational numbers and b) the density of rational numbers. While difficulties with rational number concepts have been seen in students' of all ages, including educated adults, little is known about the developmental trajectories of the separate sub-concepts. We measured 10- to 12-year-old students' conceptual knowledge of rational numbers at three time points over a one-year period and estimated models of their conceptual knowledge using latent variable mixture models. Knowledge of magnitude representations is necessary, but not sufficient, for knowledge of density concepts. A Latent Transition Analysis indicated that few students displayed sustained understanding of rational numbers, particularly concepts of density. Results confirm difficulties with rational number conceptual change and suggest that latent variable mixture models can be useful in documenting these processes.     

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

Assessment 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.     

The Influence of Number‐sense on Children's Ability to Estimate Measures

This study aimed at clarifying the relative developmental influences of age, number‐sense and context on primary school children's ability to estimate measures. Children (6‐11 years) were first assessed on three aspects of number‐sense (mental computation, understanding of relative number magnitude and understanding of relations between numbers) and were then assessed on their estimation of length and area within each of two task contexts (a ‘story’ frame and a ‘textbook’ frame). While number‐sense was found to improve with age, estimation did not. However, an ability to use and perceive number relations, together with an understanding of the relative magnitudes of larger numbers, were found to influence children's ability to estimate area. Children of all ages were also found to estimate more accurately in the ‘textbook’ context than in the ‘story’ context. These findings are discussed with reference to the notion of estimation as a situated activity     

Transitive inference in pigeons: Simplified procedures and a test of value transfer theory

Minimal procedures for the demonstration of transitive inference (TI) in animals have involved the training of four simultaneous discriminations: for example, A+B?, B+C?, C+D?, and D+E?, followed by the demonstration of a preference for B over D on test trials. In Experiment 1, we found that TI in pigeons can be found with successive training involving A+B?, B+C?, A+C?, C+D?, D+E?, C+E?, and A+E?. In Experiment 2, we found that demonstration of TI did not require inclusion of experience with the nonadjacent stimulus pairs (A+C?, C+E?, A+E?). Experiment 3 provided a test of value transfer theory (VTT; Fersen, Wynne, Delius, & Staddon, 1991). When pigeons were trained with stimulus pairs that did not permit the transitive ordering of stimuli, but did permit the differential transfer of value (e.g., A+B?, C?E+, C+D?, & A+E?), preference for B over D was still found. Analyses of the relation between direct experiences with reinforced and nonreinforced responding and stimulus preferences on test trials failed to support a reinforcement-history account of TI.     

The Stability of Individual Differences in Basic Mathematics‐Related Skills in Young Children at the Start of Formal Education

The current study investigated the development of children's performance on tasks that have been suggested to underlie early mathematics skills, including measures of cardinality, ordinality, and intelligence. Eighty‐seven children were tested in their first (T1) and second (T2) school year (at ages 5 and 6). Children's performance on all tasks demonstrated good reliability and significantly improved with age. Correlational analyses revealed that performance on some mathematics‐related tasks were nonsignificantly correlated between T1 and T2 (number line and number comparison), showing that these skills are relatively unstable. Detailed analyses also indicated that the way children solve these tasks show qualitative changes over time. By contrast, children's performance on measures of intelligence and nonnumerical ordering abilities were strongly correlated between T1 and T2. Additionally, ordering skills also showed moderate to strong correlations with counting procedures both cross‐sectionally and longitudinally. These results suggest that, initially, mathematics skills strongly rely on nonmathematical abilities.     

Misconceptions about Motion: Development and Training Effects

2 experiments were performed to examine whether children and adults possess a single-object/single-motion intuition. This intuition involves the view that because all parts of a rigid object move together, they all must move at the same speed. We found that third graders, sixth graders, ninth graders, and adults all responded in accord with this intuition. On problems where the intuition led to errors, a large majority of subjects of all ages answered incorrectly. On problems where it led to successful performance, subjects answered more accurately than on problems where the intuition was not applicable. In addition, the specific errors were those that the intuition yielded, and the intuition fit individual patterns of performance. The intuition not only influenced speed judgments but also judgments of distance traveled. Experiment 2 demonstrated that making sixth graders aware of the intuition and providing them kinesthetic experience that contradicted it produced significant improvements in the children's understanding. Implications for how scientific misconceptions in general can be overcome were discussed.     

Relationship between cooperation in an iterated prisoner's dilemma game and the discounting of hypothetical outcomes

A number of authors have proposed that preference for a larger, delayed reward in delay discounting is similar to cooperation in a repeated prisoner's dilemma game versus tit-for-tat. This proposal was examined by correlating delay-discounting (Experiment 1) and probability-discounting (Experiment 2) rates for hypothetical monetary gains and losses with performance in a repeated prisoner's dilemma game. Correlations between rate of delay discounting (discounting parameters and area under the curve) and proportion of cooperation in the repeated prisoner's dilemma game versus tit-for-tat were significant across three magnitudes, and correlations were generally higher with discounting for losses than with that for gains. As was expected, correlations between rate of delay discounting and performance versus a random strategy in the prisoner's dilemma game were not significant. Correlations between rate of probability-discounting and cooperation rate in a repeated prisoner's dilemma game versus neither a tit-for-tat nor a random strategy were significant.     

Technikons and Higher Education Restructuring

The paper examines technikons in South Africa with respect to some of their internal characteristics, and in relation to debates about their place in the system of higher education as a whole, currently being assessed by the new National Commission on Higher Education. Internally it is shown that technikons lag the universities by 1:3 in terms of student enrolments, and nearly half of the 100,000 (in 1991) students were enrolled at the distance learning Technikon RSA. The past decade has seen a shift away from science and technology (S&T) fields of study at technikons, despite their establishment as 'institutes of technology'. African and female students are shown to be especially under-represented in S&T fields; staff are significantly underqualified in comparison with the universities. The profile of technikons poses a number of questions in relation to the restructuring of the higher education system (HES). Issues of national and regional governance of the HES, articulation between universities-technikons-technical colleges, and ways of enhancing S&T fields and establishing staff development programmes are considered.     

Inference and Association in Children's Early Numerical Estimation

How do children map number words to the numerical magnitudes they represent? Recent work in adults has shown that two distinct mechanisms—structure mapping and associative mapping—connect number words to nonlinguistic numerical representations (Sullivan & Barner, 2012 ). This study investigated the development of number word mappings, and the roles of inference and association in children's estimation. Fifty‐eight 5‐ to 7‐year‐olds participated, and results showed that at both ages, children possess strong item‐based associative mappings for numbers up to around six, but rely primarily on structure mapping—an inferential process—for larger quantities. These findings suggest that children rely primarily on an inferential mechanism to construct and deploy mappings between number words and large approximate magnitudes.     

Resource location and structural properties of the nestbox as determinants of nest-site selection in the golden hamster

Nest-site selection, an ecologically relevant behavior, was studied in the golden hamster in a model environment where the animals could choose between nestboxes differing in distance from resource or in structural features (size and illumination). Experiment 1 showed that hamsters can decrease foraging costs by setting their nests in the nestbox nearest to the food and/or nestmaterial sources, and that hoarding costs, as distinct from simple procurement costs, are taken into account in the choice process. Preferences for darker and larger nestboxes were obtained in Experiment 2. Experiment 3 revealed an internally coherent hierarchical pattern in preferences, with illumination being more important than size and resource distance, and size more important than resource distance. The results suggest that nest-site selection and foraging behaviors are parts of an integrated causal system.