Kristy vanMarle

Attentive Tracking of Objects Versus Substances

Recent research in vision science, infant cognition, and word learning suggests a special role for the processing of discrete objects. But what counts as an object? Answers to this question often depend on contrasting object-based processing with the processing of spatial areas or unbound visual features. In infant cognition and word learning, though, another salient contrast has been between rigid cohesive objects and nonsolid substances. Whereas objects may move from one location to another, a nonsolid substance must pour from one location to another. In the study reported here, we explored whether attentive tracking processes are sensitive to dynamic information of this type. Using a multiple-object tracking task, we found that subjects could easily track four items in a display of eight identical unpredictably moving entities that moved as discrete objects from one location to another, but could not track similar entities that noncohesively “poured” from one location to another—even when the items in both conditions followed the same trajectories at the same speeds. Other conditions revealed that this inability to track multiple “substances” stemmed not from violations of rigidity or cohesiveness per se, because subjects were able to track multiple noncohesive collections and multiple nonrigid deforming objects. Rather, the impairment was due to the dynamic extension and contraction during the substancelike motion, which rendered the location of the entity ambiguous. These results demonstrate a convergence between processes of midlevel adult vision and infant cognition, and in general help to clarify what can count as a persisting dynamic object of attention.

Infants use different mechanisms to make small and large number ordinal judgments

Previous research has shown indirectly that infants may use two different mechanisms--an object tracking system and an analog magnitude mechanism--to represent small (<4) and large (≥4) numbers of objects, respectively. The current study directly tested this hypothesis in an ordinal choice task by presenting 10- to 12-month-olds with a choice between different numbers of hidden food items. Infants reliably chose the larger amount when choosing between two exclusively small (1 vs. 2) or large (4 vs. 8) sets, but they performed at chance when one set was small and the other was large (2 vs. 4) even when the ratio between the sets was very favorable (2 vs. 8). The current findings support the two-mechanism hypothesis and, furthermore, suggest that the representations from the object tracking system and the analog magnitude mechanism are incommensurable.

Quantitative Deficits of Preschool Children at Risk for Mathematical Learning Disability

The study tested the hypothesis that acuity of the potentially inherent approximate number system (ANS) contributes to risk of mathematical learning disability (MLD). Sixty-eight (35 boys) preschoolers at risk for school failure were assessed on a battery of quantitative tasks, and on intelligence, executive control, preliteracy skills, and parental education. Mathematics achievement scores at the end of 1 year of preschool indicated that 34 of these children were at high risk for MLD. Relative to the 34 typically achieving children, the at risk children were less accurate on the ANS task, and a one standard deviation deficit on this task resulted in a 2.4-fold increase in the odds of MLD status. The at risk children also had a poor understanding of ordinal relations, and had slower learning of Arabic numerals, number words, and their cardinal values. Poor performance on these tasks resulted in 3.6- to 4.5-fold increases in the odds of MLD status. The results provide some support for the ANS hypothesis but also suggest these deficits are not the primary source of poor mathematics learning.

Attaching meaning to the number words: contributions of the object tracking and approximate number systems.

Childrens understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems - the approximate number system (ANS) and the object tracking system (OTS) - but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to childrens early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.

Children’s early understanding of number predicts their later problem-solving sophistication in addition

Previous studies suggest that the sophistication of the strategies children use to solve arithmetic problems is related to a more basic understanding of number, but they have not examined the relation between number knowledge in preschool and strategy choices at school entry. Accordingly, the symbolic and nonsymbolic quantitative knowledge of 134 children (65 boys) was assessed at the beginning of preschool and in kindergarten, and the sophistication of the strategies they used to solve addition problems was assessed at the beginning of first grade. Using a combination of Bayes and standard regression models, we found that childrens understanding of the cardinal value of number words at the beginning of preschool predicted the sophistication of their strategy choices 3 years later, controlling for other factors. The relation between childrens early understanding of cardinality and their strategy choices was mediated by their symbolic and nonsymbolic quantitative knowledge in kindergarten. The results suggest that sophisticated strategy choices emerge from childrens developing understanding of the relations among numbers, in keeping with the overlapping waves model.

Controlling for continuous variables is not futile: What we can learn about number representation despite imperfect control

Leibovich et al. argue that because it is impossible to isolate numerosity in a stimulus set, attempts to show that number is processed independently of continuous magnitudes are necessarily in vain. I propose that through clever design and manipulation of confounding variables, we can gain deep insight into number representation and its relationship to the representation of other magnitudes.

Number Processing and Arithmetic

Decades of research investigating the origins and nature of mathematical cognition suggest that humans possess a sophisticated network of neural systems for representing and manipulating symbolic and nonsymbolic numerical information, some of which we appear to share with other animal species. This article reviews behavioral and neurocognitive research describing these systems and their development in humans, highlighting the ways in which we are similar to, and different from, other species with regards to mathematical cognition.

Growth of symbolic number knowledge accelerates after children understand cardinality

Children who achieve an early understanding of the cardinal value of number words (cardinal knowledge) have a superior understanding of the relations among numerals at school entry, controlling other factors (e.g., intelligence). We tested the hypothesis that this pattern emerges because an understanding of cardinal value jump starts childrens learning of the relations among numerals. Across two years of preschool, the cardinal knowledge of 179 children (85 boys) was assessed four times, as was their understanding of the relative quantity of Arabic numerals and competence at discriminating nonsymbolic quantities. Children were more accurate on nonsymbolic than numeral comparisons before they understood cardinality, but showed more rapid growth for numeral than nonsymbolic comparisons once they understood cardinality. Moreover, and with the possible exception of very small numerals (<5), before they understood cardinality children were no better than chance in their numeral comparisons, but greatly exceeded chance once they understood cardinality. These patterns were independent of the age at which children became cardinal principle knowers and independent of intelligence, executive function, and preliteracy skills. More broadly, the results provide a developmental bridge between cardinal knowledge and school-entry number knowledge.

Analog Magnitudes Support Large Number Ordinal Judgments in Infancy

Few studies have explored the source of infants’ ordinal knowledge, and those that have are equivocal regarding the underlying representational system. The present study sought clear evidence that the approximate number system, which underlies children’s cardinal knowledge, may also support ordinal knowledge in infancy; 10 - to 12-month-old infants’ were tested with large sets (>3) in an ordinal choice task in which they were asked to choose between two hidden sets of food items. The difficulty of the comparison varied as a function of the ratio between the sets. Infants reliably chose the greater quantity when the sets differed by a 2:3 ratio (4v6 and 6v9), but not when they differed by a 3:4 ratio (6v8) or a 7:8 ratio (7v8). This discrimination function is consistent with previous studies testing the precision of number and time representations in infants of roughly this same age, thus providing evidence that the approximate number system can support ordinal judgments in infancy. The findings are discussed in light of recent proposals that different mechanisms underlie infants’ reasoning about small and large numbers.