Exploring the mental number line via the size congruity effect.

To address the ongoing debate about the origins of the size effect (faster comparison time for smaller than larger numbers, given a fixed intrapair distance), an indication of automatic number processing was searched for. Participants performed a quantity comparison task in which they had to decide which of two sketched cups contained more liquid, while ignoring the number superimposed on each cup. In the congruent condition, the larger number appeared on the cup containing more liquid, while in the incongruent condition the larger number appeared on the cup containing less liquid. The size effect was found in a numerical comparison task, while in the quantity comparison task the size congruity effect decreased as the magnitude of the irrelevant numbers increased. Together, these results suggest that the size effect reflects a basic feature of the mental number line. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Number games, magnitude representation, and basic number skills in preschoolers.

The effect of 3 intervention board games (linear number, linear color, and nonlinear number) on young children's (mean age = 3.8 years) counting abilities, number naming, magnitude comprehension, accuracy in number-to-position estimation tasks, and best-fit numerical magnitude representations was examined. Pre- and posttest performance was compared following four 25-min intervention sessions. The linear number board game significantly improved children's performance in all posttest measures and facilitated a shift from a logarithmic to a linear representation of numerical magnitude, emphasizing the importance of spatial cues in estimation. Exposure to the number card games involving nonsymbolic magnitude judgments and association of symbolic and nonsymbolic quantities, but without any linear spatial cues, improved some aspects of children's basic number skills but not numerical estimation precision. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The mental representation of numerical fractions: Real or integer?

Numerical fractions are commonly used to express ratios and proportions (i.e., real numbers), but little is known about how they are mentally represented and processed by skilled adults. Four experiments employed comparison tasks to investigate the distance effect and the effect of the spatial numerical association of response codes (SNARC) for fractions. Results showed that fractions were processed componentially and that the real numerical value of the fraction was not accessed, indicating that processing the fraction's magnitude is not automatic. In contrast, responses were influenced by the numerical magnitude of the components and reflected the simple comparison between numerators, denominators, and reference, depending on the strategy adopted. Strategies were used even by highly skilled participants and were flexibly adapted to the specific experimental context. In line with results on the whole number bias in children, these findings suggest that the understanding of fractions is rooted in the ability to represent discrete numerosities (i.e., integers) rather than real numbers and that the well-known difficulties of children in mastering fractions are circumvented by skilled adults through a flexible use of strategies based on the integer components. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Early math matters: Kindergarten number competence and later mathematics outcomes.

Children’s number competencies over 6 time points, from the beginning of kindergarten to the middle of 1st grade, were examined in relation to their mathematics achievement over 5 later time points, from the end of 1st grade to the end of 3rd grade. The relation between early number competence and mathematics achievement was strong and significant throughout the study period. A sequential process growth curve model showed that kindergarten number competence predicted rate of growth in mathematics achievement between 1st and 3rd grades as well as achievement level through 3rd grade. Further, rate of growth in early number competence predicted mathematics performance level in 3rd grade. Although low-income children performed more poorly than their middle-income counterparts in mathematics achievement and progressed at a slower rate, their performance and growth were mediated through relatively weak kindergarten number competence. Similarly, the better performance and faster growth of children who entered kindergarten at an older age were explained by kindergarten number competence. The findings show the importance of early number competence for setting children’s learning trajectories in elementary school mathematics. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Distance and magnitude effects in sequential number discrimination by pigeons.

Number discrimination experiments with humans and monkeys have revealed distance and magnitude effects. When required to choose the more frequently occurring stimulus between two stimuli presented repeatedly in sequence, accuracy improves as the distance between number increases (distance effect) and decreases as distance is held constant and the size of the numbers increases (magnitude effect). These effects were shown in three experiments reported with pigeons as subjects. It was shown that a single model based on discrimination between noisy numerical representations could account for both the primate and bird findings. To model the pigeon data, an additional decay parameter was necessary to account for strong recency effects found for the influence on choice of terminal stimuli presented in a sequence. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Numerical estimation in preschoolers.

Children’s sense of numbers before formal education is thought to rely on an approximate number system based on logarithmically compressed analog magnitudes that increases in resolution throughout childhood. School-age children performing a numerical estimation task have been shown to increasingly rely on a formally appropriate, linear representation and decrease their use of an intuitive, logarithmic one. We investigated the development of numerical estimation in a younger population (3.5- to 6.5-year-olds) using 0–100 and 2 novel sets of 1–10 and 1–20 number lines. Children’s estimates shifted from logarithmic to linear in the small number range, whereas they became more accurate but increasingly logarithmic on the larger interval. Estimation accuracy was correlated with knowledge of Arabic numerals and numerical order. These results suggest that the development of numerical estimation is built on a logarithmic coding of numbers—the hallmark of the approximate number system—and is subsequently shaped by the acquisition of cultural practices with numbers. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Numbers and space: A computational model of the SNARC effect.

The SNARC (spatial numerical associations of response codes) effect reflects the tendency to respond faster with the left hand to relatively small numbers and with the right hand to relatively large numbers (S. Dehaene, S. Bossini, & P. Giraux, 1993). Using computational modeling, the present article aims to provide a framework for conceptualizing the SNARC effect. In line with models of spatial stimulus-response congruency, the authors modeled the SNARC effect as the result of parallel activation of preexisting links between magnitude and spatial representation and short-term links created on the basis of task instructions. This basic dual-route model simulated all characteristics associated with the SNARC effect. In addition, 2 experiments tested and confirmed new predictions derived from the model. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Transfer of magnitude and spatial mappings to the SNARC effect for parity judgments.

Left–right keypresses to numerals are faster for pairings of small numbers to left response and large numbers to right response than for the opposite pairings. This spatial numerical association of response codes (SNARC) effect has been attributed to numbers being represented on a mental number line. We examined this issue in 3 experiments using a transfer paradigm. Participants practiced a number magnitude-judgment task or spatial stimulus–response compatibility task with parallel or orthogonal stimulus–response dimensions prior to performing a parity-judgment task. The SNARC effect was enhanced following a small–left/large–right magnitude mapping but reversed following a small–right/large–left mapping, indicating that associations between magnitude and response defined for the magnitude-judgment task were maintained for the parity-judgment task. The SNARC effect was unaffected by practice with compatible or incompatible spatial mapping for the parallel spatial task but was larger following up–right/down–left mapping than up–left/down–right mapping for the orthogonal spatial task. These results are inconsistent with the SNARC effect being due to a horizontal number line representation but consistent with a view that correspondence of stimulus and response code polarities contributes to the effect. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Representations of the magnitudes of fractions.

We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However, atypical characteristics of the presented fractions might have provoked the use of atypical comparison strategies in that study. In our 3 experiments, university and community college students compared more balanced sets of single-digit and multi-digit fractions and consistently exhibited a logarithmic distance effect. Thus, adults used integrated, analog representations, akin to a mental number line, to compare fraction magnitudes. We interpret differences between the past and present findings in terms of different stimuli eliciting different solution strategies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Verbal-spatial and visuospatial coding of number–space interactions.

A tight correspondence has been postulated between the representations of number and space. The spatial numerical association of response codes (SNARC) effect, which reflects the observation that people respond faster with the left-hand side to small numbers and with the right-hand side to large numbers, is regarded as strong evidence for this correspondence. The dominant explanation of the SNARC effect is that it results from visuospatial coding of magnitude (e.g., the mental number line hypothesis). In a series of experiments, we demonstrated that this is only part of the story and that verbal-spatial coding influences processes and representations that have been believed to be purely visuospatial. Additionally, when both accounts were directly contrasted, verbal-spatial coding was observed in absence of visuospatial coding. Relations to other number–space interactions and implications for other tasks are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The developmental relations between conceptual and procedural knowledge: A multimethod approach.

Interactions between conceptual and procedural knowledge influence the development of mathematical competencies. However, after decades of research, these interrelations are still under debate, and empirical results are inconclusive. The authors point out a source of these problems. Different kinds of knowledge and competencies only show up intertwined in behavior, making it hard to measure them validly and independently of each other. A multimethod approach was used to investigate the extent of these problems. A total of 289 fifth and sixth graders’ conceptual and procedural knowledge about decimal fractions was measured by 4 common hypothetical measures of each kind of knowledge. Study 1 tested whether treatments affected the 2 groups of measures in consistent ways. Study 2 assessed, across 3 measurement points, whether conceptual and procedural knowledge could be modeled as latent factors underlying the measures. The results reveal substantial problems with the validities of the measures, which might have been present but gone undetected in previous studies. A solution to these problems is essential for theoretical and practical progress in the field. The potential of the multimethod approach for this enterprise is discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Sources of group and individual differences in emerging fraction skills.

Results from a 2-year longitudinal study of 181 children from 4th through 5th grade are reported. Levels of growth in children's computation, word problem, and estimation skills by means of common fractions were predicted by working memory, attentive classroom behavior, conceptual knowledge about fractions, and simple arithmetic fluency. Comparisons of 55 participants identified as having mathematical difficulties to those without mathematical difficulties revealed that group differences in emerging fraction skills were consistently mediated by attentive classroom behavior and conceptual knowledge about fractions. Neither working memory nor arithmetic fluency mediated group differences in growth in fraction skills. It was also found that the development of basic fraction skills and conceptual knowledge are bidirectional in that conceptual knowledge exerted strong influences on all 3 types of basic fraction skills, and basic fraction skills exerted a more modest influence on subsequent conceptual knowledge. Results are discussed with reference to how the identification of potentially malleable student characteristics that contribute to the difficulties that some students have with fractions informs interventions. Also, results will contribute to a future theoretical account concerning how domain-general and domain-specific factors influence the development of basic fraction skills. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A generalized fraction: An entity smaller than one on the mental number line.

The representation of fractions in long-term memory (LTM) was investigated by examining the automatic processing of such numbers in a physical comparison task, and their intentional processing in a numerical comparison task. The size congruity effect (SiCE) served as a marker of automatic processing and consequently as an indicator of the access to the primitives of numerical representation in LTM. Mixed pairs composed of a natural number and a fraction showed both a SiCE and a distance effect. The SiCE for mixed pairs was stable across relative sizes of natural numbers compared to the fraction digits (Experiment 4). However, comparing pairs of fractions revealed a strong influence of fractional components: An inverse SiCE was found for pairs of unit fractions (Experiment 1), while no SiCE was found for pairs of non-unit fractions (Experiments 2–3). This leads to the conclusions that: (1) there are no unique representations of distinct fraction values in LTM, and (2) there is a representation of a “generalized fraction” as an “entity smaller than one” that emerges from the notational structure common to all fractions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Developing conceptual understanding and procedural skill in mathematics: An iterative process.

The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, children's initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Instructional psychology: Past, present, and future.

Traces the history of instructional psychology from the early part of this century to recent developments. Currently, instructional psychology, with the methods and concepts of cognitive psychology, focuses on the acquisition of intellectual competence. Researchers are examining such areas as the knowledge and cognitive processes required in advanced levels of reading and text comprehension, the computational and problem-solving skills needed in mathematics, the skills of learning assessed by aptitude tests, and the effects on school achievement of the interaction between initial ability and classroom variables. The emerging field of instructional psychology can be described in terms of 4 major components: the nature of the competence to be attained, the initial state of the learner, the transition processes between these 2 stages, and ways of assessing and monitoring performance changes in the acquisition of competence. Principles that should guide the development of a psychology of instruction are discussed. (110 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding.

A theoretical analysis of the development of numerical representations indicated that playing linear number board games should enhance preschoolers’ numerical knowledge and ability to acquire new numerical knowledge. The effect on knowledge of numerical magnitudes was predicted to be larger when the game was played with a linear board than with a circular board because of a more direct mapping between the linear board and the desired mental representation. As predicted, playing the linear board game for roughly 1 hr increased low-income preschoolers’ proficiency on the 2 tasks that directly measured understanding of numerical magnitudes—numerical magnitude comparison and number line estimation—more than playing the game with a circular board or engaging in other numerical activities. Also as predicted, children who had played the linear number board game generated more correct answers and better quality errors in response to subsequent training on arithmetic problems, a task hypothesized to be influenced by knowledge of numerical magnitudes. Thus, playing linear number board games not only increases preschoolers’ numerical knowledge but also helps them learn from future numerical experiences. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The social-cognitive model of achievement motivation and the 2 × 2 achievement goal framework.

Two studies examined hypotheses drawn from a proposed modification of the social-cognitive model of achievement motivation that centered on the 2 × 2 achievement goal framework. Implicit theories of ability were shown to be direct predictors of performance attainment and intrinsic motivation, and the goals of the 2 × 2 framework were shown to account for these direct relations. Perceived competence was shown to be a direct predictor of achievement goals, not a moderator of relations implicit theory or achievement goal effects. The results highlight the utility of attending to the approach-avoidance distinction in conceptual models of achievement motivation and are fully in line with the hierarchical model of achievement motivation. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Developmental and individual differences in pure numerical estimation.

The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

What counts in the development of young children's number knowledge?

[Correction Notice: An erratum for this article was reported in Vol 47(1) of Developmental Psychology (see record 2011-00627-019). A coding error resulted in incorrect item-level data being reported on the point-to-x task (not the children‘s overall performance on this task) in Table 2 and in the section of the Results headed Point-to-X Task Performance (second column, p. 1314). In the first paragraph in the section, the correct average score for knowledge of cardinal meanings of the number words. In the second paragraph in the section, there is an example illustrating children’s greater performance on items involving a target and a distractor that were one digit apart. An additional adjustment in the second paragraph involves the finding that children performed better when at least one of two choice sets was a small number (1–3) than when both choice sets were greater than or equal to 4. More information for the corrections and the corrected table are given in the erratum.] Prior studies indicate that children vary widely in their mathematical knowledge by the time they enter preschool and that this variation predicts levels of achievement in elementary school. In a longitudinal study of a diverse sample of 44 preschool children, we examined the extent to which their understanding of the cardinal meanings of the number words (e.g., knowing that the word “four” refers to sets with 4 items) is predicted by the “number talk” they hear from their primary caregiver in the early home environment. Results from 5 visits showed substantial variation in parents' number talk to children between the ages of 14 and 30 months. Moreover, this variation predicted children's knowledge of the cardinal meanings of number words at 46 months, even when socioeconomic status and other measures of parent and child talk were controlled. These findings suggest that encouraging parents to talk about number with their toddlers, and providing them with effective ways to do so, may positively impact children's school achievement. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Sequential effects in number comparison.

The modular framework of number processing (e.g., S. Dehaene & R. Akhavein, 1995) was applied to study sequential trial-to-trial effects in a number comparison task. In Experiment 1, numbers were always presented as digits. Responses were faster when the same number was repeated, but this effect was additive with the numerical distance effect. In Experiment 2, numbers were presented either as digits or as words. The authors found significant effects of repeating (a) the same physical stimulus, (b) the same number but in a different notation, and (c) the same notation but a different number. Again, all 3 effects were additive with the numerical distance effect. The authors' results provide strong evidence against accounts according to which, on stimulus repetition trials, the comparison stage is bypassed (as proposed by S. Dehaene, 1996), and the results clearly favor an early, precomparison locus of repetition effects. (PsycINFO Database Record (c) 2010 APA, all rights reserved)